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transcript
Methods and Applications to
DSGE Models
2016-5
Anders Kronborg
PhD Thesis
DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS
AARHUS UNIVERSITY DENMARK
METHODS AND APPLICATIONS TO DSGEMODELS
By Anders Kronborg
A dissertation submitted to
Business and Social Sciences, Aarhus University,
in partial fulfilment of the requirements of
the PhD degree in
Economics and Business Economics
JANUARY 2016
CREATESCenter for Research in Econometric Analysis of Time Series
PREFACE
This dissertation is the tangible result of my PhD studies at the Department of Eco-
nomics and Business Economics at Aarhus University from February 2013 to January
2016. I am grateful to the department and to Center for Research in Econometric
Analysis of Time Series (CREATES) for providing facilities and an excellent research
environment during my studies. The generous financial support has allowed me
to participate in numerous courses and conferences both in Denmark and abroad.
Further, financial support received from Augustinus Fonden, Knud Højgaards Fond,
Konsul Axel Nielsens Mindelegat, Norges Bank, and Oticon Fonden is gratefully ac-
knowledged.
I would like to take the opportunity to thank a number of people for contributing
to this dissertation. Special thanks go to my two supervisors Torben M. Andersen and
Martin M. Andreasen for encouragement, helpful comments and valuable insights. It
has been a privilege to have such inspiring advisors. I am particularly pleased to have
worked together with Martin on one of the chapters in this dissertation.
I am thankful to Solveig N. Sørensen for help of all sorts and for proof-reading the
dissertation chapters.
From February to June 2015 I had the privilege of visiting Professor Frank Schorf-
heide at the Department of Economics at University of Pennsylvania. I would like to
thank both him and the department for the hospitality during my stay. The discus-
sions I have had with Frank have been enormously helpful and have contributed to
the third chapter of this thesis. Further, getting the opportunity to present my research
to several leading researchers in my field was inspiring and greatly appreciated.
I have been blessed with many great colleagues and a friendly work environment
during my time as a PhD student at Aarhus University. In particular, all my fellow PhD
students deserve a big thanks. First and foremost, I would like to thank Simon for the
years of laughs and interesting conversations, both related to economics and not. It
is my hope and expectation that we will continue to do so. Further, special thanks
go to Anne, Carsten, Jakob, Jonas, Mikkel, Morten, Niels, Palle, Sanni, and Silvana for
making it enjoyable to come to work every day.
A deep thank you to my family for supporting me and for taking an interest in
i
ii
my work even if it was hard to comprehend or not particularly interesting. Finally,
and most importantly, I would like to express my heartfelt gratitude to Josefine. Your
unconditional love and support has truly been invaluable.
Anders Kronborg
Copenhagen, January 2016
UPDATED PREFACE
The predefense took place on March 7. The assessment committee consists of Pro-
fessor Wouter den Haan from London School of Economics, Jens Iversen Head of
Modelling Division at Sveriges Riksbank, and Associate Professor Allan Sørensen from
Aarhus University. I am grateful to the members of the assessment committee for
their careful reading of the thesis, and their constructive comments and insightful dis-
cussion of my work. Some of the suggestions have been incorporated in the current
version of the thesis, while others remain for future research.
Anders Kronborg
Copenhagen, March 2016
iii
CONTENTS
Summary vii
Danish summary ix
1 The Extended Perturbation Method 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Extended Perturbation Method . . . . . . . . . . . . . . . . . . . 3
1.3 Stability Properties of the Extended Perturbation Method . . . . . . 8
1.4 The Neoclassical Growth Model . . . . . . . . . . . . . . . . . . . . . . 12
1.5 A New Keynesian Model . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Efficient Implementation of Extended Perturbation . . . . . . . . . . 21
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 New Evidence on Downward Nominal Wage Rigidity and the Implicationsfor Monetary Policy 55
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2 The DSGE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.3 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Econometric Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.5 Data and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.6 Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.7 Optimal Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
v
vi CONTENTS
3 Forecasting Using a DSGE Model with a Fixed Exchange Rate 853.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2 The DSGE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3 Econometric Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5 Prior Distributions and Calibrated Parameters . . . . . . . . . . . . . 101
3.6 Posterior Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 102
3.7 DSGE Model-based Forecasting . . . . . . . . . . . . . . . . . . . . . . 106
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
SUMMARY
Dynamic stochastic general equilibrium (DSGE) models are macroeconomic models
which try to explain aggregate economic movements based on an explicit micro-
economic foundation such as utility maximization. As the name suggests, they are
dynamic meaning they are concerned with intertemporal decisions which are subject
to stochastic shocks, for example through productivity changes. This uncertainty
implies that the economic agents have to form expectations about the future when
making decisions today. General equilibrium means that they aim to describe quanti-
ties and prices for both the supply and demand side simultaneously.
This dissertation comprises three self-contained chapters that all relate to me-
thods and applications to DSGE models. Since each chapter can be read indepen-
dently this implies that some repetition of arguments is impossible to avoid. Further,
it might beneficial to read the first chapter if one wants to get a more thorough
understanding of the methodology later applied in the second chapter.
Generally, the solution to DSGE models is unknown and we need to use numerical
techniques to approximate the equilibrium dynamics resulting from optimal behavior.
Chapter 1 in this dissertation proposes a solution method that generally improves
the accuracy relative to some of the most widely applied methods while still being
feasible for estimation purposes.
DSGE models are true multi-tools with many possible applications. These include
the estimation of the structural parameters in a model in order to answer some em-
pirical questions or to analyze the effects of changes in macroeconomic policy. Both
are the subject of chapter 2. Chapter 3 considers using a DSGE model for predictions
of comovements in aggregate time series.
The first chapter "The Extended Perturbation Method" (joint with Martin M. An-
dreasen) proposes a new solution method to DSGE models. Even though the nonlin-
ear solution to DSGE models is frequently approximated by higher-order perturbation
it is well known that they often generate explosive sample paths and struggle to pre-
serve underlying characteristics of the true solution, such as convexity or monotonic-
ity. Our solution method combines perturbation with the Extended Path. Using the
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viii SUMMARY
neoclassical growth model and a New Keynesian model with Calvo pricing, we show
that this extended perturbation method improves the accuracy relative to standard
perturbation when using third order approximations. Further, we are able to gen-
erate stable approximations even when standard perturbation explodes. The main
drawback of this method is that it may be computationally demanding, as it requires
solving a rather large fixed-point problem. We analyze these costs and suggest several
ways to improve the algorithm without loosing substantial precision.
The second chapter "New Evidence on Downward Nominal Wage Rigidity and the
Implications for Monetary Policy" revisits the question of wage rigidity. Downward
nominal wage rigidity has received increased attention during the recent crisis. To
examine this a simple New Keynesian DSGE model with potential asymmetric wage
rigidity is estimated using U.S. data. The model solution is approximated by the
extended perturbation method which is particularly suitable for strong model non-
linearities as in this application. In line with previous studies I find that downward
nominal wage rigidity is present, meaning nominal wages are more downwardly that
upwardly rigid. However, the parameter governing this asymmetry is found to be
orders of magnitude smaller than in the literature, which can be attributed to the
change in solution methodology. The estimated model is subsequently used to find
the optimal inflation target when implemented in a Taylor rule. For the U.S., the
optimal inflation is approximately 0.25 percent per year which provides support for a
low but positive inflation target.
The third and final chapter is "Forecasting Using a DSGE Model with a Fixed Ex-
change Rate". While macroeconomic forecasters can choose between a wide range
of models to generate predictions, DSGE models are increasingly being applied by
economic institutions and policy makers. Using Danish data, this chapter examines
the forecasting ability of a DSGE model in which the exchange rate is fixed. While
previous studies document the performances of DSGE models in a closed-economy
setting and where the exchange rate is flexible, it is not clear a priori that the model
will produce similar results under a fixed exchange rate regime. The forecasting per-
formance of the DSGE model is examined by generating recursive out-of-sample
forecasts of several time series from one to eight quarters ahead. The accuracy of the
DSGE model is generally comparable to an AR(1) model and better than the random
walk. Consistent with previous research, the DSGE model largely underestimates the
severity of the recent crisis that hit Denmark in the third quarter of 2008. However,
the model correctly predicts a continued fall in GDP growth and a subsequent slow
recovery when forecasting from the beginning of 2009.
DANISH SUMMARY
Dynamisk stokastiske generelle ligevægtsmodeller (DSGE-modeller) er makro-
økonomiske modeller, der forsøger at forklare udviklingen i aggregerede økonomiske
variable baseret på et eksplicit mikroøkonomisk fundament som f.eks. nyttemak-
simering. Som navnet antyder, er modeltypen dynamisk, hvilket betyder, at de be-
tragter intertemporale beslutninger, der påvirkes af stokastiske stød for eksempel
via produktivitetsændringer. Denne usikkerhed medfører, at de økonomiske aktører
må danne forventninger om fremtiden, når de træffer beslutninger i dag. Generel
ligevægt betyder, at der søges at beskrive mængder og priser simultant for både
udbuds- og efterspørgselssiden.
Denne afhandling består af tre selvstændige kapitler, der alle relaterer sig til
metoder og anvendelser vedrørende DSGE-modeller. Idet hvert kapitel kan læses
uafhængigt, betyder det, at det er uungåeligt, at nogle argumenter går igen. Man kan
med fordel læse det første kapitel først, såfremt man ønsker en mere dybdegående
forståelse af den metodologi, der senere anvendes i andet kapitel.
Som oftest er løsningen til DSGE-modeller ikke kendt og vi er derfor nødt til
at anvende numeriske teknikker til at approksimere den ligevægtsdynamik, der er
resultatet af optimal adfærd. Kapitel 1 i denne afhandling foreslår en løsningsmetode,
der generelt set forbedrer nøjagtigheden i forhold til nogle af de mest anvendte
metoder imens den stadigt kan anvendes til estimationsformål.
En DSGE-model er et multiværktøj med mange anvendelsesmuligheder. Disse
inkluderer estimation af modellens strukturelle parametre for at besvare empiriske
spørgsmål eller at analysere effekten af ændringer i den makroøkonomiske politik.
Begge er temaet i kapitel 2. Kapitel 3 betragter, hvorvidt man kan bruge en DSGE-
model til at forudsige bevægelserne i aggregerede tidsserier.
Det første kapitel "The Extended Perturbation Method" (skrevet med Martin M. An-
dreasen) foreslår en ny løsningsmetode til DSGE-modeller. Selvom den ikke-lineære
løsning ofte approksimeres ved perturbation af højere orden, er det velkendt, at
de ofte genererer eksplosive stikprøver og har svært ved at bevare underliggende
karakteristika fra den sande løsning såsom konveksitet eller monotonicitet. Vores
ix
x DANISH SUMMARY
løsning kombinerer perturbation med Extended Path. Ved at anvende den på den
neoklassiske vækstmodel og en Ny-Keynesiansk model med Calvo priser viser vi, at
denne extended perturbation-metode forbedrer nøjagtigheden i forhold til standard
perturbation, når der anvendes tredjeordens-approksimationer. Derudover er vi i
stand til at generere stabile stikprøver, selv i de tilfælde hvor standard perturbation
eksploderer. Den primære ulempe ved metoden er de omkostninger, der er forbundet
med den krævede beregningskraft, idet den forudsætter, at der løses et temmeligt
stort fikspunktsproblem. Vi analyserer disse omkostninger og foreslår flere måder,
hvorpå de kan nedbringes betragteligt, uden at det får betydelige konsekvenser for
præcisionen.
Det andet kapitel "New Evidence on Downward Nominal Wage Rigidity and the Im-
plications for Monetary Policy" betragter spørgsmålet om lønrigiditet. Nedadgående
nominel lønrigiditet betyder, at lønninger tilpasser sig langsommere når de mindskes
end når de øges og har tiltrukket sig stigende opmæksomhed under den nylige krise.
For at betragte dette estimeres en simpel Ny-Keynesiansk DSGE-model ved brug af
amerikanske data. Løsningen approksimeres ved extended perturbation-metoden,
som er særlig brugbar ved stærke ikke-lineariteter som i denne anvendelse. I ov-
erensstemmelse med tidligere studier finder jeg, at lønningerne er mere rigide i
nedadgående retning end omvendt. Dog findes, at den parameter der styrer graden
af asymmetri er af en anden størrelsesorden end i literaturen, hvilket kan tilskrives
ændringen i løsningsmetodologien. Den estimerede model benyttes herefter til at
bestemme det optimale inflationsmål, når dette implementeres i en Taylor-regel.
For USA findes den optimale inflation at være omkring 0,25 procent årligt, hvilket
dermed støtter et lavt, men positivt inflationsmål.
Det tredje og sidste kapitel er "Forecasting Using a DSGE Model with a Fixed Ex-
change Rate". Mens makroøkonomer kan vælge mellem en lang række modeller til at
generere fremskrivninger, bliver DSGE-modeller stadigt mere populære. Ved brug af
dansk data undersøger dette kapitel præcisionen af fremskrivningerne for en DSGE-
model, hvor valutakursen er underlagt en fastkurspolitik. Mens tidligere studier
dokumenterer fremskrivningsevnen for DSGE-modeller, hvori økonomien antages at
være lukket eller hvor valutakursen er flydende, er det ikke klart a priori, at modellen
vil fremvise lignende resultater, når valutakursen er fast. Fremskrivningsevnen un-
dersøges ved rekursivt at generere out-of-sample fremskrivninger for flere tidsserier
fra et til otte kvartaler frem. DSGE-modellens præcision kan generelt sammenlignes
med en AR(1) model og klarer sig bedre end random walk. I overensstemmelse med
litteraturen på området findes, at DSGE-modellen kraftigt undervurderer omfanget
af krisen, der ramte Danmark i tredje kvartal i 2008. Dog forudsiger modellen korrekt
xi
BNP-vækstens fortsatte fald samt det langsomme opsving, når fremskrivningerne
startes fra primo 2009.
C H A P T E R 1THE EXTENDED PERTURBATION METHOD
Martin M. Andreasen
Aarhus University and CREATES
Anders Kronborg
Aarhus University and CREATES
Abstract
The exact solution to a broad class of DSGE models can be decomposed into a compo-
nent under perfect foresight and a component containing the effects of uncertainty.
We therefore propose to compute the perfect foresight component with arbitrary
precision by the Extended Path whereas the stochastic part of the solution is approxi-
mated by the perturbation method. Using the neoclassical growth model and a New
Keynesian model, we show that this alternative approximation is more accurate than
standard third-order perturbation and delivers stable dynamics even in cases where
standard perturbation explodes.
1
2 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
1.1 Introduction
The solution to dynamic stochastic general equilibrium (DSGE) models is frequently
approximated by the perturbation method to obtain higher-order Taylor series expan-
sions of the policy function. The popularity of this approximation is mainly explained
by its ability to i) preserve non-linearities in the model such as asymmetries, ii) im-
prove parameter identification compared to a linearized solution, and iii) capture
effects of uncertainty to explore determinants of risk premia and implications of
uncertainty shocks (see An and Schorfheide (2007), Kim and Ruge-Murcia (2009),
Fernandez-Villaverde et al. (2011), Rudebusch and Swanson (2012), among others).
Despite the widespread use of second- and third-order perturbation approxima-
tions, it is well known that they often generate explosive sample paths even when
the corresponding linearized solution is stable. The perturbation approximation may
also struggle to preserve key properties of the true solution such as monotonicity and
convexity as emphasized by den Haan and de Wind (2012). These findings suggest
that the second- and third-order perturbation approximations currently applied in
the literature may not always be sufficiently accurate. Obtaining fourth- or even
fifth-order expansions is often computationally infeasible and may even in some
cases be insufficient to get an accurate approximation as shown by den Haan and de
Wind (2012) . A tractable alternative that preserves stability of the true solution, but
not necessarily monotonicity, is to apply a pruning scheme as proposed by Kim et al.
(2008) for models approximated to second order and extended to higher order by den
Haan and de Wind (2012), Andreasen et al. (2013), and Lombardo and Uhlig (2014).
The contribution of the present paper is to improve accuracy and stability of
the perturbation approximation by combining it with the Extended Path of Fair and
Taylor (1983). This is done based on a simple, yet powerful, decomposition of the
policy function into i) a deterministic component under perfect foresight and ii) a
component containing the effects of uncertainty, also referred to as the stochastic
component. The perturbation method is currently applied to approximate both parts
of the policy function, although the perfect foresight component may be approxi-
mated with arbitrary precision by the Extended Path. Based on this observation, we
propose to compute the perfect foresight component by the Extended Path, whereas
the stochastic part of the policy function is approximated by the standard perturba-
tion method. We name this combined solution procedure the extended perturbation
method and it improves accuracy and stability of standard perturbation by remov-
ing approximation errors under perfect foresight. The approximation order in the
extended perturbation method is thus determined by the order of the polynomial
used to approximate the stochastic part of the policy function.
For a second-order approximation, we show that extended perturbation always
gives a stable solution without explosive sample paths, provided the model is stable
1.2. THE EXTENDED PERTURBATION METHOD 3
under perfect foresight. This result does not generalize beyond second order and we
therefore present a stability test to numerically evaluate if a given approximation is
stable.
Using the neoclassical growth model and a New Keynesian model with Calvo
pricing, we show that extended perturbation achieves higher accuracy than standard
perturbation when using third-order approximations. We also show that extended
perturbation generates stable approximations even when standard perturbation
explodes. For the neoclassical model, the explosive behavior is caused by a lack of
monotonicity and convexity of the approximated consumption function, whereas
standard perturbation in the New Keynesian model generates an explosive price-
inflation spiral because the approximation does not account for the upper bound on
inflation induced by Calvo pricing.
A potential drawback of extended perturbation relates to the repeated use of the
Extended Path which may be computationally demanding. We address this poten-
tial concern by presenting three modifications of the Extended Path to improve its
efficiency: i) deriving good starting values using a third-order perturbation approx-
imation under perfect foresight, ii) appropriately setting the terminal condition in
the Extended Path, and iii) occasionally using the perturbation approximation of the
perfect foresight component in the policy function if it is sufficiently accurate. When
adopting the third improvement, extended perturbation reduces to the standard per-
turbation approximation close to the steady state, but otherwise applies the Extended
Path to improve accuracy and obtain a stable approximation. We show that each of
the three improvements may be combined to substantially reduce the computational
cost of extended perturbation. Depending on the required degree of precision, we
are able to simulate 1,000 draws from a medium-sized New Keynesian model with
nine state variables in 10 to 20 seconds using MATLAB on a standard desktop.
The remaining part of the paper is structured as follows. Section 1.2 derives
the extended perturbation method and compares it to existing solution methods.
The stability properties of extended perturbation is analyzed in Section 1.3 which
also presents a stability test. Numerical evidence on the performance of extended
perturbation is provided for the neoclassical growth model in Section 1.4 and a New
Keynesian model in Section 1.5. We discuss how to efficiently implement extended
perturbation in Section 1.6, while Section 1.7 concludes.
1.2 The Extended Perturbation Method
We start by presenting the considered class of DSGE models in Section 1.2.1. The
extended perturbation method is derived in Section 1.2.2 and compared to standard
perturbation in Section 1.2.3 and a stochastic version of the Extended Path in Section
1.2.4.
4 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
1.2.1 DSGE Models
We consider the broad class of DSGE models which can be expressed as
Et
[f(xt ,xt+1,yt ,yt+1
)]= 0, (1.1)
where Et denotes the conditional expectation given information available in time
period t . The state vector xt with dimension nx ×1 belongs to the set Xx , denoting
Borel subsets of Rnx . The control variables are stored in yt with dimension ny ×1
and yt ∈Xy , where Xy refers to Borel subsets of Rny . We further let nx +ny = n. The
function f maps elements from Xx ×Xx ×Xy ×Xy into Rn , and we assume that this
mapping is at least m times differentiable, where m will be used below to indicate
the approximation order of the DSGE model.
It is useful to consider the partitioning xt ≡[
x′1,t x′2,t
]′, where x1,t contains the
endogenous state variables and x2,t denotes the exogenous states. The dimensions
of these vectors are nx1 ×1 and nx2 ×1, respectively, with nx1 +nx2 = nx . We further
assume that the dynamics of the exogenous state variables belong to the general class
x2,t+1 = Γ(x2,t
)+σηεt+1, (1.2)
where εt+1 ∈Xε and has dimension nε×1. We also assume εt+1 to be independent
and identically distributed with zero mean and a unit covariance matrix, i.e. εt+1 ∼IID
(0,I
). The function Γmaps elements from Xx2 into Xx2 and is required to be at
least m times differentiable. We further assume that Γ generates a stable process
for x2,t .1 In linear systems, this corresponds to requiring that all eigenvalues of the
Jacobian ∂Γ/∂x′2,t lie inside the unit circle. For non-linear systems, Γmust satisfy the
general stability condition for nonlinear first-order Markov processes provided in
Section 1.3.1.
As in much of the perturbation literature, we focus on models with a unique
solution. The exact solution may then be expressed as (see Schmitt-Grohe and Uribe
(2004))
yt = g(xt ,σ
)(1.3)
xt+1 = h(xt ,σ
)+σηεt+1 (1.4)
η≡[
0nx1×nε
η
].
The assumption that innovations enter linearly in (1.2) and (1.4) is without loss of
generality, because the state vector may be extended to account for non-linearities
1This implies that trends may only be included in the class of DSGE models considered if a givenmodel after re-scaling has an equivalent representation without trending variables. A similar requirementis needed to apply the standard perturbation method. The procedure of re-scaling a DSGE model withtrends is carefully described in King and Rebelo (1999).
1.2. THE EXTENDED PERTURBATION METHOD 5
between xt and εt+1, as shown by Andreasen (2012b). The perturbation parameter
σ ≥ 0 scales the square root of the covariance matrix for the innovations η with
dimension nx ×nε and enables us to capture effects of uncertainty in the policy
functions. In particular, when σ= 0 we get a model under perfect foresight, i.e.
gPF (xt
) ≡ g(xt ,σ= 0
)(1.5)
hPF (xt
) ≡ h(xt ,σ= 0
),
whereas the model with uncertainty is obtained by letting σ = 1. Unfortunately,
the policy functions g and h in (1.3) and (1.4) are generally unknown and must be
approximated.
1.2.2 The Extended Perturbation Method
Our paper builds on the key observation that the policy functions can be decomposed
into
g(xt ,σ
) ≡ gPF (xt
)+gstoch (xt ,σ
)(1.6)
h(xt ,σ
) ≡ hPF (xt
)+hstoch (xt ,σ
),
where gstoch and hstoch capture effects of uncertainty when the perfect foresight
component is removed from the policy function. We also refer to gstoch and hstoch
as the stochastic part of the policy function, as indicated by the superscript. The
decomposition in (1.6) shows that the stochastic part of the policy function is zero
under perfect foresight, i.e. gstoch(xt ,σ= 0
)= 0 and hstoch(xt ,σ= 0
)= 0 for all values
of xt . This in turn implies that all derivatives of g and gPF solely with respect to the
states are identical at σ= 0, and similarly for h and hPF . That is,
g(xt ,σ= 0
)xm = gPF (
xt)
xm for all xt ∈Xx (1.7)
h(xt ,σ= 0
)xm = hPF (
xt)
xm for all xt ∈Xx
for m = 0,1,2, ...
where subscripts refer to partial derivatives taken m times with
respect to xt . We also note that all derivatives involving the perturbation parameter σ
are identical for g and gstoch because σ does not appear in gPF , and similarly for hand hstoch . That is,
g(xt ,σ
)xmσ j = gstoch (
xt ,σ)
xmσ j for all xt ∈Xx ,σ ∈R+ (1.8)
h(xt ,σ
)xmσ j = hstoch (
xt ,σ)
xmσ j for all xt ∈Xx ,σ ∈R+
for m = 0,1,2, ...
and j =
1,2, ...
, where subscripts refer to partial derivatives taken
m times with respect to xt and j times with respect to σ. Thus, our observations in
(1.7) and (1.8) imply that the standard perturbation method can be used to compute
6 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
gstoch(xt ,σ
)xmσ j and hstoch
(xt ,σ
)xmσ j at the deterministic steady state, i.e. at xss =
xt+1 = xt and σ= 0.
Inserting the decomposition in (1.6) into (1.3) and (1.4), the exact solution may
be expressed as
yt = gPF (xt
)+gstoch (xt ,σ
)(1.9)
xt+1 = hPF (xt
)+hstoch (xt ,σ
)+σηεt+1.
Following the work of Guu and Judd (1997), the perturbation method is usually ap-
plied to approximate both(gstoch ,hstoch
)and
(gPF ,hPF
)at the deterministic steady
state. However, a finite Taylor series expansion of gPF and hPF may generate unneces-
sary approximation errors, given that gPF and hPF can be approximated to arbitrary
precision by the Extended Path. We therefore suggest to compute the perfect foresight
components gPF and hPF by the Extended Path, whereas the stochastic part of the
policy function, i.e. gstoch and hstoch , remains approximated by the standard pertur-
bation method at the steady state. We name this combined solution procedure the
extended perturbation method, and it improves accuracy and stability of standard
perturbation by removing approximation errors in the perfect foresight component
of the policy function.
The approximation order in the extended perturbation method is determined
by the order of the Taylor series expansion used to approximate gstoch and hstoch .
Hence, a first-order approximation simply reproduces the perfect foresight solution,
whereas the second-order approximation is
yt = gPF (xt
)+ 1
2gσσ (1.10)
xt+1 = hPF (xt
)+ 1
2hσσ+σηεt+1.
The third order approximation reads
yt = gPF (xt
)+ 1
2gσσ+ 3
6gσσx
(xt −xss
)+ 1
6gσσσ (1.11)
xt+1 = hPF (xt
)+ 1
2hσσ+ 3
6hσσx
(xt −xss
)+ 1
6hσσσ+σηεt+1.
In (1.10) and (1.11) derivatives of gstoch and hstoch known to be zero are omitted for
simplicity (see Schmitt-Grohe and Uribe (2004) and Ruge-Murcia (2012)). Thus, it is
straightforward to form the m’th-order approximation by the extended perturbation
method. The required steps are:
Step 1: Run the standard perturbation method to obtain all required derivatives
of gstoch(xt ,σ
)and hstoch
(xt ,σ
)at the steady state to order m. Use these
derivatives to construct the perturbation approximations of gstoch(xt ,σ
)and
hstoch(xt ,σ
), denoted gstoch
(xt ,σ
)and hstoch
(xt ,σ
).
1.2. THE EXTENDED PERTURBATION METHOD 7
Step 2: In every time period, use the Extended Path to compute gPF(xt
)and hPF
(xt
)and approximate gstoch
(xt ,σ
)and hstoch
(xt ,σ
)by gstoch
(xt ,σ
)and hstoch
(xt ,σ
),
respectively.
Appendix A summarizes the Extended Path and explains how we numerically
obtain the perfect foresight solution. It should be emphasized that all derivatives of gand h solely with respect to the state variables obtained in Step 1 are not redundant,
as we use these derivatives to efficiently implement the Extended Path as described
in Section 1.6.
1.2.3 Comparing Extended and Standard Perturbation Approximations
To see how extended perturbation differs from standard perturbation, consider an
infinite Taylor series expansion of the perfect foresight solution at the steady state,
i.e.
gPF (xt
) = g(xt ,0
)= ∞∑k=0
g(xss ,0
)xk
k !
(xt −xss
)⊗k
hPF (xt
) = h(xt ,0
)= ∞∑k=0
h(xss ,0
)xk
k !
(xt −xss
)⊗k
where (xt −xss
)⊗k ≡ (xt −xss
)⊗ ...⊗ (xt −xss
)︸ ︷︷ ︸k times
.
Here, g(xss ,0
)xk is expressed as an ny ×
(nx
)k matrix and h(xss ,0
)xk as an nx ×
(nx
)k
matrix. We next analyze a third-order approximation by the extended perturbation
method, which may be written as (for σ= 1)
yt =∞∑
m=0
g(xss ,0
)xm
m!
(xt −xss
)⊗m + 1
2gσσ+ 3
6gσσx
(xt −xss
)+ 1
6gσσσ (1.12)
xt+1 =∞∑
m=0
h(xss ,0
)xm
m!
(xt −xss
)⊗m + 1
2hσσ+ 3
6hσσx
(xt −xss
)+ 1
6hσσσ+σηεt+1
Equation (1.12) shows that our new approximation may be interpreted as adding the
higher-order terms∑∞
m=4g(xss ,0)xm
m!
(xt −xss
)⊗m and∑∞
m=4h(xss ,0)xm
m!
(xt −xss
)⊗m to a
standard third-order perturbation solution. It is also evident that we only include
some of the additional terms in a higher-order Taylor series expansion. Comparing
to a fourth-order approximation for the control variables, we include g(xss ,0
)x4 but
not the additional terms gσ4 , gx2σ2 , gx,σ3 correcting for uncertainty. These terms and
further risk corrections at fifth and sixth order are typically small in DSGE models,
8 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
and we therefore conjecture that a relative low approximation order of gstoch and
hstoch will be sufficient for most models.
A further inspection of (1.12) reveals that extended perturbation at third order
has at least two attractive properties. Firstly, the policy functions include gσσx and
hσσx from standard perturbation that capture time-varying precautionary saving
and state-dependent risk premia in equity and bond prices. Secondly, the linear
approximation to the stochastic part of the policy function implies that extended
perturbation at third order is likely to preserve monotonicity and convexity of the true
policy function. This is because the approximation always captures these properties
for the perfect foresight component, and the linear approximation to the stochastic
component (obtained at the steady state) has the same monotonicity and convexity
properties for all values of xt . Thus, it is only if changes in monotonicity and/or
convexity in the exact solution arise from the stochastic part of the policy function
that extended perturbation at third order is unable to reproduce these properties of
the policy function.
1.2.4 Comparing Extended Perturbation to the Stochastic Extended Path
The extended perturbation method is also related to the stochastic version of the
Extended Path in Fair and Taylor (1983), where the Extended Path is computed S
times in every time period using sample paths for the structural innovations. That
is, the innovationsεt+i
∞i=1 are not restricted to zero as under the perfect foresight
solution, and this generates a distribution for the control and state variables, de-
noted
y(s)t ,x(s)
t+1
Ss=1
. An approximation that accounts for uncertainty is then given by
yt = 1S
∑Ss=1 y(s)
t and xt+1 = 1S
∑Ss=1 x(s)
t+1, which may be termed the Stochastic Extended
Path. This solution is numerically very demanding and therefore rarely applied. Ex-
tended perturbation may be viewed as an efficient approximation to the Stochastic
Extended Path, as it captures effects of uncertainty by derivatives with respect to the
perturbation parameter σ instead of computing the Extended Path S times in every
time period.2
1.3 Stability Properties of the Extended Perturbation Method
We next analyze the stability properties of the process for xt as implied by extended
perturbation. Given that the control variables are functions of the states, the stability
properties of yt follow from those of xt . We proceed by first describing sufficient
2 Another way to reduce the computational burden of the Stochastic Extended Path is providedin Adjemian and Julliard (2013) by only accounting for uncertainty in the first few time periods in theStochastic Extended Path after which all innovations are set to zero.
1.3. STABILITY PROPERTIES OF THE EXTENDED PERTURBATION METHOD 9
conditions for stability in first-order nonlinear Markov systems in Section 1.3.1,
which we apply in Section 1.3.2 to analyze the stability properties of the extended
perturbation approximation. Given that stability can not be guaranteed when using
extended perturbation beyond a second order approximation, we finally present a
numerical test for stability in Section 1.3.3.
1.3.1 Stability in First-order Nonlinear Markov Systems
Stability of a first-order nonlinear Markov system as in (1.4) is typically ensured by
requiring h(xt ,σ
)to be contracting. However, Potscher and Prucha (1997) argue that
this contraction condition is too strong and often violated for stable systems. The
prime example is the companion representation of a stable VAR(p) model, where
the contraction condition never holds. One way to obtain a less restrictive stability
condition is to iterate the system in (1.4) forward in time and instead impose the
contraction restriction on the iterated system. To formally present the condition,
iterate (1.4) forward by k time periods to obtain
xt+k = h(k) (xt ,εt+1,εt+2, ...εt+k−1,σ
)+σηεt+k ,
where h(2)(xt ,εt+1,σ
) ≡ h(h
(xt ,σ
)+σηεt+1,σ)
and so forth. Potscher and Prucha
(1997) then show that the system in (1.4) is stable if only h(k) is contracting. Two
sufficient conditions ensure that this contraction property holds. The first states that
there must exist an integer k ≥ 1 at which
sup
∣∣∣∣∣∣stacnx
j=1
[i′j∂h(k)
∂x′
(x j ,
ε
jd
k−1
d=1,σ
)]∣∣∣∣∣∣< 1, (1.13)
given x j ∈Xx and ε j ∈Xε. Here, ∂h(k)∂x′
(x j ,
ε
jd
k−1
d=1,σ
)is an nx ×nx Jacobian matrix
evaluated at
(x j ,
ε
jd
k−1
d=1
), and |A| denotes the norm given by the square root of
the largest eigenvalue of the matrix product A′A. The vector i j is the j ’th column in
the nx ×nx identity matrix, and the stac-operator creates a matrix using the rows
shown as arguments to the operator.3 Hence, the condition in (1.13) states that for
a sufficiently large integer k, the largest norm of ∂h(k)/∂x′ must be strictly smaller
than one for all values of xt and εt in their feasible domains. The second condition
for h(k) to display the contraction property is much weaker than (1.13) and given by
sup
∣∣∣∣∣∂h(k)
∂ε′l
(x,
εd
k−1d=1 ,σ
)∣∣∣∣∣<∞, (1.14)
3For instance, let a j denote the j ’th row of an m×n matrix A, then stacmj=1a j = A. The stac-operator is
used in (1.13) to allow rows in ∂h(k)/∂x′ to be evaluated at different points, as indicated by the superindex
j on the arguments at which ∂h(k)/∂x′ is evaluated.
