Essays on Fractional Filters and Co-Integration - AU...

Essays on Fractional Filters and

Co-Integration

2017-7

Federico Carlini

PhD Dissertation

DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS

AARHUS BSS AARHUS UNIVERSITY DENMARK

ESSAYS ON FRACTIONAL FILTERS ANDCO-INTEGRATION

By Federico Carlini

A dissertation submitted to

School of Business and Social Sciences, Aarhus University,

in partial fulfilment of the requirements of

the PhD degree in

Economics and Business Economics

December 2016

CREATESCenter for Research in Econometric Analysis of Time Series

PREFACE

This thesis was written in the period from September 2011 to December 2016 duringmy graduate studies at the Department of Economics and Business Economics at AarhusUniversity and during my visit to the Department of Economics at Queen’s University,Canada in the spring 2014. I am grateful for financial support throughout the PhD studiesprovided by Department of Economics and Business Economics at Aarhus University, andby the Center for Research in Econometric Analysis of Time Series (CREATES), fundedby Danish National Research Foundation.

I would like to thank my supervisor, Prof. Niels Haldrup, for his support and in-sightful contributions and suggestions to all chapters of this thesis. Also, I would like tothank Prof. Søren Johansen because the discussions with him improved the expositionof Chapters 2 and 3 in particular, and he stressed the importance of being precise inmathematical terms. I am also grateful to Prof. Rocco Mosconi who introduced me to thetopic of fractional integration in my master thesis at Politecnico di Milano and for his wiseadvices. A special thanks goes to Prof. Morten Nielsen for making my stay at Queen’sUniversity a good opportunity to talk about my research projects.

I extend my deepest gratitude to Associate Professor Paolo Santucci de Magistriswho has collaborated with me on the Chapter 2 of this dissertation and having with him alot of brilliant econometrics and funny non-econometrics discussions.

Special thanks go to Lorenzo and Orimar for sharing with me the passion of Econo-metrics and Mathematics and for the laughs when we spoke about Burzum. I would alsolike to thank Vladimir for the interesting discussions we had about the fractional world.Other thanks go to Ugo, Diego, Ricca, Benny and Romina. In these years you were reallyhelpful to me.

I would like to thank my parents for all their efforts, love and helps they donated tome in very difficult moments of my life. Finally, thanks Silvia for your love and patienceand for sharing with me everything you have in your mind and in your heart.

Federico CarliniAarhus, December 2016

i

UPDATED PREFACE

The pre-defence meeting with the assessment committee consisting of Massimo Franchi,University of Rome La Sapienza, Eric Hillebrand (Chair), Aarhus University, and MortenNielsen, Queen’s University, took place on February 17th, 2017 in Aarhus. I am thankfulto the committee for their careful reading and their comments and suggestions. Some ofthe suggestions have been incorporated into the present version of the dissertation whileothers remain for future research.

Federico CarliniAcqui Terme, March 2017

iii

CONTENTS

Summary vii

Danish summary xi

1 Fractional difference algorithms: a comparison 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Fractional difference algorithms . . . . . . . . . . . . . . . . . . . . . 31.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 ARFIMA(1, d, 1) model . . . . . . . . . . . . . . . . . . . . . 81.3.2 FCVARd,b model . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 On the identification of fractionally cointegrated VAR models 292.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 The Identification Problem . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Identification and Inference . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.1 Identification in Finite Samples . . . . . . . . . . . . . . . . . . 392.3.2 Constrained Likelihood . . . . . . . . . . . . . . . . . . . . . . 41

2.4 Unknown cointegration rank . . . . . . . . . . . . . . . . . . . . . . . 472.4.1 Model selection under unknown rank and lag-length . . . . . . 50

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.5.1 Proof of Proposition 2.2.2 . . . . . . . . . . . . . . . . . . . . 552.5.2 Proof of Lemma 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . 552.5.3 Proof of Proposition 2.4.1 . . . . . . . . . . . . . . . . . . . . 562.5.4 Proof of Corollary 2.4.2 . . . . . . . . . . . . . . . . . . . . . . 572.5.5 Proof of Lemma 2.4.3 . . . . . . . . . . . . . . . . . . . . . . . 57

2.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3 A new estimation method of a fractional cointegrated model 673.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2 Model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2.1 Johansen’s FCVAR model . . . . . . . . . . . . . . . . . . . . 69

v

vi CONTENTS

3.2.2 Avarucci’s FECM model . . . . . . . . . . . . . . . . . . . . . 713.3 Statistical identification of the FCVAR model . . . . . . . . . . . . . . 73

3.3.1 Identification issues when the lag length is unknown . . . . . . 743.3.2 Identification issue when αβ′ = 0 . . . . . . . . . . . . . . . . 753.3.3 The nesting structure of the FECM model . . . . . . . . . . . . 75

3.4 Estimation of the FECM model . . . . . . . . . . . . . . . . . . . . . . 763.4.1 The switching algorithm . . . . . . . . . . . . . . . . . . . . . 773.4.2 Estimation and Identification issues . . . . . . . . . . . . . . . 783.4.3 Initial values . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.5 Simulation experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 793.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

SUMMARY

In recent decades, scientific disciplines such as econometrics, geology and hydrology havefocused on time series processes that exhibit long range dependence. Long range depen-dence refers to the feature that the autocorrelation function decays at a slow hyperbolicrate rather than a geometric rate as is usually the case for ARMA processes for instance.For a general exposition of mathematical definitions of long range dependence (or longmemory), see e.g. Guégan (2005), Palma (2007) and Beran et al. (2013). Empirical studiesthat support the presence of long memory in real data is overwhelming and can be found inmacroeconomics, finance, electoral studies as well as in numerous other studies that are toomany to cite here. In this thesis we focus on some methodological and theoretical issues inthe analysis of parametric fractional time series models. The emphasis is mainly on thestudy and representation of multivariate models with fractional cointegration restrictions.The thesis also suggests a method to approximate the fractional difference filter (1− L)d.

In Chapter 1, I consider a comparison of four fractional difference algorithms.Furthermore, I introduce a new fractional difference algorithm (fast fractional filter) thatapproximates the fractional filter using two main ideas. Firstly, the time series is filteredwith M0 autoregressive weights of the expansion of the dth−difference operator (1− L)d.

Secondly, it approximates the last T −M0 weights by a step function that controls thedecline profile of the remaining terms. It is shown that the approximation of the algorithmperforms well even in big samples when compared with the fractional filter (1 − L)d.For comparison, I study the behaviour of different fractional filters in two examples: theARFIMA model and the FCVAR model of Johansen and Nielsen (2012). It is shownin a Monte Carlo experiment that the distributions of the estimated parameters of themodels with the fast fractional filter and the complete filter are similar, indicating a goodapproximation of the new algorithm for the d−difference of the series. However, thefractional filter described in this chapter is outperfomed by the Jensen and Nielsen (2014)algorithm, based on the convolution with the Fast Fourier Transform.

In Chapter 2 (co-authored with Paolo Santucci de Magistris) we focus on the FCVARmodel discussed in Johansen (2008) and Johansen and Nielsen (2012). The paper examinestwo different features of the model. Firstly, we show that if the number of autoregressivelags k is unknown and the cointegration rank r is known, then the fractional parameters andthe short term parameters are not uniquely identified. The properties of these multiple non-identified models are studied and a necessary and sufficient condition for the identification

vii

viii CONTENTS

of the fractional parameters of the system is provided. The condition is called the "F(d)

condition". This appears to be a generalization of the I(1) condition to the fractionalmodel. The second problem we examine is concerned with the joint indeterminacy of thecointegration rank and the lag-length k. It is shown that the model with rank zero andk lags may be given an equivalent re-parametrization of the model with full rank andk − 1 lags. This precludes the possibility to test for the cointegration rank unless a properrestriction on the fractional integration parameter is imposed.

In the final Chapter 3 (co-authored with Katarzyna Łasak) we consider the FractionalCointegration model of Avarucci (2007), which is characterized by a different lag structurethan the model proposed in Johansen (1995). Both models generate the same class ofprocesses, but their properties differ significantly. The model of Avarucci has a convenientnesting structure, which allows for testing the number of lags and the cointegration rank asin the standard cointegration framework of Johansen (1995). The identification problems inthe model of Avarucci (2007) are less severe than in the model of Johansen (2008) and forthis reason the model of Avarucci (2007) might be a convenient model representation foranalysis. However, it turns out that due to a more advanced parametrization, the estimationis in fact more complicated in the Avarucci model specification. We propose a 4-stepestimation procedure for this model that is based on the switching algorithm employedin Carlini and Mosconi (2014) and the GLS procedure of Mosconi and Paruolo (2014).Finally, we check the performance of our estimation procedure in finite samples by meansof a Monte Carlo experiment.

References

Avarucci, M. (2007). Three essays on fractional cointegration. PhD Thesis University ofRome Tor Vergata.

Beran, J., Feng, Y., Ghosh, S., and Kulik, R. (2013). Long-memory processes. Monographson Statistics and Applied Probability, (61).

Carlini, F. and Mosconi, R. (2014). Twice integrated models. Technical report, Politecnicodi Milano.

Guégan, D. (2005). How can we define the concept of long memory? An econometricsurvey. Econometric reviews, 24(2):113–149.

Jensen, A. N. and Nielsen, M. Ø. (2014). A fast fractional difference algorithm. Journal ofTime Series Analysis, 35(5):428–436.

Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector AutoregressiveModels. Oxford University Press, Oxford.

CONTENTS ix

Johansen, S. (2008). A representation theory for a class of vector autoregressive modelsfor fractional processes. Econometric Theory, Vol 24, 3:651–676.

Johansen, S. and Nielsen, M. Ø. (2012). Likelihood inference for a fractionally cointegratedvector autoregressive model. Econometrica, 80(6):2667–2732.

Mosconi, R. and Paruolo, P. (2014). Rank and order conditions for identification insimultaneous system of cointegrating equations with integrated variables of order two.

Palma, W. (2007). Long-memory time series: theory and methods, volume 662. John Wiley& Sons.

DANISH SUMMARY

De seneste år har der i økonometri været en del interesse for tidsserieprocesser, der udviserlang hukommelse (”long memory”) i den forstand, at observationer, der tidsmæssigt erlangt fra hinanden, udviser afhængighed. ”Long memory” kan måles på flere måder. Oftebetragtes autokorrelationsfunktionen som i tilfældet med ”long memory” aftager meden langsom hyperbolsk rate, hvilket er langsommere end det man f.eks. ser for ARMAprocesser. Mange empiriske studier dokumenterer sådanne egenskaber, f.eks. i finansielledata såsom processer for finansiel volatilitet, rente-spreads, og lignende, men også i mangemakroøkonomiske variable og variable fra valg-studier. En klasse af parametriske modeller,der ofte benyttes til modellering af ”long memory” er fraktionelt integrerede processer,FI(d), der er en generalisering af den sædvanlige differens operator ∆ = (1− L), idet dangiver den fraktionelle differens (1− L)d, hvor d kan antage reelle værdier, og altså ikkekun de heltallige værdier. I denne afhandling fokuserer jeg på forskellige metodemæssigeog teoretiske egenskaber ved fraktionelle processer.

I kapitel 1 af afhandlingen foreslår jeg en ny algoritme til at approksimere denfraktionelle differensoperator (1− L)d . I empiriske studier og i Monte Carlo studier erdet vigtigt at kunne benytte præcise algoritmer til beregning af fraktionelle differenser.Selvom den foreslåede metode viser sig ikke at være den foretrukne sammenlignet med detfraktionelle filter beskrevet i Jensen og Nielsen (2014), så giver den foreslåede algoritmeen nyttig indsigt i hvordan autoregressive processer med mange lags og ”long memory”kan beregnes og simuleres i praksis.

I kapitel 2 af afhandlingen (med Paolo Santucci de Magistris som medforfatter) be-tragter vi fraktionelle VAR processer, der tillader kointegration mellem system-variablene.Specielt betragter vi FCVAR modellen foreslået af Johansen (2010) og Johansen og Ni-elsen (2012) og vi viser, at denne model potentielt kan have et identifikationsproblem,når antallet af lags i modellen er ukendt og kointegrationsrangen er kendt. Vi foreslår enbestemt betingelse, der skal være opfyldt, for at modellens parametre er identificeret. Etandet problem vi beskriver er, hvordan man simultant kan bestemme kointegrationsrangenog lag-længden af VAR modellen.

Kapitel 3 i afhandlingen (med Katarzyna Łasak som medforfatter) fokuserer ogsåpå fraktionelle VAR modeller med kointegration. Vi betragter den model, der er foreslåeti Avarucci (2007) som har en anderledes lag-struktur end den der anvendes i JohansensFCVAR model. Selvom de to modelrepræsentationer genererer den samme klasse af

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xii CONTENTS

processer, så har de to modeller alligevel forskellige egenskaber. Forskellene mellemmodellerne bliver beskrevet og specielt vises det, at det potentielle identifikationsproblem,der beskrives i kapitel 2 af afhandlingen, er mindre alvorligt i Avaruccis repræsentation.Der argumenteres for, at Avaruccis formulering kan være mere attraktiv som udgangspunktfor den statistiske analyse. Men samtidig er modellen dog mere kompliceret i andresammenhænge på grund af en mere avanceret parametrisering.

CH

AP

TE

R

1FRACTIONAL DIFFERENCE

ALGORITHMS: A COMPARISON

Carlini FedericoAarhus University and CREATES

Abstract

This paper provides a comparison of four fractional difference algorithms. The trun-cated fractional filter and the fast fractional filter are compared in terms of executiontime and correlation with the exact fractional filters of type II, calculated as a con-volution in the time domain or passing trough the frequency domain as in Jensenand Nielsen (2014). The truncated fractional filter is a convolution of the series withM fractional weights. The fast fractional filter algorithm approximates the type IIfractional operator in two steps. In the first step, we filter the series with a trun-cated fractional filter with M0 weights. In the second step, we sum the truncatedfiltered series with the cumulated time series multiplied by appropriate functions

of the generalized binomial coefficients

(d

k

). Finally, we conclude that the fast

fractional difference algorithm developed by Jensen and Nielsen (2014) has the bestperformance compared to the other filters.

1

2 CHAPTER 1. FRACTIONAL DIFFERENCE ALGORITHMS: A COMPARISON

1.1 Introduction

In the past thirty years new time series models have been developed based on the conceptof long memory. Long memory of a time series refers to the particular feature that theautocorrelation function decays at an hyperbolic rate.

The first models proposed in the statistical literature regarding these particularfeatures were developed in the same year by two econometricians and a hydrologist, i.e.Granger and Joyeux (1980) who modelled US DGP and Consumption time series andHosking (1981) who modelled the Nile River flows series.

They both provided a new mathematical setting where a time series, instead of beinginteger valued integrated, is fractionally integrated. The fractional difference operator, aweighted sum of infinitely many observations with the size of the weights depending on aunique real number often denoted d (the fractional order of a time series), is the innovationof their model.

The applied econometrics literature has experienced a vast increase in the amountof data available for instance for different time series financial assets. The datasets arecharacterized by a large number of observations (typically sampled at high frequency) andoften the fractionally integrated model describes the autocorrelation features of these seriesrather well.

For example Baillie, Bollerslev, and Mikkelsen (1996) applied the fractional inte-grated GARCH (FIGARCH) model to model return volatility of stock prices and Bollerslevand Mikkelsen (1996) applied the exponential version of the FIGARCH to exchange ratesdata.

Other examples are proposed in finance as Baillie and Bollerslev (1994) where theobject of the study was to find a dynamic relationship between nominal spot and forwardexchange rates in a fractional cointegration set-up. The fractional cointegration relationshipis a statistical property regarding linear combinations of two or more series having afractional order lower than all the individual series considered in the analysis.