10 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
where x ∈ Xx and εd ∈ Xε for l = 1,2, ...,k −1. It is clear that this second condition
holds for basically all smooth approximations to DSGE models if xt is finite, meaning
that the essential stability condition is given by (1.13). We therefore focus on (1.13) in
our subsequent discussion and leave (1.14) as a technical regularity condition.
1.3.2 Stability of the Extended Perturbation Approximation
Before analyzing the extended perturbation approximation, it is useful to study the
stability properties of the perfect foresight solution. As emphasized by Boucekkine
(1995), the perfect foresight solution can only be obtained for DSGE models that
are stable under perfect foresight. An assumption which may be tested using the
procedure in Boucekkine (1995). This stability requirement means that the state
process under perfect foresight xPFt is stable, where xPF
t evolves as xPFt+1 = hPF
(xPF
t
)+
σηεt+1. In other words, hPF satisfies condition (1.13) and explosive sample paths for
xPFt do not appear.
We next analyze the stability of extended perturbation when gradually increasing
the approximation order, i.e. the Taylor expansion of hstoch . For this analysis, it is
useful to write the extended perturbation approximation as xt+1 = hE xPer(xt ,σ
)+σηεt+1, where hE xPer
(xt ,σ
)≡ hPF(xt
)+ hstoch(xt ,σ
). In a first-order approximation,
there is no correction for uncertainty because hstoch = 0, meaning that extended
perturbation reduces to the stable perfect foresight solution.
In a second-order approximation, there is a constant correction for uncertainty
as hstoch = 12 hssσ
2. This means that partial derivatives of hE xPer with respect to the
state variables are equal to those of hPF for all values of xt , implying that the sta-
bility condition (1.13) also holds for hE xPer . Accordingly, the extended perturbation
method at second order guarantees a stable approximation because the uncertainty
correction only re-centers the stable perfect foresight solution.
In a third-order approximation, the uncertainty correction is a linear function
of the state variables as seen in (1.11). This implies that partial derivatives of hE xPer
differ from those of hPF and the stability condition (1.13) can not be guaranteed to
hold for hE xPer , although it is satisfied for hPF . In other words, the process for xt does
not necessarily inherit stability from the perfect foresight solution, because hσσx may
generate instability if the linear approximation of hstoch is insufficiently accurate.
Given that the uncertainty correction typically is small in most DSGE models, we
expect that most approximations by extended perturbation will be stable. In general,
any instability in the extended perturbation method and explosive sample paths only
occur if the approximation of the stochastic part of the policy function is insufficiently
accurate.
Going beyond third order, the stochastic component of the policy function is
approximated more accurately and this reduces the risk of getting unstable state
1.3. STABILITY PROPERTIES OF THE EXTENDED PERTURBATION METHOD 11
dynamics with the extended perturbation method. We can not guarantee a stable ap-
proximation for the same reason as provided at third order, because partial derivatives
of hstoch may violate the stability condition in (1.13) for hE xPer although satisfied for
hPF .
1.3.3 Testing for Stability
Given that extended perturbation does not necessarily provide a stable approxima-
tion, it seems useful to have a test to determine if a given approximation is stable or
not. The test we propose applies two simplifying assumptions to get an operational
version of the stability condition in (1.13). We first propose to only evaluate (1.13) on
a sparse grid containing extreme state values since unstable state dynamics are most
likely to appear at such points. To construct the grid, let Si =
l xi ,ux
i
for i = 1,2, ...,nx
contain the lower bound l xi and the upper bound ux
i of the i th state variable. The
values of
l xi ,ux
i
nx
i=1should cover the region where the approximation is used. Guid-
ance on how to set these bounds may be obtained from unconditional moments or
extreme values in a simulated sample path for the extended perturbation approxima-
tion. We then form the Cartesian set Sx≡ S1 ×S2 × ...×Snx having 2nx elements. Our
second simplifying assumption is only to consider the stability condition in (1.13)
when the rows in ∂h(k)/∂x′ are evaluated at the same point.4
Given these simplifying assumptions, the stability condition in (1.13) reduces to
the testable requirement that h(k) is contracting if there exists an integer k ≥ 1 such
that
max
∣∣∣∣∣∂h(k)
∂x′
(x,
ε(v)
d
k−1
d=1,σ
)∣∣∣∣∣ , for all x ∈ Sx and v = 1,2, ...,M
< 1. (1.15)
Here, each point in Sx is evaluated usingM sample paths of the structural innovationsε(v)
d
k−1
d=1to avoid that a non-stable system may satisfy the contraction condition
given a fortunate sample path for the innovations. The test may be carried out for dif-
ferent values of k and M. Some guidance on a reasonable value of k may be obtained
by implementing the test on a stable linear solution.5 We generally recommend using
a fairly large value of k, say 100 or 500, because it is easier for h(xt ,σ
)to display the
contraction property when iterated many periods forward in time. It should finally
be emphasized that this stability test is not limited to the extended perturbation
4At the expense of increasing the computational cost of the test, it is obvious that a finer grid for the
state variables may be considered and that rows in ∂h(k)/∂x′ could be evaluated at different points.5Given that the Jabocian ∂h(k)/∂x′ is computed by numerical differentiation, the most efficient
implementation of the test is to evaluate∣∣∣∂h
(jx
)/∂x′
∣∣∣ by gradually increasing jx and then stop for a
given x ∈ Sx when the condition is met, even though jx may be less than a pre-determined value of k. If
max
jx
x∈Sx
< k, then the stability condition in (1.15) is satisfied.
12 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
approximation, but may also be used for other approximations, including standard
perturbation as shown in Section 1.4 and 1.5.
1.4 The Neoclassical Growth Model
This section studies accuracy and stability of standard and extended perturbation us-
ing the neoclassical growth model, which constitutes the core of many DSGE models.
Throughout this section, a log-transformation is adopted to obtain approximations in
percentage deviation from steady state, as typically done in the literature. Our focus
is devoted to the performance of standard and extended perturbation at third order,
because they capture a time-varying uncertainty correction and remain computa-
tionally tractable for most DSGE models. We proceed by describing the neoclassical
growth model in Section 1.4.1 before analyzing accuracy in Section 1.4.2 and stability
in Section 1.4.3.
1.4.1 Model Description
A representative agent obtains utility from consumption ct and optimizes
Et
[∑∞l=0
βl
1−γc1−γt+l
], where β ∈ (
0,1)
is the subjective discount factor and γ controls
risk aversion. The optimization is subject to the real budget constraint ct + it = at kαt ,
where it denotes investment and α ∈ [0,1
]describes the production function. The
capital stock kt evolves as kt+1 = (1−δ)kt +it , where δ ∈ [0,1
]is the depreciation rate.
Total factor productivity at is exogenous and given by log at+1 = ρa log at +σaεa,t+1,
where εa,t+1 ∼NID(0,1). The optimality condition for the representative agent is
given by Et
[β
(ct+1/ct
)−γ (αat+1kαt+1 +1−δ
)]= 1.
1.4.2 Accuracy Analysis
Table 1.1 provides parameters for the neoclassical model.6 The left column of Figure
1.1 considers the benchmark specification and plots the approximated policy func-
tions for consumption when capital kt ≡ log(kt /kss
)ranges from −5 to +5 standard
deviations in a log-linearized solution. When technology is at its steady state level
(the middle row), all perturbation methods perform very well compared to a highly
accurate 10th order projection approximation, considered as a stand-in for the true
6In contrast to the paper by den Haan and de Wind (2012), we do not consider the specificationwith δ= 1 and γ= 1, where the closed-form solution is certainty equivalent and given by kt+1 =αβat kαtand ct = (1−αβ)at kαt . The reason being that the Extended Path and also extended perturbation for this
specification recover the true solution with gstoch (xt ,σ
)= 0 and hstoch (xt ,σ
)= 0.
1.4. THE NEOCLASSICAL GROWTH MODEL 13
solution (see Appendix B for further details).7 The top and bottom row of Figure 1.1
show that we draw the same conclusion, when the technology level is altered to −3
and +3 standard deviations, respectively. A more detailed inspection of the approxi-
mation errors in the right column of Figure 1.1 reveals two interesting findings. Firstly,
extended perturbation and standard perturbation at third order are highly accurate
and perform equally well. Secondly, extended perturbation improves substantially on
the accuracy provided by a first-order approximation and a perfect foresight solution,
where the latter case documents the benefit of correcting for uncertainty in the policy
function.
Our second specification labeled ’High risk aversion’ increases γ from 5 to 25 to
more clearly illustrate the benefit of removing approximation errors in the perfect
foresight component of the policy function. The middle chart in Figure 1.2 to the left
shows that standard perturbation at third order is unable to maintain monotonicity
and convexity of consumption for this specification - even when the technology level
is at the steady state. Note also how rapidly the accuracy of standard perturbation
deteriorates when moving away from steady state. In contrast, extended perturbation
is very close to the true solution and preserves both monotonicity and convexity of
consumption. The satisfying performance of extended perturbation is also evident
from the reported errors in the policy function which remain low even far from the
steady state. These findings suggest that the unsatisfying performance of standard
perturbation at third order is related to large errors in the perfect foresight component
of the policy function, whereas these errors are eliminated by extended perturbation.
To see where the perfect foresight solution struggles, we finally consider a speci-
fication labeled ’High variance’ with an ordinary degree of risk aversion(γ= 5
)but
with highly volatile technology shocks(σa = 0.1
). Charts to the left in Figure 1.3 show
that ignoring the uncertainty correction in the perfect foresight solution pushes con-
sumption above its true level. On the other hand, third order approximations using
either standard or extended perturbation perform much better due to the included
uncertainty correction.
We summarize the performance of standard and extended perturbation in Table
1.2 by reporting root mean squared errors (RMSEs) on a grid with 20 points uniformly
spaced along each dimension of the state variables. The RMSEs are computed using
log(ct /c tr ue
t
)= ct − c tr ue
t , where c tr uet denotes consumption in percentage deviation
from steady state in the true solution (i.e. in the projection approximation).
Extended and standard perturbation deliver broadly the same performance in
7See also Aruoba et al. (2006) who document the high accuracy of the projection method for theneoclassical growth model.
14 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
the benchmark specification, whereas extended perturbation dominates in the two
other specifications, as indicated by the bold figures in Table 1.2. The table also shows
that extended perturbation outperforms the perturbation +1 and +2 approximations
by den Haan and de Wind (2012) and a modified version of their algorithm named
perturbation +13r d and +23r d , where the system is closed using a third-order pertur-
bation approximation instead of the linearized solution as in den Haan and de Wind
(2012).8
The RMSEs from a simulated sample path of 20,000 observations (with a burn-in
of 1,000 observations) using the same set of innovations for technologyεa,t
20,000t=1 in
the approximations are reported in Table 1.3. Observations of capital and technol-
ogy close to the steady state appear frequently in this simulation and are therefore
assigned a higher weight than points far from the steady state. This is contrary to
the accuracy results computed in Table 1.2, where all points in the grid are weighted
equally. For our benchmark specification, standard perturbation at third order deliv-
ers the best performance with a RMSE of 7.57×10−6 and is marginally more accurate
than extended perturbation having a RMSE of 1.63×10−5. For our high risk aversion
specification, extended perturbation delivers the best performance, whereas the
standard perturbation approximation explodes. Extended perturbation also delivers
the lowest RMSEs in the high variance specification. We also note that extended per-
turbation is more accurate than the pruned third-order perturbation approximation
of Andreasen et al. (2013), which always ensures stability in contrast to extended
perturbation at third order. The second part of Table 1.3 reports the standard de-
viation of quarterly consumption ct = log(ct /css
)in the simulated samples. In our
benchmark specification we find that standard perturbation, pruning, and extended
perturbation all generate the same variability as the true solution (i.e. the projec-
tion approximation). In the two other specifications with stronger non-linearities,
extended perturbation displays the best performance with variability in consumption
closest to the true solution, as indicated by the bold figures in Table 1.3.
1.4.3 Stability Analysis
To analyze the stability properties of standard and extended perturbation beyond
simple inspection of sample paths, Figure 1.4 plots future capital in deviation from
the diagonal line dt , i.e. kt+1 −dt , as a function of kt .9 For the benchmark speci-
fication in the first row of Figure 1.4, the two perturbation methods induce stable
approximations for all values of technology, as decreasing lines correspond to kt+1
8We are in debt to Joris de Wind for suggesting this modifcation of their solution method to us.9The approximated functions of kt+1 are very close to the diagonal line for all values of kt , and
simply plotting kt+1 as a function of kt would therefore make it nearly impossible to analyze the stabilityproperties of the approximated functions.
1.4. THE NEOCLASSICAL GROWTH MODEL 15
first being above the diagonal line and then below this line when kt+1 −dt turns
negative. Hence, the equilibrium level of capital for a given technology level is pro-
vided by the intersection with the horizontal axis. We draw the same conclusion from
running our stability test, where standard and extended perturbation at third order
pass the test with k = 150, M = 50, and Sx constructed by setting the bounds for
capital and technology equal to ±4 and ±3 standard deviations in a log-linearized
solution, respectively. Note that we use wider bounds for capital compared to the
technology process, because non-linearities in the approximated law of motion for
capital may generate a wider distribution for capital than found in the log-linear
solution.
The corresponding stability plots with high risk aversion are given in the second
row of Figure 1.4. Extended perturbation once again returns a stable approximation,
but the inability of standard perturbation to maintain monotonicity and convexity of
consumption in this specification affects the law of motion for capital with kt+1 =at kαt +(1−δ)kt −ct and generates an unstable approximation. To realize this, consider
the case where technology is at the steady state. Although the function for kt+1 attains
positive values of kt+1 −dt when kt < 0 and intersects the horizontal axis around
0.5, the problematic feature relates to the second crossing with the horizontal axis
around 1.7. If kt exceeds this value, kt+1 is incorrectly increasing in kt because the
approximated consumption function does not maintain monotonicity and decreases
for higher values of kt , as shown in Figure 1.2. Accordingly, the law of motion for
capital diverts if kt exceeds this second crossing with the horizontal axis. The situation
is even more problematic when the technology level is either high or low, because
the function for kt+1 does not even intersect the horizontal axis and hence diverts.
Given that the process for technology moves between the high and low level in
Figure 1.4, this explains why we experienced an explosive sample path for standard
perturbation in Section 1.4.2. Indeed, using our test from Section 1.3.3, we reject
stability of standard perturbation at third order, because the system explodes when
we attempt to iterate it sufficiently many periods forward in time as required to
verify the contraction property. On the other hand, extended perturbation passes the
stability test with k = 350, M= 50, and Sx constructed using the same procedure as
for our benchmark specification.
The last row in Figure 1.4 shows that standard and extended perturbation generate
stable approximations in our specification with high volatility. The same conclusion
follows from our stability test with k = 150, M = 50, and Sx constructed using the
same procedure as in the two previous specifications.
16 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
1.5 A New Keynesian Model
We next explore accuracy and stability of standard and extended perturbation using a
New Keynesian model with price stickiness as in Calvo (1983). Two reasons motivate
our choice of model. Firstly, the New Keynesian model with Calvo pricing is one
of the most popular DSGE models. Secondly, and perhaps somewhat surprisingly,
some dimensions of this New Keynesian model are highly non-linear even for a
standard calibration. The strong non-linearities in the model also imply that standard
perturbation at third order easily generate explosive sample paths, suggesting that
unstable approximations are not only obtained at extreme calibrations, as found in
the neoclassical growth model.
To make the global projection method of our New Keynesian model computation-
ally feasible, we initially only consider technology shocks and defer the inclusion of
additional disturbances to Section 1.6. As in our analysis of the neoclassical growth
model, we adopt a log-transformation and study the performance of standard and
extended perturbation at third order. We proceed by describing the New Keynesian
model in Section 1.5.1, before studying accuracy in Section 1.5.2 and stability in
Section 1.5.3.
1.5.1 Model Description
A representative household maximizes
Ut = Et
∞∑l=0
βl
c1−φ2
t+l1−φ2
+φ0
(1−ht+l
)1−φ1
1−φ1
, (1.16)
where ct is consumption and ht is labor supply. In addition to a no-Ponzi-game
condition, the optimization is subject to the real budget constraint
ct +bt + it = ht wt + r kt kt + Rt−1bt−1
πt+di vt , (1.17)
where resources are allocated to consumption, one-period nominal bonds bt , and
investment it . Letting wt denote the real wage and r kt the real price of capital kt ,
the household receives i) labor income wt ht , ii) income from capital services sold
to firms r kt kt , iii) payoffs from bonds purchased in the previous period Rt−1bt−1/πt ,
and iv) dividends from firms di vt . Here, πt ≡ Pt /Pt−1 is gross inflation and Rt is the
gross nominal interest rate. The optimization of (1.16) is also subject to the law of
motion for capital kt+1 =(1−δ)
kt +it − κ2
(itkt
−ψ)2
kt , where κ≥ 0 introduces capital
adjustment costs based on ii /kt as in Jermann (1998). The constant ψ ensures that
these adjustment costs are zero in the steady state.
1.5. A NEW KEYNESIAN MODEL 17
We consider a perfectly competitive representative firm that produces final output
using yi ,t and the production function yt =(∫ 1
0 y(η−1)/ηi ,t di
)η/(η−1)with η > 1. This
generates the demand function yi ,t =(
Pi ,tPt
)−ηyt , with aggregate price level Pt =[∫ 1
0 P 1−ηi ,t di
]1/(1−η).
The intermediate goods are produced by monopolistic competitors using the
production function yi ,t = at kθi ,t h1−θi ,t , where technology at evolves according to
log at+1 = ρa log at +σaεa,t+1 with εa,t+1 ∼NID(0,1
). The i th firm sets Pi ,t , hi ,t , and
ki ,t by maximizing the present value of dividends, i.e.
Et
∞∑l=0
D t ,t+l Pt+l
(Pi ,t+l
Pt+l
)yi ,t+l − r k
t+l ki ,t+l −wt+l hi ,t+l
,
where D t ,t+l is the nominal stochastic discount factor for payments between time
period t and t + l . Beyond a no-Ponzi-game condition, the firm must satisfy demand
for the i th good. When setting prices, we follow Calvo (1983) and assume that only a
fraction α ∈ [0,1
)of firms set their prices optimally, with the remaining firms letting
Pi ,t = Pi ,t−1.
Finally, monetary policy is determined by the Taylor-rule
log
(Rt
Rss
)= ρR log
(Rt−1
Rss
)+ (
1−ρR)κπ log
(πt
πss
)+κy log
(yt
yss
) . (1.18)
The model’s equilibrium conditions are provided in Appendix C.10 We adopt a
relative standard parametrization for a quarterly model, where households have an
intertemporal elasticity of substitution of 0.5 (φ2 = 2), a Frisch labor supply elasticity
of 0.5(φ1 = 2
), and allocate one third of their time endowment to labor in steady
state (hss = 0.33). Firms optimally reset prices once a year on average (α= 0.75) and
impose a mark-up of 20% (η= 6). The main objective of the central bank is to stabilize
inflation (κπ = 1.5, κy = 0.125), subject to a desire to smooth changes in the policy
rate(ρR = 0.8
). Our parametrization is summarized in Table 1.4.
1.5.2 Accuracy Analysis
Our New Keynesian model can be summarized by the four control variables(ct , it ,πt , x2
t
), where x2
t denotes an auxiliary variable for the recursive representation
of firms’ first-order condition with respect to the optimal price. These control vari-
ables are a function of the states(Rt−1,kt , st , at
), with st denoting the price dispersion
10These conditions are derived in a technical appendix accompanying the paper. The technical ap-pendix is available from the homepage of the corresponding author.
18 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
index linked to the Calvo pricing. One way to display these policy functions is to con-
dition on representative values for the first three state variables, and plot the control
variables as a function of the remaining state variable, i.e. technology. This exercise
reveals that some of the largest differences between the approximation methods
considered appear for a low nominal interest rate, a low capital stock, and a high
value of the price dispersion index as the model here displays strong non-linearities.
To conserve space, we therefore focus on this state configuration in Figure 1.5, before
studying accuracy on a grid covering the entire state space.
The first chart of Figure 1.5 shows that extended perturbation captures most
of the non-linear pattern in the policy function for consumption and outperforms
standard perturbation for low levels of technology. Extended perturbation therefore
displays smaller errors with low values of technology (charts to the right in Figure
1.5), whereas the two methods display roughly similar performance for higher levels
of technology. The accuracy of the two perturbation solutions is evaluated using a
highly accurate 12th-order projection approximation, considered as a stand-in for
the true solution (see Appendix C for further details). The plots for the remaining
control variables also reveal that there generally is a significant gain in accuracy from
using extended instead of standard perturbation, in particular for investment and
the auxiliary control variable x2t .
To analyze accuracy on the entire state space, Table 1.5 reports RMSEs for the con-
sidered approximation methods on a grid with 20 points uniformly spaced along
each dimension of the state variables, giving a total of 204 = 160,000 points. As
in Section 1.4.2, we compute the RMSEs using log(zt /z tr ue
t
)= zt − z tr ue
t , where
zt ≡
ct , it ,πt , x2t
and the true solution is given by the projection approximation. We
find that standard perturbation at third order is more accurate than a log-linearized
approximation, but is generally outperformed by the perfect foresight solution.
Adding an uncertainty correction to the perfect foresight solution further improves ac-
curacy, and extended perturbation therefore delivers the best overall approximation
to the four control variables. Notable improvements in RMSEs from using extended
instead of standard perturbation appear for investment (0.0088 vs. 0.0172) and the
auxiliary control variable (0.0387 vs. 0.0558).
We also study accuracy on a simulated sample path of 20,000 observations (with
a burn-in of 1,000 observations) using the same set of innovations for technologyεa,t
20,000t=1 in all of the approximations. Table 1.6 shows that extended perturbation at
third order also in this setting is more accurate than standard perturbation. To explain
where some of this gain in accuracy comes from, Table 1.6 also reports the RMSEs for
a modified version of the standard perturbation solution, where the approximated
1.5. A NEW KEYNESIAN MODEL 19
transition function for st in the simulation is replaced by the exact expression, i.e.
st+1 = (1−α)1
1−η[
1−απη−1t
] ηη−1 +απηt st . (1.19)
That is, the price dispersion index is computed using (1.19) with πt =π3r dt , i.e. infla-
tion from the third-order perturbation approximation. Under the label "Perturbation:
3r d order, exact st " in Table 1.6, we find that using the exact transition function for
st lowers the RMSEs for all control variables, because we more accurately track the
non-linear evolution in st .11 Extended perturbation also includes the exact transition
function for st and additional non-linearities for the control variables (although only
under perfect foresight), and this explains why extended perturbation outperforms
this modified perturbation approximation. We also note that extended perturbation
is more accurate than the pruned third-order perturbation approximation which
always ensures stability. The second part of Table 1.6 reports quarterly standard devi-
ations in the simulated samples. We find minor negative biases for all approximations
compared to the projection method with extended perturbation having a lower bias
than standard perturbation.
1.5.3 Stability Analysis
The graphical stability analysis used in Section 1.4.3 is not applicable to our New
Keynesian model with three endogenous state variables. Instead, we use our stability
test from Section 1.3.3 to study the dynamic properties of standard and extended
perturbation and to understand why a given approximation may be unstable.
To run the stability test for the New Keynesian model, we first construct the set Sx .
The bounds for at are given by ±3 standard deviations of technology, while bounds
for the two endogenous state variables(Rt−1, kt
)are set to ±4 standard deviations in
a log-linearized solution, and hence slightly wider than for technology to account for
effects of non-linearities. The price dispersion index st is constant in a log-linearized
solution without steady state inflation, and we therefore use a simulated sample
path of extended perturbation to guide our bounds of −0.005 and 0.05. Using this
specification of Sx , we find that standard perturbation at third order passes our
stability test with k = 150 and M= 50. We also find that the extended perturbation
approximation is stable with k = 100 and M= 50.
Although standard perturbation at third order displays stable dynamics for the
considered specification, the approximation is fragile to even minor modifications.
We illustrate this point by increasing πss from 1.00 to 1.0015, giving an annual steady
11Unreported results show that using the exact transition function for all endogenous state variables incombination with the standard third-order perturbation approximation of the control variables do notdeliver a further improvement in accuracy.
20 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
state inflation rate of 0.6%. The bounds for(Rt−1, kt , at
)in the set Sx are determined
using the same procedure as in our benchmark specification, and we increase the
upper bound of st to 0.10. Our stability test reveals that extended perturbation re-
mains stable, whereas standard perturbation now induces unstable dynamics.12
Accordingly, the explosive behavior of standard perturbation is due to inaccuracies in
the perfect foresight component of the policy function. An inspection of the 24 = 16
state configurations in Sx reveals that it is only when we simultaneously have a low
nominal interest rate, a low capital stock, a high price dispersion index, and a low
technology level that the approximation explodes when iterated forward in time to
evaluate the contraction condition.13
To understand why standard perturbation displays unstable dynamics with πss =1.0015, consider Figure 1.6 plotting the policy function for inflation and the transition
equation for st at this state configuration. The bottom chart to the left shows that
inflation in a standard perturbation approximation increases sharply for higher
values of the price dispersion index st . This in turn leads to even higher values of
the price dispersion index in the next period st+1, as seen in the middle chart. This
then increases πt+1, which in turn increases st+2 and so on. In contrast, inflation
increases only slowly in st with extended perturbation, because this approximation
accounts for the upper bound(1/α
)1/(η−1) on inflation. This bound follows from
(1.19), as the term[
1−απη−1t
] ηη−1
implies complex numbers for inflation beyond(1/α
)1/(η−1). Accounting for the upper bound on inflation ensures broadly the same
moderate increase in st+1 for higher values of st as in the true solution (i.e. the
projection approximation) and explains why extended perturbation generates stable
dynamics.14
A careful inspection of Figure 1.6 reveals that the policy functions for πt and st+1
in standard perturbation are nearly identical for πss = 1.00 and πss = 1.0015 when
considered at a given value of st . The same applies for extended perturbation. Hence,
the main effect from introducing steady state inflation is that the distribution of st
moves to the right and attains an even longer right tail, as shown in the third column
of Figure 1.6. It is these extreme values of st that start a price-inflation spiral in the
standard perturbation approximation and generates explosive dynamics.
12This result is confirmed by simulating repeated sample paths using a standard third order perturba-tion approximation.
13At this critical state configuration, we interestingly find for extended perturbation both withπss = 1.00and πss = 1.0015 that h is contracting without iterating this function forward in time. This means thatthe number of considered sample paths M in the stability test is irrelevant at this state configuration forextended perturbation.
14We conjecture that the presence of the upper bound on inflation explains the relative high approxi-mation order needed in the projection method to obtain a sufficiently accurate approximation.
1.6. EFFICIENT IMPLEMENTATION OF EXTENDED PERTURBATION 21
1.6 Efficient Implementation of Extended Perturbation
Having documented the gain in accuracy and stability from extended perturbation,
we next address its computational costs. The first step of extended perturbation
involves computing derivatives of the model at the steady state, as outlined in Sec-
tion 1.2.2. This can be done within a few seconds for third order approximations
when using the optimized MATLAB codes of Binning (2013).15 A computationally
more demanding aspect of extended perturbation is to obtain the perfect foresight
component of the policy function by the Extended Path, as it requires solving a large
fixed-point problem. Although this fixed-point problem typically is solved within a
few iterations using the Newton-Raphson algorithm, the computational burden may
nevertheless be substantial if the perfect foresight solution is called repeatedly, for
instance when simulating a long sample path.
This section therefore analyzes the computational costs of the Extended Path
and presents several ways to reduce the time spent computing the perfect foresight
component of the policy function. That is, the focus of this section is entirely devoted
to the perfect foresight solution. We proceed as follows. Section 1.6.1 extends our New
Keynesian model to get a medium-sized DSGE model. Section 1.6.2 derives efficient
starting values for the fixed-point problem in the Extended Path, and Section 1.6.3 and
1.6.4 discuss how to efficiently set the terminal condition in the Extended Path. We
finally explore the possibility of occasionally using the perturbation approximation
of the perfect foresight component in the policy function in Section 1.6.5.
1.6.1 Expanding the New Keynesian Model
Given that many non-linear solution methods either perform poorly or become
infeasible in large models, we first extend our model from Section 1.5 to study the
computational complexity of extended perturbation on a fairly large model. We
therefore replace the utility function in (1.16) by
Ut = Et
∞∑l=0
βl dt+l
(ct+l −bct−1+l
)1−φ2
1−φ2+φ0
(1−ht+l
)1−φ1
1−φ1
,
where b introduces external habit formation and dt are preference shocks evolving
according to logdt+1 = ρd logdt +σdεd ,t with εd ,t ∼ NID(0,1
). We also augment
(1.18) with monetary policy shocks σRεR,t where εR,t ∼ NID(0,1
), and introduce
15Obtaining a third-order perturbation approximation to the model described below in Section 1.6.1with nine state variables takes only one second on a standard desktop in MATLAB 2014a using an Intel(R)Core(TM) i5-4200 CPU with 2.50 GHz.
22 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
investment specific shocks et by replacing (1.17) with
ct +bt + it /et = ht wt + r kt kt + Rt−1bt−1
πt+di vt ,
where loget+1 = ρe loget +σeεe,t+1 and εe,t+1 ∼NID(0,1
). Finally, trends in technol-
ogy are introduced through zt in the production function, i.e. yi ,t = at kθi ,t
(zt hi ,t
)1−θ
where log zt+1 = logµz,ss + log zt +σzεz,t+1 and εz,t+1 ∼NID(0,1
). As a result, this
New Keynesian model has five shocks and nine state variables, making its size com-
parable to many of the DSGE models typically used in the literature when studying
the business cycle (see for instance Christiano et al. (2005), Fernandez-Villaverde and
Rubio-Ramirez (2007), among others).16
For the simulation experiments below, we let b = 0.3, ρd = 0.98, ρe = 0.90, σd =0.015,σe = 0.01,σR = 0.0025, µz = 1.005, and πss = 1.005. Given the additional shocks
and steady state inflation of 2% per year, we initially eliminate the price-inflation spi-
ral in the standard perturbation approximation by letting κπ = 2.0 to make the central
bank more aggressive to deviations in the inflation gap. With the remaining parame-
ters as in Table 1.4, we obtain a calibration where standard third-order perturbation
generates stable dynamics. This specification is therefore referred to as the ’stable
perturbation calibration’. For our simulation experiments, it is also useful to consider
a setting where standard third-order perturbation explodes. We therefore lower κπ to
1.95 in our second calibration (which otherwise is identical to the first calibration)
and refer to this second specification as the ’explosive perturbation calibration’.
1.6.2 Efficient Starting Values for the Extended Path
It is essential to have good starting values to obtain fast convergence when solving the
fixed-point problem implied by the Extended Path. These starting values are typically
derived based on a first-order approximation. However, when the Extended Path is
used in the extended perturbation method, higher-order derivatives of the functions
g and h are already available as they are required to compute the uncertainty correc-
tions. It therefore seems natural to use these higher-order derivatives to improve the
starting values from a linearized solution. To minimize the computational burden
of derivingEt
[xt+i
],Et
[yt+i−1
]N
i=1, we use the perturbation method of Andreasen
and Zabczyk (2015).17 The method is outlined in Appendix D and illustrated for a
16The computational cost of running the Extended Path is largely unaffected by the number of exoge-nous shocks because they are concentrated out when solving the fixed-point problem, as explained inAppendix A. Hence, the execution time reported below for the perfect foresight component of the policyfunction should be representative of the computational costs implied by models with more than fivestructural shocks.