A very recent empirical application in Carlini, Manzoni, and Mosconi (2010) con-cerns the possibility of finding a dynamic relationship between high-frequency prices andcumulated order flows, i.e. the cumulated sum of the difference between buyer and sellerinitiated trades. Firstly, they have adopted the fractional difference operator of type II tofilter the series. This operator is a trimmed version of the fractional difference operatorsuggested in Hosking (1981) and in Granger and Joyeux (1980) and it involves longweighted partial sums of the series. Due to the extreme computational burden to transformthe data with the type II filter when a large dataset is employed, a particular fractionalcointegration model developed by Johansen (2008) and subsequently by Johansen andNielsen (2012), wasn’t feasible to estimate.

This paper compares four different fractional difference algorithms and analyzes anew approximated fractional difference operator called the fast fractional filter. Section 2

1.2. FRACTIONAL DIFFERENCE ALGORITHMS 3

defines the four fractional difference algorithms and their properties in terms of executiontime and in terms of approximation. Section 3 describes the results of Monte Carlosimulations of ARFIMA(1, d, 1) models and fractionally cointegrated models to comparethe performances of the fast fractional filter against the truncated version of the fractionaldifference operator. Section 4 concludes.

1.2 Fractional difference algorithmsConsider the time series

Yt t = 1, . . . , T

and the transformation∆dYt = (1− L)dYt

where d ∈ R and L is the lag operator such that LYt = Yt−1 and R denotes the set of realnumbers. It is well known that

∆d = (1− L)d =∞∑j=0

(−1)j(d

j

)Lj =

∞∑j=0

πj(d)Lj

Therefore, the filter implies an infinite summation

∆dYt = (1− L)dYt =∞∑j=0

πj(d)Yt−j (1.1)

The power series associated to the process ∆dYt is∑∞

j=0 πj(d)zj . This series is definedfor d > −1/2 and it converges in the set |z| ≤ 1. Hence, the parameter space of (1.1) isd > −1/2.

The formula (1.1) is not feasible for observed time series, where t = 1, 2, . . .. In thisframework, the equation (1.1) is replaced by the type II difference operator

∆d+Yt =

t−1∑j=0

πj(d)Yt−j (1.2)

We calculate ∆d+Yt as a convolution in time domain. This means that the number of

operations needed to generate the filtered series ∆d+Yt are O(T 2).

Sometimes, the filter is approximated by the truncated version (or truncated filter)

∆dMYt =

min(t−1,M−1)∑j=0

πj(d)Yt−j M < T (1.3)

The number of operations needed to calculate ∆dMYt are O(M2) where M < T . This

means that the execution time of ∆dMYt is faster than ∆d

+Yt.


We propose an alternative algorithm, called the fast fractional filter, given by

∆dFYt =

min(M0−1,t−1)∑j=0

πj(d)Yt−j + It>M0π1(d)

min(M1−1,t−1)∑j=M0

Yt−j +

+ It>M1π2(d)

min(M2−1,t−1)∑j=M1

Yt−j + . . .+ It>Mnπk(d)

(t−1)∑j=Mn

Yt−j (1.4)

with t ≤ T , where 1 < M0 < . . . < Mn ≤ T are n natural numbers and IA(x) denotes theindicator function, πj(d) are the weights described in equation (1.1) and

πj(d) =

∑Mj+1−1i=Mj

πi(d)

Mj+1 −Mj

j = 0, . . . , n (1.5)

With some manipulation the equation (1.4) becomes

∆dFYt =

min(M0−1,t−1)∑j=0

πj(d)Yt−j + It>M0π1(d)t−1∑i=M0

Yt−i + (1.6)

+ It>M1(π2(d)− π1(d))t−1∑i=M1

Yt−i + · · ·+ It>Mn(πn(d)− πn−1(d))t−1∑j=Mn

Yt−j

or equivalently,

∆dFYt =

min(M0−1,t)∑j=0

πj(d)Yt−j +n∑j=1

It>MjδjYt−Mj(1.7)

where δ1 = π1(d) , δj = (πj − πj−1) and Yt =∑t−1

i=0 Yi. The number of operations neededto calculate ∆d

FYt are M20 + T + 2n.

In Appendix the codes implemented in MATLAB for the fast fractional filter, calledLbfffilter.m, are provided. This filtering procedure is programmed by means of two quanti-ties. The first one, defined as M0, controls the weights needed for the truncated fractionalfilter ∆d

M0Yt, i.e. the first term in the right hand-side of the equation (1.6).

The second parameter, called in the code γ, a real number such that 0 < γ < 1,calculates the values indicated in (1.7) byMj for j = 1, . . . , n and it selects the appropriaten with an algorithm that evaluates the steepness of the sequence of weights πj(d) forj > M1. This algorithm is programmed as follows:

Step 0 Conditioning on M0, we calculate the quantity ξt =∣∣∣πM0

−πtπM0

∣∣∣ for each t > M0.

Step 1 We find the first index t1 such that ξt1 is greater than γ and the condition t1 ≤ T

is satisfied. Hence, we define M1 := t1.


Step k We find the first index tk such that ξtk is greater than

γ × (1− (1− γ)k) (1.8)

and the condition tk ≤ T is satisfied. Hence, we define Mk := tk for k =

2, . . . , n. The index n is found in correspondence to the last tn such that theconditions (1.8) and tk ≤ T are both satisfied.

Furthermore, we considered the fast fractional difference algorithm proposed byJensen and Nielsen (2014). This filter calculates the fractional difference in (1.2) passingthrough the frequency domain. It is based on the application of the circular convolutioncombined with the discrete Fast Fourier Transform of a time series and the linear operatorinduced by ∆d.

Given a vector of observations Y = (Y1, Y2, . . . , YT )′, where ′ indicates the transpo-sition operator, we construct the vector X = (Y1, Y2, . . . , YT , 0, . . . , 0)′ with dim(X) =

2s × 1 where s is the smallest integer number such that T ≤ 2s. The d−difference of Y ,∆dSY , is calculated as the first T observations of

F−1[F(X) F(π)

](1.9)

where F(·) and F−1(·) denote respectively the discrete Fourier transform and the inversediscrete Fourier transform, π = (π0, π1, . . . , πT−1, 0, . . . , 0)′ is the 2s-dimensional vectorof the linear operator induced by (1.1) and is the Hadamard product. The Fast FractionalDifference Algorithm is an exact algorithm and its high speed is due to the application ofthe fast Fourier transform instead of the discrete Fourier transforms and the number ofoperations is shown to be O(T log T ).

To show the main differences of the four kind of filters, we plot the impulse responseof ∆d

+Yt, ∆dMYt, ∆d

FYt and ∆dSYt, respectively plotted in Figure 1.1. The impulse response

is defined as the filter applied to the vector (1, 0, . . . , 0)′, i.e. we plot the function of theweights of the different fractional filters considered above.

Figure 1.1(a) shows the impulse response of the type II filter ∆d+Yt. In this case the

weights decay continuously and hyperbolically to zero. Figure 1.1(b) plots the impulseresponse of the truncated filter ∆d

MYt. This plot shows that the impulse response suddenlyjumps to zero for t = M . Therefore the number of weights used to filter the series areexactly M . Figure 1.1(c) displays the impulse response of ∆d

FYt with a particular choiceof (M0, γ). This plot reveals the nature of the fast fractional filter. It is an approximation ofthe filter in Figure 1.1(a) where the weights decay to zero like a step function. The lengthof these steps is controlled by γ and the first step starts exactly at observation t = M0.Figure 1.1(d) plots the impulse response of ∆d

SYt. This plot has no differences with theplot in Figure 1.1(a) because it is a type II fractional filter as ∆d

+Yt.


To understand the properties of the fast fractional filter we have conducted somesimulation studies. The first one is shown in Tables 1.1 and 1.2. We have generated N = 10

times the series Xt = ∆−d0+ εt or Xt = ∆−d0S εt where εt ∼ N(0, 1) for t = 1, . . . , 30, 000,with the values indicated on the top row. Then, we have calculated respectively ∆d

+ or ∆dS ,

∆dMXt and ∆d

FXt, where d are the values indicated on the left column. The cells of thesetables refer respectively to corr(∆d

FXt,∆d+Xt) = corr(∆d

FXt,∆dSXt), see Table 1.1, and

corr(∆dMXt,∆

d+Xt) = corr(∆d

MXt,∆dSXt), see Table 1.2, where corr(X, Y ) denotes the

sample correlation between X and Y based on 10 replications. The results show that thecorrelation between the truncated filter and the type II filter is lower compared to thecorrelation between the fast fractional filter and the type II filter. This means that the fastfractional filter ∆d

FYt can better approximate ∆d+Xt or ∆d

SXt compared to the truncatedfilter ∆d

MXt.The second step is to clock the filtering execution time of ∆d

+Yt, ∆dMYt, ∆d

FYt and∆dSYt. The results are shown in Tables 1.3-1.6. Table 1.3 indicates the seconds employed by

MATLAB to filter a series ∆d+Yt with t = 1, . . . , T for different values of T . The execution

time expressed in seconds is an exponential function of T . Table 1.4 displays the time-spellinduced by the filter ∆d

MYt with t = 1, . . . , T for different values of M and T . Fixing aT and increasing M the time execution raises linearly because the moving averages arecalculated with an increasing number of weights πj(d) for j = 0, . . . ,M − 1 in (1.3).

Table 1.5 shows the execution time to filter a series ∆dFYt as a function of γ and M0

when T = 5 × 105. It shows that the fast fractional filter accelerate the execution timefor every choice of γ and M0. In fact when T = 5 × 105 the execution time of ∆d

+ is94.42 seconds against the 9.73 seconds for filtering the series with ∆d

F when γ = 0.01

and M0 = 500. The gain in execution time is due to the second part of the equation in(1.7). In fact, to cumulate a time series Yt is an easy operation in terms of operations(order of milliseconds for T = 5, 000, 000). Secondly, the execution time to calculate aproduct of a vector by a scalar (

∑nj=0 δj × Yt−Mj

) is small. Fixing M0 and decreasing γthe execution time increases because the number of steps induced by γ and the numberof products involved in

∑nj=1 δjYt−Mj

raises. Fixing γ and increasing M0 the executiontime is non-monotonic because there exists a trade-off on time execution between the firsttruncated filter

∑min(M1−1,t)j=1 πj(d)Yt−j and the scalar-by-vector products

∑nj=1 δjYt−Mj

.Finally in Table 1.6 are reported the execution times of the fast fractional difference

algorithm of Jensen and Nielsen (2014). The fast fractional difference algorithm beats allthe other algorithms in terms of time execution.

The last exercise is to quantify the gap of the approximation of ∆d+ or ∆d

S with (1.3)and (1.7). We have simulated a time series Yt = ∆−d0+ εt = ∆−d0S εt with εt ∼ N(0, 1) andt = 1, . . . , T with the fractional filter calculated as in (1.2). Thereafter, we have calculatedthe quantity

RMSE =

√√√√ T∑t=1

(∆dxYt −∆d

+Yt)2


where x = M,F are the d-differences filtered by using the algorithms (1.3), (1.7) andimposing d ≤ d0. We have chosen this measure to calculate the magnitude of the errorinvolved using the different approximated fractional difference algorithms.

Table 1.8 shows the results. The root mean square error of the fast fractional filter islower than the truncated filter of an order of magnitude between 102 to 104. The magnitudedepends on the parameter M of the truncated filter in (1.3) and the parameters M0 and γ ofthe fast fractional filter in (1.7). Furthermore, in Table 1.8 we have tabulated the executiontime of ∆d

F and ∆dM . It is evident that the execution times depend on the choice of γ, M0

and M .

0 10 20 30 40 50 60 70 80 90−0.01

−0.005

0(a) Type II filter and d=0.2

0 10 20 30 40 50 60 70 80 90−0.01

−0.005

0(b) Truncated filter and d=0.2

0 10 20 30 40 50 60 70 80 90−0.01

−0.005

0(c) Fast Fractional filter and d=0.2

0 10 20 30 40 50 60 70 80 90−0.01

−0.005

0(d) Fourier based filter and d=0.2

Figure 1.1: Impulse response of (a) ∆d+Yt ,(b) ∆d

MYt, (c) ∆dFYt and (d) ∆d

SYt.


1.3 Simulation Study

1.3.1 ARFIMA(1, d, 1) model

The ARFIMA model is the generalization of an ARIMA model where the integer difference∆ := (1− L) is replaced by the fractional difference ∆d = (1− L)d.

The Gaussian ARFIMA(p, d, q) model is represented by

Φ(L)∆dYt = Θ(L)εt, εt ∼ N(0, σ2), t = 1, . . . , T (1.10)

where Φ(L) = 1 − φ1L − φ2L2 − . . . φpL

p is the autoregressive polynomial, Θ(L) =

1+θ1L+θ2L2 + . . .+θqL

q is the moving average polynomial, and Yt is the observation ofthe univariate time series at time t. The estimation method employed is the maximization ofthe likelihood function induced by 1.10. The estimation strategy we are going to describeis developed in Li and McLeod (1986).

To simplify the explanation, we restrict the attention to an ARFIMA(1, d, 1) given by

(1− φ1L)∆d+Yt = (1 + θ1L)εt εt ∼ i.i.d.N(0, σ2) t = 1, . . . , T (1.11)

where ∆d+ is the fractional difference operator of type II. Given a sample Y1, Y2, . . . , YT ,

the idea of Li and McLeod (1986) is to truncate the fractional difference as in (1.11) and towrite the truncated series of the residuals as follows

ηt = ηt(d, φ1, θ1) = Yt −t−1∑j=1

τjYt−j t = 1, . . . , T

where τj is a function depending on d, φ1, θ1.Hence, the Gaussian log-likelihood is given by

L(d, φ1, θ1, σ2) = −T

2log(2π)− T

2log(σ2)− 1

2

T∑t=1

η2t

σ2

Defining the vector of the parameters ν = vec(d, φ1, θ1, σ2), the maximum likelihood

estimator is:νML = arg max

νL(ν)

The following simulation exercises are conducted to find the distributions of theparameters of an ARFIMA(1, d, 1) when the filters ∆d

+Yt or ∆dSYt, ∆d

MYt and ∆dFYt are

applied during the maximization of the likelihood.The data generating process of the ARFIMA(1, d, 1) selected in the first simulation

exercise, are d0 = 0.4, θ01 = 0.4, φ0

1 = 0.9 and (σ0)2 = 1, where µ0 denotes the DGPvalue of a generic parameter µ. The number of Monte Carlo experiments conducted areN = 1000 and the number of observations is T = 10, 000. The smoothed distributions1 of

1We use a Gaussian smoother

1.3. SIMULATION STUDY 9

ν of the first experiment using ∆d+ or ∆d

S , ∆dM with the parameter M = 100 and ∆d

F withthe parameters M0 = 100 and γ = 0.3 are plotted in Figure 1.2.

The discussion of the approximation error induced by the selection of M in thetruncated filter is analysed by Chan and Palma (1998) and more recently by Grassi andde Magistris (2014) for estimators based on the Kalman filter. The same problem ofoptimal selection of M for classical maximum likelihood estimators is discussed slightlyin Bollerslev and Mikkelsen (1996) for FIGARCH and FIEGARCH models.

In our setting, we have selected the parameters of ∆dMYt to be M = 100 and M =

1000. Obviously, increasing M the execution time raises and the expected ψ distributionswith the truncated filter get closer and closer to the type II ones. Instead, the parameterschosen for the fast fractional filter are the pairs (M0, γ) which balance the approximationof the type II filter and the execution time.

The plot in Figure 1.2 shows that the distributions of ∆d+(∆d

S) and ∆dF are normally

distributed and they coincide, while ∆dM has a distribution that is skewed and with fat tails.

This indicates that the fast fractional filter approximates very well the type II one; instead,if we truncate with such M the fractional filter, the approximation is poor and it can causeproblems in inference.