17Given a third order perturbation approximation and N = 200, it takes only 0.04 and 0.40 seconds tocompute the required loadings for the conditional expectations up to second and third order, respectively,in our New Keynesian model.
1.6. EFFICIENT IMPLEMENTATION OF EXTENDED PERTURBATION 23
third-order approximation but generalizes easily to any orders.
We evaluate the effect of different starting values in the Extended Path by simu-
lating a sample path of 5,000 observations. For the stable perturbation calibration it
takes 2,465 seconds to generate this sample path when using starting values from
a linearized solution, but only 1,926 seconds when the starting values are obtained
from a third-order approximation. This is equivalent to an efficiency gain of 21.9%.18
The corresponding execution times for the explosive perturbation calibration are
2,878 and 2,478 seconds, respectively, and imply an efficiency gain of 13.9%. The
execution times are slightly higher in this second calibration, because it is time-
consuming to obtain convergence in the Extended Path for state values far from
the steady state where standard perturbation explodes.19 Based on this finding, we
therefore use starting values from a third-order approximation in the remaining part
of this section when testing the performance of the Extended Path.
1.6.3 The Terminal Condition in the Extended Path
The execution time of the Extended Path is also affected by the length of the horizon
N considered before closing the DSGE model with a terminal condition for the
control variables yt+N . As the size of the fixed-point problem in the Extended Path
grows linearly in N , it is desirable to use a relatively low value of N (see Appendix
A). Adjemian and Julliard (2010) note that N can be lowered if the standard terminal
condition of yt+N = yss is replaced by yt+N = Et
[y1st
t+N
], where y1st
t+N denotes the
control variables at time t+N in a first-order approximation. When using this terminal
condition, N should be set such that Et
[y1st
t+N
]is within the radius of convergence for
the linearized solution, and this clearly is a weaker requirement on N than imposing
yt+N = yss . However, higher-order approximations to the conditional expectation of
yt+N are easily obtained from the process of computing efficient starting values in
Section 1.6.2, and it therefore seems natural to consider the terminal condition yt+N =Et
[y3r d
t+N
], with y3r d
t+N denoting the control variables at time t + N in a third-order
approximation. Compared to using Et
[y1st
t+N
], this alternative terminal condition
imposes even weaker requirements on N , because the radius of convergence is larger
for a third-order approximation than a linearized solution.
Table 1.7 analyzes the accuracy of the Extended Path with respect to the hori-
zon N and the choice of terminal condition on a simulated sample path of 5,000
observations. For both calibrations, all terminal conditions imply the same level of
18The computations are carried out in MATLAB 2014a using an Intel(R) Core(TM) i5-4200 CPU with2.50 GHz and a horizon of N = 200 in the Extended Path.
19For state values where Extended Path struggles to converge using starting values from a third-orderperturbation approximation, we re-run the Extended Path using the solution in the previous period asstarting values to obtain convergence. These alternative starting values are used for 0.24% and 0.36% of theobservations when using initial starting values from a first- and third-order approximation, respectively.
24 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
accuracy with a long horizon of N = 250, which serve as the benchmark for com-
puting the RMSEs. When we gradually reduce the horizon, the standard terminal
condition of yt+N = yss is clearly less accurate compared to using the terminal condi-
tion from a linearized solution. The final column in Table 1.7 shows that a third-order
approximation for the terminal condition is even more accurate than the linearized
solution, except when N = 50 in the explosive perturbation calibration. Hence, we
generally obtain the highest level of accuracy in the Extended Path by using a third-
order approximation to compute the terminal condition, and we therefore adopt this
specification in the remainder of the section.
1.6.4 A State-dependent Horizon in the Extended Path
We have so far assumed that the same horizon applies to all state values in the
Extended Path. But if xt is close to the steady state, a relative short horizon should
be sufficient to ensure that Et
[y3r d
t+N
]is within the radius of convergence. We next
exploit this observation to introduce a state-dependent horizon N∗, where the aim is
to dynamically adjust the horizon to reduce the computational cost of the Extended
Path. We implement this idea by the rule
N∗ (Dss ,xt
)= min
N ∈N : max
∣∣∣∣Et
[y3r d
t+N
]∣∣∣∣≤ Dss st. N ∈ [Nmin, Nmax
],
(1.20)
where Dss denotes the tolerated distance of Et
[y3r d
t+N
]from the steady state. That
is, we use the shortest horizon where the largest element in Et
[y3r d
t+N
]is within the
distance Dss from the steady state, subject to N ∈ [Nmin, Nmax
]. This implies that N∗
depends on Dss and the current state xt through Et
[y3r d
t+N
], as indicated in (1.20).
Table 1.8 evaluates the performance of N∗ with Nmin = 20 and Nmax = 200 on
a simulated sample path of 5,000 observations. The execution time with a fixed
horizon of N = 200 corresponds to Dss = 0 and requires 385 seconds per 1,000 draws.
Introducing the state-dependent horizon with Dss = 0.01 implies hardly any loss of
accuracy (RMSE = 7.91×10−5), but reduces the execution time to just 196 seconds
per 1,000 draws due to an average horizon of just 87 time periods. For larger values of
Dss , the average horizon and the execution time fall even further but it also affects
the precision of the Extended Path, which only outperforms the standard third-order
perturbation approximation when Dss ≤ 0.02.
Turning to the explosive perturbation calibration in the lower part of Table 1.8, we
once again find that a state-dependent horizon lowers the execution time of the Ex-
tended Path with only a small loss in accuracy. For instance, with Dss = 0.02 we have
a RMSE of 2.00×10−4, and it only takes 263 seconds per 1,000 draws compared to
496 seconds with a fixed horizon of N = 200. Table 1.8 also shows that Dss should not
1.6. EFFICIENT IMPLEMENTATION OF EXTENDED PERTURBATION 25
exceed 0.05 for the Extended Path to outperform the pruned third-order perturbation
approximation, which is a natural benchmark given that the unpruned perturbation
approximation explodes for this calibration. Compared to the Extended Path, we
also note that the pruned approximation in this case displays an impressive high
precision with a RMSE of 0.0072, given that it only uses 0.12 second per 1,000 draws.
Based on these findings we conclude that a state-dependent horizon in the Extended
Path may substantially reduce execution time with only a minimal loss of accuracy
for sufficiently low values of Dss .
1.6.5 Using the Perturbation Approximation of the Perfect Foresight
Component
We have so far used the Extended Path to compute the perfect foresight component of
the policy function for all state values. But a third-order perturbation approximation
of this component may for some state values be sufficiently accurate. This may be
the case if xt is close to the steady state or if the model is nearly linear along some
dimensions of the state space. Exploiting this observation should substantially reduce
the execution time for extended perturbation as it no longer relies on the Extended
Path for all state values.
To formalize this idea, let y3r dt denote the third-order perturbation approximation
of the control variables at xt . The perturbation approximation of Et
[y3r d
t+1
]is derived
in Section 1.6.2, and x3r dt+1 follows directly from the perturbation approximation of the
state equations. We then use (1.1) to compute the Euler-equation residuals under
perfect foresight of the standard perturbation approximation at time t , i.e. Ψt ≡f(
xt ,x3r dt+1,y3r d
t ,Et
[y3r d
t+1
]), where f is expressed in unit-free terms. Next, let EE denote
the tolerated Euler-equations errors, and consider the approximation
yt = 1max|Ψt |≤EE
g3r d (xt
)+ (1−1
max|Ψt |≤EE)gPF (
xt)
xt+1 = 1max|Ψt |≤EE
h3r d (xt
)+ (1−1
max|Ψt |≤EE)hPF (
xt)+σηεt+1
where 1max|Ψt |≤EE
is the indicator function. Here, g3r d(xt
)denotes the standard
perturbation approximation of the perfect foresight component of the policy function,
and similarly for h3r d(xt
). That is, we use standard perturbation when it is sufficiently
accurate, i.e. max∣∣Ψt
∣∣≤ EE , otherwise the Extended Path is used. When using the
Extended Path in this context, we also apply EE to determine convergence of the fixed-
point solver and stop iterating the Newton-Rahpson algorithm when max∣∣Ψt
∣∣≤ EE .
This is a weaker convergence criteria than previously used where all Euler-equation
residuals until horizon N should be less than 10−10. We adopt this modification of the
26 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
Extended Path to make its convergence criteria comparable to the accuracy condition
used for the perturbation approximation.20
The performance of this combined approximation to the perfect foresight compo-
nent of the policy function is shown in Table 1.9 for the stable perturbation calibration
using a simulated sample path of 5,000 observations. The first part of the table im-
poses Dss = 0 and uses a fixed horizon of N = 200 in the Extended Path. Letting
EE = 5×10−5, the Extended Path is only used for 27% of the observations, and this
allows us to generate 1,000 draws in just 58 seconds compared to 385 seconds with
EE = 0. This sizable reduction in execution time is achieved with nearly no loss
in accuracy as seen from the RMSEs of 3.28× 10−5. For larger values of EE , even
more observations are computed by the perturbation approximation and this further
reduces the computational cost at the expense of a small loss in accuracy.21 The
remaining part of Table 1.9 adds a state-dependent horizon to the Extended Path and
this further lowers the execution time. For instance, using Dss = 0.01 and EE = 0.0001,
we are able to generate 1,000 draws in just 18 seconds with nearly no loss in accuracy
(RMSEs= 6.24×10−5).
The results for the explosive perturbation calibration in Table 1.10 also document
a large reduction in execution costs with only a small loss in accuracy by occasion-
ally using the perturbation approximation. Contrary to the previous calibration, the
highest level of EE = 0.01 does not imply that all observations are computed by the
perturbation approximation, because the Extended Path is used for roughly 1% of
the observations, where standard perturbation would generate explosive dynamics.
Hence, in a parametrization of extended perturbation with a relatively high value of
EE , we only use the Extended Path to avoid explosive sample paths and otherwise rely
on the standard perturbation approximation. This is illustrated in Figure 1.7, where
we plot the distance of the state variables from steady state∥∥xt
∥∥=√∑nx
i=1 x2i ,t for the
part of our simulated sample where standard perturbation explodes. With a high
value of EE as considered in the bottom chart, extended perturbation only relies on
the Extended Path (marked by gray areas) from observation 4125 to 4250 where stan-
dard perturbation explodes, as shown by the diverting blue line. Hence, the suggested
rule for combining the perfect foresight component of standard perturbation and
the Extended Path may also be interpreted as a way to ensure stability of a standard
perturbation approximation. The case with a relatively low value of EE is considered
in the top chart of Figure 1.7, where the Extended Path is used to avoid explosive
20The exception is when convergence can not be obtained by the standard or extended Newton-Raphson algorithm and we instead use the Levenberg-Marquardt routine to minimizeΣt+N−1
i=t Ψ′i Ψi using
standard convergence criteria.21When EE = 0.01 all observations are computed by the perturbation approximation. Its execution
time is here higher than reported in Table 1.8, mainly due to the computational costs of obtainingΨt .
1.7. CONCLUSION 27
dynamics but also to improve accuracy when standard perturbation does not explode.
Based on these findings we conclude that the computational costs of obtaining
the perfect foresight component of the policy function can be greatly reduced by
occasionally using the perturbation approximation. Adopting a state-dependent hori-
zon in the Extended Path reduces the execution time further and makes extended
perturbation sufficiently fast to incorporate the approximation in estimation routines.
The most obvious estimators are probably simulated method of moments following
Duffie and Singleton (1993) or Indirect Inference with moments obtained from an
auxiliary model (Smith (1993)). Another appealling alternative is to consider the
quasi-maximum likelihood approach suggested by Andreasen (2013), as it requires
relatively few evaluations of the policy function.
1.7 Conclusion
This paper introduces the extended perturbation method which improves accuracy
and stability of standard perturbation by using a better approximation to the per-
fect foresight component of the policy function. For the neoclassical growth model
and a New Keynesian model with Calvo pricing, we find that extended perturbation
achieves higher accuracy than standard perturbation. We also find that the gain in
accuracy is sufficient to generate stable approximations by extended perturbation
when standard perturbation explodes. Our results therefore suggest that the explo-
sive behavior of standard perturbation reported in the literature may be related to
inaccuracies in the perfect foresight component of the policy function and may be
eliminated with the extended perturbation method. To reduce the computational
costs of implementing extended perturbation, we also introduce several improve-
ments of the Extended Path which substantially lowers execution costs and makes
our method feasible for estimation purposes.
Acknowledgments
We thank Rhys Bidder, Lawrence Christiano, Vasco Curdia, Anders B. Kock, Eric Swan-
son, Joris de Wind, Konstantinos Theodoridis, and participants at the econometrics
workshop at University of Pennsylvania for useful comments and discussions. M.
Andreasen greatly acknowledges financial support from the Danish e-Infrastructure
Coorporation (DeIC). We also appreciate financial support to CREATES - Center for
Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish
National Research Foundation.
28 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
References
ADJEMIAN, S. AND M. JULLIARD (2010): “Dealing with ZLB in DSGE models: An
application to the Japanese economy,” ESRI Discussion Paper.
——— (2013): “Stochastic Extended Path Approach,” Working Paper.
AN, S. AND F. SCHORFHEIDE (2007): “Bayesian Analysis of DSGE Models,” Econometric
Review, 26.
ANDREASEN, M. M. (2012a): “An Estimated DSGE Model: Explaining Variation in
Nominal Term Premia, Real Term Premia, and Inflation Risk Premia,” European
Economic Review, 56.
——— (2012b): “On the Effects of Rare Disasters and Uncertainty Shocks for Risk
Premia in Non-Linear DSGE Models,” Review of Economic Dynamics, 15.
——— (2013): “Non-Linear DSGE Models and the Central Difference Kalman Filter,”
Journal of Applied Econometrics, 28.
ANDREASEN, M. M., J. FERNANDEZ-VILLARVERDE, AND J. F. RUBIO-RAMIREZ (2013):
“Calculating and Using Second-order Accurate Solutions of Discrete Time Dynamic
Equilibrium Models,” NBER Working Paper.
ANDREASEN, M. M. AND P. ZABCZYK (2015): “Efficient Bond Price Approximations
in Non-Linear Equilibrium-Based Term Structure Models,” Studies in Nonlinear
Dynamics and Econometrics, 19.
ARUOBA, S. B., J. FERNANDEZ-VILLAVERDE, AND J. F. RUBIO-RAMIREZ (2006): “An
alternative methodology for solving nonlinear forward-looking models,” Journal of
Economic Dynamics & Control, 30.
BINNING, A. (2013): “Solving second and third-order approximations to DSGE models:
A recursive Sylvester equation solution,” Norges Bank, Working Paper.
BOUCEKKINE, R. (1995): “An alternative methodology for solving nonlinear forward-
looking models,” Journal of Economic Dynamics & Control, 19.
CALVO, G. A. (1983): “Staggered Prices in a Utility-Maximizing Framework,” Journal
of Monetary Economics, 12.
CHRISTIANO, L. J., M. EICHENBAUM, AND C. L. EVANS (2005): “Nominal Rigidities and
the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy,
113.
1.7. CONCLUSION 29
DEN HAAN, W. AND J. DE WIND (2012): “Nonlinear and stable perturbation-based
approximations,” Journal of Economic Dynamics & Control, 36.
DUFFIE, D. AND K. J. SINGLETON (1993): “Simulated Moments Estimation of Markov
Models of Asset Prices,” Econometrica, 61.
FAIR, R. C. AND J. B. TAYLOR (1983): “Solution and Maximum Likelihood Estimation
of Dynamic Nonlinear Rational Expectations Models,” Econometrica, 51.
FERNANDEZ-VILLAVERDE, J., P. GUERRON-QUINTANA, J. F. RUBIO-RAMIREZ, AND
M. URIBE (2011): “Risk Matters: The Real Effects of Volatility Shocks,” American
Economic Review, 101.
FERNANDEZ-VILLAVERDE, J. AND J. F. RUBIO-RAMIREZ (2007): “Estimating Macroeco-
nomic Models: A Likelihood Approach,” Review of Economic Studies, 74.
GUU, S.-M. AND K. L. JUDD (1997): “Asymptotic methods for aggregate growth mod-
els,” Journal of Economic Dynamics & Control, 21.
JERMANN, U. J. (1998): “Asset Pricing in Production Economies,” Journal of Monetary
Economics, 41.
JUDD, K. L. (1992): “Projection Method for Solving Aggregate Growth Models,” Journal
of Economic Theory, 58.
KIM, J., S. KIM, E. SCHAUMBURG, AND C. A. SIMS (2008): “Calculating and Using
Second-order Accurate Solutions of Discrete Time Dynamic Equilibrium Models,”
Journal of Economic Dynamics & Control, 32.
KIM, J. AND F. J. RUGE-MURCIA (2009): “How much inflation is necessary to grease
the wheels?” Journal of Monetary Economics, 56.
KING, R. G. AND S. T. REBELO (1999): “Resuscitating Real Business Cycles,” Handbook
of Macroeconomics.
LOMBARDO, G. AND H. UHLIG (2014): “A Theory of Pruning,” European Central Bank,
Working Paper Series.
POTSCHER, B. M. AND I. PRUCHA (1997): “Dynamic Nonlinear Econometric Models:
Asymptotic Theory,” Springer.
RUDEBUSCH, G. D. AND E. SWANSON (2012): “The Bond Premium in a DSGE Model
with Long-Run Real and Nominal Risks,” American Economic Journal: Macroeco-
nomics, 4.
30 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
RUGE-MURCIA, F. (2012): “Estimating nonlinear DSGE models by the simulated
method of moments: With an application to business cycles,” Journal of Economic
Dynamics & Control, 35.
SCHMITT-GROHE, S. AND M. URIBE (2004): “Solving Dynamic General Equilibrium
Models Using a Second-order Approximation to the policy function,” Journal of
Economic Dynamics & Control, 28.
SMITH, J. A. A. (1993): “Estimating Nonlinear Time-Series Models Using Simulated
Vector Autoregressions,” Journal of Applied Econometrics, 8.
1.7. CONCLUSION 31
Appendix A: The Extended Path
Obtaining the Perfect Foresight Solution
Uncertainty about future shocks is absent under perfect foresight, implying that (1.1)
reduces to
f(xt ,xt+1,yt ,yt+1
)= 0 for all t = 1,2, ... (1.21)
The system contains an infinite number of equations and can not be solved without
simplifying assumptions. The approach in the Extended Path of Fair and Taylor (1983)
is to truncate the problem at some finite horizon N , after which the variables are
assumed to be constant, for instance at their steady state values. Hence, the approxi-
mation errors from the truncation decrease in N and can be made arbitrary small
for an appropriately chosen horizon N (see Fair and Taylor (1983) and Boucekkine
(1995)). Thus, the Extended Path closes the infinite system in (1.21) by a terminal
value for yt+N , giving rise to the finite dimensional system
f(xt ,xt+1,yt ,yt+1
)= 0n×1 (1.22)
f(xt+1,xt+2,yt+1,yt+2
)= 0n×1
f(xt+2,xt+3,yt+2,yt+3
)= 0n×1
...
f(xt+N−1,xt+N ,yt+N−1,yt+N
)= 0n×1.
The initial state xt and yt+N are known by assumption, whereas we solve for the n∗N
unknowns(yt ,xt+1,yt+1,xt+2,yt+2, ...,xt+N−1,yt+N−1,xt+N
)using the same number
of equations in (1.22). Hence, the perfect foresight solution is obtained by the fixed-
point problem in (1.22), provided the solution is unaffected when increasing N (see
Fair and Taylor (1983)).
To reduce the computational burden when solving (1.22), we concentrate outx2,t+ j
N
j=1as they can be computed directly by iterating on (1.2). The concentrated
fixed-point problem is given by
f1(x1,t ,x1,t+1,yt ,yt+1
)= 0n1×1 (1.23)
f1(x1,t+1,x1,t+2,yt+1,yt+2
)= 0n1×1
f1(x1,t+2,x1,t+3,yt+2,yt+3
)= 0n1×1
...
f1(x1,t+N−1,x1,t+N ,yt+N−1,yt+N
)= 0n1×1,
where we introduce the partitioning
f(xt ,xt+1,yt ,yt+1
)≡ [f1
(x1,t ,x1,t+1,yt ,yt+1
)f2
(x2,t ,x2,t+1
) ].
32 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
The function f2 (·) has dimensions n2 ×1, where n2 ≡ nx2 , and contains the law of
motions for the exogenous variables, whereas the function f1 (·) has dimensions n1×1
with n1 ≡ ny +nx1 and contains all the remaining equilibrium conditions. Note that
we suppress the dependence of x2,t in f1 (·) to reduce the notational burden. The
concentrated system in (1.23) has only n1 ∗N equations with the same number of
unknowns. For numerical stability of the proposed routines below, we recommend
expressing the residuals of f in terms of unit-free errors.
Computing the Perfect Foresight Solution Numerically
To solve the fixed-point problem in (1.23), let
Z ≡[
yt yt+1 ... yt+N−2 yt+N−1
x1,t+1 x1,t+2 ... x1,t+N−1 x1,t+N
],
meaning that (1.23) can be condensely expressed as
F (Z)≡
f1
(x1,t ,x1,t+1,yt ,yt+1
)f1
(x1,t+1,x1,t+2,yt+1,yt+2
)f1
(x1,t+2,x1,t+3,yt+2,yt+3
)...
f1(x1,t+N−1,x1,t+N ,yt+N−1,yt+N
)
= 0.
Linearizing this system around the point Z∗ gives
F (Z) ≈ F(Z∗)+ J
(Z∗)(
vec (Z)− vec(Z∗))
, (1.24)
where J(Z∗)
denotes the Jacobian evaluated at Z∗. That is, J(Z∗)≡ ∂F(Z)
∂vec(Z)′∣∣∣
Z=Z∗ , hav-
ing dimensions(n1 ∗N
)× (n1 ∗N
), and vec (Z) is the stacked vector, i.e.
vec (Z) =
yt
x1,t+1
yt+1
x1,t+2
...
yt+N−1
x1,t+N
,
with dimension(n1 ∗N
)×1.
The Standard Newton-Raphson Routine
The most efficient way to solve (1.23) is the Newton-Raphson routine, i.e. to iterate
on
vec(Zi+1
)= vec
(Zi
)− J
(Zi
)−1F
(Zi
)
1.7. CONCLUSION 33
until convergence, where Zi denotes the value of Z at iteration i . This routine is
sometimes referred to as a Newton-Raphson relaxation algorithm (see Boucekkine
(1995)), and we compute J(Zi
)−1F
(Zi
)using the efficient method of Boucekkine
(1995).
The Extended Newton-Raphson Routine
The standard Newton-Raphson routine may not converge if the problem is very
non-linear. In this case an extended Newton-Raphson routine is considered
vec(Zi+1 (
δ))= vec
(Zi
)−δ×
(J(Zi
))−1
F(Zi
), (1.25)
where the scaling parameter δ ∈ R is determined by minδ F(Zi+1
(δ))′
F(Zi+1
(δ))
,
solved by a rough grid search. That is, δ accounts for the possibility that a linear
approximation of F (Z) may be insufficiently accurate. We refer to this algorithm as
an extended Newton-Raphson relaxation routine.
Minimizing the Squared Model-Residuals by the LM Optimizer
If the standard and extended Newton-Raphson routine are unsuccessful in solving
the fixed-point problem in (1.23), then the Levenberg-Marquardt (LM) optimizer is
used to minimize F (Z)′ F (Z) across Z.
Appendix B: The Neoclassical Growth Model: A ProjectionApproximation
We approximate consumption using a 10th order Chebyshev polynomial constructed
by the tensor of 10th order polynomials for capital and technology. Capital in period
t +1 is obtained by using the law of motion for capital. The conditional expectation
in the consumption Euler-equation is evaluated using Gauss-Hermite quadratures
with five points. The grid for the state variables is determined from the unconditional
standard deviations in a log-linearized approximation, multiplied by ±5 for capital
and by ±3 for technology. We determine the loadings in the Chebyshev polynomial for
consumption by the collocation method (see Judd (1992)). To evaluate the accuracy
of the approximation, we consider a grid using 20 uniformly spaced points along
each of the state dimensions, i.e. a total of 400 points. The largest absolute errors are:
i) 4.86×10−15 in the benchmark specification, ii) 1.87×10−11 with high risk aversion,
and iii) 4.99×10−6 with high variance where the residuals are expressed in unit-free
errors.
34 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
Appendix C: The New Keynesian Model
Equilibrium Conditions
The Households
Eq. 1 λt = c−φ2t
Eq. 2 qtλt =βλt+1[r kt+1 +qt+1
(1−δ)−qt+1
κ2
(it+1kt+1
− Isskss
)2 +qt+1κ(
it+1kt+1
− Isskss
)it+1kt+1
]
Eq. 3 φ0(1−ht
)−φ1 =λt wt
Eq. 4 1 = qt
(1−κ
(iikt
− Isskss
))Eq. 5 λt =βRt
[λt+1πt+1
]The Firms
Eq. 6 mct at(1−θ)( ht
kt
)−θ = wt
Eq. 7 at mctθ(
htkt
)1−θ = r kt
Eq. 8 (η−1)x2t
η = yt mct p−η−1t +
[αβλt+1
λt
(pt
pt+1
)−η−1πηt+1
(η−1)x2t+1
η
]Eq. 9 x2
t = yt p−ηt +
[αβλt+1
λt
(pt
pt+1
)−ηπη−1t+1 x2
t+1
]Eq. 10 1 = (1−α) p1−η
t +α(
1πt
)1−η
The Central Bank
Eq. 11 logRt − logRss = ρR(logRt−1 − logRss
)+ (1−ρR
)(κπ log
(πtπss
)+κy log
(ytyss
))Other relations
Eq. 12 at kθt h1−θt = yt st+1
Eq. 13 st+1 = (1−α) p−ηt +απηt st
Eq. 14 kt+1 =(1−δ)
kt + it − κ2
(iikt
− Isskss
)2kt
Eq. 15 yt = ct + it
Exogenous processesEq. 16 log at+1 = ρa log at +σaεa,t+1
Here, the expectation operator has been omitted for notational simplicity. In
addition to the variables introduced in Section 1.5.1, λt is the household’s lagrange
multiplier for the budget constraint, qt is the lagrange multiplier for the law of motion
for capital, mct is firms’ marginal costs, and x2t is an auxiliary control variable needed
to obtain an exact recursive representation of the first-order condition for the optimal
real price pt of firms that adjust their prices in a given time period.
A Projection Approximation
When implementing the projection approximation, it is convenient to reduce the
number of control variables to minimize the number of unknown coefficients in
1.7. CONCLUSION 35
the Chebyshev polynomials. We therefore observe that the number of control vari-
ables can be greatly reduced by substituting the following expressions into the key
equations given below:
• Eq 1: λt = c−φ2t
• Eq 15: yt = ct + it
• Eq 10: pt = 1−α
(1πt
)1−η
1−α
11−η
• Eq 12: at kθt(ht
)1−θ = yt
((1−α) p−η
t +απηt st
)m
at kθt(ht
)1−θ = yt st+1
m
ht =[
yt
((1−α)p
−ηt +απηt st
)at kθt
] 11−θ
• Eq 3: wt =φ0(1−ht
)−φ1 /λt
• Eq 6: mct = wt
at (1−θ)(
htkt
)−θ
• Eq 7: r kt = at mctθ
(htkt
)1−θ
• Eq 8: qt = 1
1−κ2
(iikt
− Isskss
)
• Eq 11: Rt = explogRss +ρR
(logRt−1 − logRss
)+(
1−ρR)(κπ log
(πtπss
)+κy log
(ytyss
))
Thus we can write the system condensely as:
36 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
The Households
Eq. 2 qtλt =βλt+1[r kt+1 +qt+1
(1−δ)−qt+1
κ22
(it+1kt+1
− Isskss
)2
+qt+1κ2
(it+1kt+1
− Isskss
)it+1kt+1
]
Eq. 5 λt =βRt
[λt+1πt+1
]The Firms
Eq. 8 (η−1)x2t
η = yt mct p−η−1t +
[αβλt+1
λt
(pt
pt+1
)−η−1 (1
πt+1
)−η (η−1)x2t+1
η
]Eq. 9 x2
t = yt p−ηt +
[αβλt+1
λt
(pt
pt+1
)−η (1
πt+1
)1−ηx2
t+1
]Other relations
Eq. 13 st+1 = (1−α) p−ηt +απηt st
Eq. 14 kt+1 =(1−δ)
kt + it − κ22
(iikt
− Isskss
)2kt
Exogenous processesEq. 16 log at+1 = ρa log at +εa,t+1
The reduced system has four states(kt , st ,Rt−1, at
)and only four control variables
(ct , it ,πt , x2t ). We approximate the controls by Chebyshev polynomials using complete
polynomials, and obtain the control variables in period t +1 from Eq 13, 14, and
16 to get states in period t +1. The conditional expectations in Eq 2, 5, 8, and 9 are
evaluated using Gauss-Hermite quadratures with five points. The grid for the state
variables is constructed using 11 points along each dimension, where the points
are determined as the Chebyshev nodes. That is, we use a total of 114 = 14,641 grid
points. The upper and lower bounds along each dimension is determined from the
unconditional standard deviation of the states in a log-linearized approximation,
multiplied by ±3 for Rt−1, kt , and at . For the price dispersion index, st , the range is
set to −0.005 to 0.05. Given that we have more nodes than unknown coefficients in
the Chebyshev polynomials, we determine these coefficients by least squares (see
Judd (1992)). The optimization was implemented in FORTRAN 90 and executed on a
computer cluster using about 80 to 100 CPUs. Multiprocessing was exploited when
computing the Jacobian numerically in the Levenberg-Marquardt optimizer. In the
optimization, the solution at approximation order m was used when starting the
optimization at order m + 1. On the grid used for the approximation, the largest
residual is 1.39×10−5 when measured in terms of unit-free errors for the 12th order
approximation.
Appendix D: Conditional Expectations in DSGE Models
Consider the case where the DSGE model reports the endogenous variable rt and
we want to compute conditional expectations of this variable, i.e. r1,t ≡ Et[rt+1
],
1.7. CONCLUSION 37
r2,t ≡ Et[rt+2
], r3,t ≡ Et
[rt+3
], etc.22 The law of iterated expectations implies r2,t ≡
Et[rt+2
]= Et
[Et+1
[rt+2
]]= Et[r1,t+1
]and so on. Hence, only a formula for comput-
ing pt ≡ Et[rt+1
]is needed because all other expectations can be found be iterating
on this formula. We therefore consider the problem
p(xt
)= Et
[r(xt+1
)],
where we omit the perturbation parameter σ, given our focus on the perfect foresight
solution.23 We then observe that
F(xt
)≡ Et
[−p
(xt
)+ r(h
(xt
)+σηεt+1
)]= 0, (1.26)
because xt+1 = h(xt
)+σηεt+1. Note then that (1.26) must hold for all values of xt . This
allow us to compute all derivatives of p with respect to xt around the deterministic
steady state, i.e. xt = xss and σ= 0, given derivatives of h(xt
)and r
(xt+1
)around the
same point. For the indices we adopt the convention that the subscript indicates the
order of differentiation. I.e. a subscript 1 is for the first time we take derivatives and
so on. Thus,
α1,α2,α3 = 1,2, ...,nx γ1,γ2,γ3 = 1,2, ...,nx .
To compute the first-order terms, straightforward differentiation of (1.26) implies
[px
]α1
= [rx
]γ1
[hx
]γ1α1
,
or in standard matrix notation
px(1; :
)= rx(1, :
)hx.
The second-order terms are given by
[pxx
]α1α2
= [rxx
]γ1γ2
[hx
]γ2α2
[hx
]γ1α1
+ [rx
]γ1
[hxx
]γ1α1α2
,
or in the standard matrix notation
pxx = h′xrxxhx +
nx∑γ1=1
rx(1,γ1
)hxx
(γ1, :, :
),
22If the variable of interest is a control variable, then the function r(xt+1
)follows from the function
g (·). If the variable of interest is a state variable, then we let rt ≡ i′xt to obtain moments for the i ’th statevariable with i
(k,1
)= 1 for k = i , otherwise i(k,1
)= 0.23Derivatives of the conditional expectation with respect to the perturbation parameter are derived in
a technical appendix accompanying Andreasen (2012a).
38 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
where hxx has dimensions nx ×nx ×nx and contains all second order derivatives of
h (·) with respect to (xt ,xt ). Finally, the third-order terms are given by[pxxx
]α1α2α3
= [rxxx
]γ1γ2γ3
[hx
]γ3α3
[hx
]γ2α2
[hx
]γ1α1
+[rxx
]γ1γ2
[hxx
]γ2α2α3
[hx
]γ1α1
+[rxx
]γ1γ2
[hx
]γ2α2
[hxx
]γ1α1α3
+[rxx
]γ1γ3
[hx
]γ3α3
[hxx
]γ1α1α2
+[rx
]γ1
[hxxx
]γ1α1α2α3
,
or in the standard matrix notation
pxxx(α1,α2,α3
) =nx∑γ3=1
hx(:,α1
)′ rxxx(:, :,γ3
)hx
(:,α2
)hx
(γ3,α3
)+hx
(:,α1
)′ rxxhxx(:,α2,α3
)+
nx∑γ1=1
rxx(γ1, :
)hx
(:,α2
)hxx
(γ1,α1,α3
)+
nx∑γ1=1
rxx(γ1, :
)hx
(:,α3
)hxx
(γ1,α1,α2
)+rx
(1, :
)hxxx
(:,α1,α2,α3
).
Here, hxxx has dimensions nx ×nx ×nx ×nx and contains all third order derivatives of
h (·) with respect to (xt ,xt ,xt ). Similarly, rxxx and pxxx have dimensions nx ×nx ×nx
and contain all third order derivatives of the r - and p-functions, respectively.
1.7. CONCLUSION 39
Figure 1.1. The Neoclassical model: Accuracy Plots using the Benchmark Specification
Charts in the left column plot consumption in percentage deviation from steady state, i.e.
ct , as a function of kt = log(kt /kss
), ranging from −5 to +5 standard deviations in a log-
linearized solution. Charts in the right column report the log10 errors in consumption using
the projection approximation as the true solution. The conditioning level of technology in
the top and bottom charts equal −3 and +3 standard deviations in a log-linearized solution,
respectively.
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.3
−0.2
−0.1
0
0.1
Capital, kt
Con
sumption
,c t
Policy function: Low technology level
Perturbation: 1st-order Perturbation: 3rd-order Perfect foresight Extended Perturbation: 3rd-order Projection: 10th-order
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−5
−4
−3
−2
Capital, kt
Log10-errors
Errors: Low technology level
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.2
−0.1
0
0.1
0.2
Capital, kt
Con
sumption
,c t
Policy function: Technology at steady state
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−7
−6
−5
−4
−3
−2
Capital, kt
Log10-errors
Errors: Technology at steady state
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.1
0
0.1
0.2
0.3
Capital, kt
Con
sumption
,c t
Policy function: High technology level
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−5
−4
−3
−2
Capital, kt
Log10-errors
Errors: High technology level
40 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
Table 1.1. The Neoclassical Model: The Structural Parameters
Benchmark High risk aversion High varianceβ 0.99 0.99 0.99δ 0.025 0.025 0.025α 0.36 0.36 0.36γ 5.00 25 5.00ρa 0.98 0.98 0.98σa 0.01 0.01 0.10
Table 1.2. The Neoclassical Model: Accuracy of Consumption on Grid
The RMSEs are computed based on log(ct /c tr ue
t
)= ct − c tr ue
t , where c tr uet denotes
consumption in percentage deviation from the steady state in the true solution (i.e. theprojection approximation). The grid is constructed using 20 points uniformly spaced alongeach dimension of the state space, giving a total of 400 grid points. The bounds for capital andtechnology in the grid range from -3 to +3 standard deviations in a log-linearized solution.Bold figures highlight the approximation with the lowest RMSEs.
Benchmark High risk aversion High varianceRMSEsPerturbation: 1st order 0.0044 0.0393 0.3959Perturbation: 3r d order 1.19×10−5 0.0183 0.1242Perfect foresight 0.0033 0.0262 0.3141Perturbation +1 0.0125 0.0304 N aNPerturbation +2 0.0117 0.0296 N aNPerturbation +13r d 8.75×10−5 0.0251 N aNPerturbation +23r d 7.97×10−5 0.0251 N aNExtended Perturbation: 3r d order 2.46×10−5 0.0024 0.0822
1.7. CONCLUSION 41
Table 1.3. The Neoclassical Model: Accuracy of Consumption in Simulation
The RMSEs are computed based on log(ct /c tr ue
t
)= ct − c tr ue
t , where c tr uet denotes
consumption in percentage deviation from the steady state in the true solution (i.e. theprojection approximation). The reported standard deviations are for consumption inpercentage deviation from the steady state. The RMSEs and the standard deviations arecomputed using simulated paths based of 20,000 observations with a burn-in of 1,000observations. The symbol N aN indicates that consumption explodes using thisapproximation. Bold figures highlight the best performing approximation method(s).
Benchmark High risk aversion High varianceRMSEsPerturbation: 1st order 0.0033 0.0329 0.2865Perturbation: 3r d order 7.57×10−6 N aN 0.1290Perturbation pruned: 3r d order 6.46×10−5 0.0353 0.3624Perfect foresight 0.0024 0.0309 0.1936Perturbation +1 0.0029 0.0311 N aNPerturbation +2 0.0028 0.0303 N aNPerturbation +13r d 1.09×10−5 N aN N aNPerturbation +23r d 1.04×10−5 N aN N aNExtended Perturbation: 3r d order 1.63×10−5 0.0022 0.0921
Standard deviations for ct
Perturbation: 1st order 0.0635 0.0744 0.6351Perturbation: 3r d order 0.0625 N aN 0.3339Perturbation pruned: 3r d order 0.0625 0.0404 0.1838Perfect foresight 0.0628 0.0654 0.5311Perturbation +1 0.0630 0.0643 N aNPerturbation +2 0.0629 0.0639 N aNPerturbation +13r d 0.0630 N aN N aNPerturbation +23r d 0.0629 N aN N aNExtended Perturbation: 3r d order 0.0625 0.0497 0.3484Projection: 10th order 0.0625 0.0487 0.3698
Table 1.4. The New Keynesian Model: The Structural Parameters
β 0.99 α 0.75hss 0.33 ρR 0.80φ1 2.00 κπ 1.50φ2 2.00 κy 0.125κ 2.00 πss 1.00δ 0.025 ρa 0.95θ 0.36 σa 0.006η 6.00
42 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
Table
1.5.Th
eN
ewK
eynesian
Mo
del:A
ccuracy
on
Grid
Th
eR
MSE
sare
com
pu
tedb
asedo
nlo
g (zt /z
true
t
)=z
t −z
true
t,w
here
zt ≡ c
t ,it ,πt ,x
2t an
dth
etru
eso
lutio
nis
givenb
yth
ep
rojectio
n
app
roximation
.Th
egrid
iscon
structed
usin
g20
poin
tsu
niform
lysp
acedalon
geach
dim
ension
ofthe
statesp
ace,giving
atotalof20
4=160,000
gridp
oin
ts.Th
eb
ou
nd
sfo
rth
ein
terestrate,capital,an
dtech
no
logy
inth
egrid
range
from
-3to
+3stan
dard
deviatio
ns
ina
log-lin
earizedso
lutio
n.Fo
rth
ep
riced
ispertio
nin
dex
the
gridran
gesfro
m−
0.005to
0.05.Bo
ldfi
gures
for
eacho
fthe
variables
high
lightth
eap
proxim
ation
with
the
lowestR
MSE
.
Stand
ardPertu
rbatio
n:
Perfectfo
resight
Exten
ded
Perturb
ation
:1
sto
rder
3rd
ord
er3
rdo
rder
RM
SEs
Co
nsu
mp
tion
:ct
0.002530.00170
0.001440.00124
Investm
ent:it
0.068130.01718
0.009570.00884
Infl
ation
:πt
0.001900.00058
0.000600.00049
Au
xiliaryco
ntro
lvariable:x
2t0.18741
0.055780.03863
0.03866
1.7. CONCLUSION 43
Table 1.6. The New Keynesian Model: Accuracy in Simulation
The RMSEs are computed based on log(zt /z tr ue
t
)= zt − z tr ue
t , where zt ≡
ct , it ,πt , x2t
and
the true solution is given by the projection approximation. The reported standard deviationsare in percentage deviation from the steady state. The RMSEs and the standard deviations arecomputed using simulated paths based of 20,000 observations with a burn-in of 1,000observations. Bold figures highlight the best performing approximation method(s).
Consumption Investment Inflation Aux. controlct it πt x2
tRMSEsPerturbation: 1st order 0.00263 0.00811 0.00091 0.01284Perturbation: 3r d order 0.00116 0.00431 0.00043 0.00499Perturbation: 3r d order, exact st 0.00103 0.00415 0.00038 0.00401Perturbation pruned: 3r d order 0.00134 0.00457 0.00049 0.00579Perfect foresight 0.00137 0.00371 0.00045 0.00234Extended Perturbation: 3r d order 0.00094 0.00362 0.00038 0.00313
Standard deviationsPerturbation: 1st order 0.01517 0.05083 0.00579 0.05874Perturbation: 3r d order 0.01570 0.05202 0.00595 0.06114Perturbation: 3r d order, exact st 0.01577 0.05222 0.00598 0.06158Perturbation pruned: 3r d order 0.01564 0.05182 0.00593 0.06088Perfect foresight 0.01562 0.05185 0.00594 0.06237Extended Perturbation: 3r d order 0.01579 0.05215 0.00599 0.06210Projection: 12th order 0.01617 0.05328 0.00618 0.06368
44 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
Table 1.7. Extended Path: The Terminal Condition
All three terminal conditions imply the same RMSEs for the control variables (denoted RMSEy)using N = 250, where N denotes the horizon in the Extended Path. The RMSEs are computedin a simulated sample path of 5,000 observations. Starting values for the Extended Path arecomputed from a third-order perturbation approximation. Bold figures highlight the bestperforming method for a given value of N .
RMSEs for terminal conditions
yt+N = yss yt+N = Et
[y1st
t+N
]yt+N = Et
[y3r d
t+N
]Stable perturbation calibrationN = 200 6.41×10−9 1.32×10−10 1.29×10−10
N = 175 6.06×10−8 1.25×10−9 1.21×10−9
N = 150 5.63×10−7 1.18×10−8 1.11×10−8
N = 125 5.22×10−6 1.10×10−7 9.91×10−8
N = 100 7.01×10−5 1.04×10−6 8.59×10−7
N = 75 5.63×10−4 1.02×10−5 7.08×10−6
N = 50 0.0035 1.32×10−4 5.54×10−5
Explosive perturbation calibrationN = 200 2.77×10−8 2.45×10−10 2.16×10−10
N = 175 2.71×10−7 6.37×10−9 3.62×10−9
N = 150 2.67×10−6 1.44×10−7 9.49×10−8
N = 125 2.67×10−5 2.91×10−6 2.08×10−6
N = 100 0.0074 6.00×10−5 4.46×10−5
N = 75 0.0165 0.0040 4.83×10−4
N = 50 0.0120 0.0183 0.0197
1.7. CONCLUSION 45
Table 1.8. State-Dependent Horizon in the Extended Path
The RMSEs for the control variables are computed using the horizon N = 200 as thebenchmark in a simulated sample path of 5,000 observations. The optimal horizon N∗ isrestricted to the interval from 20 to 200. Starting values and the terminal condition areobtained from a third-order perturbation approximation. The perturbation approximation atthird order and the pruned version are computed without the uncertainty correction to get theperfect foresight approximation. The computations are carried out in MATLAB 2014a using anIntel(R) Core(TM) i5-4200 CPU with 2.50 GHz.
RMSEs Mean(seconds) Mean(N∗)per 1,000 draws
Stable perturbation calibrationExtended Path: Dss = 0 0 385 200Extended Path: Dss = 0.01 7.91×10−5 196 87Extended Path: Dss = 0.02 2.50×10−4 136 58Extended Path: Dss = 0.03 3.92×10−4 101 43Extended Path: Dss = 0.05 5.92×10−4 74 29Extended Path: Dss = 0.08 7.43×10−4 62 21Extended Path: Dss = 0.10 7.89×10−4 51 20Perturbation: 1st order 0.0071 0.004 −Perturbation: 3r d order 2.87×10−4 0.05 −Perturbation pruned: 3r d order 5.05×10−4 0.12 −
Explosive perturbation calibrationExtended Path: Dss = 0 0 496 200Extended Path: Dss = 0.01 6.31×10−5 361 100Extended Path: Dss = 0.02 2.00×10−4 263 69Extended Path: Dss = 0.03 3.40×10−4 208 52Extended Path: Dss = 0.05 7.84×10−4 137 35Extended Path: Dss = 0.08 0.0127 75 25Extended Path: Dss = 0.10 0.0176 66 22Perturbation: 1st order 0.0134 0.004 −Perturbation: 3r d order N aN − −Perturbation pruned: 3r d order 0.0072 0.12 −
46 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
Table 1.9. Combining Perturbation and Extended Path: Stable Perturbation Calibration
The RMSEs for the control variables are computed using N = 200 and EE = 0 as thebenchmark in a simulated sample path of 5,000 observations. The optimal horizon N∗ isrestricted to the interval from 20 to 200. Starting values and the terminal condition areobtained from a third-order perturbation approximation. The perturbation approximation atthird order is computed without the uncertainty correction to get the perfect foresightapproximation. The computations are carried out in MATLAB 2014a using an Intel(R)Core(TM) i5-4200 CPU with 2.50 GHz.
RMSEs Mean(seconds) Pct of times Extendedper 1,000 draws Path is used
Dss = 0 EE = 0 0 385 100EE = 0.00005 3.28×10−5 57 27EE = 0.0001 6.20×10−5 35 15.3EE = 0.001 2.38×10−4 1.4 0.36EE = 0.01 2.87×10−4 0.5 0
Dss = 0.01 EE = 0 7.91×10−5 196 100EE = 0.00005 3.65×10−5 33 27EE = 0.0001 6.24×10−5 18 15.3EE = 0.001 2.38×10−4 1.0 0.36EE = 0.01 2.87×10−4 0.5 0
Dss = 0.03 EE = 0 3.92×10−4 101 100EE = 0.00005 1.61×10−4 16 27EE = 0.0001 1.11×10−4 12 15EE = 0.001 2.38×10−4 1.0 0.36EE = 0.01 2.87×10−4 0.5 0
Dss = 0.05 EE = 0 5.92×10−4 74 100EE = 0.00005 3.33×10−4 11 27EE = 0.0001 2.38×10−4 9 15EE = 0.001 2.38×10−4 1.0 0.36EE = 0.01 2.87×10−4 0.5 0
Dss = 0.08 EE = 0 7.43×10−4 62 100EE = 0.00005 4.63×10−4 7 27EE = 0.0001 3.49×10−4 5 15EE = 0.001 2.38×10−4 0.7 0.36EE = 0.01 2.87×10−4 0.5 0
1.7. CONCLUSION 47
Table 1.10. Combining Perturbation and Extended Path: Explosive Perturbation Calibra-tion
The RMSEs for the control variables are computed using N = 200 and EE = 0 as thebenchmark in a simulated sample path of 5,000 observations. The optimal horizon N∗ isrestricted to the interval from 20 to 200. Starting values and the terminal condition areobtained from a third-order perturbation approximation. The perturbation approximation atthird order is computed without the uncertainty correction to get the perfect foresightapproximation. The computations are carried out in MATLAB 2014a using an Intel(R)Core(TM) i5-4200 CPU with 2.50 GHz.
RMSEs Mean(seconds) Pct of times Extendedper 1,000 draws Path is used
Dss = 0 EE = 0 0 498 100EE = 0.00005 0.0052 151 45EE = 0.0001 0.0058 119 34EE = 0.001 0.0068 42 5EE = 0.01 0.0079 27 0.9
Dss = 0.01 EE = 0 6.31×10−5 361 100EE = 0.00005 0.0052 101 45EE = 0.0001 0.0058 82 34EE = 0.001 0.0068 29 5EE = 0.01 0.0079 18 0.9
Dss = 0.03 EE = 0 3.40×10−4 208 100EE = 0.00005 0.0044 78 45EE = 0.0001 0.0050 62 34EE = 0.001 0.0069 18 5EE = 0.01 0.0080 11 0.9
Dss = 0.05 EE = 0 7.84×10−4 137 100EE = 0.00005 0.0050 55 45EE = 0.0001 0.0061 39 34EE = 0.001 0.0073 13 5EE = 0.01 0.0079 8 0.9
Dss = 0.08 EE = 0 0.0127 75 100EE = 0.00005 0.0106 31 45EE = 0.0001 0.0106 28 34EE = 0.001 0.0127 10 5EE = 0.01 0.0145 7 1.2
48 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
Figure 1.2. The Neoclassical Model: Accuracy Plots with High Risk Aversion
Charts in the left column plot consumption in percentage deviation from steady state, i.e.
ct , as a function of kt = log(kt /kss
), ranging from −5 to +5 standard deviations in a log-
linearized solution. Charts in the right column report the log10 errors in consumption using
the projection approximation as the true solution. The conditioning level of technology in
the top and bottom charts equal −3 and +3 standard deviations in a log-linearized solution,
respectively.
−1 −0.5 0 0.5 1
−0.4
−0.2
0
0.2
Capital, kt
Con
sumption
,c t
Policy function: Low technology level
Perturbation: 1st-order Perturbation: 3rd-order Perfect foresight Extended Perturbation: 3rd-order Projection: 10th-order
−1 −0.5 0 0.5 1
−4
−3
−2
−1
Capital, kt
Log10-errors
Errors: Low technology level
−1 −0.5 0 0.5 1
−0.3
−0.2
−0.1
0
0.1
0.2
Capital, kt
Con
sumption
,c t
Policy function: Technology at steady state
−1 −0.5 0 0.5 1
−5
−4
−3
−2
−1
Capital, kt
Log10-errors
Errors: Technology at steady state
−1 −0.5 0 0.5 1
−0.2
−0.1
0
0.1
0.2
0.3
Capital, kt
Con
sumption
,c t
Policy function: High technology level
−1 −0.5 0 0.5 1
−3.5−3
−2.5−2
−1.5−1
−0.5
Capital, kt
Log10-errors
Errors: High technology level
1.7. CONCLUSION 49
Figure 1.3. The Neoclassical Model: Accuracy Plots with High Variance
Charts in the left column plot consumption in percentage deviation from steady state, i.e. ct ,as a function of kt = log
(kt /kss
), ranging from −5 to +5 standard deviations in a
log-linearized solution. Charts in the right column report the log10 errors in consumptionusing the projection approximation as the true solution. The conditioning level of technologyin the top and bottom charts equal −3 and +3 standard deviations in a log-linearized solution,respectively.
−3 −2 −1 0 1 2 3−2.5−2
−1.5−1
−0.50
0.5
Capital, kt
Con
sumption
,c t
Policy function: Low technology level
Perturbation: 1st-order Perturbation: 3rd-order Perfect foresight Extended Perturbation: 3rd-order Projection: 10th-order
−3 −2 −1 0 1 2 3
−3
−2.5
−2
−1.5
−1
−0.5
0
Capital, kt
Log10-errors
Errors: Low technology level
−3 −2 −1 0 1 2 3
−1
−0.5
0
0.5
1
Capital, kt
Con
sumption
,c t
Policy function: Technology at steady state
−3 −2 −1 0 1 2 3
−2.5
−2
−1.5
−1
−0.5
0
0.5
Capital, kt
Log10-errors
Errors: Technology at steady state
−3 −2 −1 0 1 2 3
0
0.5
1
1.5
2
Capital, kt
Con
sumption
,c t
Policy function: High technology level
−3 −2 −1 0 1 2 3
−2
−1
0
1
Capital, kt
Log10-errors
Errors: High technology level
50 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
Figure 1.4. The Neoclassical Model: Stability Plots
The low and high conditioning level of technology equal −3 and +3 standard deviations in alog-linearized solution, respectively.
−1 −0.5 0 0.5 1−0.02
−0.01
0
0.01
0.02
0.03
kt+
1−
dt
Capital, kt
Benchmark: Standard 3rd order perturbation
Low technology level Steady state technology level High technology level
−1 −0.5 0 0.5 1−0.02
−0.01
0
0.01
0.02
0.03
kt+
1−
dt
Capital, kt
Benchmark: Extended 3rd order perturbation
−2 −1 0 1 2−0.02
−0.01
0
0.01
0.02
0.03
kt+
1−
dt
Capital, kt
High risk aversion: Standard 3rd order perturbation
−2 −1 0 1 2−0.02
−0.01
0
0.01
0.02
0.03
kt+
1−
dt
Capital, kt
High risk aversion: Extended 3rd order perturbation
−10 −5 0 5 10−0.5
0
0.5
1
kt+
1−
dt
Capital, kt
High variance: Standard 3rd order perturbation
−10 −5 0 5 10−0.5
0
0.5
1
kt+
1−
dt
Capital, kt
High variance: Extended 3rd order perturbation
1.7. CONCLUSION 51
Figure 1.5. New Keynesian Model: Accuracy Plots
Charts in the left column plot the control variables as a function of at , ranging from −3 to +3standard deviations. Charts in the right column report the log10 errors in the control variablesusing the projection approximation as the true solution. The conditioning level of the nominalinterest rate and the capital stock equals −3 standard deviations in a log-linearized solution.The conditional level of st is 0.04.
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05
−0.05
−0.04
−0.03
−0.02
−0.01
0
Technology, at
Deviation
from
SS
Consumption, ct
Perturbation: 1st-order Perturbation: 3rd-order Perfect foresight Extended Perturbation: 3rd-order Projection: 12th-order
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
Errors in Consumption, ct
Log10-errors
Technology, at
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05
0.4
0.5
0.6
0.7
0.8
0.9
1
Technology, at
Deviation
from
SS
Investment, it
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
Errors in Investment, it
Log10-errors
Technology, at
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05
0.02
0.03
0.04
0.05
0.06
Technology, at
Deviation
from
SS
Inflation, πt
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
Errors in Inflation, πt
Log10-errors
Technology, at
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
Technology, at
Deviation
from
SS
Auxiliary control variable, x2t
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05
−3
−2.5
−2
−1.5
−1
−0.5
0
Errors in Auxiliary control variable, x2t
Log10-errors
Technology, at
52 CHAPTER 1. THE EXTENDED PERTURBATION METHOD
Figure 1.6. The New Keynesian Model: Stability Plots
The policy function for inflation and the transition function for the price dispertion index areplotted as a function of st . The conditioning level of the nominal interest rate and the capitalstock equals −4 standard deviations in a log-linearized solution. The conditioning level oftechnology equals −3 standard deviations. The histogram for the distribution of st iscomputed from a simulated sample path of 20.000 observations using the projectionapproximation.
0 0.01 0.02 0.03 0.04 0.050.054
0.056
0.058
0.06
0.062
0.064
0.066
0.068
0.07
Inflation (πss = 1.00)
πt
Price dispersion index, st
Perturbation: 3rd-order Extended Perturbation: 3rd-order Projection: 12th-order
0 0.01 0.02 0.03 0.04 0.05
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
Price dispersion index (πss = 1.00):
s t+1
Price dispersion index, st0.01 0.02 0.03 0.04
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Distribution of st (πss = 1.00)
Price dispersion index, st
0 0.02 0.04 0.06 0.08 0.1
0.055
0.06
0.065
0.07
0.075
0.08
0.085Inflation (πss = 1.0015)
Price dispersion index: st
πt
0 0.02 0.04 0.06 0.08 0.1
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
Price dispersion index (πss = 1.0015)
Price dispersion index: st
s t+1
0 0.02 0.04 0.06 0.080
0.05
0.1
0.15
0.2
Distribution of st (πss = 1.0015)
Price dispersion index, st
1.7. CONCLUSION 53
Figure 1.7. Combining Perturbation and Extended Path: Plot of Sample Path
This figure shows part of the simulated sample where standard third-order perturbationexplodes. That is, the simulation is for the explosive perturbation calibration and with Dss = 0.The extended perturbation approximation is computed by standard perturbation, except atthe areas shaded gray where the Extended Path is used. The y-axis reports the distance of the
state variables from the steady state, i.e. xt from the steady state, i.e.∥∥xt
∥∥=√∑nx
i=1 x2i ,t .
4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 45000
0.1
0.2
0.3
0.4
Tolerated Euler Error, EE = 0.0001
∥xt∥
Extended Path Perturbation 3rd-order Extended Perturbation: 3rd-order
4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 45000
0.1
0.2
0.3
0.4
Tolerated Euler Error, EE = 0.01
∥xt∥
C H A P T E R 2NEW EVIDENCE ON DOWNWARD NOMINAL WAGE
RIGIDITY AND THE IMPLICATIONS FOR
MONETARY POLICY
Anders Kronborg
Aarhus University and CREATES
Abstract
This paper examines the degree of downward nominal wage rigidity and its macroe-
conomic implications. For this purpose, a simple dynamic stochastic general equilib-
rium model is estimated where the nominal wage rigidity is allowed to be asymmetric.
As a novelty, the nonlinear model equilibrium is approximated using the extended
perturbation method in Andreasen and Kronborg (2016) which improves the accu-
racy relative to standard perturbation when the model is characterized by strong
nonlinearities. The estimates show that wages are more downwardly than upwardly
rigid, which generates asymmetric responses to shocks. However, the asymmetries
generated are generally less pronounced than what is found in the literature. The
estimated model is subsequently used to compute the welfare implications of differ-
ent inflation targets. For the U.S., the optimal inflation target is approximately 0.25
percent per year which provides support for low but positive inflation.
55
56
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
2.1 Introduction
Downward nominal wage rigidity (henceforth DNWR), a phenomenon where wages
are more downwardly than upwardly rigid, has received increased attention during
the recent Great Recession. Schmitt-Grohe and Uribe (2013) show that during the
recent economic downturn, the average hourly nominal wage has stayed largely the
same as before the crisis in several European economies even as unemployment rates
have risen dramatically. In a speech at the Federal Reserve Bank of Kansas City in
2014, the Chair of the Federal Reserve Janet Yellen suggested that the sluggish wage
growth during the recovery of the U.S. economy might be a result of ’pent-up wage
deflation’ because wages did not adjust sufficiently during the crisis.1
This paper looks at DNWR and its macroeconomic implications for the busi-
ness cycle and monetary policy. This is done by estimating a simple New Keynesian
dynamic stochastic general equilibrium (DSGE) model similar to that in Kim and
Ruge-Murcia (2009). A key feature in the economy is that the wage rigidity is allowed
to be asymmetric, in the sense that it is more costly for the households to cut their
nominal wages than to increase them. This differs from most of the DSGE literature,
in which sticky prices and wages are usually modeled in a symmetric fashion (see
Blanchard (2009) for a discussion), partly reflecting the use of linearization as the
preferred solution technique. Thus, to capture the asymmetric features of DNWR in
the DSGE model it is necessary to apply a nonlinear approximation.
As a novelty, the nonlinear model solution is approximated using the extended
perturbation method of Andreasen and Kronborg (2016). This approach combines the
perfect foresight solution obtained from the Extended Path (Fair and Taylor (1983))
with higher-order risk corrections from standard perturbation. This approach gener-
ally improves the accuracy relative to a higher-order perturbation approximation by
removing error terms in the model solution under perfect foresight. Further, it is likely
to preserve characteristics of the true solution, such as monotonicity or convexity,
something that standard perturbation may struggle to obtain, as emphasized by Den
Haan and De Wind (2012). These improvements can be substantial if the model is
characterized by strong nonlinearities. As shown in this paper, this is in fact the case
for the applied model, even for modest levels of DNWR.
Using U.S. data from 1964Q2-2015Q1 the structural parameters of the model
are estimated by the simulated method of moments (SMM) as described in Duffie
and Singleton (1993). The choice of econometric methodology reflects the nonlinear
nature of the approximated solution, which does not allow for the moments to be
computed in closed-form, making GMM infeasible. Instead, as shown in Ruge-Murcia
(2012), SMM can be a feasible way of estimating DSGE models by minimizing the
1The remarks, titled "Labor Market Dynamics and Monetary Policy", can be found at the webpage ofthe Board of Govenors of the Federal Reserve System.
2.1. INTRODUCTION 57
weighted difference between the data moments and those from model generated
sample paths.
The main findings in this paper are as follows: First, and in line with previous
studies, both prices and wages are found to be nominally rigid. For wages, the data
suggest that the adjustment to shocks is asymmetric, meaning DNWR is present in
the U.S. economy. Hence, the estimated model is characterized by asymmetric propa-
gation of shocks, which is especially pronounced for nominal wages and inflation but
feed through to the real economic variables where contractions are generally larger
than expansions. While the results confirm the qualitative findings in the literature
(see below), the degree of asymmetry found in this paper is generally less pronounced.
Specifically, the parameter estimate relating to the asymmetry in wage adjustments
is orders of magnitude smaller than what is found in previous studies, which can
attributed to the change in solution methodology. Second, as argued by Tobin (1972),
the presence of DNWR can help bridge the apparent gap between much of the classic
monetary literature prescribing price stability or deflation (examples include Fried-
man (1969) and Woodford (2003)) and the fact that this is not the monetary policy
pursued by central banks. In the DSGE model in this paper, the central bank faces
a trade-off: A positive steady state inflation reduces the need for nominal wages to
decline in face of adverse shocks, so it may be prudent to operate with a positive infla-
tion target in lieu of price stability. On the other hand, price rigidity implies that this is
associated with systematic costs. Based on the estimates, the optimal inflation target
is found to be approximately 0.25 percent per year when implemented as a strict
inflation target in a Taylor rule which provides support for low but positive inflation
in the presence of DNWR. The welfare improvements of a higher inflation target are
decreasing in the level of price rigidity and increasing in the level of macroeconomic
volatility.
The finding that wages are more downwardly than upwardly rigid is well docu-
mented at the micro level (see for example Dickens et al. (2007)). Some researchers
suggest that this may be caused by the tendency of firm managers to cut wage only
in periods of severe financial distress, as they worry about adverse effects on worker
morale (Kahneman et al. (1986) and Akerlof et al. (1996)). Others point to institu-
tional or legal factors such as minimum wages, wage indexation, restrictions on firing
workers and collective bargaining (Babecky et al. (2010)). It is not clear however, to
which extent DNWR at the micro level translates into the aggregated level, e.g. due
to composition or business cycle effects (Holden and Wulfsberg (2009)). Further, in
order to examine how nominal rigidities affect the business cycle and what the impli-
cations are for monetary policy it is necessary to use a macroeconomic framework.
An agnostic approach to DNWR is taken in this paper, implying that the reason(s)
behind this friction is not modeled explicitly. This is done to ensure tractability and
to keep the analysis as simple as possible.
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CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
This work is related to a series papers in the DSGE literature which use a similar
framework. In Kim and Ruge-Murcia (2009) and Kim and Ruge-Murcia (2011) the
authors estimate a model with asymmetric wage costs similar to the one in this paper
using a second-order pruned perturbation approximation. They find that DNWR
implies that the optimal inflation target is between 0.75 and 1 percent per year. Fahr
and Smets (2010) extend this framework to calibrate a model with two countries in
a monetary union and show that the region with DNWR adjusts slowly to adverse
shocks due to a persistent loss of competitiveness. Abbritti and Fahr (2013) calibrate a
version of this model to match the positive skewness of annual growth rates of wages
and the negative skewness of employment, investment, and output found for most
economies. Other studies find the optimal inflation target in economies with DNWR
to be higher than the above. For example, Akerlof et al. (1996) consider a model in
which wages can never fall and find the optimum to be 3 percent per year.
The methodological choice in this paper has several implications for the analysis:
First, given that the approximated solution is more accurate, the estimates are likely
to be closer to the "true" values as well, provided that the model is correct. Second,
the interpretation of the parameter estimates of the approximated equilibrium is
closer to that of the underlying model. For example, since the optimality conditions
includes taking the first-order derivative of a linex function, second-order perturba-
tion implies that the approximated equilibrium relies on its third-order derivative. As
a result, when using second- and third-order perturbation the resulting approximated
adjustment costs are no longer convex and become negative when wages increase
beyond a certain point, even for moderate amounts of DNWR. With the extended
perturbation method this does not happen as it includes higher-order terms of the
perfect foresight component of the policy functions. Third, since the extended pertur-
bation does not rely on a local approximation of the perfect foresight component its
precision is relatively high even far from the steady state. This allows us to study the
nonlinearities of the model far from the non-stochastic steady state more accurately,
for example when the economy is hit by large adverse shocks.
The remainder of the paper is structured as follows. Section 2.2 presents the
DSGE model. Section 2.3 introduces the extended perturbation method used to ap-
proximate the model. The estimation methodology used is described in Section 2.4.
Section 2.5 briefly lists the data and moments used in the estimation. The model
properties are analyzed in Section 2.6. Section 2.7 provides an estimate of the op-
timal inflation target and discusses the robustness of the results while Section 2.8
concludes.
2.2. THE DSGE MODEL 59
2.2 The DSGE Model
This section presents the DSGE model to be estimated and used for the analysis. The
model is identical to the one found in Kim and Ruge-Murcia (2009). A key feature
of the model is the labor market. The representative household is a monopolistic
supplier of differentiated labor and hence has some market power through the setting
of wages. Nominal wage rigidity is allowed to be asymmetric in the model by assuming
a more general cost function than the usual quadratic costs in Rotemberg (1982). The
economy is assumed to feature shocks to preferences and productivity.