The second Monte Carlo simulation is conducted selecting the following parameters:d0 = 0.2, θ0

1 = −0.5, φ01 = −0.4 and (σ0)2 = 1. The parameter of the truncated filter is

M = 100 and the parameters of ∆dF are γ = 0.3 and M0 = 100. Figure 1.3 depicts the

distributions of the simulated model and the results are similar to the first Monte Carlosimulations.

Finally, the third Monte Carlo experiment is conducted with the following parameters:d0 = 0.45 , θ0

1 = 0.6 , φ01 = 0.5 and (σ0)2 = 1. In this setting the parameter of the truncated

filter is M = 1000 and the parameters of ∆dF are γ = 0.3 and M0 = 100. Figure 1.4 shows

the results. In the last Monte Carlo it is remarkable that all the distributions coincide. Thisconfirms that increasing M , the distribution of the parameters ν, adopting (1.2) or (1.9) ,(1.3) and (1.7), is very similar.

1.3.2 FCVARd,b model

In this section we consider the model proposed and investigated by Johansen (2008) forco-fractional time series (FCVARd,b model). This model is thought as an extension ofthe well-known cointegration VECM model, see Johansen (1995), adapted to fractionalprocesses.

The system of equations involved in the FCVARd,b model is given by

∆dYt = αβ′∆d−bLbYt +k∑j=1

Γj∆dLjbYt + εt, d ≥ b > 0, t = 1, . . . , T (1.12)

where Yt is a p-dimensional vector, α and β are matrices with rank r such that 0 ≤ r ≤ p,Γj for j = 1, . . . , k are p× p matrices and εt is a p-dimensional i.i.d process with mean


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

14

16

18

d hat

∆d

F Filter

∆d

+ and ∆d

S Filter

∆d

M Filter, M=100

Normal

0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

θ hat

∆d

F Filter

∆d

+ and ∆d

S Filter

∆d

M Filter, M=100

Normal

0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

5

10

15

20

25

30

35

40

φ hat

∆d

F Filter

∆d

+ and ∆d

S Filter

∆d

M Filter, M=100

Normal

Figure 1.2: Monte Carlo distribution of d, φ, θ when the DGP is generated with d0 = 0.4,θ1 = 0.4, φ1 = 0.9 and σ2 = 1. The distributions of ∆d

F and ∆d+(∆d

S) coincide.

zero and variance Ω, a positive definite matrix. The parameter d is the fractional orderof the system and b is the parameter responsible for the reduction of memory along thedirections of β.

The maximum likelihood estimator, studied in Johansen and Nielsen (2012), is


0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

2

4

6

8

10

12

14

16

18

d hat

∆d

F Filter

∆d

+ and ∆d

S Filter

∆d

M Filter, M=100

Normal

−0.65 −0.6 −0.55 −0.5 −0.45 −0.4 −0.35 −0.30

5

10

15

θ hat

∆d

F Filter

∆d

+ and ∆d

S Filter

∆d

M Filter, M=100

Normal

−0.46 −0.44 −0.42 −0.4 −0.38 −0.36 −0.340

5

10

15

20

25

30

35

φ hat

∆d

F Filter

∆d

+ and ∆d

S Filter

∆d

M Filter, M=100

Normal

Figure 1.3: Monte Carlo distributions of d, φ, θ when the DGP is generated with d = 0.2 ,θ1 = −0.5 , φ1 = −0.4 and σ2 = 1. The distributions of ∆d

F and ∆d+(∆d

S) coincide.

calculated by means of the profile likelihood that depends exclusively on d and b. Theprofile likelihood is given by:

logL(ψ) = log det(S00(ψ)) +r∑i=1

log(1− λi(ψ)) (1.13)


0.35 0.4 0.45 0.5 0.55 0.60

2

4

6

8

10

12

14

16

18

d hat

∆d

F Filter

∆d

+ and ∆d

S Filter

∆d

M Filter, M=1000

Normal

0.55 0.56 0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.640

5

10

15

20

25

30

35

40

θ hat

∆d

F Filter

∆d

+ and ∆d

S Filter

∆d

M Filter, M=1000

Normal

0.35 0.4 0.45 0.5 0.55 0.60

2

4

6

8

10

12

14

16

φ hat

∆d

F Filter

∆d

+ and ∆d

S Filter

∆d

M Filter, M=1000

Normal

Figure 1.4: Monte Carlo distributions of d, φ, θ when the DGP is generated with d = 0.45 ,θ1 = 0.6 , φ1 = 0.5 and σ2 = 1.The distributions of ∆d

F and ∆d+(∆d

S) coincide.

where ψ = (d, b)′. First of all, to define λ(ψ) and S00(ψ), we need the terms Rit(ψ)

for i = 1, 2. They are defined as the residuals, respectively, of the regressions ∆dYt on(∆dLbYt, . . . ,∆

dLkbYt) and ∆d−bLbYt on∑k

j=1 ∆dLjbYt.Hence, we define the matrices Sij(ψ) := T−1

∑Tt=1 Rit(ψ)R′jt(ψ) for i, j = 1, 2 and


λi(ψ) for i = 1, . . . , p are the solution of the eigenvalue problem

det(λS11(ψ)− S10(ψ)S−100 (ψ)S01(ψ)) (1.14)

which gives, as result, the estimated eigenvalues 1 > λ1(ψ) > · · · > λp(ψ) > 0.The estimates α, β, Γj for j = 1, . . . , k and Ω can be easily found analytically as

in Johansen and Nielsen (2012). We have performed three simulation exercises to studythe empirical distribution of the estimated parameters in a FCVARd,b model. We havegenerated synthetic observations with a co-fractional FCVARd,b with 0 lags given by

[∆d0

+ Y1t

∆d0+ Y2t

]=

[α1

α2

] [1 β

] [∆d0−b0+ Lb0Y1t

∆d0−b0+ Lb0Y2t

]+

[ε1t

ε2t

] [ε1t

ε2t

]∼ i.i.d.N

[0

0

],

[1 0

0 1

](1.15)

with t = 1, . . . , 10, 000 and ∆d+ is defined in (1.2).

The Monte Carlo experiments are based on the estimation of three different models.We estimate the FCVARd,b fixing the number of lags k = 0, the rank r = 12 and imposingthe restriction d ≥ b > 0.01 in the maximization routines. Then, we calculate the filteredseries ∆d

+Yt or ∆dS defined in (1.2) and (1.9), the fast fractional filter ∆d

FYt defined in (1.7)and the truncated filter ∆d

M defined in (1.3).We define the vector of parameters θ = (d, b, α1, α2, β)′ and θi for i = +, F,M

as the estimates of the parameters when we apply respectively the filters ∆d+, ∆d

F and ∆dM .

The aim of the simulation study is again to compare the empirical distributions of θi fori = +, S, F,M , considering the distribution of θ+ or θS as a benchmark.

The first experiment generates N = 1000 Monte Carlo replications and we set theparameters3 to be α0

1 = −1.5, α02 = 1.3, β0 = −0.25, d0 = 0.830 and b0 = 0.600. The

parameter of ∆dM is M = 100 while the parameters of ∆d

F is γ = 0.3 and M0 = 0.3.For each run of the Monte Carlo, we generate a time series and we estimate θi withi = +, S, F,M. With this set-up we study a FCVARd,b with d > 0.5, d− b < 0.5 andb ≥ 0.5. In the literature, if the cointegration gap is b > 0.5, we are referring to strongcointegration.

The vector Yt = [Y1t, Y2t]′ in equation (1.15) is Yt ∈ F(0.830) and [1,−0.25] · Yt ∈

F(0.230). Figure 1.5 and 1.6 show the smoothed distributions of the estimated parameters θifor i = +, F,M. The distributions for θ+(θS) and θF coincide, whereas the distributionsθM are different because they are skewed and leptokurtic. The average execution times toestimate θ+,θS , θM and θF are respectively 5.98 sec, 0.01 sec, 0.10 sec. and 0.12 sec.

The second experiment generatesN = 1000 time series with T = 10, 000 and we setthe parameters to be α0

1 = −1.5, α02 = 1.3 and β0 = −0.25 , d0 = 0.830 and b0 = 0.400.