2.2.1 Households
Preferences
The households are assumed to be infinitely lived and indexed by n ∈ [0,1]. They
maximize the expected discounted utility
Ut = Et
∞∑s=0
βs dt+s
c1−ρn,t+s −1
1−ρ −χhn,t+s
, (2.1)
where Et is the conditional expectation at time t . Consumption of household n at
time t is cn,t and hn,t is the labor supply. The discount factor is denoted β ∈ (0,1),
ρ > 0 controls the utility curvature of consumption andχ> 0 is a parameter governing
the disutility from hours worked. The preference shock, dt , shifts the intertemporal
utility of the household and is assumed to evolve as
log dt+1 = ρd log dt +σdεdt+1, εd
t+1 ∼N.i .d .(0,1),
where ρd ∈ (0,1). Consumption is a basket of differentiated goods indexed by i ∈ [0,1],
and aggregated by
cn,t =[∫ 1
0
(cn,i ,t
) η−1η di
] ηη−1
,
where η> 1 is the elasticity of substitution between consumption goods, implying a
steady state markup of ηη−1 . Cost minimization gives the following demand for good i
by household n
cn,i ,t =(
Pi ,t
Pt
)−ηcn,t , (2.2)
where Pi ,t is the nominal price of good i and Pt =[∫ 1
0
(Pi ,t
)1−ηdi
] 11−η
is the aggregate
price index at time t . Hence, total consumption expenditure is Pt cn,t .
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CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
Wage Setting
The household enters in a market with monopolistic competition for its labor ser-
vices. Thus, the household has some market power through the determination of its
nominal wage, Wn,t , subject to the firm demand for labor. The labor market is subject
to frictions in the form of nominal rigidity in the wage adjustment so that changing
the nominal wage is associated with the following adjustment costs
Φ
(Wn,t
Wn,t−1
)=φ
[exp
−ψ
(Wn,t
Wn,t−1−1
)+ψ
(Wn,t
Wn,t−1−1
)−1
ψ2
]. (2.3)
The parameter φ≥ 0 determines the general degree of convexity while ψ controls the
asymmetry in wage adjustments. When ψ> 0 the exponential term will dominate
the linear term whenWn,t
Wn,t−1< 1 and hence nominal wage decreases are more costly
than increases. Thusψ> 0 corresponds to the case of DNWR. Asψ→∞, the function
obtains an "L-shape" implying that nominal wages can never fall. The function is
convex and differentiable. The latter is necessary to apply perturbation and the
extended perturbation method.
Finally, note that the linex function nests quadratic costs in the limiting case,
l i mψ→0
Φt = φ
2
(Wn,t
Wn,t−1−1
)2
.
This implies that we can test for DNWR econometrically by testing the significance of
ψ in a one-side test.
Budget Constraint and Optimality Conditions
In every period t , the budget constraint for household n is given by (in consumption
units)
cn,t +Et
[Qt ,t+1 An,t
Pt
]+ Bn,t
Pt= Rt−1Bn,t−1
Pt+ An,t−1
Pt+ Wn,t
Pt
(1−Φn,t
)hn,t +
Dn,t
Pt,
(2.4)
where An,t is a portfolio of state-contingent Arrow-Debreu assets, i.e. An,t (ω j ) pays
out one dollar if state ω j is reached in period t +1. The price of this portfolio is deter-
mined using the nominal stochastic discount factor, Qt ,t+1. Further, the household
can buy one-period nominal bonds, Bn,t , which carry the gross nominal interest rate
Rt from period t to t+1. The household uses resources for consumption and acquires
assets for next period. Expenditures are financed by the payoff from assets carried
2.2. THE DSGE MODEL 61
from the previous period, wages net of adjustment costs2, and dividends received
from firms, Dn,t .
After imposing a symmetric equilibrium, the optimality conditions for the repre-
sentative household for consumption and nominal wages are, respectively3
0 =βRtEt
[λt+1
λtΠ−1
t+1
]−1, (2.5)
0 = λt ht
Pt
[(ν−1)
(1−Φt
)+ ∂Φt
∂ωtωt
]−dtχν
ht
Wt−βEt
[∂Φt+1
∂ωt+1
λt+1
Pt+1ω2
t+1ht+1
]. (2.6)
Here the gross nominal wage growth is denoted ωt ≡ WtWt−1
and gross inflation is
Πt = PtPt−1
. The stochastic discount factor is given as Qt ,t+1 = βλt+1λtΠ−1
t+1 and λt =c−ρt dt is the marginal utility of consumption. The parameter ν> 1 is the elasticity of
demand for labor, implying a steady state markup of νν−1 . Equation (2.5) is the Euler
equation which balances the expected marginal rate of substitution for intertemporal
consumption with the relative price of consumption between periods, given by the
nominal interest rate and the inflation rate. Equation (2.6) is the wage Phillips curve.
The household equals the marginal cost of increasing the nominal wage with its
marginal benefit. The costs include lower demand for its labor when firms substitute
towards cheaper labor input as well as the adjustment costs. The benefits consist
of a higher hourly wage, lower disutility as the hours worked are reduced, and next
period’s expected net utility gain from lower adjustment costs.
2.2.2 Firms
Production Technology and Labor Demand
The production side of the economy is populated by a continuum of firms, indexed by
i ∈ [0,1]. Each firm i produces a differentiated good by using labor as input, according
to the following production function
yi ,t = at h1−θi ,t , (2.7)
where θ ∈ (0,1) is a production parameter. The variable at refers to an economy wide
exogenous productivity level and is assumed to evolve as follows
log at+1 = ρa log at +σaεat+1, εa
t+1 ∼N.i .d .(0,1),
2Φn,t gives the fraction of gross real wage income,Wn,t hn,t
Pt, wasted due to adjustment costs. This in
fact implies that the asymmetry in total wage adjustment costs depends on both ν and ψ.3These optimality conditions can be derived by combining the first-order conditions for consumption,
hours worked, demand for state-contingent claims and bonds, and the nominal wage level. Detailedderivations used in this paper can be found in a technical appendix, available upon request. A modeloverview is given in Appendix A.
62
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
where ρa ∈ (0,1). The labor used in the production by firm i is a composite of the
differentiated labor types
hi ,t =[∫ 1
0
(hn,i ,t
) ν−1ν dn
] νν−1
.
The demand by firm i for each labor unit n is derived from the cost minimization
problem
mi nhn,i ,t
∫ 1
0Wn,t hn,i ,t dn, s.t . hi ,t =
[∫ 1
0
(hn,i ,t
) ν−1ν di
] νν−1 ≥ hi ,t ,
where the nominal wage index paid for the labor composite at time t is given as Wi ,t =[∫ 10
(Wn,t
)1−νdn
] 11−ν
and the aggregated quantities are defined as hn,t =∫ 1
0 hn,i ,t di
and ht =∫ 1
0 hi ,t di . The resulting demand for type n labor by firm i is given as
hn,i ,t =(
Wn,t
Wt
)−νhi ,t . (2.8)
The aggregate demand for labor faced by household n is then given by aggregating
(2.8) over all of the firms
hn,t =∫ 1
0
(Wn,t
Wt
)−νhi ,t di =
(Wn,t
Wt
)−νht . (2.9)
Price Setting
Output prices are nominally sticky in the Rotemberg sense, i.e. adjusting prices
implies a cost for firm i at time t given by4
Γ
(Pi ,t
Pi ,t−1
)= γ
2
(Pi ,t
Pi ,t−1−1
)2
. (2.10)
The parameter γ≥ 0 controls the degree of convexity in adjustment costs.
At time t , firm i maximizes the expected discounted profits (in real terms)
maxPi ,t+s ,hi ,t+s
∞s=0
Et
∞∑s=0
βs λt+s
λt
1
Pt+s
[(1−Γi ,t+s
)Pi ,t+s yi ,t+s −Wt+s hi ,t+s
],
subject to the production technology in (2.7) and the demand yi ,t = ci ,t . Using (2.2)
and aggregating over the households we can obtain the total demand faced by firm i
ci ,t =∫ 1
0
(Pi ,t
Pt
)−ηcn,t dn =
(Pi ,t
Pt
)−ηct . (2.11)
4Like in Kim and Ruge-Murcia (2009), price adjustment costs are assumed to be quadratic to econo-mize with the number of parameters in the estimation. Preliminary estimates using pruning suggest thatthe asymmetry parameter in the generalized function is in fact insignificant.
2.2. THE DSGE MODEL 63
After imposing a symmetric equilibrium, the optimality conditions for the represen-
tative firm for labor demand and price are, respectively
Wt
Pt= mct (1−θ)at h−θ
t , (2.12)
0 = (η−1)(1−Γt
)+ ∂Γt
∂PtΠt −ηmct −βEt
[λt+1
λt
ct+1
ct
∂Γt+1
∂Pt+1Πt+1
], (2.13)
where mct is the real marginal cost of production (i.e. the Lagrangian multiplier
for the optimization problem). Equation (2.12) gives the labor demand by equating
the marginal product of labor with the cost. The optimal price in (2.13) is the price
Phillips-curve and can be interpreted in similar fashion as (2.6), where the firm bal-
ances the marginal costs and benefits of a price increase. The costs include reduced
demand for the firm’s goods as consumers substitute towards cheaper ones and the
adjustment costs. The benefits include a higher unit price, a reduction in marginal
production costs (a result of lower demand), and next period’s expected net profit
gain from lower adjustment costs.
2.2.3 Aggregation
In equilibrium, all households supply the same amount of labor and set the same
nominal wage. Further, the net supply of all financial assets is zero. Using this, the
budget constraint in (2.4) simplifies to
ct = Wt
Pt(1−Φt )ht + D t
Pt. (2.14)
As all firms set the same price and produce the same quantity in equilibrium, aggre-
gate dividends are given by D t = (1−Γt )Pt yt −Wt ht . Combining this with (2.14), the
economy resource constraint simplifies to
yt =ct + Wt
PtΦt ht
1−Γt. (2.15)
Hence, nominal price and wage rigidity lowers the aggregate consumption level
relative to its frictionless counterpart. Further, if the economy is characterized by
DNWR, this gap increases when nominal wages decrease.
2.2.4 Monetary Policy
The model is closed by a monetary policy rule. The central bank is assumed to set the
interest rate to stabilize inflation and hours worked in accordance with the following
Taylor rule
l og
(Rt
Rss
)= ρR l og
(Rt−1
Rss
)+κπlog
(Πt
Πss
)+κh l og
(ht
hss
), (2.16)
64
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
where ρR ∈ (0,1) and κπ,κh ≥ 0 are constant policy parameters.Πss is the (quarterly)
gross inflation target and Rss and hss are the steady state values of the interest rate
and labor, respectively.
2.3 Solution Methodology
This section presents the extended perturbation method used to approximate the
solution of the model in Section 2.2. For further details, the reader is referred to
Andreasen and Kronborg (2016). Let the ny ×1 vector yt denote the control variables
and let the state variables be denoted by the nx ×1 vector xt =[
x′1,t x′2,t
]′, where
x1,t and x2,t are the endogenous and exogenous state variables with dimensions
nx1 ×1 and nx2 ×1, respectively. The vectors yt and xt belong to the sets χy ⊂Rny and
χx ⊂Rnx , respectively.
Consider a broad class of DSGE models, which can be represented as
Et[f(yt ,yt+1,xt ,xt+1
)]= 0, (2.17)
where f :χy ×χy ×χx ×χx 7→Rn and n = ny +nx . It is assumed that this mapping is
at least m times differentiable, where m will denote the order of approximation. Now,
let εt be an nε×1 vector of i.i.d. shocks to the exogenous state variables. Provided
that the DSGE model in (2.17) has a unique solution, this can be expressed as (see
Schmitt-Grohe and Uribe (2004))5
yt = g(xt ,σ), (2.18)
xt+1 = h(xt ,σ)+σηεt+1, (2.19)
η≡[
0nx1×nε
η
].
The perturbation parameter σ≥ 0 scales the square root of the covariance matrix, η,
which has dimensions nx ×nε. This enables us to capture the effects of uncertainty
in the policy functions g and h. For most DSGE models, however, we do not know the
true policy functions.
Now, let (2.18) and (2.19) be decomposed into
g(xt ,σ) = gPF (xt )+gstoch(xt ,σ), (2.20)
h(xt ,σ) = hPF (xt )+hstoch(xt ,σ), (2.21)
where gstoch(xt ,σ) and hstoch(xt ,σ) contain the effects of uncertainty when the per-
fect foresight component is removed from the policy functions. Thus, gstoch(xt ,σ=5In the following estimation, the parameter space will be restricted to the determinacy region.
2.3. SOLUTION METHODOLOGY 65
0) = 0 and hstoch(xt ,σ = 0) = 0. Note trivially that all derivatives of g and gPF with
respect to the state variables xt are identical atσ= 0. Further, all derivatives involving
σ are identical for g and gstoch (similarly for h).
In standard perturbation, a Taylor series expansion around the non-stochastic
steady state is applied to both components in (2.20) and (2.21). However, this may
not be necessary as gPF and hPF can be approximated to arbitrary precision, using
the Extended Path (see Fair and Taylor (1983)). In Andreasen and Kronborg (2016)
we therefore suggest that the perfect foresight components are approximated in this
fashion whereas the stochastic parts of the policy functions gstoch and hstoch , remain
approximated by perturbation. The order of approximation for the extended pertur-
bation is defined by the order of the Taylor series expansion used to approximate the
stochastic components.
Note that the extended perturbation can be thought of as adding the higher-order
terms,∑∞
i=m+1g(xss ,0)xi
i !
(xt −xss
)⊗i and∑∞
i=m+1h(xss ,0)xi
i !
(xt −xss
)⊗i , to an m’th-order
perturbation approximation. To the extent that the approximation errors of gPF
and hPF from perturbation are large this is likely to be a significant improvement
for several reasons. First, given that the approximated solution is more accurate,
the estimates will be closer to the "true" values as well, provided that the model is
correct, as discussed in Fernandez-Villaverde and Rubio-Ramirez (2005) and An and
Schorfheide (2007). Thus, the choice of solution method will feed through to the
estimation of the parameters. Second, the improved accuracy is likely to mitigate the
tendency of higher-order perturbation to generate exploding sample paths.6 Third,
the interpretation of the parameter estimates is closer to that of the underlying model,
e.g. by preserving convexity of the adjustment costs.
In the model equilibrium both the function and its first-order derivative enter the
equilibrium dynamics, implying that a second-order perturbation approximation
depends on a third-order Taylor expansion hereof. Figure 2.1 shows the function in
(2.3) when the degree of asymmetry is low (left graph) as well as when it is high (right
graph). Both the actual function and the third-order Taylor expansion are shown. As
seen from the figure, the approximated cost function is no longer convex for positive
wage inflation. Further, as wages increase beyond a certain point the adjustment
costs become negative under the standard perturbation approximation. This point is
found to lie well within the ergodic distribution for the estimates by a second-order
perturbation approximation, which questions whether this is sufficiently accurate. Fi-
nally, it is clear that the approximation struggles more when the model nonlinearities
are strong, i.e. the above problems are exaggerated in the right graph of Figure 2.1. As
the extended perturbation method does not rely on a Taylor expansion of finite order
6A tractable alternative that preserves stability is pruning (see Kim et al. (2008) and Andreasen et al.(2013)). This approximation however may still struggle to preserve monotonicity and may not be suffi-ciently accurate.
66
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
for the perfect foresight components in (2.20) and (2.21) these approximation errors
should be significantly smaller than under standard perturbation.
In this application, the model will be estimated using the extended perturbation
method for both second and third order. Using a second-order approximation im-
plies there will be a constant correction for risk, given by the terms gσσ and hσσ,
respectively. As shown in Andreasen and Kronborg (2016) this ensures that the ex-
tended perturbation is stable for any state xt , provided the same holds for the model
under perfect foresight. Under a third-order approximation, the risk correction is
a linear function of the state variables. In addition to the risk correction under the
second-order approximation this implies that the terms gσσσ, gσσx and hσσσ, hσσx
are added to the respective approximated policy functions. Although stability can
no longer be guaranteed, these terms are generally quite small and do not seem to
induce instability in the resulting approximation.
A simple description of the extended perturbation algorithm used in the simula-
tions is the following: For a given set of parameters, θ, and a given order m: First, run
the standard perturbation method to compute all relevant derivatives of gstoch(xt ,σ)
and hstoch(xt ,σ). Second, for each period, t , use the Extended Path to compute
gPF (xt ) and hPF (xt ) and approximate the policy functions by adding the stochastic
terms. This approach can be relatively demanding in terms of computational costs,
especially if the algorithm is called repeatedly, e.g. through long simulated sample
paths since the Extended Path requires solving a large fixed-point problem for every
period. To improve computational efficiency, a state-dependent truncation as well
as a combination with perturbation, dependent on some tolerance parameters is
applied.7 Intuitively, this means that standard perturbation is used when xt is suffi-
ciently close to the steady state, where the Taylor expansion of the perfect foresight
component is accurate.
2.4 Econometric Methodology
Since a nonlinear approximation is used to solve the model in Section 2.2, the like-
lihood function can not be obtained from the Kalman filter. Instead, Ruge-Murcia
(2012) shows that the Simulated Method of Moments (SMM), first described in Duffie
and Singleton (1993), is a feasible way of estimating the structural parameters in
nonlinear DSGE models. This estimation method is well suited for the purpose of this
7The model is solved by standard perturbation if, for xt , the unit free Euler equation errors are lessthan a specified tolerance level, EE . The truncation of the perfect foresight problem is determined by aspecified radius of convergence, Dss , based on a third-order approximation of the solution. Specifically,EE = 0.001 and Dss = 0.005 are set. The boundaries of the truncation length is set as N ∈
20,200
. For theestimated parameter set, this implies that the fraction of periods in which the Extended Path is applied isapproximately 8 percent.
2.4. ECONOMETRIC METHODOLOGY 67
paper since closed-form expressions of the model moments are not obtained from
the extended perturbation method. Instead, the model moments are approximated
by simulating sample paths, Y1:τT , minimizing the weighted distance of these to the
moments in the data, Y1:T .
Let θ ∈Θbe the q×1 vector of variables to be estimated, whereΘ⊂Rq is a compact
set. Let gt be a p ×1 vector of data transformations at time t , where p ≥ q , for which
we are interested in the unconditional expectation (the moment conditions), and let
gt (θ) be the corresponding series generated by using the DSGE model in simulation.
The size of this simulated sample path is given as τT , where τ≥ 0 is an integer. Finally,
let WT be some positive-definite weighting matrix with dimension p ×p. The SMM
estimate, θ, is found by minimizing the weighted distance between the data moments
and those of the model. Formally, the estimator is given by
θ = ar g . mi n.θ∈Θ
QT (θ) = GT (θ,τ)′WT GT (θ,τ), (2.22)
where
GT (θ,τ) = 1
T
T∑t=1
gt − 1
τT
τT∑t=1
gt (θ), (2.23)
denotes the difference in sample moments. Local identification of θ requires that
the matrix J = E[∂Gt (θ)∂θ
]has rank q . While it is hard to prove global identification, the
parameters can be confirmed to be locally identified around the estimated values.
Further, when simulating a series of artificial data from the model, this rank condition
is found to be satisfied for all samples.
The optimal weighting matrix is found from the two-step procedure: In the first
step, the parameter estimates are found using the inverse standard errors of the
moments. In the second step, based on these estimates, the weighting matrix is set as
WT = S−1, where S is the sample estimate of
S0 =∞∑
j=−∞E
([g t −E
(g t (θ)
)][g t+ j −E
(g t+ j (θ)
)]′). (2.24)
The matrix S is found non-parametrically, using the Newey-West estimator (see
Appendix C). This weighting matrix implies that the moments with the smallest
variance in the simulated sample are given a relatively higher weight in the objective
function in (2.23) and hence, choosing the optimal WT improves the efficiency of the
SMM estimator as is also a well-known result from the GMM literature. Further, as
shown in Ruge-Murcia (2012), these efficiency gains are likely to be increasing in the
degree of nonlinearity in the DSGE model.
68
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
Under the regularity conditions given in Duffie and Singleton (1993), the asymp-
totic distribution of the SMM estimator in (2.22) is
pT
(θ−θ0
)→N
(0,(1+1/τ)(J′S−1J)−1
).
In the application τ= 10 is set, which is similar to or slightly larger than what is used
in the literature. This choice obviously reflects a balance between computational
and statistical efficiency. Since the extended perturbation can be computationally
demanding, and since a new sample path is generated for every function evaluation
in the numerical optimization, one would like to keep τ as small as possible. Further,
for both the pruning and extended perturbation approximation, the estimates are
found to be fairly stable when increasing τ beyond 5.
2.5 Data and Moments
The model is estimated using quarterly seasonally adjusted U.S. data series from
1964Q2 to 2015Q1 giving a total of 204 observations. The variables used are real con-
sumption per capita (Personal Consumption Expenditures divided by the quarterly
average of monthly Civilian Noninstitutional Population), hours worked (Aggregate
Weekly Hours Index: Total Private Industries), quarterly CPI inflation (Consumer
Price Index for All Urban Consumers: All items), quarterly wage inflation (Average
Hourly Earnings: Total Private Industries), and the nominal interest rate (Effective
Federal Funds Rate). All series can be downloaded from the FRED database at the
Federal Reserve Bank of St. Louis web page. Prior to estimation, the log was taken to
all variables and they have been detrended by a linear deterministic trend to conform
to the stationarity of the DSGE model.
Similar to Kim and Ruge-Murcia (2009), the moments used in the estimation are
the 15 variances and covariances as well as the five first-order autocovariances of the
data series.
2.6 Model Properties
This section presents the parameter estimates and examines the resulting model
properties. I follow the literature and calibrate a number of parameters prior to the
estimation, using fairly standard values. An overview of the calibrated parameters is
given in Table 2.1.
This leaves 11 structural parameters for the estimation. Table 2.2 reports the es-
timated values, the standard errors, and the value of the objective function for both
second and third-order pruning and extended perturbation.
2.6. MODEL PROPERTIES 69
Consider first the extended perturbation estimates. The sets of parameter esti-
mates are similar for most parameters which indicates that the third-order terms
for the risk corrections are small. Consequently, the quantitative properties of the
two approximations will be quite similar as well and thus, for ease of exposition,
only the third-order extended perturbation approximation will be considered in the
remainder of the paper. For the price and wage rigidity, both γ and φ are found to be
positively significant at the 5 percent level with estimates of 44.13 and 20.94, respec-
tively. Thus, the data clearly reject the hypothesis that the economy is characterized
by fully flexible goods and factor prices. Further, in the case of wages, the rigidity is
asymmetric since ψ is estimated to be 51.04. Again, the hypothesis that ψ = 0 can
be rejected at the 5 percent level which means that the data support the presence of
DNWR in the U.S. economy.
It is instructive to compare these estimates to those obtained by a pruned per-
turbation approximation. Table 2.2 shows that both second and third-order pruning
estimates imply a higher utility curvature of consumption since ρ is found to be
more than twice as large. Further, with perturbation the price rigidity is found to
be less pronounced especially for the second-order approximation. The parameters
relating to the Taylor rule and exogenous shocks are remarkably similar for stan-
dard perturbation and extended perturbation. However, there are large differences
found in the estimates for the parameters in the wage adjustment cost function in
(2.3). First, the φ has an estimated value of 1,314.03 and 4,330.10 for second and
third-order perturbation, respectively, while it is only 20.94 for extended perturbation.
Second, the ψ estimates are 6,654.26 and 6,395.19 for perturbation but only 51.04 for
extended perturbation. While the estimated value of ψ using perturbation is compa-
rable to what is found or used in previous studies (for example Kim and Ruge-Murcia
(2009), Fahr and Smets (2010), Kim and Ruge-Murcia (2011), and Abbritti and Fahr
(2013)), where this parameter ranges from 3,844 to 26,000, the estimate is orders of
magnitude smaller when using extended perturbation. Hence, while the results in
this paper qualitatively confirm the findings in previous studies, it can be seen from
Table 2.2 that the model is able to match the data moments with a smaller degree of
asymmetry in the wage rigidity when using this solution method.
2.6.1 Impulse Response Functions
It has become common in the DSGE literature to study the endogenous propagation
of exogenous shocks by considering impulse response functions. However, when the
model is solved using the extended perturbation method, closed-form expressions
for the IRFs are not available. Below, let the vector ut consist of the control and state
variables of interest. Now let the i ’th exogenous shock be hit by a disturbance of
size vi in period t +1. The generalized impulse response function (GIRF) for ut+l
70
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
proposed by Koop et al. (1996) is then defined as
G I RFu(l , vi ) = E[
ut+l
∣∣∣∣εi ,t+1 = vi ,xt
]−E
[ut+l
∣∣∣∣xt
], (2.25)
for l = 1, ...,L. Note that the GIRF is only conditioned on the i ’th shock and only for l =1. The impulse responses generated from (2.25) are not scaleable (i.e. G I RFu(l ,c vi ) 6=c G I RFu(l , vi )) nor symmetric (i.e. G I RFu(l , vi ) 6= −G I RFu(l ,−vi )), and they are de-
pendent on the current state of the economy. The expectations in (2.25) are found by
simulation, i.e. by drawing ε j ,t+1 for j 6= i and ε j ,t+l for all j shocks when l = 2, ...,L.
5,000 simulations are used for both shock types. The state vector xt from which the
GIRF is computed is set to its unconditional expectation.
Figure 2.2 shows the generalized impulse responses of consumption, hours
worked, inflation, nominal wage inflation, real wages, and the nominal interest rate to
productivity shocks of difference sizes. Both positive and negative shocks (solid and
dashed lines, respectively) of 2 and 3 standard errors (smaller and larger linewidth,
respectively) are shown. An increase in the productivity level increases output and
thus should be considered as an expansionary shock. Following (2.25), the figure
depicts the expected percentage deviations of the variables from a path without
conditioning on the shocks as a function of the horizon l .
The impulse responses show asymmetries for most variables considered. The
obvious case is nominal wage inflation where initial response is for nominal wages to
increase for both expansionary and contractionary shocks. However, as the marginal
product of labor decreases following an adverse productivity shock this implies that
real wages must fall in accordance with (2.12). This is instead obtained through higher
inflation which responds asymmetrically since nominal wage adjust more flexibly
for expansionary shocks. Thus, when DNWR is present we note how an increase
in inflation can serve as a mean of restoring equilibrium after adverse productivity
shocks. The asymmetric inflation response in Fig. 2.2 is an example of how DNWR
effectively acts as an additional cost push shock when productivity decreases. This
in turn affects the response of the nominal interest rate: Since the central bank is
targeting inflation, the interest rate decreases following an expansionary shock and
vice versa. However, the monetary policy response is asymmetric, reflecting the in-
flation impulse response. Hence, the initial response of the nominal interest rate
following contractionary productivity shocks is larger than expansionary ones. In
accordance with the Euler equation in (2.5), aggregate consumption responds to the
change in interest rate and wealth effect by deviating persistently from the uncondi-
tional expectation. The changes in consumption for contractionary shocks are larger
than for their positive counterpart, partly reflecting the asymmetric monetary policy.
Thus, the accumulated difference over the depicted horizon is approximately 0.8 and
2.2 percent relative to the unconditional expectation for shocks of 2 and 3 standard
2.7. OPTIMAL MONETARY POLICY 71
errors, respectively.
Figure 2.3 shows the impulse responses to preference shocks. A positive preference
shock increases marginal utility of consumption today relative to future periods.
Hence, a positive shock should be considered as an expansionary demand shock and
it temporarily increases aggregate consumption, in accordance to the Euler equation
in (2.5). Following an expansionary shock, final goods equilibrium implies that hours
worked increase to satisfy the increased demand. Increased employment lowers the
marginal product of labor and thus increases the real marginal cost of production
in (2.12). As a result, inflation increases for expansionary shocks and decrease for
contractionary. Again, nominal wages are clearly more restricted downwardly than
upwardly. The larger change in nominal wages following an expansionary shock
means that, in equilibrium, inflation increases more than it decreases following a
contractionary shock. The asymmetric inflation response implies an asymmetric
interest rate response by the inflation targeting central bank which increases the
interest rate more following expansionary preference shocks than it decreases it
following contractionary ones. As opposed to the productivity shocks this coun-
tercyclical response actually reduces the real economic effects of DNWR. Overall,
DNWR implies that consumption falls more following contractionary shocks than it
increases in the opposite case. The accumulated difference over the depicted horizon
is approximately 1.5 and 4.1 percent for shocks of 2 and 3 standard errors, respectively.
The impulse responses in Figure 2.2 and 2.3 show how nominal rigidities cause
real economic effects as consumers and firms adapt to exogenous shocks. In the
presence of DNWR, changes in nominal wages are particularly restricted when con-
tractionary shocks hit the economy. When the nominal wage adjustment is more
sluggish, optimal behavior implies that other variables must compensate to obtain
output and labor market equilibrium. Intuitively, this results in a flattening of the
wage Phillips curve as employment decreases. As a results, the model with DNWR
implies asymmetric business cycles in line with the stylized facts in macroeconomic
time series. Finally, the graphs show that the asymmetry is more pronounced for
larger shocks. Hence, the economic effects of DNWR might be small in tranquil peri-
ods but on the other hand have important implications in states far from the steady
state such as in severe recessions.
2.7 Optimal Monetary Policy
This section examines the optimal inflation target when the economy is characterized
by DNWR. As in Kim and Ruge-Murcia (2009), this is done by allowing the central
bank to conduct its monetary policy in (2.16) as a strict inflation target. Thus, the
72
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
nominal interest rate is set to stabilize inflation only, i.e. irrespective of the state of
the economy and with no preference for smoothing or stabilizing employment. The
optimal policy is then defined as the inflation target that maximizes the expected
discounted utility of the representative household in (2.1).
When choosing its inflation target the central bank faces a trade-off: On one
hand, inflation generates systematic costs since both output prices and wages are
rigid. Hence, with little or no macroeconomic volatility it would be suboptimal to
operate with an inflation target much above price stability. On the other hand, higher
steady state inflation reduces the probability that the economy ends up in a state
where adverse shocks create a large downward pressure on nominal wages. As dis-
cussed earlier, inflation will serve as a mean of adjustment to a new equilibrium
when nominal wages can not adjust. This will also increase the changes in the real
variables such as consumption and employment, something which is disliked by
the households. Hence, it might be prudent for the central bank to operate with a
positive inflation target and to incur small systemic costs so as to reduce overall
macroeconomic volatility.
Figure 2.4 depicts the change in unconditional welfare of different inflation targets
relative to price stability (solid black line). The expectation is found by simulation,
using 5,000 sample paths of 500 observations with a burn-in of 100 observations.8
The welfare is measured as percentage changes in consumption equivalents, i.e. the
change in (2.1) scaled with the inverse of marginal utility of consumption. Given the
estimates in Table 2.2, the optimal inflation target is found to be approximately 0.25
percent per year. The results show that, under uncertainty and with an economy
characterized by DNWR, it is optimal to reduce the probability of a high-cost event
by incurring small but systematic costs. As the inflation target is increased, further
lowering the probability of downward pressure on nominal wages is associated with
diminishing welfare gains. Furthermore, higher steady state inflation is increasingly
associated with systematic costs due to price rigidity and this latter effect will eventu-
ally dominate. The costs are found to outweigh the benefits once the inflation target
is raised beyond 0.50 percent per year in the benchmark case. Hence, overall the
results in this paper lend support to the notion of a small but positive inflation target
in lieu of price stability but is not able to fully justify the typically observed inflation
targets of most central banks, solely based on DNWR. The optimal inflation target
is lower than the 3 percent per year suggested by Akerlof et al. (1996) who assume
that nominal wages can never fall as well as the estimates of 0.75 to 1 percent per
year found in Kim and Ruge-Murcia (2009) and Kim and Ruge-Murcia (2011) where
DNWR is more pronounced than found in this paper. This underlines the importance
of an accurate assessment of the degree of DNWR for monetary policy. To the extent
8The same sets of innovations,εd
t ,εat
500
t=1, are used to compare across different inflation targets.
2.7. OPTIMAL MONETARY POLICY 73
that this paper underestimates the potential asymmetry in the wage rigidity so will it
underestimate the optimal inflation target.
To examine the robustness of the results, the welfare gains are computed for
different levels of volatility (by scaling σd and σa , see the left column of Figure 2.4)
and price rigidity (by scaling γ, see the right column of Figure 2.4). As seen from the
graph, increasing volatility generally shifts the curve up, implying that the optimal
inflation target increases. The reason is fairly intuitive: When volatility increases,
the benefits of a precautionary buffer against nominal wage deflation also increases
since, for any level of inflation, it is now more likely that the economy will be in a
state where DNWR is binding. Thus, in the case where the standard deviations of
both shocks are increased by 50 percent, the optimal inflation target increases to
approximately 0.40 percent per year. Lowering the volatility by 50 percent implies
that the optimal target falls to approximately 0.1 percent per year. Increasing the price
rigidity generally shifts the curve down. As γ increases this means that the systematic
costs of higher steady state inflation increase, counteracting the benefits of the buffer.
As a result, when price rigidity is increased by 50 percent this lowers the optimal
target to 0.15 percent per year whereas it increases to 0.35 percent per year when
price rigidity is 50 percent higher. Further, a higher value of γ has a significant effect
on the rate at which the costs will eventually outweigh the benefits of inflation.
How general are the results? Obviously, the results will depend on the model
specification. For example, Fagan and Messina (2009) show that if heterogeneity
across workers is assumed, this raises the optimal inflation level to avoid resource
misallocation in the presence of DNWR when real wages vary cross-sectionally. Simi-
larly, heterogeneity across firms or sectors may also increase the optimal inflation
target. On the other hand, the cost of inflation would be raised further if households
needed to hold money for economic transactions (this could be obtained e.g. through
a money-in-utility specification). It might also be beneficial to consider the effects
shocks to the disutility of labor, something which is not considered in this paper. Fur-
ther, the central bank can obtain a smoother and more symmetric adjustment of real
variables to supply shocks by targeting inflation less strictly than what is assumed in
the above. This is likely to lower the optimal inflation target below the estimate found
in this paper. Other factors, such as the zero lower bound restriction on monetary
policy, will tend to increase the optimal inflation rate, as discussed in Blanchard et al.
(2010). Previous micro studies (see for example Dickens et al. (2007)) suggest that
the degree of DNWR may vary a lot from country to country. In that case, so will the
real economic costs of deflationary wage pressure and the inflation buffer needed.
Finally, the model (like most New Keynesian DSGE models) is exposed to the Lucas
(1976) critique. While the nominal frictions might be a fairly good approximation of
the aggregate economy, the implicit assumption that the parameters are completely
policy invariant is, of course, questionable. Specifically, the in-sample inflation level
74
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
used to estimate the model is significantly higher than what is found as the optimal
level. If the mechanism by which monetary policy is transmitted to the economy
changes as a result this might result in misleading conclusions.
2.8 Conclusion
This paper examines the extent of downward nominal wage rigidity in the U.S. econ-
omy and its implications for monetary policy. This is done by specifying a simple
dynamic stochastic general equilibrium model in which wage rigidity is allowed to
be asymmetric. The model estimates show that nominal rigidities are important for
both prices and wages. For wages, it is found that nominal rigidities are asymmetric,
i.e. that DNWR is present in the U.S. economy.
The model equilibrium is approximated using the extended perturbation method
in Andreasen and Kronborg (2016) instead of a standard perturbation-based approxi-
mation. This solution method generally improves the accuracy of the approximation
and is more likely to preserve characteristics of the underlying model such as con-
vexity and monotonicity. The change in the solution methodology is found to have a
substantial impact on the parameter estimates associated with the wage rigidity. In
particular, the parameter governing the asymmetry in wage adjustments is estimated
to be orders of magnitude smaller than in previous studies. While the estimated
model is characterized by asymmetric wage rigidity and thus confirms the qualitative
findings in the literature, the asymmetric propagation of shocks is generally less
pronounced than what has been found previously.
Based on the estimated model, the optimal inflation target is computed when
implemented as a strict inflation target in a Taylor rule. This choice is governed by a
trade-off between systematic inflation costs and the benefits of reducing the probabil-
ity of ending in a state with deflationary pressure on nominal wages. I find the optimal
net inflation target to be approximately 0.25 percent per year. Hence, the findings in
this paper lend support to the notion of a small but positive inflation target albeit
less than what is implemented by most central banks. Increasing macroeconomic
volatility increases the optimal inflation target whereas higher price rigidity lowers it.
Acknowledgments
The author gratefully acknowledges support from Aarhus University, Department
of Economics and Business Economics and from CREATES - Center for Research in
Econometric Analysis of Time Series (DNRF78), funded by the Danish National Re-
search Foundation. Comments from participants at the Danish Graduate Programme
in Economics Workshop and CREATES seminars are also gratefully acknowledged.
2.8. CONCLUSION 75
References
ABBRITTI, M. AND S. FAHR (2013): “Downward wage rigidity and business cycle
asymmetries,” Journal of Monetary Economics, 60.
AKERLOF, G., W. T. DICKENS, AND G. PERRY (1996): “The Macroeconomics of Low
Inflaiton,” Brookings Papers on Economic Activity, 27.
AN, S. AND F. SCHORFHEIDE (2007): “Bayesian Analysis of DSGE Models,” Econometric
Reviews, 26.
ANDREASEN, M. M., J. FERNANDEZ-VILLAVERDE, AND J. RUBIO-RAMIREZ (2013): “The
Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical
Applications,” NBER Working Paper Series.
ANDREASEN, M. M. AND A. KRONBORG (2016): “The Extended Perturbation Method,”
Aarhus University and CREATES Research Paper.
BABECKY, J., P. DU CAJU, T. KOSMA, M. LAWLESS, J. MESSINA, AND T. ROOM (2010):
“Downward Nominal and Real Wage Rigidity: Survey Evidence from European
Firms,” The Scandinavian Journal of Economics, 112.
BLANCHARD, O., G. DELL’ARICCIA, AND P. MAURO (2010): “Rethinking Macroeco-
nomic Policy,” IMF Staff Position Note.
BLANCHARD, O. J. (2009): “The State of Macro,” Annual Review of Economics, 1.
DEN HAAN, W. J. AND J. DE WIND (2012): “Nonlinear and stable perturbation-based
approximations,” Journal of Economic Dynamics & Control, 36.
DICKENS, W. T., L. GOETTE, E. L. GROSHEN, S. HOLDEN, J. MESSINA, M. E.
SCHWEITZER, J. TURUNEN, AND M. E. WARD (2007): “How Wages Change: Mi-
cro Evidence from the International Wage Flexibility Project,” Journal of Economic
Perspectives, 21.
DUFFIE, D. AND K. J. SINGLETON (1993): “Simulated Moments Estimation of Markov
Models of Asset Prices,” Econometrica, 61.
FAGAN, G. AND J. MESSINA (2009): “Downward Wage Rigidity and Optimal Steady-
state Inflation,” ECB Working Paper Series.
FAHR, S. AND F. SMETS (2010): “Downward Wage Rigidities and Optimal Monetary
Policy in a Monetary Union,” Scandinavian Journal of Economics, 112.
FAIR, R. C. AND J. B. TAYLOR (1983): “Solution and Maximum Likelihood Estimation
of Dynamic Nonlinear Rational Expectations,” Econometrica, 51.
76
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
FERNANDEZ-VILLAVERDE, J. AND J. F. RUBIO-RAMIREZ (2005): “Estimating Dynamic
Equilibrium Economies: Linear Versus Nonlinear Likelihood,” Journal of Applied
Econometrics, 20.
FRIEDMAN, M. (1969): The Optimum Quantity of Money, Macmillan.
HOLDEN, S. AND F. WULFSBERG (2009): “How strong is the macroeconomic case for
downward wage rigidity?” Journal of Monetary Economics, 56.
KAHNEMAN, D., J. L. KNETSCH, AND R. THALER (1986): “Fairness as a Constraint on
Profit Seeking: Entitlements in the Market,” The American Economic Review, 76.
KIM, J., H. KIM, E. SCHAUMBURG, AND C. A. SIMS (2008): “Calculating and Using
Second-Order Accurate Solutions of Discrete Time Dynamic Equilibrium Models,”
Journal of Economic Dynamics & Control, 32.
KIM, J. AND F. J. RUGE-MURCIA (2009): “How much inflation is necessary to grease
the wheels?” Journal of Monetary Economics, 56.
——— (2011): “Monetary policy when wages are downwardly rigid: Friedman meets
Tobin,” Journal of Economic Dynamics & Control, 35.
KOOP, G., M. H. PESARAN, AND S. M. POTTER (1996): “Impulse response analysis in
nonlinear multivariate models,” Journal of Econometrics, 74.
LUCAS, R. E. (1976): “Econometric Policy Evaluation: A Critique,” Carnegie-Rochester
Conference Series on Public Policy, 1.
ROTEMBERG, J. J. (1982): “Monopolisitic Price Adjustment and Aggregate Output,”
Review of Economic Studies, 49.
RUGE-MURCIA, F. J. (2012): “Estimating nonlinear DSGE models by the simulated
method of moments: With an application to business cycles,” Journal of Economic
Dynamics & Control, 35.
SCHMITT-GROHE, S. AND M. URIBE (2004): “Solving Dynamic General Equilibrium
Models Using a Second-order Approximation to the policy function,” Journal of
Economic Dynamics & Control, 28.
——— (2013): “Downward Nominal Wage Rigidity and the Case for Temporary Infla-
tion in the Eurozone,” Journal of Economic Perspectives, 27.
TOBIN, J. (1972): “Inflation and Unemployment,” The American Economic Review, 62.
WOODFORD, M. (2003): Interest and Prices: Foundations of a Theory on Monetary
Policy, Princeton University Press.
2.8. CONCLUSION 77
Appendix A: Model Overview
1. λt = dt c−ρt ,
2. 1 =βRtEt
[λt+1λt
],
3. 0 =λt ht
[(ν−1)
(1−Φt
)+ ∂Φt∂ωt
wtwt−1
Πt
]−dtχν
htwt
−βEt
[∂Φt+1∂ωt+1
λt+1Πt+1
(wt+1
wtΠt+1
)ht+1
],
4. mct (1−θ)at h−θt = wt ,
5. 0 = (η−1)(1−Γt
)+ ∂Γt∂PtΠt −ηmct −βEt
[λt+1λt
ct+1ct
∂Γt+1∂Pt+1
Πt+1
],
6. yt(1−Γt
)= ct +wtΦt ht ,
7. yt = at h1−θt ,
8. l og(
RtRss
)= ρR log
(Rt−1Rss
)+ (1−ρR )
[κπlog
(ΠtΠss
)+κh log
(hthss
)],
9. l og dt+1 = ρd log dt +σdεdt+1,
10. l og at+1 = ρa l og at +σaεat+1.
The model has 10 equations + 2 link equations (for Rt−1 and wt−1). The 4 state
variables are Rt−1, wt−1, dt , and at . The 8 control variables are λt , ct , Rt ,Πt , ht , mct ,
yt , and wt (where wt ≡ WtPt
).
Appendix B: Model Steady State
LetΠ be the steady state inflation and use the normalization for the shocks: d = a = 1.
R = Πβ
,
mc = 1
η
[(η−1)(1−Γ)+ (1−β)π
∂Γ
∂P
],
h =[
χν
(ν−1)(1−Φ)+ (1−β)Π ∂Φ∂P
1[1−Γ−mc(1−θ)Φ
]−ρmc(1−θ)
] 1−ρ(1−θ)−θ
,
w = mc(1−θ)h−θ,
c = h1−θ[1−Γ−mc(1−θ)Φ],
y = h1−θ,
λ= c−ρ .
78
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
Appendix C: Sample Estimate of (2.24)
Let θ1 be the step 1 estimates:
θ1 = ar g . mi n.θ∈Θ
QT (θ) = GT (θ,τ)′Ip GT (θ,τ).
Denote the sample mean of the outer product of the moment distance as
Γ j = 1
T −1
T− j∑t=1
[gt − 1
τT
τT∑s=1
gs (θ1)
][gt+ j − 1
τT
τT∑s=1
gs (θ1)
]′, j = 0, ...,T −1.
Note that Γ j = Γ− j . The non-parametric estimate of the covariance matrix is then
S =T−1∑
j=−T+1κ
(j
CT
)Γ j , κ (x) =
1−|x|, |x| ≤ 1
0 other wi se,
where CT is a bandwidth parameter.
2.8. CONCLUSION 79
Table 2.1. Calibrated parameters
Parameter Description Valueα (1−α) is the wage share of income 0.3333β Discount factor 0.9900η Elasticity of substitution, intermediate goods 11.0000ν Elasticity of substitution, labor input 4.3333χ Disutility of labor 1.5000Πss Average gross quarterly inflation 1.0112
Nominal wage growth0.96 0.98 1 1.02 1.04
Adju
stm
ent c
osts
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5φ = 100, ψ = 100
Quadratic costsDNWRDNWR, 3rd-order approximation
Nominal wage growth0.96 0.98 1 1.02 1.04
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5φ = 100, ψ = 1,000
Figure 2.1. Wage adjustment costsThe figure depicts the adjustment cost function in (2.3). Both plots show the function for
quadratic costs (ψ= 0) and with DNWR (ψ> 0). In the latter, case both the true function and
the third-order Taylor expansion are shown. The left and right plots show a case of relatively
low and high asymmetry, respectively.
80
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
Table 2.2. Parameter estimates
The estimates are shown for both pruned perturbation (first two columns) and extended
perturbation (last two columns). The figures in parenthesis denote the standard errors of the
parameter estimates.
Pruning Extended perturbation2nd order 3r d order 2nd order 3r d order
ρ 2.883 2.467 1.161 1.159(0.418) (0.541) (0.112) (0.084)
φ 1,314.03 4,330.10 20.91 20.94(431.24) (454.79) (5.40) (4.53)
ψ 6,654.26 6,395.19 51.15 51.04(249.08) (1,440.75) (8.82) (16.28)
γ 23.292 38.289 44.238 44.134(6.835) (11.301) (11.900) (10.892)
ρR 0.809 0.861 0.824 0.822(0.031) (0.024) (0.043) (0.039)
κπ 1.584 1.968 1.563 1.563(0.203) (0.214) (0.346) (0.405)
κh 0.073 0.049 0.125 0.124(0.014) (0.010) (0.107) (0.062)
ρa 0.962 0.957 0.957 0.955(0.007) (0.007) (0.007) (0.008)
σa 0.0107 0.0128 0.0120 0.0121(0.0012) (0.0013) (0.0013) (0.0013)
ρd 0.8725 0.8960 0.8186 0.7837(0.0264) (0.0311) (0.0510) (0.0445)
σd 0.0428 0.0412 0.0265 0.0258(0.0057) (0.0056) (0.0024) (0.0016)
2.8. CONCLUSION 81
0 5 10 15 20−4
−3
−2
−1
0
1
2
3
4Consumption
+2 σ−2 σ+3 σ−3 σ
0 5 10 15 20−1
−0.5
0
0.5
1
1.5Hours worked
0 5 10 15 20−1
−0.5
0
0.5
1Inflation
0 5 10 15 20−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3Nominal wage inflation
0 5 10 15 20−4
−3
−2
−1
0
1
2
3
4Real wages
0 5 10 15 20−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4Nominal interest rate
Figure 2.2. Generalized impulse responses: Productivity shockThe GIRFs are shown in percentage deviations from the expected value. The expectations are
approximated numerically from 5,000 simulated paths. Extended perturbation of third order
is shown.
82
CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE
IMPLICATIONS FOR MONETARY POLICY
0 5 10 15 20−4
−3
−2
−1
0
1
2
3
4Consumption
+2 σ−2 σ+3 σ−3 σ
0 5 10 15 20−6
−4
−2
0
2
4
6
8Hours worked
0 5 10 15 20−1.5
−1
−0.5
0
0.5
1
1.5Inflation
0 5 10 15 20−3
−2
−1
0
1
2
3
4Nominal wage inflation
0 5 10 15 20−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5Real wages
0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8Nominal interest rate
Figure 2.3. Generalized impulse responses: Preference shockThe GIRFs are shown in percentage deviations from the expected value. The expectations are
approximated numerically from 5,000 simulated paths. Extended perturbation of third order
is shown.
2.8. CONCLUSION 83
0 0.1 0.2 0.3 0.4 0.5 0.6
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Inflation target
Con
sum
ptio
n eq
uiva
lent
s (in
%)
0.5σ1.0σ1.5σ
0 0.1 0.2 0.3 0.4 0.5 0.6
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
Inflation target
0.5γ1.0γ1.5γ
Figure 2.4. Optimal inflation targetThe figure depicts the welfare change for various yearly (net) inflation targets compared to
the case of price stability. Welfare is measured as the percentage change in consumption
equivalents. Each line shows the welfare changes for different levels of volatility (left column)
or price rigidity (right column). Extended perturbation of third order is shown.
C H A P T E R 3FORECASTING USING A DSGE MODEL WITH A
FIXED EXCHANGE RATE
Anders Kronborg
Aarhus University and CREATES
Abstract
Dynamic stochastic general equilibrium (DSGE) models are increasingly being used
when conducting forecasting of macroeconomic time series. This paper examines the
forecasting accuracy of a small open economy DSGE model in which the exchange
rate is fixed. As shown, the fixed exchange rate has implications for the relative
importance of the structural shocks in the model. Using Danish data the model is
estimated recursively to assess the out-of-sample forecast accuracy of several time
series from one to eight quarters ahead. The DSGE model is generally comparable to
an AR(1) model in terms of root mean square error while it outperforms the random
walk. Consistent with previous literature, the DSGE model largely underestimates the
severity of the Great Recession. However, the model correctly predicts a continued
fall in GDP growth and a subsequent slow recovery when forecasting from 2009Q1.
85
86 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
3.1 Introduction
Macroeconomic forecasters can choose between a wide range of models to generate
predictive distributions. Increasingly, dynamic stochastic general equilibrium (DSGE)
models are applied by economic institutions and policy makers when conducting
such quantitative forecasts at business cycle frequencies. The popularity of DSGE
models can at least partially be credited to research that has shown that New Keyne-
sian models with nominal and real frictions are comparable to VAR models in terms
of marginal likelihood and forecast ability, for example as shown in the well-known
paper by Smets and Wouters (2007).
Using Danish data, this paper investigates the forecasting performance of an open
economy DSGE model in which the exchange rate is fixed, as to reflect the currency
peg of the Danish Krone against the Euro. The model builds on the open economy
model by Adolfsen et al. (2007b) in which the exchange rate is flexible. However, it
is not clear a priori that the fixed exchange rate DSGE model will produce similar
results as its flexible exchange rate counterpart in terms of forecasting performance.
As discussed in Pedersen and Ravn (2013), the choice of exchange rate regime is likely
to have significant effects on the DSGE model, since both the relative importance of
shocks and their transmission are likely to differ. First, as monetary policy in a fixed
exchange rate regime is devoted to maintain the currency peg it must abandon other
objectives such as targeting domestic output and inflation. Second, shocks to the
foreign economy to which the small open economy has fixed its currency transmit
more forcefully through domestic variables since the nominal exchange rate can
not serve as a buffer, e.g. by allowing a nominal depreciation improve the domestic
competitiveness following an adverse foreign demand shock.
The structural parameters of the model are estimated using a Bayesian econo-
metric approach on a dataset from 1990Q2-2010Q4. Following an initial estimation
sample, the model is estimated and evaluated recursively to assess its predictive
performance for out-of-sample forecasting horizons between 1 and 8 quarters. The
evaluation is based on actual forecasts and root mean square errors (RMSE) of point
forecasts, using the series for GDP, consumption, investment, real wages, imports,
exports, the output deflator and the consumer price index. The forecasting accuracy
of the DSGE model is compared to an AR(1) model and the random walk.
The main findings of this paper are as follows: First, the estimated DSGE model
is able to produce unconditional second moments that are generally in line with
the data. Second, the accuracy of point forecasts generated by the DSGE model are
generally comparable to those of the AR(1) model and better than the random walk
when measured by RMSEs. Third, special attention is given to the DSGE model’s
prediction prior to and during the Great Recession which for Denmark I define as
starting in 2008Q3. In line with the findings in Del Negro and Schorfheide (2013), the
3.1. INTRODUCTION 87
DSGE model underestimates the severity of the downturn at the outset of the crisis.
However, when increasing the information set to include 2009Q1 the DSGE model
correctly predicts a further decline in GDP growth as well as a relatively slow recovery.
By performing a historical decomposition of the filtered output gap in the model the
causes of the crisis are analyzed. The main drivers are found to be domestic demand
and foreign variables at the outset of the crisis while domestic demand and markup
shocks explain the continued suppressed output.
Why choose a DSGE model for forecasting? Because DSGE models deliver a set of
dynamic equations that are based on equilibrium conditions and the optimizing be-
havior of forward-looking agents they deliver forecasts that have a strong theoretical
coherence. This model class allows the researcher to give a structural interpretation
of the state of the economy as well as attribute business cycle fluctuations to under-
lying structural shocks. However, as discussed in Pagan (2003), there might exist a
trade-off between theoretical and empirical coherence. For DSGE models, parameter
restrictions on the resulting state space representation of the model might lead to a
poor empirical fit if they are not a good description of the data. To the practitioner,
examining the ability of a fixed exchange rate DSGE model to predict future paths of
macroeconomic time series is in itself an interesting topic of research. Further, if the
model is to be trusted to deliver quantitative credible answers to more theoretically
based questions like effects of policy initiatives, counterfactuals, etc., it must first be
able to adequately explain and predict the comovements in the data.
This work contributes to an expanding body of literature on DSGE models and
their empirical performances. Much of the recent work is based on the New Keynesian
models with frictions such as habit formation, investment adjustment costs, price
and wage rigidities as well as various exogenous shocks proposed by Christiano et al.
(2005) and Smets and Wouters (2003, 2007). The forecasting performance of these
closed economy DSGE models have been examined extensively (see for example Del
Negro and Schorfheide (2013) and Amisano and Geweke (2013)). By extending the
model framework to include open economy aspects, Adolfsen et al. (2007a) examine
the predictive abilities of a DSGE model with a flexible exchange rate. The general
finding in this literature is that the DSGE models can compete with reduced-form
statistical models such as AR or V AR models in terms of out-of-sample forecasting
but that it is possible to generate better forecasts through more sophisticated models
such as dynamic factor models. Pedersen and Ravn (2013) suggest a model that
incorporates some of the characteristics of the Danish economy, including the fixed
exchange rate regime. However, little research has been conducted to examine the
forecasting performance of open economy DSGE models with fixed exchange rates.
Hence, the main contribution of this paper is to take a first look at the forecasting
ability of such a model. As a result, the model presented below will be kept as simple
as possible, laying the groundwork for future extensions. This implies that interesting
88 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
aspects such as modeling financial frictions or a more detailed fiscal policy framework
is left out.
The remainder of the paper is structured as follows. Section 3.2 presents the DSGE
model. Section 3.3 briefly presents the estimation methodology while Section 3.4
outlines the data series used. Section 3.5 gives the prior distributions and calibrated
parameters. Some posterior properties of the estimated model are examined in
Section 3.6. Section 3.7 performs the forecasting exercise while Section 3.8 concludes.
3.2 The DSGE Model
This section presents the open economy DSGE model used in the forecasting exercise.
The model contains several nominal and real frictions in the New Keynesian tradition
with open economy elements similar to those found in Adolfsen et al. (2007b). Impor-
tantly, as monetary policy is assumed to be characterized by a credible fixed exchange
rate regime this implies that - up to a risk premium - the central bank will set the
domestic interest rate equal to that of the economy to which the exchange rate is fixed.
The domestic economy is assumed to be small compared to its foreign counterpart
which implies that the latter can be perceived as approximately exogenous.
To model the trends in the macroeconomic time series, a unit root is introduced
in the model through a non-stationary productivity shock. Together with steady state
inflation, this introduces a real and a nominal trend in the model which is subse-
quently stationarized. An overview of the log-linearized model is given in Appendix
A.
3.2.1 Firms
There are three types of firms in the model. Domestic goods are produced by inter-
mediate firms in a differentiated fashion and subsequently aggregated into a final
homogenous good. The domestic firm demand labor and capital for production,
which is subject to a stochastic productivity level. The importing firms buy the ho-
mogenous goods at the world market and differentiate them before selling them
to domestic households. The imported good enters both aggregate consumption
and investment. This helps the model in explaining the growth in imports which are
more volatile than domestic consumption. Similarly, the exporting firms buy the final
domestic goods and differentiate them, before selling them at the world market. All
firm types set prices subject to nominal rigidities according to a Calvo model with
indexation.
3.2. THE DSGE MODEL 89
Domestic Goods Firms
At time t , the final domestic output, Yt , consists of an aggregation of the intermediate
goods, Yi ,t , indexed by i ∈ [0,1], by the following production function
Yt =[∫ 1
0Y
1
λdt
i ,t di
]λdt
,
where 1 ≤λdt <∞ is the the stochastic time-varying markup over marginal cost in the
domestic intermediate goods market, whereλd
t
λdt −1
is the elasticity of demand. Taking
the prices, Pi ,t and Pt , as given, costs minimization leads to the demand for each
intermediate good i
Yi ,t =(
Pi ,t
Pt
)− λdt
λdt −1
Yt .
Each intermediate goods producer i operates subject to the following production
function
Yi ,t = εt Kαi ,t
(zt Hi ,t
)1−α−φzt , (3.1)
where Ki ,t is the capital stock and Hi ,t is the labor input in production. A covariance-
stationary productivity shock is denoted εt , whereas zt is a permanent labor-augmenting
productivity shock. Introducing productivity growth induces a common stochastic
trend in the model, implying that the real variables will be cointegrated with zt . The
parameter φ captures the fixed cost in production and set to ensure zero profits in
the steady state. It is assumed to be proportional to the permanent productivity level
to ensure that it grows with the real variables of the economy. The capital share in
production is given by α.
The optimal capital and labor demand of firm i solves the following costs mini-
mization problem
mi nKi ,t ,Hi ,t
Rt−1Wt Ht +Rkt Kt , s.t . εt Kα
i ,t
(zt Hi ,t
)1−α−φzt ≥ Yi ,t .
Nominal wages are denoted Wt , Rt is the gross nominal interest rate on one-period
zero-coupon bonds, and Rkt is the gross nominal rental rate of capital. To allow for
working capital, wages are assumed to be payed one period ahead, which implies
that the wage bill has to be financed at the risk free rate. After imposing a symmetric
equilibrium, the optimality conditions for capital and labor demand are given as,
respectively
r kt = α
1−αwt Rt−1Ht
ktµz,t , (3.2)
mcdt =
(1
1−α)1−α (
1
α
)α (r k
t
)α (wt Rt−1
)1−α 1
εt, (3.3)
90 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
where µz,t = ztzt−1
is the growth rate of zt and real marginal cost is denoted mcdt . The
optimality conditions balance the marginal cost and value of marginal product for
factor demands, where each firm takes the factor prices as given. Here and for the
remainder of the paper, small letters are used to indicate that the variables have been
appropriately detrended to account for the nominal and real trends in the model as
to obtain a stable equilibrium. Here, r kt ≡ Rk
tPt
, wt ≡ WtPt
, mcdt ≡ MC d
tPt
, and kt+1 ≡ Kt+1zt
.
The domestic firms face Calvo-style nominal rigidities. In each period t , firm i is
allowed to set a new price, P d ,newt , with probability 1−ξd . Further, prices are allowed
to be partly indexed to previous periods’ inflation, captured by the parameter κd .
This implies that the price put forth by non-reoptimizing firms will evolve according
to P dt+1 = (πd
t )κdπ1−κd P dt , where π denotes the steady state level of inflation. The
resulting profit optimizing price-setting today is characterized by the expected future
aggregated price movements.
maxP d ,new
t
= Et
∞∑s=0
(βξd )s D t+s
[(πd
t ...πdt+s−1
)κd(π...π)1−κd Pt
d ,new
−MC di ,t+s
]Yi ,t+s −MC d
i ,t+sφzt
,
where Et is the conditional expectation at time t , D t+s is the marginal utility of
household derived from consumption between period t and t + s. The parameter β
is the household discount factor. Note that the effective stochastic discount factor
applied between period t and t + s is (βξd )s D t+s . High levels of price stickiness imply
that the forward-looking element in the optimal price behavior gets more pronounced
as more weight is attached to future cash flows and less on setting the intra period
optimal markup over marginal cost. The log-linearized first-order condition yields
the familiar New Keynesian Phillips-curve
πdt = β
1+κdβEt π
dt+1 +
κd
1+κdβπd
t−1 +(1−ξd )(1−βξd )
ξd (1+κdβ)(mcd
t + λdt ). (3.4)
Here and for the remainder of the paper, variables denoted with a hat are in log-
deviations from their steady state, xt ≡ log xt − log xss . Three factors determine the
current level of inflation for domestic goods: First, higher future inflation raises
current inflation since this implies higher future marginal costs. As a result, the
representative firm raises prices in anticipation of these costs due to the possibility
that it might not be able to due so in future periods. Second, lagged inflation carries
over to the current period through price indexation. Third, an increase in the current
marginal cost or the mark-up will increase inflation, the latter since a lower degree of
competitiveness between domestic firms allow them to increase prices.
3.2. THE DSGE MODEL 91
Importing Firms
The import sector consists of two types of producers, each indexed by i ∈ [0,1]. The
importing firm buys a homogenous good from foreign producers at price P∗t (in
local currency), differentiates and imports them to the domestic market (e.g. through
branding). The homogeneous foreign goods are imported such as to transform it into
consumption, C mi ,t , and investment, I m
i ,t , respectively. The importing consumption
and investment firm faces Calvo-style nominal price rigidities. When allowed to set a
new price, firm i solves the following maximization problem
maxP m,c,new
t
= Et
∞∑s=0
(βξm,c )s D t+s
[(πm,c
t ...πm,ct+s−1
)κm,c(π...π)1−κm,c C m
i ,t+s P m,c,newt
−St+s P∗t+s
[C m
i ,t+s +φm,c zt+s]]
,
maxP m,i ,new
t
= Et
∞∑s=0
(βξm,i )s D t+s
[(πm,i
t ...πm,it+s−1
)κm,i(π...π)1−κm,i I m
i ,t+s P m,i ,newt
−St+s P∗t+s
[I m
i ,t+s +φm,i zt+s]]
,
where the nominal exchange rate (domestic currency units per foreign currency units)
is denoted St . The differentiated import goods at time t are aggregated using the
following CES functions
C mt =
[∫ 1
0(C m
i ,t )1
λm,ct di
]λm,ct
, I mt =
[∫ 1
0(I m
i ,t )1
λm,it di
]λm,it
, (3.5)
where λm,ct and λm,i
t are time-varying markups in the import sectors. Hence, in the
flexible price equilibrium, the representative firm will set the price as a markup over
marginal costs, P m,ct =λm,c
t St P∗t and P m,i
t =λm,it St P∗
t .
Taking the aggregate prices as given, costs minimization then implies the follow-
ing demand for the imported consumption and investment good i , respectively
C mi ,t =
P m,ci ,t
P m,ct
− λm,ct
λm,ct −1
C mt , I m
i ,t =P m,i
i ,t
P m,it
− λm,it
λm,it −1
I mt .
The resulting Phillips curve is
πm, jt = β
1+κm, jβπ
m, jt+1 +
κm, j
1+κm, jβπ
m, jt−1 +
(1−ξm, j )(1−βξm, j )
ξm, j (1+κm, jβ)(mcm, j
t + λm, jt ),
92 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
where mcm, jt = P∗
t
St Pm, jt
is the real marginal cost of the importing firm in sector j ∈ c, i .
Time-varying markups and price rigidity imply that terms of trade deviations happen
in the short run. However, since the domestic economy does not affect the foreign
economy and because of the fixed exchange rate, this implies that the relative prices
must eventually return to their long-run equilibrium.
Exporting Firms
The exporting sector consists of a continuum of firms, indexed by i ∈ [0,1]. Firm i buys
the homogenous domestic good, differentiates it and sells it for P xi ,t (denominated in
local currency). Exports are aggregated according to
X t =[∫ 1
0(Xi ,t )
1λx
t di
]λxt
,
where λxt is the time-varying markup in the export sector. Firm i takes the aggregate
price as given and faces the following demand
Xi ,t =(
P xi ,t
P xt
)− λxt
λxt −1
X t .
The exporter faces Calvo-style nominal rigidities and non-reoptimized prices are
indexed in the same fashion, P xt+1 = (πx
t )κxπ1−κx P xt . When the exporting firm i is able
to reset its price, P new,xt , it solves the following optimization problem
maxP x,new
t
= Et
∞∑s=0
(βξx )s D t+s
[(πx
t ...πxt+s−1
)κx (π...π)1−κx Xi ,t+s P x,newt
− Pt+s
St+s
[Xi ,t+s +φx zt+s
]].