2This means that Y1t and Y2t are co-fractional3The values of the parameters chosen are such that the dynamic system is non-explosive and is fractionally

cointegrated


The parameters of ∆dM and ∆d

F are the same of the first Monte Carlo. This is a case whered > 0.5, d− b < 0.5 and b < 0.5. The fractional parameters define another set-up for thefractional cointegration called weak cointegration. The vector Yt = [Y1t, Y2t]

′ in equation(1.15) is Yt ∈ F(0.830) and the linear combination [1,−0.25] · Yt ∈ F(0.430). Figure 1.7and 1.8 display the distributions of the estimated parameters. The evidence is the sameas the first experiment. The average execution times to estimate θ+, θS , θM and θF arerespectively 5.54 sec, 0.01 sec., 0.07 sec. and 0.10 sec.

Finally, the third experiment generatesN = 1000 series with T = 10, 000 and we setthe parameters to be α0

1 = −1.5, α02 = 1.3 and β0 = −0.25 , d0 = 0.830 and b0 = 0.400.

In this case, the truncation parameter in ∆dM is M = 1, 000 and the parameters in ∆d

F areM0 = 100 and γ = 0.3. Figure 1.9 and 1.10 show some peculiarities with respect thedistribution obtained in the previous Monte Carlo simulations. In fact, the distribution ofα1, α2, d and b are normally distributed and they coincide for all the three type of filter.The distribution of β is normally distributed, as the asymptotic theory suggests, just for thefilters ∆d

+(∆dS) and ∆d

F whilst for ∆dM is still not normal. This suggests that the use of the

truncated filter in the FCVARd,b model can be dangerous in terms of inference.We conduct other Monte Carlo experiments to study the characteristics of the rank

tests of the FCVAR in a bivariate system when we fix the rank r = 1. We have generatedthe FCVAR model with d = b. The DGP is given by:

∆d0+

[Y1t

Y2t

]=

[α1

α2

] [1 β

] [ Ld0Y1t

Ld0Y2t

]+

[ε1t

ε2t

]εt ∼ N(0,Ω) t = 1, . . . , 104

where α01 = −1.5 , α0

2 = 1.3, β0 = −0.025, Ω0 =

(1 0

0 1

)and d0 = 0.9, 0.3. We

choose d0 = 0.9 and d0 = 0.3 because the asymptotic distribution of the cointegrationrank test differ. In fact, when d0 = 0.9 the test is asymptotically the trace of productsof functionals of type II fractional Brownian Motion and when d0 = 0.3 the test isasymptotically chi squared distributed, see Johansen and Nielsen (2012).

We generate N = 1000 FCVAR paths and we estimate the models with cointegrationrank r = 0, r = 1 and r = 2 using the filters ∆d

+(∆dS),∆d

F (M0 = 100, γ = 0.3),∆dM1

and ∆dM2

(M1 = 1000 and M2 = 100). In this experiment we are interested to studythe empirical power and the empirical size of the test. The 5% critical value for the nullhypothesis H0

1 : r = 1 (size of the test) against the alternative hypothesis HA2 : r = 2

when d0 = 0.9(d0 = 0.3), is CV1|2 = 3.9844(3.841)4. If the null hypothesis is given byH0

0 : r = 0 against the alternativeHA2 : r = 2 (power of the test) when d0 = 0.9(d0 = 0.3),

the 5% critical value is CV0|2 = 11.667(9.488).The empirical size and the empirical power of the tests are reported in Table 1.7.

This table shows that the empirical size of ∆d+(∆d

S),∆dF ,∆

dM1

are almost the same, whilst4The critical values for the case d0 = 0.9 are calculated by means of the programs developed by

MacKinnon and Nielsen (2014)

1.4. CONCLUSION 15

∆dM2

has a big size distortion. Hence, increasing M , the distributions under the null ofr = 1 get closer and closer to the exact ones given by the filters ∆d

+ or ∆dS . Instead, the

empirical power is the same as when we adopt the four algorithms proposed.

1.4 ConclusionIn this paper a new algorithm to fractionally difference a time series has been developed.

We use Monte Carlo simulations to compare the behaviour of the distributions ofthe estimators when we apply 4 different fractional filters, namely ∆d

+Yt in (1.2), ∆dFYt in

(1.7), ∆dSYt in (1.9) and ∆d

MYt in (1.3).Firstly, if we adopt the type II filter ∆d

+Yt with large datasets, the estimation ofthe FCVAR model is not feasible due to the computational burden. Secondly, if the filter∆dMYt is employed, the distributions of the estimated parameters change for small values

of M with respect to our benchmark, and the execution time for the estimation dependslinearly on the truncation parameter. Thirdly, in terms of execution time, the fast fractionalfilter ∆d

FYt is faster than ∆d+Yt and slower than ∆d

MYt for small values of M . In terms ofapproximation, ∆d

FYt is better than ∆dMYt for small values of M . Finally, the evidence

suggests that the filter ∆dSYt in Jensen and Nielsen (2014) is the best option(it is an exact

algorithm) and computational efforts. The computational gain is so high that it allows toestimate fractional models with 5,000,000 observations in a few minutes without any kindof approximation.

AcknowledgementsThe author is grateful to Niels Haldrup and Søren Johansen for their suggestions thatimproved the quality of this paper.


0.8 0.82 0.84 0.86 0.88 0.9 0.920

10

20

30

40

50

60

d hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 0.720

5

10

15

20

25

30

b hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

−1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3 −1.20

1

2

3

4

5

6

α1 hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

Figure 1.5: First Monte Carlo Simulation. Empirical distribution for d b and α1 whenwe apply ∆d

+Yt , ∆dFYt and ∆d

MYt. The DGP values are α01 = −1.5, α0

2 = 1.3, β0 =−0.25, d0 = 0.830 and b0 = 0.6. The distributions of ∆d

F and ∆d+(∆d

S) coincide.

1.4. CONCLUSION 17

1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.50

2

4

6

8

10

12

14

α2 hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

−0.26 −0.255 −0.25 −0.245 −0.24 −0.2350

50

100

150

200

250

β hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

Figure 1.6: First Monte Carlo Simulation. Empirical distribution for α2 β, when we apply∆d

+Yt , ∆dFYt and ∆d

MYt. The DGP values are α01 = −1.5, α0

2 = 1.3, β0 = −0.25, d0 =0.830 and b0 = 0.6.The distributions of ∆d

F and ∆d+(∆d

S) coincide.


0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.920

5

10

15

20

25

30

35

40

45

50

d hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

2

4

6

8

10

12

14

16

18

20

b hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

−2.1 −2 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3 −1.20

1

2

3

4

5

6

α1 hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

Figure 1.7: Second Monte Carlo Simulation. Empirical distribution for d b and α1 whenwe apply ∆d

+Yt, ∆dFYt and ∆d

MYt filters. The DGP values are α01 = −1.5, α0


F and ∆d+(∆d

S) coincide.

1.4. CONCLUSION 19

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

1

2

3

4

5

6

α2 hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

−0.28 −0.27 −0.26 −0.25 −0.24 −0.23 −0.22 −0.21 −0.20

10

20

30

40

50

60

70

β hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

Figure 1.8: Second Monte Carlo Simulation. Empirical distribution for α2 and β, whenwe apply ∆d




F and ∆d+(∆d

S) coincide.


0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.870

5

10

15

20

25

30

35

40

45

50

d hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.50

2

4

6

8

10

12

14

16

18

20

b hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

−1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3 −1.20

1

2

3

4

5

6

α1 hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

Figure 1.9: Third Monte Carlo Simulation. Empirical distribution for d b and α1 whenwe apply ∆d




F and ∆d+(∆d

S) coincide.

1.4. CONCLUSION 21

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

1

2

3

4

5

6

α2 hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

−0.28 −0.27 −0.26 −0.25 −0.24 −0.23 −0.220

10

20

30

40

50

60

70

β hat

∆d

+ Filter

∆d

F and ∆d

S Filter

∆d

M Filter

Normal

Figure 1.10: Third Monte Carlo Simulation. Empirical distribution for α2 and β, whenwe apply ∆d




F and ∆d+(∆d

S) coincide.