The resulting Phillips curve is
πxt = β
1+βκxπx
t+1 +κx
1+βκxπx
t−1 +(1−ξx )(1−βξx )
ξx (1+βκx )(mcx
t + λxt ),
where the real marginal cost of the exporter is given as mcxt = Pt
St P xt
. Note that the law
of one price does not hold in the short run when export prices are sticky (ξx > 0).
Since the domestic economy is of negligible size, the domestic price level and the
consumer prices coincide in the foreign economy, implying that P xt = P∗
t . Further,
by assuming that foreign consumption and investment are CES aggregated with the
same elasticity of substitution, the total export demand can be written as
X t =C xt + I x
t =(
P xt
P∗t
)−η f
Y ∗t , (3.6)
3.2. THE DSGE MODEL 93
where Y ∗t and P∗
t denote the foreign output and price level, respectively. Hence it
is not necessary to model foreign consumption and investment separately. Instead,
exports move proportionally to aggregate foreign output.
3.2.2 Households
The household sector is characterized by infinitely lived households, indexed by
n ∈ [0,1]. The representative household maximizes the expected discounted utility,
given by
Et
∞∑s=0
βt
ζc
t+s log (Cn,t+s −bCn,t−1+s )−ζht+s AL
hσLn,t+s
1+σL
, (3.7)
where Cn,t is a consumption good composite and hn,t denotes the hours worked
by the household. The parameter b governs the level of internal habit formation in
consumption and σL is the labor supply elasticity. There are two preference shocks,
ζct and ζh
t , that shift the intertemporal margins of utility of consumption and labor.
These are common for all households and can be interpreted as aggregate demand
and labor supply shocks, respectively. Consumption is assumed to be aggregated by
a CES function that combines domestically produced consumption goods, C dt , and
imported consumption goods, C mt .
Ct =[
(1−ωc )1ηc
(C d
t
) ηc−1ηc +ω
1ηcc
(C m
t
) ηc−1ηc
] ηcηc−1
.
The parameter ηc denotes the elasticity of substitution between domestic and for-
eign goods in consumption. Hence, a high value of ηc implies a high willingness
to shift the composition of aggregated consumption which becomes smooth even
if domestic or imported consumption are more volatile. The degree of home bias
in consumption given as (1−ωc ), which affects the steady state level of domestic
consumption. Cost minimization implies the following demand function for the two
types of consumption goods
C dt = (1−ωc )
(Pt
P ct
)−ηc
Ct , C mt =ωc
(P m,c
t
P ct
)−ηc
Ct . (3.8)
Similarly, investment is assumed to be a CES aggregate of domestic and imported
investment goods, I dt and I m
t , respectively.
It =[
(1−ωi )1ηi
(I d
t
) ηi −1ηi +ω
1ηii
(I m
t
) ηi −1ηi
] ηiηi −1
,
94 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
with the demand functions
I dt = (1−ωi )
(Pt
P it
)−ηi
It , I mt =ωi
P m,it
P it
−ηi
It . (3.9)
The consumer and investment price indices are given as
P ct =
[(1−ωc )P 1−ηc
t +ωc (P m,ct )1−ηc
] 11−ηc , P i
t =[
(1−ωi )P 1−ηit +ωi (P m,i
t )1−ηi] 1
1−ηi .
It is important to note the close relationship between the substitution elasticity
between domestic and foreign goods and the markup shocks within each import
sector, given in (3.5). A higher elasticity of substitution between domestic and foreign
goods will lower the markup for all importing firms. In fact, changes in the elasticity
are observationally equivalent to changes in the exogenous markups in each sector.
Thus, in this model, the markup shocks cover both changes in the optimal pricing
behavior of firm i (for example due to changes in the degree of competition) and
the extend to which the consumers are willing to substitute between domestic and
foreign goods.
Investment is used to accumulate capital by the following law of motion
Kt+1 = (1−δ)Kt +Υt F (It , It−1)+∆t ,
where δ is the rate of which capital depreciates.Υt is a stationary investment-specific
productivity shock, which affects the intertemporal margin of the investment deci-
sion of firms. The function F (It , It−1) =(1− F
(It
It−1
))It summarizes the relationship
between current and past investment and next period’s physical capital. It is assumed
to be cost free to invest at the steady state growth rate but increasingly costly as
investment moves away from the balanced growth path1. The variable ∆t reflects
that capital can be traded frictionlessly between households at price Pk ′,t . Although
the equilibrium condition ∆t = 0 holds for all periods t , this is included in the model
to derive the market value of capital.
In each period t , household n faces the the following intertemporal budget con-
straint
Bn,t+1 +St B∗n,t+1 + (1+τc )P c
t Ci ,t +P it In,t +Pt
[Pk ′,t∆n,t
]=Rt−1Bn,t + (1−τk )Tt + (1−τy
t )Wn,t hn,t + (1−τk )Rkt Kn,t +R∗
t−1Φ
(At−1
zt−1, φt−1
)St B∗
n,t
−τk[
(Rt−1 −1)Bn,t +(
R∗t−1Φ
(At−1
zt−1, φt−1
)−1
)St B∗
n,t +B∗n,t (St −St−1)
]+Qn,t .
(3.10)
1Only the second order derivative, F ′′ is identified in the log-linearized model and will be treated as aparameter.
3.2. THE DSGE MODEL 95
The household spends resources on domestic and foreign zero-coupon bond pur-
chases, Bn,t+1 and B∗n,t+1 which carry the gross interest rate, Rt and R∗
t , from period
t to t +1, respectively. Further, consumption and investment goods are purchased at
prices P ct and P i
t , respectively. Households gain resources from their bond holdings,
form profits transferred by the intermediate good producers, Tt , from labor income,
and from renting out capital to firms. The real net foreign asset position is given as
At = St B∗t+1
Pt. There is a risk premium, Φ
(At−1zt−1
, φt−1
)> 0, to holding foreign bonds if
the domestic economy as a whole is a net borrower (At < 0). Further, each household
has a negligible size, implying that increased borrowing has no aggregate effects on
the risk premium and thus does not internalize the effects on the net asset position.
Note that the households face idiosyncratic risks as suppliers of labor as the pricing
signal arrives stochastically with probability 1−ξw . To avoid ex post heterogeneity
and to preserve the representative agents framework the assumption is imposed that
the households can freely trade the entire set of Arrow-Debreu securities. This allows
the households to enter into an insurance scheme with perfect risk sharing, making
them ex post homogeneous. The net income from this portfolio is denoted Qn,t .
After imposing a symmetric equilibrium, the first-order conditions for Ct , ∆t ,
Kt+1, It , Bt+1, and B∗t+1, respectively are
0 = ζct
ct −b ct−1µz,t
−βEtbζc
t+1
µz,t+1ct+1 −bct− λt (1+τc )
P ct
Pt, (3.11)
0 =− λt
ztPk ′,t +qt , (3.12)
0 =−λt Pk ′,t +βEt
λt+1
µz,t+1
[(1−τk )r k
t+1 + (1−δ)Pk ′,t+1
], (3.13)
0 = λt
[Pk ′,tΥt F1 −
P it
Pt
]+βEt
λt+1
µz,t+1Pk ′,t+1Υt+1F2, (3.14)
0 =−λt +βEt
λt+1
µz,t+1πt+1
[Rt −τk (Rt −1)
], (3.15)
0 =−λt St +βEt
λt+1
µz,t+1πt+1
[R∗
t Φ(at , φt )St+1 −τk(R∗
t Φ(at , φt )−1)
St+1
−τk (St+1 −St )
]. (3.16)
The Lagrangian multiplier to the household budget constraint, λt , has been appro-
priately scaled as λt = Pt ztλt . Equation (3.11) is the Euler equation of consumption
which balances intertemporal marginal utility of consumption with the relative price.
The variable qt is the real Tobin’s Q, i.e. the marginal value of an extra unit of capital.
The equations (3.12)-(3.14) give the equilibrium price and optimal demand for capi-
tal, either through purchases or by investing in new capital. The relative demands of
96 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
domestic and foreign bonds in (3.15) and (3.16) are determined by the interest rate
spread and the expected capital gains from changes in the nominal exchange rate.
To model wage setting it is assumed that each household supplies monopolisti-
cally differentiated labor. The labor aggregate, ht , is constructed from the following
function
ht =[∫ 1
0h
1λw
i ,t di
]λw
, 1 ≤λw <∞,
where λw is the wage markup. Hence, the demand for labor of type n is
hn,t =(
Wi ,t
Wt
)− λw
λw −1ht . (3.17)
Wage setting is subject to Calvo-style nominal rigidities so that with probability ξw
the household is not able to set a new wage in the subsequent period which follows
an indexation rule instead, Wn,t+1 = πκwt π1−κwµz,t+1Wn,t . When the household is
able to reset its wage it solves the following optimization problem
maxW new
t
= Et
∞∑s=0
(βξw )s[−ζh
t+s ALh1+σL
n,t+s
1+σL
+ vt+s (1−τyt+s )(πt ...πt+s−1)κw (π...π)1−κw (µz,t+1...µz,t+s )W new
t hn,t+s
],
subject to (3.17).
3.2.3 UIP and Monetary Policy
The uncovered interest rate parity (UIP) relates expected changes in the nominal
exchange rate with the interest rate spread between the domestic and foreign econ-
omy. To ensure that the DSGE model is stationary, there is a risk premium on foreign
bonds which is strictly decreasing in the real net foreign asset position, at = St B∗t+1
Pt zt.2
The premium on foreign bonds is assumed to be described as
Φ(at , φt ) = exp
−φa(at −a)+ φt
, (3.18)
where the parameter φa denotes the sensitivity of net foreign asset holdings on the
risk premium and φt is a risk premium shock. As will be seen below, in an economy
with a fixed exchange rate this shock has a similar role as a monetary policy shock for
economies with a floating exchange rate.
2If this was not the case, the domestic households could borrow infinitely to finance consumption,violating the transversality condition. See Schmitt-Grohe and Uribe (2003).
3.2. THE DSGE MODEL 97
By combining the household demand for domestic and foreign bonds in (3.15)
and (3.16), a no arbitrage condition can be derived. In log-linearized form the UIP
condition is given as
Rt − R∗t = Et∆St+1 −φa at + ˆφt . (3.19)
Hence, in equilibrium, if the domestic interest rate is lower than its foreign counter-
part this must be reflected in either nominal exchange rate appreciation or the risk
premium. Similarly, an increase in the foreign interest rate must be met by a similar
domestic increase as if to avoid currency depreciation, which underlines the close
link between monetary policy and the exchange rate.
Assume now that the monetary policy can be described by the following Taylor
rule
Rt = ρR Rt−1 + (1−ρR )(κππ
ct−1 +κy yt−1 +κs∆St
). (3.20)
A credible and fixed exchange rate policy implies that Et∆St+1 = 0 for every period t ,
which requires that the central bank responds forcefully to changes in the nominal
exchange rate, κs →∞. Combining this with (3.19) gives the following log-linearized
relationship for the domestic interest rate
Rt = R∗t −φa at + ˆφt , (3.21)
implying that monetary policy is now completely endogenous in the sense that
changes in the foreign interest rate will affect the domestic interest rate one-to-one.
Any interest rate spread between the domestic and foreign policy rates is due to
the risk premium that domestic consumers have to pay on foreign bonds. Hence,
a risk premium shock, φt , is similar in nature to a monetary in closed-economy
models or models where the exchange rate is flexible. However, where the latter has
the interpretation of an exogenous deviation from the policy rule, a shock to (3.21)
reflects an exogenous increase in the perceived risk of foreign investors of holding
domestic bonds, e.g. due to fear of devaluation. Note finally, that φt is not a true
structural shock, in the sense that it is derived from first principles such as utility
maximization but instead a residual that captures UIP deviations.
3.2.4 Equilibrium Conditions and Shock Processes
Final goods market equilibrium is given as
C dt +C x
t + I dt + I x
t +Gt = εt z1−αt Kα
t H 1−αt −φzt .
Inserting the demand functions (3.6), (3.8), and (3.9) and stationarizing variables
yields
(1−ωc )(γc,dt )ηc ct + (1−ωi )(γi ,d
t )ηi it + (γx,∗t )−η f y∗
t z∗t + g t = εtµ
−αz,t kαt H 1−α
t −φ.
98 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
The variable z∗t ≡ z∗t
zthas been used to detrend foreign output and allows the domestic
and foreign productivity to evolve in a stationary asymmetric fashion.
The evolution of net foreign assets is described by the balance of payments
St B∗t+1 = St P x
t (C xt + I x
t )−St P∗t (C m
t + I mt )+St R∗
t−1Φ(at−1, φt−1)B∗t .
The balance of payments gives the equilibrium dynamics between foreign debt and
the trade balance. If imports exceed exports this must be financed by an increase in
foreign debt (corresponding to a decrease in net foreign assets). This in turn increases
the interest rate that domestic households must pay on foreign bond in the next
period.
Inserting the demand functions and stationarizing variables yields
at =(mcxt )−1(γx,∗
t )−η f y∗t
z∗t
zt− (mcx
t γx,∗t )−1
[ωc
γmc,dt
γc,dt
−ηc
ct +ωi
γmi ,dt
γi ,dt
−ηi
it
]
+ St
St−1R∗
t−1Φ(at−1, φt−1)at−1
πtµz,t.
The exogenous shocks in the model are all assumed to adhere to an AR(1) process
xt+1 = ρx xt +σxεxt+1, εx
t+1 ∼N(0,1),
where x = λd ,λmc ,λmi ,λx ,µz ,ε, z∗,ζc ,ζh ,Υ, φ,τy , g .
Finally, a set of relative prices is used to define the model equilibrium. They are
given in Appendix A.
3.2.5 Foreign Economy
By assuming that the domestic economy is of negligible size compared to the foreign
economy, the latter can be modeled exogenously. This significantly simplifies the
modeling task and allows for a more flexible specification. The foreign economy is
assumed to have the following structural VAR representation
B0Xt = B1Xt−1 + ...+Bp Xt−p +εt , εt ∼N(0,Ik ). (3.22)
The vector Xt =(π∗
t , y∗t ,R∗
t
)′ consists of HP-filtered Euro area inflation and output3
and the demeaned ECB nominal interest rate. To obtain identification of the structural
3One concern of using HP-filtered data for the foreign economy is that it might make the foreignvariables "too smooth" compared the their unfiltered domestic counterparts. Preliminary results show thatusing demeaned growth rates in the foreign VAR tend to increase the relative importance of foreign shocksin the model, however the forecasting implications of this change was not investigated. This remains animportant question for further research.
3.3. ECONOMETRIC METHODOLOGY 99
shocks it is assumed that B0 is lower triangular. Thus, a Cholesky decomposition
of the estimated covariance matrix can be applied to obtain identification from
the reduced-form representation of (3.22). The parameters of the VAR model are
estimated before those of the DSGE model using a Bayesian approach with prior
distributions represented by dummy observations.4
3.3 Econometric Methodology
The solution to the DSGE model in Section 3.2 has to be approximated before it
can be estimated. In this paper, the model equilibrium will be approximated by a
log-linearization around the non-stochastic steady state. Provided that the model
has a unique and stable solution this can be expressed as
st+1 =Φ1(θ)st +Φε(θ)εt+1, (3.23)
where st is a vector of appropriately defined model variables and θ is vector contain-
ing the parameters of the DSGE model. The observables are related to the model by a
set of measurement equations
yt =Ψ0(θ)+Ψ1(θ)st +ut . (3.24)
The matrices Φ1, Φε, Ψ0, and Ψ1 of the reduced-form system depend on the un-
derlying structural model parameters, θ. Measurement errors, ut , are added to all
observables except the interest rates. This is not necessary for the applications in
this paper in the sense that the model contains enough structural shocks to avoid
stochastic singularity for the chosen number of data series. However, including mea-
surement errors can be a beneficial way of relaxing the restrictions imposed by the
model during the estimation. Further, it is well known that macroeconomic data
series are subject to revisions because their "true" values are unknown.
The equations (3.23) and (3.24) constitute the state-space representation of the
(linearized) DSGE model. It is assumed that ut ∼N(0,Σu), where Σu is a diagonal
matrix, and since εt ∼N(0,Σε) follows from the model the exact likelihood of the
model can be evaluated directly, using the Kalman filter.
In the following, let the nobs ×T matrix Y1:T matrix denote the sequence of ob-
servables
y1, ...,yT
. The structural parameters are estimated using Bayesian econo-
metrics, which is described in detail in An and Schorfheide (2007). Thus, the object
of interest is the posterior distribution
p(θ|Y1:T ) = p(Y1:T |θ)p(θ)
p(Y1:T ), (3.25)
4See Sims and Zha (1998) for a detailed description. These so-called Minnesota priors are dependenton several hyperparameters that controls the correlation structure of the dummy observations and arewell known to increase the forecasting performance, relative to a standard VAR model.
100 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
where p(Y1:T ) is the model likelihood, p(θ) is the prior distribution, and p(Y1:T ) is
the data density. For DSGE models it is generally not possible to obtain closed-form
expressions of (3.25). This is instead approximated by simulating Markov chains that
converges to this posterior ergodically. Specifically, a numerical optimizer is applied
to approximate the Hessian at the posterior mode which then guides the step size for
the Random-Walk Metropolis Hastings algorithm using a chain with 500,000 draws,
discarding the first 250,000 draws to better ensure convergence. The Hessian is scaled
such that the acceptance rate in the chain is approximately 20-30 percent, close
to what is generally recommended in the literature (see for example Roberts et al.
(1997)). Convergence of the chain was checked by using the convergence diagnostics
in Geweke (1999) which in standard in the literature (not shown).
3.4 Data
The model is estimated using quarterly seasonally adjusted Danish data from 1990Q2-
2010Q4. As the model allows for trends the raw data series can in general be used in
the estimation procedure as opposed to using pre-filtered data. However, following
Adolfsen et al. (2007b), some of the series are altered prior to estimation: First, due
to increased globalization, both imports and exports grow at a faster rate than the
overall economy in the sample period. This is at odds with the model that imposes
constant steady state ratios of imports and export to output, determined by ωc and
ωi . Instead of making these coefficients time-varying the "excess" linear trend of the
two series relative to that of GDP is removed. Second, Euro area output and inflation
are HP-filtered prior to estimation while the foreign interest rate is demeaned.
Throughout this paper, the question of data vintages is set aside. Hence, all data
series used are the latest vintage. Since the model performance is not compared with
real-time forecasts this is not likely to affect the assessment of the relative forecasting
performances.
The set of observables used in the measurement equations in (3.24) are (real) GDP,
consumption, investment, wages, exports, imports, the Danish policy rate, the GDP
deflator, the consumer price index, and Euro area output, inflation, and interest rate.
The set of measurement equations and how they relate to the model variables is given
in Appendix C. As noted in Adolfsen et al. (2007b) the foreign economy variables
can still be included in the estimation to the modeler’s advantage, even though this
part is modeled exogenously and estimated prior to the estimation of the DSGE
model. The reason is that they contain information about the transmission of foreign
shocks through the domestic economy and enables identification of the risk premium
and asymmetric productivity shocks. Further, the foreign variables contain crucial
information about the current state of the economy and hence for the forecasts.
The choice of data series in the estimation turns out to be crucial for the forecast-
3.5. PRIOR DISTRIBUTIONS AND CALIBRATED PARAMETERS 101
ing performance. On one hand, including more data series will help identifying the
structural parameters in the model. For example, including another price deflator
will sharpen the parameter estimate of that particular nominal rigidity. On the other
hand, if too many series are included this will have adverse effects on the filtering of
the unobserved state, st . This will in turn affect the mean of the forecasts through the
transition equation in (3.23). Specifically, it is found that the medium and long term
forecasts suffer if too many inflation series are included.
Finally, for the measurement equations in (3.24) a new set of model variables is
defined to consistently match the data specifications. Aggregate consumption and
investment are CES aggregated in the model but enter the national income identity
linearly. Hence, to get consistent measures, the consumption, investment, imports,
and consumer price series are transformed (see Appendix B for details).
3.5 Prior Distributions and Calibrated Parameters
The prior distributions used in the estimation are given in Table 3.2. Note that, in
the model estimation, the Kalman filter generates a sequence of probability den-
sities which form the model likelihood p(Y1:T |θ) = ∏Tt=1 p(yt |Y1:t−1,θ). Hence, the
estimation routine which uses the posterior distribution in (3.25) is based on the
1-step ahead prediction error, while we are interested in a high density at longer
horizons as well. Further, with diffuse priors DSGE models have a tendency to be
multi-modal with deep valleys of low likelihood in between. It is well known that the
Random Walk Metropolis Hastings algorithm has very low efficiency in this case. As a
result, somewhat tight priors is generally necessary to ensure a decent forecasting
performance.
The priors are broadly in line with previous literature and take into account
the a priori restrictions imposed on the parameter space. For the nominal price
and wage rigidity, the beta distribution is specified since the domain of this density
function is [0,1]. The mean is set to imply a reset probability of 25 percent, implying
an average duration of contracts of four quarters. The indexation parameters are
also beta distributed with mean 0.25, since most previous studies find relatively
low degrees of indexation. For the elasticity of foreign demand and the substitution
elasticity of domestic and foreign investment, the inverse gamma distribution with
mean 1.5 is used since this assigns probability mass only to positive parameter values.
The price markups also adhere to an inverse gamma distribution with mean 1.2,
which corresponds to a 20 pct. markup over marginal cost and a demand elasticity of
6. The prior distribution of F ′′ is a normal distribution with mean 5.0 and standard
error 1.5. The habit parameter is given a beta distribution with mean 0.5.
Since the exogenous shock processes assert great influence on the dispersion
of the endogenous variables, the magnitude of persistence and standard errors will
102 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
greatly affect the posterior predictive density of the observables. Hence, relatively
uninformative priors are generally specified for parameters governing the shocks in
the model so as mostly to let the data guide the posterior estimates. For the persis-
tence coefficients the beta distribution is used while the inverse gamma distribution
is used for the standard errors. All prior distributions are kept constant during the
sequential estimation of the model.
A subset of the parameters in the DSGE model is calibrated instead of estimated,
i.e. they are assigned a prior with infinite mass at the calibrated value. This is done
mainly because these parameters are weakly identified in the data, implying that
the model likelihood is flat in these dimensions or because they determine the so-
called "great ratios" (e.g. imports as share of total output). Examples include capital
depreciation, the discount factor, and the labor supply elasticity. Secondly, due to the
sequential estimation used in the forecasting exercise in this paper, some parameters
are calibrated rather than estimated to avoid unstable estimates with poor out of
sample properties as a result. Tihs includes the elasticity of substitution in consump-
tion is fixed at ηc = 2.5. Thirdly, the monetary policy parameters are calibrated to
reflect the fixed exchange rate regime of Denmark. Specifically, κs = 100,000 is set
while keeping the remaining parameters at standard values. Finally, the fiscal shocks
are calibrated with persistency parameter of 0.5 and a standard error of 1 percent.
The measurement errors in (3.24) are given standard errors of 10 percent of that of
the corresponding data series. Table 3.1 provides an overview of the calibration.
3.6 Posterior Model Evaluation
The estimated parameters are reported in Table 3.2 and 3.3. This section presents
some of the properties of the estimated DSGE model.
As a posterior predictive check Table 3.4 shows the model’s ability to match the
unconditional second-order moments in the data. Specifically, the standard devi-
ations, first-order autocorrelations, and the correlations with GDP growth will be
considered for the set of variables used in the estimation (all in quarterly growth
rates). The model-implied moments are the analytical moments based on the mode
of the posterior distribution.
Considering first the standard deviations. It can be seen that these are slightly
higher in the model than in the data for output, wages, and the price deflators. How-
ever, for the remaining variables the model is relatively successful in replicating the
volatility found in the data. For the first-order autocorrelations the model generally
does a good job at matching the persistency in the data. However, the model is unable
to match the low persistency found in the wage, consumption, and consumer price
series. Finally, for the cross-correlations with output growth it can be seen that wages
3.6. POSTERIOR MODEL EVALUATION 103
and consumption are somewhat too procyclical in the DSGE model. Since it is gen-
erally hard for DSGE models to generate acyclical real wages (see for example King
and Rebelo (1999)), it can be concluded that the model generally does very well in
capturing the correlation with output growth. Overall, for a DSGE model this size the
level of conformity to the moments in the data can be characterized as satisfactory.
3.6.1 Impulse Response Functions
It has become common in the DSGE literature to study the endogenous propagation
of exogenous shocks by looking at impulse response functions (IRFs). The figures
are based on the posterior mode of the estimated parameters. For ease of exposition
only a subset of the model shocks will be considered in this section: A stationary
productivity shock, a shock to the domestic price markup, and a shock to foreign
output.
Figure 3.1 shows the impulse response of output, real wage, consumption, in-
vestment, imports, exports, the output deflator, and consumer price inflation to an
expansionary shock to the stationary productivity level of one standard error. From
(3.3) it is seen that this lowers the real marginal costs of domestic producers. This
in turn feeds through to prices and reduces domestic inflation which follows from
the Phillips curve in (3.4). Aggregate consumption increases since forward looking
consumers spend parts of their increased future income today. The increased produc-
tivity raises the marginal product of capital which spurs an increase in investment.
The increase in aggregate demand leads to higher marginal production costs which
eventually cancel out the fall in inflation. Lower domestic inflation implies an im-
provement in the terms of trade which leads to an increase in demand for domestic
goods instead of imports which declines initially. Eventually, increased domestic
demand for consumption and investment increases imports above its steady state
level. Exports increase as well, due to an improvement in the terms of trade through
the real exchange rate. As pointed out by Pedersen and Ravn (2013), all shocks in
the model that affect the different inflation measures in the domestic economy will
cause overshooting. Due to the small open economy assumption (and since all goods
are tradeable), the terms of trade has to return to its long-run level. This adjustment
can only happen through the adjustment of domestic prices in a fixed exchange rate
regime.5 Note that the nominal interest rate remains constant instead of falling, as
usually implied in DSGE models where the central bank seeks to stabilize inflation.
Thus, the fixed exchange rate mitigates the real effects of productivity shocks.
Next, consider the impulse responses to an increase in the domestic price markup,
depicted in Figure 3.2. This can be interpreted as an increase in the degree of market
5From a more technical perspective, this inflation overshooting is necessary for solution determinacy.
104 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
power for the representative firm, thus allowing it to set a higher price, ceteris paribus.
As a result, domestic and consumer price inflation goes up. As markups are only
weakly correlated over time this effect only lasts a few quarters until inflation falls
again. Higher prices lower demand for domestic goods, resulting in a fall in output
as well as exports due to a real exchange rate appreciation which on the contrary
boosts imports. The reduced demand for domestic goods results in lower real wages
and hence marginal cost of production. Eventually inflation goes down to restore
competitiveness so exports return to the steady state. Again, we see the inflation
overshooting for both the output deflator and consumer prices to obtain balance of
trade equilibrium. As with the productivity shock, the exchange rate peg works coun-
tercyclical in this case as the interest rate is not increased in face of higher inflation.
Lower real interest rates support domestic aggregate consumption and investment
hence mitigates the real downturn of the shock.
Figure 3.3 shows the impulse response to an increase in the foreign output. A di-
rect effect of the increase in foreign demand is a sharp increase in exports. This
increase in aggregate demand causes output, consumption, and investment to in-
crease and remain above their steady state values for a long time. Forward looking
firms respond to the increase in current and expected future aggregate demand by
raising price so both domestic and consumer price inflation increase. Rising income
and domestic inflation cause imports to increase, the latter because of a real exchange
rate appreciation. The fixed exchange rate policy works procyclical in face of foreign
demand shocks. If instead the exchange rate was floating, a nominal depreciation
could serve as a buffer against adverse foreign shocks because it would support
exports by restoring competitiveness. Further, as monetary policy does not target
inflation the interest rate is kept constant in face of increased inflation. Hence, foreign
shocks are transmitted more forcefully and persistent through the domestic economy
because of the fixed exchange rate.
3.6.2 Forecast Error Variance Decomposition
It is useful to assess which structural shocks are important for forecasting which
variables and at different horizons. Such a forecast error variance decomposition
is closely related to the impulse response functions. LetΩhj ≡ ∂yt+h
∂ε j ,tbe the impulse
response for a given variable at horizon h to shock j with standard errorσ j , where h =1, ..., H and j = 1, ..., J . The contribution to the H-step ahead forecast error variance
from shock i can then be expressed as ΣHi =∑H
h=1
(Ωh
i
)2σ2
i /∑J
j=1
∑Hh=1
(Ωh
j
)2σ2
j .
Table 3.5 and 3.6 show the forecast error variance decomposition for the endoge-
nous variables used in estimation at the 1- and 8-step ahead horizons, respectively.
The parameters are fixed at the posterior mode for the initial estimation sample.
3.6. POSTERIOR MODEL EVALUATION 105
In line with Smets and Wouters (2003) the investment specific preference shock
is found to be a major contributor to output variations. Like the shock to consump-
tion preferences this can be interpreted as a domestic aggregate demand shock, as
both output and inflation increase. Together, these two shocks account for more than
30 percent of the variation in GDP growth for both horizons.
Productivity shocks account for approximately 5.5 and 7.5 percent of GDP growth
at the 1 and 8 quarter horizon, respectively. Shocks to labor preferences account
for between 3 and 6 percent of output variations approximately. It is characteristic
for both of these capacity shocks that their relative importance increase with the
forecasting horizon for most of the observed variables.
With approximately 40 percent in total, the largest contributors to output vari-
ations at both 1 and 8 quarters are the markup shocks, in particular the markup in
the export sector. This might be necessary for the model to explain the joint evolve-
ment of a relatively large set of observables included in the estimation. As discussed
previously, the fixed exchange rate imposes restrictions not found in models with
a flexible exchange rate, as all variations in the real exchange rate must come from
the relative prices. Thus, the somewhat large contribution to several variables by the
markup shocks might potentially reflect some model misspecification or inability to
explain comovements of both prices and quantities.
Not surprisingly, aggregate consumption growth is driven to a large extend by
shocks to consumption preferences. Similarly, for investment, almost all of the varia-
tion comes from the investment specific shock while real wage variation comes from
shocks to labor preferences and the domestic price markup.
The importance of foreign shocks6 is seen for both GDP, consumption, and ex-
ports growth, where these account for approximately 15 to 21 percent of fluctuations.
This is considerably less than what is found in Pedersen and Ravn (2013) but still
more than what is found in most of the open economy DSGE literature, for example
in Justiniano and Preston (2010). Thus, as we would expect from economic theory,
this suggests that the fixed exchange rate tends to increase the relative importance of
shocks originating from the foreign economy.
Turning to the inflation series, the tables show that the markup shocks account
for most of the variability. However, going from 1 to 8 quarters other shocks become
more important in explaining the variation in inflation, whereas the importance of
weakly correlated markup shocks diminishes (a finding that echoes that in Adolfsen
et al. (2007b)).
6"Foreign shocks" here and below are defined as shocks to the foreign VAR model, UIP deviations, andthe asymmetric productivity shock, zt .
106 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
3.7 DSGE Model-based Forecasting
This sections briefly explains how to obtain forecasts from the estimated DSGE model
(see Del Negro and Schorfheide (2013) for a more detailed discussion hereof). For
forecasting purposes we are interested in the h-step ahead predictive density of the
vector yT+h , given the past observations, Y1:T . This can be expressed as
p(YT+1:T+h |Y1:T ) =∫θ
p(YT+1:T+h |θ,Y1:T )p(θ|Y1:T )dθ
=∫
(sT ,θ)
∫ST+1:T+h
p(YT+1:T+h |ST+1:T+h)
p(ST+1:T+h |sT ,θ,Y1:T )dST+1:T+h
×p(sT |θ)p(θ|Y1:T )d(sT ,θ),
(3.26)
where ST+1:T+h denotes the sequence sT+1, ...,sT+h. Hence, when conducting fore-
casts with a DSGE model there are four sources of uncertainty affecting the predictive
distribution in (3.26). First, there is parameter uncertainty reflected in the posterior
distribution p(θ|Y1:T ). Second, since the vector of state variables contains variables
that are unobservable to the econometrician, there is uncertainty about the current
state of the economy, captured by p(sT |θ,Y1:T ). Third, there is uncertainty about the
future states of the economy and the realization of shocks as reflected by p(ST+1:T+h).
Fourth, since the observables are being measured with error there is uncertainty
about the true values of the observed time series, p(YT+1:T+h |ST+1:T+h). Bayesian
inference provides predictive distributions that take into account all these sources
of uncertainty. Since a closed-form expression of (3.26) is not available this must
instead be approximated by drawing y(i )T+h (based on a parameter draw θi ), where
i = 1, ..., Nsi m .
Often, the primary object of interest is the mean of the posterior predictive dis-
tribution. In fact, if the loss function associated with forecast errors is quadratic the
mean will minimize the expected loss. Using (3.23) and (3.24), the h-step ahead point
forecast based on the mean is given as
E[yt+h |θ,Y1:T
]=Ψ0(θ)+Ψ1(θ)[Φ1(θ)
]hE[st |θ,Y1:T
], (3.27)
where E[st |θ,Y1:T
]is found using the Kalman filter and numerical integration with
respect to the parameter vector θ. Alternatively, a plug-in estimate of θ can be used
(e.g. the posterior mode or mean), which speeds up the computation but ignores the
true Bayesian distribution, not taking into account parameter uncertainty. Amisano
and Geweke (2013) find that using the full Bayesian predictive distribution substan-
tially improves the model predictability relative to the plug-in estimate. However, this
is will be dependent of the specific application, for example how much density of the
3.7. DSGE MODEL-BASED FORECASTING 107
posterior distribution is near the plug-in estimate. Overall, any plug-in estimate will
underrate the dispersion of p(YT+1:T+h |Y1:T ) and therefore, in the section below, the
full Bayesian forecasts will be used.
3.7.1 Forecasting Performance
The forecasting performance of the DSGE model and the alternative models is as-
sessed by a rolling estimation and forecast procedure in the following way: First, the
model parameters are estimated using data up until time T . Second, the estimated
model is used to compute point forecasts and forecast densities H quarters ahead,
yT+1,yT+2, ...,yT+H . Then, the information set is updated as the estimation sample
is increased to Y1:T+1 and new forecasts are made. These steps are iterated for the
entire sample where the initial estimation sample is 1990Q2-2001Q4 and the hold-out
sample is 2002Q1-2008Q2 (the Great Recession episode starting in 2008Q3 will be
examined separately below). The forecast horizons h = 1,2, ...,8 will be considered.
Hence, there are 26 observations for the 1-step ahead and 19 observations for the
8-step ahead forecasts. The variables considered below are (real) GDP, wages, con-
sumption, investment, imports, exports, the output deflator and the consumer price
index.
The point forecast accuracy is based on the mean of the posterior predictive
distribution given in (3.27). As is common in the literature the root mean square errors
(RMSE) will be used as the assessment criteria. The forecasts of the DSGE model
will be compared with an AR(1) model and the random walk, i.e. constant growth
rates. Because of the simplicity of these alternative models few further assumptions
that will affect the relative model performances have to be made. Further, it is well
known that it is not an easy task for structural models to beat the AR(1) model in
out-of-sample predictions so this will be a good benchmark to measure the DSGE
model against.
Figure 3.4 and 3.5 show the RMSEs, based on yearly growth rates as a function of
the forecast horizon from one to eight quarters ahead. Generally, the DSGE model
does well in forecasting the different variables, better than the random walk and
comparable in performance to the AR(1) model. Thus, the results suggest that it
is possible to apply a DSGE model incorporating a fixed exchange rate regime for
forecasting and obtain reasonable forecasts.
For GDP, the DSGE and AR(1) model have almost the same predictive ability for
short horizons, although for forecasts four to six quarters ahead, the DSGE model is
more accurate. Both models are better than the random walk and this is especially
pronounced at longer forecast horizons. For aggregate consumption, the RMSE of the
three models follow a similar pattern. Hence, it seems that the random walk is hard
to beat for consumption at short horizons. However, at longer horizons the other
108 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
models (and especially the DSGE) clearly outperform constant growth rates predic-
tions. Similarly, the random walk is outperformed for all horizons for investment
growth. In this case, the DSGE model seems more accurate than the AR(1) model for
shorter horizons while the opposite is the case for forecasts longer than a year. For
consumer price inflation it is again hard to beat the random walk, although the DSGE
model seems to be slightly more accurate across forecasting horizons in general. It is
particularly encouraging to observe the forecasting performance of the open econ-
omy DSGE model for the growth rates of imports and exports. For these variables,
the model performs well compared to the alternatives, especially pronounced at
shorter horizons. Finally, for real wage growth and the domestic price deflator, the
DSGE model does a poor job at forecasting beyond two quarters when compared to
the other two models. However, it should be noted that the RMSEs are somewhat
contaminated by a few periods with very poor forecasts, in which the medium-run
variations in these series are too large.
3.7.2 Forecasting During the Great Recession
After having analyzed the properties and forecasting performance of the fixed ex-
change rate DSGE model, specific attention will be given to the Great Recession
period in Denmark, which I define as starting in 2008Q3. For the sake of brevity, the
subsequent analysis will focus on the model’s ability to predict GDP growth.
Figure 3.6 shows the realized yearly growth rates and forecasts for the DSGE model
(upper row) and the AR(1) model (lower row). As a general pattern we see that both
models tend to be too optimistic throughout the Great Recession by over-predicting
the growth in real GDP. Initially, both models strongly underestimate the severity
of the downturn but instead predict a steady return to the balanced growth path
when forecasting from 2008Q3 and 2008Q4. This is consistent with results found
from a similar exercise performed for the Smets-Wouters model in Del Negro and
Schorfheide (2013) and as such not surprising. In 2009Q1, the GDP forecasts of the
DSGE model start to differ from the AR(1) model. Here, it can be seen from the figure
that the DSGE model is more pessimistic. At this point, real GDP in Denmark had
already contracted for three consecutive quarters, underlining the severity of the
crisis. The DSGE model correctly predicts a continued decline in GDP growth in the
second quarter of 2009, although actual growth decreased even more. From 2009Q2,
the DSGE model captures the gradual recovery of the Danish economy with fairly
accurate growth predictions while the AR(1) model predicts a stronger recovery than
what occurred.
A major benefit of using DSGE models is that they provide a structural interpre-
tation of business cycles. Figure 3.7 shows the shock decomposition for yt . The line
depicts the deviations from the steady state of the filtered variable and the bars the
3.8. CONCLUSION 109
contribution of the smoothed structural shocks. Hence, we can interpret this as the
ex post interpretation of the crisis when seen through the lens of the estimated DSGE
model. Since Denmark is largely affected by the Euro zone economy it is interesting to
examine whether the prediction errors during the crisis were caused by domestic or
foreign sources. According to the figure, output fell below its steady state in the third
quarter of 2008. This is consistent with the fact that real GDP growth in Denmark was
negative in this quarter, compared to the previous quarter and the year before. It is
noteworthy that the foreign variables contributed positively to Danish GDP in all the
pre-crisis quarters, reflecting the boom in the Euro area which supported exports.
From the onset of the crisis however, the foreign variables make a strong negative
contribution to output. Thus, to a wide extent the crisis was initially imported from
the rest of the world. The large residuals in the foreign VAR might reflect that model-
ing of the foreign economy does not sufficiently account for nonlinearities. Domestic
demand makes a significant negative contribution throughout the downturn. In fact,
this group of shocks continue to affect output negatively, even as the Euro area recov-
ered temporarily in 2010. This underlines that the Danish economy recovered more
slowly than the Euro zone, in part because domestic consumption and investment
were suppressed. However, after the initial downturn, the main contributors to the
negative output gap in Denmark are negative markup shocks.
3.8 Conclusion
This paper estimates an open-economy DSGE model in which the nominal exchange
rate is fixed and examines its forecasting properties. While the predictive abilities of
DSGE models have previously been examined in various papers, these have focused
on either closed-economy models or models where the exchange rate is flexible.
Fixing the exchange rate has implications for the relative importance of shocks and
their propagation through the model. Specifically, foreign shocks assume a greater
role since the nominal exchange rate can not serve as a buffer and since monetary
policy can no longer be used as a macroeconomic stabilization tool.
Using Danish data from 1990Q2-2008Q2 a sequential out-of-sample forecast
evaluation of the DSGE model is constructed for (real) GDP, wages, consumption,
investment, imports, exports, the output deflator and the consumer prices index,
considering forecasting horizons of one to eight quarters. The DSGE model generally
delivers point estimates that are comparable to an AR(1) model and better than the
random walk when measured by the root mean square errors.
Finally, the Great Recession episode from 2008Q3 is examined specifically. In
line with previous research the DSGE model largely underestimates the severity of
the downturn initially. However, from 2009Q1 the DSGE model correctly predicts
a continued decline in GDP growth and subsequently relatively modest increase
110 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
in growth. This can be compared to the AR(1) model which consistently predicts a
stronger recovery than what occurred. A historical shock decomposition of output
deviations shows that the initial sharp decline in Danish GDP is primarily caused by
foreign shocks. The continued downturn however is mainly due to shocks to domestic
demand and markups. Generally, the large contribution to fluctuations in several
variables by markup shocks in the model might suggest some sort of misspecification
and is an obvious topic for future research.
Acknowledgments
The author gratefully acknowledges support from Aarhus University, Department
of Economics and Business Economics and from CREATES - Center for Research
in Econometric Analysis of Time Series (DNRF78), funded by the Danish National
Research Foundation. Comments from Frank Schorfheide are also gratefully acknowl-
edged. Further, I am grateful to Jesper Pedersen and Søren Hove Ravn for sharing the
dataset used in their paper.
3.8. CONCLUSION 111
References
ADOLFSEN, M., M. K. ANDERSSON, J. LINDE, M. VILLANI, AND A. VREDIN (2007a):
“Modern Forecasting Models in Action: Improving Macroeconomic Analyses at
Central Banks,” International Journal of Central Banking.
ADOLFSEN, M., S. LASEEN, J. LINDE, AND M. VILLANI (2007b): “Bayesian estimation
of an open-economy DSGE model with incomplete pass-through,” Journal of
International Economics, 72.
AMISANO, G. AND J. GEWEKE (2013): “Prediction Using Several Macroeconomic Mod-
els,” European Central Bank working paper series.
AN, S. AND F. SCHORFHEIDE (2007): “Bayesian Analysis of DSGE models,” Econometric
Reviews, 26.
CHRISTIANO, L. J., M. EICHENBAUM, AND C. L. EVANS (2005): “Nominal Rigidities and
the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy,
113.
DEL NEGRO, M. AND F. SCHORFHEIDE (2004): “Priors from General Equilibrium
Models for VARS,” International Economic Review, 45.
——— (2013): “DSGE Model-Based Forecasting,” Chapter in Handbook of Economic
Forecasting, Volume 2, Part A.
GEWEKE, J. (1999): “Using simulation methods for Bayesian econometric models:
Inference, development and communication,” Econometric Reviews, 18.
JUSTINIANO, A. AND B. PRESTON (2010): “Can structural small open-economy mod-
els account for the influence of foreign disturbances?” Journal of International
Economics, 81.
KING, R. G. AND S. T. REBELO (1999): “Resuscitating Real Business Cycles,” Handbook
of Macroeconomics, 1.
PAGAN, A. (2003): “Report on Modeling and Forecasting at the Bank of England,” Bank
of England Quarterly Bulletin.
PEDERSEN, J. AND S. H. RAVN (2013): “What Drives the Business Cycle in a Small
Open Economy? Evidence from an Estimated DSGE Model of the Danish Economy,”
Danmarks Nationalbank working paper series.
ROBERTS, G. O., A. GELMAN, AND W. R. GILKS (1997): “Weak Convergence And
Optimal Scaling of Random Walk Metropolis Algorithms,” The Annals of Applied
Probability, 7.
112 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
SCHMITT-GROHE, S. AND M. URIBE (2003): “Closing Small Open Economy Models,”
Journal of International Economics, 61.
SIMS, C. A. AND T. ZHA (1998): “Bayesian Methods for Dynamic Multivariate Models,”
International Economic Review, 39.
SMETS, F. AND R. WOUTERS (2003): “An Estimated Dynamic Stochastic General Equi-
librium Model of the Euro Area,” Journal of the European Economic Association,
1.
——— (2007): “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Ap-
proach,” American Economic Review, 97.
3.8. CONCLUSION 113
Appendix A: Model overview
1. πdt = β
1+κdβπd
t+1 + κd1+κdβ
πdt−1 +
(1−ξd )(1−βξd )ξd (1+κdβ) (mcd
t + λdt ),
2. r kt = wt + Rt−1 + Ht − kt + µz,t ,
3. mcdt =αr k
t + (1−α)(wt + Rt−1)− εt ,
4. πm,ct = β
1+κm,cβπm,c
t+1 +κm,c
1+κm,cβπm,c
t−1 +(1−ξm,c )(1−βξm,c )ξm,c (1+κm,cβ) (mcm,c
t + λm,ct ),
5. mcm,ct =−mcx
t − γx,∗t − γmc,d
t ,
6. πm,it = β
1+κm,iβπm,i
t+1 +κm,i
1+κm,iβπm,i
t−1 +(1−ξm,i )(1−βξm,i )ξm,i (1+κm,iβ) (mcm,i
t + λm,it ),
7. mcm,it =−mcx
t − γx,∗t − γmi ,d
t ,
8. πxt = β
1+κxβπx
t+1 + κx1+κxβ
πxt−1 +
(1−ξx )(1−βξx )ξx (1+κxβ) (mcx
t + λxt ),
9. wt =− 1η1
[η0wt−1 +η2wt+1 +η3πt +η4πt+1 +η5π
ct−1 +η6π
ct +η7ψz,t
+η8Ht +η9τyt
]+ ζht ,
10. kt+1 = 1−δµz
(kt − µz,t )+ (1− 1−δµz
)(it + Υt ),
11. ct = 1µ2
z+b2β(bβµz ct+1 +bµz ct−1 −bµz (µz,t −βµz,t+1)
+(µz −bβ)(µz −b)ψz,t + τc
(1+τc ) (µz −bβ)(µz −b)τct + (µz −bβ)(µz −b)γc,d
t
−(µz −b)(µz ζct −bβζc
t+1)),
12. it = µ2z F ′′
(µ2z F ′′)(1+β)
(it−1 +βit+1 − µz,t +βµz,t+1 + Pk,t − γi ,dt + Υt ),
13. ψz,t + µz,t+1 − ψz,t+1 − β(1−δ)µz
Pk,t+1 + Pk,t − µz−β(1−δ)µz
r kt+1 + τk
(1−τk )µz−β(1−δ)
µzτk
t+1,
14. ∆St+1 = Rt − R∗t +φa at + ˆφt ,
15. (1−ωc )(γc,d )ηc cy (ct +ηc γ
c,dt )+ (1−ωi )(γi ,d )ηi i
y (it +ηi γi ,dt )
+ gy g t + y∗
y (y∗t −η f γ
x,∗t + ˆz∗
t ) =λd (εt +α(kt − µz,t )+ (1−α)Ht ),
16. at = y∗(−mcxt −η f γ
x,∗t + y∗
t + ˆz∗t )+ (cm + i m)γ f
t
−cm(−ηc (1−ωc )(γc,d )ηc−1γmc,dt + ct )
−i m(−ηi (1−ωi )(γi ,d )ηi−1γmi ,dt + it )+ R
πµzat−1,
17. Rt = ρR Rt−1 + (1−ρR )(κππct−1 +κy yt−1 +κs∆St ),
18. γc,dt =ωc (γmc,c )(1−ηc )γmc,d ,
19. γi ,dt =ωi (γmi ,c )(1−ηi )γmi ,d ,
20. γmc,dt = γmc,d
t−1 πm,ct − πd
t ,
21. γmi ,dt = γmi ,d
t−1 πm,it − πd
t ,
22. γx,∗t = γx,∗
t−1 + πxt − π∗
t ,
23. mcxt = mcx
t−1 + πt − πxt −∆St ,
24. γft = mcx
t + γx,∗t ,
25.−37 xt+1 = ρx xt +σxεxt+1,
38.−40. Xt = A1Xt−1 +A2Xt−2 +A3Xt−3 +A4Xt−4.
114 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
The expectation operator has been omitted for notational simplicity. The relative
prices are defined as
γc,dt = P c
t
Pt,
γi ,dt = P i
t
Pt,
γmc,dt = P m,c
t
Pt,
γmi ,dt = P m,i
t
Pt,
γx,∗t = P x
t
P∗t
,
γft = Pt
St P∗t
.
Appendix B: Model consistent data measurement
For the measurement equations, some of the model variables are rewritten to be
consistent with the data. For example, consider consumption which is CES aggregated
in the model but Ct ≡C dt +C m
t in the data. This implies that (using the consumption
demands in (3.8))
Ct = (1−ωc )
(Pt
P ct
)−ηc
Ct +ωc
(P m,c
t
P ct
)−ηc
Ct .
Log-linearizing the stationary relationship yields
ct = cd
cd + cm(ηc γ
c,dt + ct )+ cm
cd + cm(−ηc (1−ωc )(γc,d )ηc−1γmc,d
t + ct ).
A similar measurement equation can be constructed for investment (It = I dt + I m
t ).
Imports are given as Mt ≡C mt + I m
t . Using (3.8) and (3.9) this relates to the model in
the following way
Mt =ωc
(P m,c
t
P ct
)−ηc
Ct +ωi
P m,it
P it
−ηi
It .
Log-linearizing the stationary relationship yields
mt = cm
cd + cm(−ηc (1−ωc )(γc,d )ηc−1γmc,d
t + ct )
+ im
id + im(−ηi (1−ωi )(γi ,d )ηi−1γmi ,d
t + it ).
3.8. CONCLUSION 115
For the consumer price index, the data definition is given as P ct ≡ Pt C d
t +P m,ct C m
t
C dt +C m
t.
This implies the following relationship with the stationarized and log-linearized
model
ˆπct =
cd
cd +λmc cmπt + λmc cm
cd +λmc cmπm,c
t
+((
cd
cd +λm,c cm− cd
cd + cm)ηcωc (γc,mc )ηc−1
− (λm,c cm
cd +λm,c cm− cm
cd + cm)ηc (1−ωc )(γc,d )ηc−1
)(γmc,d
t − γmc,dt ).
Appendix C: Measurement equations
yt =
∆lnY d at at
∆l nC d at at
∆l nI d at at
∆lnX d at at
∆lnM d at at
∆ln(
WtPt
)d at a
πy,d at at
πc,d at at
Rd at at
Y ∗,d at at
π∗,d at at
R∗,d at at
=
µz −1
µz −1
µz −1
µz −1
µz −1
µz −1
π−1
π−1
0
0
0
0
+
µz,t + yt − yt−1
µz,t + ˆct − ˆct−1
µz,t + ˆit − ˆit−1
µz,t + xt − xt−1
µz,t + ˆmt − ˆmt−1
µz,t + wt − wt−1
πtˆπc
t
Rt
y∗t
π∗t
R∗t
+ut .
116 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
Table 3.1. Calibrated parameters
Parameter Description Valueβ Discount factor 0.99δ Capital depreciation rate 0.025α Capital share in production 0.30ηc Substitution elasticity of consumption 2.50µz Mean growth rate 1.01π Mean inflation rate 1.005AL Constant in disutility of labor 7.5σL Elasticity of labor supply 1.00λw Wage markup 1.05ωc Imported consumption share 0.40ωi Imported investment share 0.40
0 5 10 15 20
xt+
h
×10-4
02468
GDP
0 5 10 15 20
×10-4
02468
Real wages
0 5 10 15 20
xt+
h
×10-4
02468
Consumption
0 5 10 15 20
×10-4
0
5
10
15Investment
0 5 10 15 20
xt+
h
×10-4
-2024
Imports
0 5 10 15 20
×10-4
0
2
4Exports
Impulse horizon, h0 5 10 15 20
xt+
h
×10-4
-6-4-20
GDP deflator
Impulse horizon, h0 5 10 15 20
×10-4
-4
-2
0
CPI
Figure 3.1. Impulse responses: Productivity shock εtThe impulse respones are shown in log-deviations from the steady state, xt = l og xt − log xss ,
and show the response of an increase in εt of one standard deviation, based on the posterior
mode.
3.8. CONCLUSION 117
Table 3.2. Prior and posterior distributions of estimated parameters
The table shows the prior distributions and posterior estimates based on the initial sample
1990:Q2-2001:Q4.
Prior distribution Posterior distribution
Parameter Density Param 1 Param 2 Mode Mean 5 pct. 95 pct.
ξw Beta 0.750 0.050 0.7569 0.7493 0.6687 0.8323
ξd Beta 0.750 0.050 0.6980 0.6956 0.6107 0.7865
ξm,c Beta 0.750 0.050 0.7646 0.7589 0.6948 0.8235
ξm,i Beta 0.750 0.050 0.8218 0.8171 0.7624 0.8729
ξx Beta 0.750 0.050 0.8243 0.8167 0.7601 0.8726
κw Beta 0.250 0.050 0.2429 0.2513 0.1684 0.3312
κd Beta 0.250 0.050 0.1949 0.2020 0.1311 0.2717
κm,c Beta 0.250 0.050 0.2050 0.2119 0.1404 0.2860
κm,i Beta 0.250 0.050 0.2194 0.2256 0.1485 0.2991
κx Beta 0.250 0.050 0.2294 0.2339 0.1558 0.3083
λd Inverse gamma 1.200 0.050 1.1949 1.1967 1.1166 1.2794
λm,c Inverse gamma 1.200 0.050 1.0221 1.0365 1.0001 1.0699
λm,i Inverse gamma 1.200 0.050 1.1949 1.2006 1.1186 1.2829
F ′′ Normal 5.000 1.500 5.5635 5.7574 3.8295 7.6838
b Beta 0.5000 0.100 0.5664 0.5485 0.3917 0.7108
ηi Inverse gamma 1.500 2.000 1.3665 1.4679 0.8411 2.0586
η f Inverse gamma 1.500 2.000 1.2236 1.2421 0.9760 1.5052
ρλd Beta 0.250 0.100 0.1018 0.1330 0.0358 0.2274
ρλm,c Beta 0.250 0.100 0.1006 0.1257 0.0325 0.2111
ρλm,i Beta 0.250 0.100 0.1715 0.2068 0.0628 0.3457
ρλx Beta 0.250 0.100 0.2601 0.2819 0.1161 0.4428
ρµz Beta 0.500 0.100 0.4212 0.4268 0.2691 0.5790
ρε Beta 0.500 0.100 0.4969 0.4922 0.3305 0.6536
ρΥ Beta 0.500 0.100 0.3804 0.3903 0.2312 0.5440
ρ z∗ Beta 0.500 0.10 0.5101 0.5095 0.3512 0.6639
ρζc Beta 0.500 0.100 0.4059 0.4231 0.2669 0.5757
ρζh Beta 0.500 0.100 0.3594 0.3616 0.2269 0.4975
ρφ Beta 0.850 0.050 0.8917 0.8801 0.8161 0.9465
118 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
Table 3.3. Prior and posterior distributions of estimated parameters
The table shows the prior distributions and posterior estimates based on the initial sample
1990:Q2-2001:Q4.
Prior distribution Posterior distribution
Parameter Density Param 1 Param 2 Mode Mean 5 pct. 95 pct.
σλd Inverse Gamma 0.01 0.001 0.0094 0.0096 0.0083 0.0109
σλm,c Inverse Gamma 0.01 0.001 0.0126 0.0130 0.0111 0.0148
σλm,i Inverse Gamma 0.05 0.001 0.0506 0.0506 0.0490 0.0523
σλx Inverse Gamma 0.01 0.001 0.0144 0.0146 0.0123 0.0168
σµz Inverse Gamma 0.005 0.001 0.0047 0.0052 0.0035 0.0069
σε Inverse Gamma 0.01 2 0.0045 0.0069 0.0024 0.0118
σΥ Inverse Gamma 0.01 2 0.0261 0.0270 0.0207 0.0335
σz∗ Inverse Gamma 0.01 2 0.0026 0.0028 0.0019 0.0036
σζc Inverse Gamma 0.01 2 0.0091 0.0098 0.0066 0.0128
σζh Inverse Gamma 0.01 2 0.0037 0.0037 0.0027 0.0047
σφ Inverse Gamma 0.01 2 0.0017 0.0018 0.0014 0.0022
3.8. CONCLUSION 119
Tab
le3.
4.U
nco
nd
itio
nal
seco
nd
mo
men
ts
All
vari
able
sar
esh
own
inq
uar
terl
ygr
owth
rate
s.T
he
mo
men
tso
fth
eD
SGE
mo
del
com
pu
ted
bas
edo
nth
ep
ost
erio
rm
od
e.T
he
dat
ase
tuse
d
for
this
esti
mat
ion
isth
ein
itia
lsam
ple
,199
0:Q
2-2
001:
Q4.
Dat
aM
od
elV
aria
ble
Std
.dev
.A
uto
corr
elat
ion
Co
rr(∆
Yt)
Std
.dev
.A
uto
corr
elat
ion
Co
rr(∆
Yt)
∆ln
Yt
1.26
0.36
1.00
1.83
0.35
1.00
∆ln
( Wt
Pt
)0.
680.
06-
0.14
1.39
0.24
0.34
∆ln
Ct
1.21
0.10
0.33
1.41
0.42
0.73
∆ln
I t7.
280.
240.
626.
930.
350.
58∆
lnM
t2.
340.
120.
553.
080.
280.
23∆
lnX
t3.
060.
320.
542.
640.
240.
60π
d t0.
800.
63-
0.43
1.12
0.37
-0.
38π
c t0.
580.
070.
080.
850.
35-
0.12
120 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
Table
3.5.1-stepah
eadfo
recasterror
variance
deco
mp
ositio
n
Th
ed
ecom
po
sition
isco
mp
uted
based
on
the
po
sterior
mo
de
ofth
ep
arameters.M
easurem
enterro
rsh
aveb
eenexclu
ded
.
GD
PR
ealwages
Co
nsu
mp
tion
Investm
ent
Imp
orts
Exp
orts
GD
Pd
eflato
rC
PI
Fiscal
1.4780.004
0.0510.000
0.0010.000
0.0100.006
Do
mestic
marku
p11.351
49.18911.474
0.5662.783
0.16386.519
49.613
Imp
orted
con
sum
ptio
nm
arkup
9.6360.064
3.6210.399
2.7620.008
0.45940.822
Imp
orted
investm
entm
arkup
5.1730.016
9.5870.092
35.9210.008
0.2290.131
Exp
ortm
arkup
15.1470.113
1.6640.095
0.41275.231
0.5230.300
Pro
du
ctivity5.495
10.22912.171
0.3672.900
2.9111.303
0.747
Investm
entsp
ecific
30.8082.731
6.26297.572
53.8910.402
2.9201.675
Co
nsu
mp
tion
preferen
ce1.546
0.02429.768
0.0670.437
0.0000.019
0.011
Labo
rp
reference
3.25537.464
8.3100.352
0.0240.192
6.3713.653
Foreign
16.1100.164
17.0920.490
0.87121.084
1.6483.042
3.8. CONCLUSION 121
Tab
le3.
6.8-
step
ahea
dfo
reca
ster
ror
vari
ance
dec
om
po
siti
on
Th
ed
eco
mp
osi
tio
nis
com
pu
ted
bas
edo
nth
ep
ost
erio
rm
od
eo
fth
ep
aram
eter
s.M
easu
rem
ente
rro
rsh
ave
bee
nex
clu
ded
.
GD
PR
ealw
ages
Co
nsu
mp
tio
nIn
vest
men
tIm
po
rts
Exp
ort
sG
DP
defl
ato
rC
PI
Fis
cal
1.65
20.
004
0.04
20.
000
0.00
10.
000
0.00
80.
005
Do
mes
tic
mar
kup
10.2
2341
.668
9.45
50.
693
3.69
50.
269
64.9
5740
.109
Imp
ort
edco
nsu
mp
tio
nm
arku
p8.
249
0.51
52.
894
0.62
63.
799
0.02
70.
664
31.5
38
Imp
ort
edin
vest
men
tmar
kup
4.77
30.
019
8.24
80.
483
32.1
230.
035
0.30
40.
188
Exp
ort
mar
kup
14.0
280.
158
1.36
70.
136
0.53
072
.250
0.70
60.
436
Pro
du
ctiv
ity
7.44
013
.972
10.6
810.
874
4.81
04.
401
1.94
41.
200
Inve
stm
ents
pec
ific
30.6
0212
.478
17.0
2395
.308
53.2
762.
486
16.4
3410
.147
Co
nsu
mp
tio
np
refe
ren
ce1.
820
0.02
924
.368
0.11
40.
615
0.00
00.
023
0.01
4
Lab
or
pre
fere
nce
5.95
930
.891
10.0
991.
056
0.21
30.
797
8.99
55.
554
Fore
ign
15.2
540.
267
15.8
240.
712
0.93
519
.734
5.96
310
.809
122 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
0 5 10 15 20
xt+
h
×10-3
-6
-4
-2
0GDP
0 5 10 15 20
×10-3
-8-6-4-20
Real wages
0 5 10 15 20
xt+
h
×10-3
-4
-2
0Consumption
0 5 10 15 20
×10-3
-8-6-4-20
Investment
0 5 10 15 20
xt+
h
×10-3
-2024
Imports
0 5 10 15 20
×10-3
-2
-1
0Exports
Impulse horizon, h0 5 10 15 20
xt+
h
×10-3
-202468
GDP deflator
Impulse horizon, h0 5 10 15 20
×10-3
0
2
4
CPI
Figure 3.2. Impulse responses: Markup shock λdt
The impulse respones are shown in log-deviations from the steady state, xt = l og xt − log xss ,
and show the response of an increase in λt of one standard deviation, based on the posterior
mode.
3.8. CONCLUSION 123
0 5 10 15 20
xt+
h
×10-3
0
1
2
GDP
0 5 10 15 20
×10-4
-2024
Real wages
0 5 10 15 20
xt+
h
×10-3
0
1
2
Consumption
0 5 10 15 20
×10-3
0
2
4
Investment
0 5 10 15 20
xt+
h
×10-3
0
1
2
Imports
0 5 10 15 20
×10-3
0
2
4Exports
Impulse horizon, h0 5 10 15 20
xt+
h
×10-4
-1012
GDP deflator
Impulse horizon, h0 5 10 15 20
×10-4
-2
0
2
CPI
Figure 3.3. Impulse responses: Foreign output shock, Y ∗t
The impulse respones are shown in log-deviations from the steady state, xt = log xt − l og xss ,
and show the response of an increase in Y ∗t of one standard deviation, based on the posterior
mode.
124 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
1 2 3 4 5 6 7 8
RMSE(h)
1.2
1.4
1.6
1.8
2
2.2
2.4
GDP
DSGE AR(1) Random Walk
1 2 3 4 5 6 7 8
0.5
0.6
0.7
0.8
0.9
1
1.1
Wages
Forecast horizon, h1 2 3 4 5 6 7 8
RMSE(h)
1.61.82
2.22.42.62.83
3.2Consumption
Forecast horizon, h1 2 3 4 5 6 7 8
7
8
9
10
11
12Investment
Figure 3.4. Out-of-sample predictive performanceThe RMSEs are shown for forecasting horizons h = 1, ...,8 and are based on yearly growth rates.
The recursive out-of-sample forecasts have been conducted for the period 2002Q1-2008Q2.
3.8. CONCLUSION 125
1 2 3 4 5 6 7 8
RMSE(h)
4
5
6
7
8
Imports
DSGE AR(1) Random Walk
1 2 3 4 5 6 7 82.53
3.54
4.55
5.56
6.5Exports
Forecast horizon, h1 2 3 4 5 6 7 8
RMSE(h)
0.550.60.650.70.750.80.850.9
GDP deflator
Forecast horizon, h1 2 3 4 5 6 7 8
0.350.40.450.50.550.60.650.70.75
CPI
Figure 3.5. Out-of-sample predictive performanceThe RMSEs are shown for forecasting horizons h = 1, ...,8 and are based on yearly growth rates.
The recursive out-of-sample forecasts have been conducted for the period 2002Q1-2008Q2.
126 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
2007Q1 Q2 Q3 Q4 2008Q1 Q2 Q3 Q4 2009Q1 Q2 Q3 Q4 2010Q1 Q2 Q3 Q4-8-6-4-20246
DSGE
Yearly GDP growthForecast
2007Q1 Q2 Q3 Q4 2008Q1 Q2 Q3 Q4 2009Q1 Q2 Q3 Q4 2010Q1 Q2 Q3 Q4-8
-6
-4
-2
0
2
AR(1)
Figure 3.6. Forecasting Danish GDP growth during the Great RecessionThe figure depicts historical values of real GDP growth in Denmark. Further, the sequential
forecasts for h = 1, ...,8 are shown for the DSGE model (upper row) and the AR(1) model (lower
row).
3.8. CONCLUSION 127
2007Q1 Q2 Q3 Q4 2008Q1 Q2 Q3 Q4 2009Q1 Q2 Q3 Q4 2010Q1 Q2 Q3 Q4
-0.15
-0.1
-0.05
0
0.05
Output gapFiscalMarkupsCapacityDomestic demandForeign
Figure 3.7. Historical shock decomposition of outputThe figure depicts the historical filtered values of deviations from the steady state of real output,
yt , as well as a decomposition of the structural shock contributions. The decomposition is
computed based on the posterior mode of the parameters and measurement errors have been
excluded.
128 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE
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