Tables

d ↓ d0 → −0.675 −0.225 0.225 0.675 1.225 1.575 2.025−0.675 0.998

(0.000)0.995(0.000)

0.998(0.000)

0.999(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

−0.225 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

0.225 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

0.675 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.225 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.575 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

2.025 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

Table 1.1: Correlation between ∆d+Xt(∆

dSXt) and ∆d

FXt. The series is generated byXt = ∆−d0+ εt. In parenthesis are calculated the standard error associated. The parametersare T = 30000, M0 = 100, γ = 0.3 and N = 10 runs.

d ↓ d0 → −0.675 −0.225 0.225 0.675 1.225 1.575 2.025−0.675 0.996

(0.000)0.930(0.010)

0.606(0.090)

0.677(0.216)

0.809(0.203)

0.687(0.367)

0.946(0.064)

−0.225 1.000(0.000)

1.000(0.000)

0.993(0.004)

0.959(0.028)

0.956(0.044)

0.962(0.046)

0.989(0.014)

0.225 1.000(0.000)

1.000(0.000)

1.000(0.000)

0.992(0.007)

0.931(0.057)

0.979(0.017)

0.974(0.034)

0.675 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

0.917(0.037)

0.853(0.103)

0.715(0.348)

1.225 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

0.902(0.104)

0.535(0.529)

1.575 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

0.996(0.004)

0.451(0.252)

2.025 1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

1.000(0.000)

0.999(0.000)

Table 1.2: Correlation between ∆d+Xt(∆

dSXt) and ∆d

MXt. The series is generated byXt = ∆−d0εt. In parenthesis are calculated the standard error associated. The parametersare M = 1000 and N = 10 runs.

Time T = 106 T = 5× 105 T = 2× 105 T = 105 T = 104

secs. 512.72 94.42 12.39 3.09 0.06

Table 1.3: Filtering time expressed in seconds of ∆d+Yt as a function of T .

1.4. CONCLUSION 23

T ↓M → 500 1, 000 2, 000 5, 000 10, 000 20, 000 50, 000 100, 0005× 104 0.024 0.071 0.121 0.282 2.116 4.628 13.842 26.3982× 104 0.014 0.049 0.111 0.429 0.835 1.789 5.071 8.695

Table 1.4: Filtering time of ∆dMYt with t = 1, . . . , T expressed in seconds as a function of

M .

γ ↓ , M0 → 500 1, 000 2, 000 50, 00 10, 000 20, 000 50, 0000.01 9.73 8.56 7.95 7.11 6.22 5.72 5.660.05 1.92 1.75 1.66 1.48 1.37 1.38 1.960.1 0.975 0.923 0.858 0.780 0.767 0.850 1.5330.15 0.649 0.605 0.565 0.535 0.548 0.667 1.3800.25 0.490 0.458 0.436 0.420 0.445 0.564 1.3070.25 0.397 0.379 0.353 0.353 0.380 0.512 1.2590.30 0.331 0.315 0.301 0.306 0.339 0.467 1.225

Table 1.5: Filtering time expressed in seconds as a function of M0 and γ when T =500, 000.

Time T = 106 T = 5× 105 T = 2× 105 T = 105 T = 104

secs. 0.235 0.156 0.099 0.041 0.018

Table 1.6: Filtering time expressed in seconds of ∆dSYt as a function of T .

DGP ↓ Filter→ ∆d+ or ∆d

S ∆dF ∆d

M1∆dM2

d0 = 0.3 Empirical Power 100% 100% 100% 100%Empirical Size 6.1% 6.1% 7.3% 47.7%

d0 = 0.9 Empirical Power 100% 100% 100% 100%Empirical Size 5.6% 5.6% 6.5% 57.7%

Table 1.7: Empirical Size and Empirical Power of the cointegration rank test when d0 = 0.3,d0 = 0.9, M1 = 1000 and M2 = 100.


d0,d,T↓

Filte

r→∆d F

∆d F

∆d F

∆d F

∆d M

∆d M

M0

=10

2M

0=

102

M0

=10

3M

0=

103

M=

102

M=

103

γ=

0.3

γ=

0.2

γ=

0.3

γ=

0.2

d0

=.9,d

=0.

9R

MSE

0.53

10.

581

0.01

30.

0083

1530

188

T=

5×

105

Tim

e(se

c.)

0.42

70.

606

0.45

70.

584

0.03

20.

161

d0

=0.

9,d

=0.

4R

MSE

13.6

15.9

3.04

11.

1291

7232

96T

=5×

105

Tim

e(se

c.)

0.32

80.

471

0.49

10.

514

0.03

20.

033

d0

=0.

3,d

=0.

3R

MSE

0.61

70.

335

0.18

90.

102

23.8

26.

64T

=5×

105

Tim

e(se

c.)

0.43

70.

524

0.38

30.

480

0.03

20.

181

d0

=0.

3,d

=0.

1R

MSE

0.83

40.

450

0.41

00.

218

32.9

514.0

3T

=5×

105

Tim

e(se

c.)

0.28

90.

400

0.35

90.

434

0.03

20.

162

d0

=0.

9,d

=0.

9R

MSE

0.14

10.

156

0.00

60.

002

40.7

24.

615

T=

105

Tim

e(se

c.)

0.04

30.

055

0.05

90.

067

0.06

70.

031

d0

=0.

9,d

=0.

4R

MSE

3.24

94.

057

1.38

90.

4780

2263.2

702.

1T

=10

5Ti

me(

sec.

)0.

041

0.06

40.

074

0.06

00.

006

0.03

1

Table 1.8: RMSE and execution time when T, d0 and d are setted as indicated by the firstcolumn of the table.

1.4. CONCLUSION 25

Codes

This code filters the data with the fast fractional filter. All the comments are inside thecode.

f u n c t i o n [fx ]=Lbfffilter (y ,d ,b ,M0 , gamma ,iK )%INPUT% gamma : i s t h e p a r a m e t e r t h a t c o n t r o l f o r t h e p a r t i a l sums ( i t ' s a ←

p e r c e n t a g e ) .% M0 : d e f i n e how many t e r m s we f i l t e r w i th a moving a v a r a g e o f M w e i g h t s

5 % d : i s t h e f r a c t i o n a l o p e r a t o r , a r e a l number% b : i s t h e f r a c t i o n a l o p e r a t o r , a r e a l number% y : i s t h e t ime s e r i e s c o n s i d e r e d ( c o u l d be p−d i m e n s i o n a l )% iK : i s t h e k−t h l a g c o n s i d e r e d on D e l t a ^d L_b^k%OUTPUT

10 %fx : t h e s e r i e s f i l t e r e d wi th t h e f a s t f r a c t i o n a l f i l t e r .

l= s i z e (y ) ; p=min (l ) ; s=max (l ) ; % P a r a m e t e r s p i s t h e number o f t ime s e r i e snobs=s ; % Number o f d a t a f o r each v a r i a b l e c o n s i d e r e d

15 i f iK>0;pesi = z e r o s (s , 1 ) ;

f o r iJ=1:iK+1;k = ( 1 :s−1) ' ; % c r e a t e a z e r o v e c t o r o f s−1pesi0 = (k − d − (iJ−1)*b − 1) . / k ;

20 pesi0= [1 ; cumprod (pesi0 ) ] ;pesi = pesi + (−1) ^ (iJ+1) * nchoosek (iK ,iJ−1)* pesi0 ;end ;

c l e a r pesi0 ;end ;

25i f iK == 0 ;

k = ( 1 :s−1) ' ; % c r e a t e a z e r o v e c t o r o f s−1pesi = (k − d − 1) . / k ;pesi= [1 ; cumprod (pesi ) ] ;

30 end ;

i f iK==−1 ;pesi = z e r o s (s , 1 ) ;f o r iJ= 1 : 2 ;

35k = ( 1 :s−1) ' ; % c r e a t e a z e r o v e c t o r o f s−1pesi0 = (k − d + (iJ−1)*b − 1) . / k ;pesi0= [1 ; cumprod (pesi0 ) ] ;pesi = pesi + (−1) . ^ ( iJ ) . * pesi0 ;

40 end ;%end ;c l e a r pesi0 ;

end ;

45 f= f i l t e r (pesi ( 1 :M0 ) , 1 ,y ) ; % F i l t e r from 1 t o M0, where M0 i s chosen by ←t h e u s e r .

begin=M0+1;start1=gamma ;weight0=0;


50 mInfoWeights= z e r o s (s , 2 ) ;prova= abs ( (pesi−pesi (M0 ) ) . / pesi (M0 ) ) ; %The p e r c e n t a g e i n c r e m e n t o f t h e ←

w e i g h t s w i th r e s p e c t t o t h e M−t h we ig h t o f D e l t a ^dL_b^K.

count=0; % s t a r t t h e c o u n t e r t o be z e r o .f o r jj=M0+1:nobs

55i f prova (jj ) >= start1 ; % I f t h e p e r c e n t u a l i n c r e m e n t o f t h e j−t h we igh t ←

i s g r e a t e r t h a n s t a r t 1count=count+1; % We change t h e c o u n t e r .weight=mean (pesi (begin :jj ) ) ; % F i r s t s t e p : We c a l c u l a t e t h e ←

a v a r a g e o f w e i g h t s from M+1 t o t h i s j j * where t h e c o n d i t i o n prova ( j j )←>=1 s t a r t 1 i s f u l l f i l l e d

60 begin=jj ; % We need t o save t h e number j j such t h a t we have a jump on t h e←a v a r a g e s .

mInfoWeights (count , : ) =[weight , jj ] ; % I w r i t e i n t o a m a t r i x t h e ←i n f o r m a t i o n r e l a t i v e t o t h e j j and t h e v a r i a b l e w e ig h t .

start1=start1+gamma*(1−start1 ) ; % I change t h e p e r c e n t a g e o f t h e ←i n c r e m e n t . . . f o r t h e f o r and i f c o n d i t i o n

end ;65

end ;c l e a r provamInfoWeights ( a l l (mInfoWeights==0 ,2) , : ) = [ ] ; % i t e r a s e s a l l z e r o rows[iI ,iJ ]= s i z e (mInfoWeights ) ; %c a l c u l a t e t h e s i z e o f t h e m a t r i x mInfoWeights

70cumyy=cumsum (y ) ;% Cumulate o f t h e s e r i e s y . We use i t f o r t h e f a s t f r a c t i o n a l f i l t e r .

i f iI>1;75 mInfoWeights=[ 0 ,M0 ; mInfoWeights ; mean (pesi (begin :s ) ) ,s ] ; %i n m a t r i x ←

mInfoWeights I p u t a l l t h e i n f o r m a t i o n s a b o u t t h e w e i g h t s o f t h e ←f r a c t i o n a l f i l t e r .

c l e a r pesi ;[iI ,iJ ]= s i z e (mInfoWeights ) ; %c a l c u l a t e t h e s i z e o f t h e m a t r i x ←

mInfoWeights

uni= z e r o s (s ,p ) ;80 uni2= z e r o s (s ,p ) ;

f o r iL=2:iIuni2=[ z e r o s (mInfoWeights (iL−1 ,2) +1 ,p ) ;cumyy ( 1 :s−mInfoWeights (iL−1 ,2) −1 , : )←

*(mInfoWeights (iL , 1 )−mInfoWeights (iL−1 ,1) ) ] ;uni=uni2+uni ;

85 ende l s e

uni= z e r o s (s ,p ) ;end

90 c l e a r mInfoWeights ;c l e a r cumyy uni2 ;fx=f+uni ; %% sum of t h e D e l t a ^d _ M and t h e cumula t ed sums ! ! !end

1.5. REFERENCES 27

1.5 ReferencesBaillie, R. T., Bollerslev, T., 1994. Cointegration, fractional cointegration, and exchange

rate dynamics. The Journal of Finance 49 (2), 737–745.

Baillie, R. T., Bollerslev, T., Mikkelsen, H. O., 1996. Fractionally integrated generalizedautoregressive conditional heteroskedasticity. Journal of econometrics 74 (1), 3–30.

Bollerslev, T., Mikkelsen, H. O., 1996. Modeling and pricing long memory in stock marketvolatility. Journal of econometrics 73 (1), 151–184.

Carlini, F., Manzoni, M., Mosconi, R., 2010. The impact of supply and demand imbalanceon stock prices: An analysis based on fractional cointegration using borsa italiana ultrahigh frequency data. In: XII Workshop on Quantitative Finance.

Chan, N. H., Palma, W., 1998. State space modeling of long-memory processes. Annals ofStatistics, 719–740.

Granger, C. W., Joyeux, R., 1980. An introduction to long-memory time series models andfractional differencing. Journal of time series analysis 1 (1), 15–29.

Grassi, S., de Magistris, P. S., 2014. When long memory meets the Kalman filter: Acomparative study. Computational Statistics & Data Analysis 76, 301–319.

Hosking, J. R., 1981. Fractional differencing. Biometrika 68 (1), 165–176.

Jensen, A. N., Nielsen, M. Ø., 2014. A fast fractional difference algorithm. Journal of TimeSeries Analysis 35 (5), 428–436.

Johansen, S., 2008. A representation theory for a class of vector autoregressive models forfractional processes. Econometric Theory Vol 24, 3, 651–676.

Johansen, S., Nielsen, M. Ø., 2012. Likelihood inference for a fractionally cointegratedvector autoregressive model. Econometrica 80 (6), 2667–2732.

Li, W. K., McLeod, A. I., 1986. Fractional time series modelling. Biometrika 73 (1), 217.

MacKinnon, J. G., Nielsen, M. Ø., 2014. Numerical distribution functions of fractionalunit root and cointegration tests. Journal of Applied Econometrics 29 (1), 161–171.

CH

AP

TE

R

2ON THE IDENTIFICATION OF

FRACTIONALLY COINTEGRATED VARMODELS

Federico CarliniAarhus University and CREATES

Paolo Santucci de MagistrisAarhus University and CREATES

Abstract

This paper discusses identification problems in the fractionally cointegratedsystem of Johansen (2008) and Johansen and Nielsen (2012). It is shown that severalequivalent reparameterizations of the model associated with different fractional inte-gration and cointegration parameters may exist for any choice of the lag-length whenthe true cointegration rank is known. The properties of these multiple non-identifiedmodels are studied and a necessary and sufficient condition for the identification of thefractional parameters of the system is provided. The condition is named F(d). This isa generalization of the well-known I(1) condition to the fractional case. Imposing aproper restriction on the fractional integration parameter, d, is sufficient to guaranteeidentification of all model parameters and the validity of the F(d) condition. Thepaper also illustrates the indeterminacy between the cointegration rank and the lag-length. It is also proved that the model with rank zero and k lags may be an equivalentreparameterization of the model with full rank and k − 1 lags. This precludes thepossibility to test for the cointegration rank unless a proper restriction on the fractionalintegration parameter is imposed.

29

30 CHAPTER 2. ON THE IDENTIFICATION OF FRACTIONALLY COINTEGRATED VAR MODELS

2.1 Introduction

The past decade has witnessed an increasing interest in the statistical definition andevaluation of the concept of fractional cointegration, as a generalization of the idea ofcointegration to processes with fractional degrees of integration. In the context of long-memory processes, fractional cointegration allows linear combinations of I(d) processesto be I(d− b), with d, b ∈ R+ with 0 < b ≤ d. More specifically, the concept of fractionalcointegration implies the existence of common stochastic trends integrated of order d,with short-period departures from the long-run equilibrium integrated of order d− b. Thecoefficient b is the degree of fractional reduction obtained by the linear combination ofI(d) variables, namely the cointegration gap.

Notable methodological works in the field of fractional cointegration are Robinsonand Marinucci (2003) and Christensen and Nielsen (2006) that develop regression-basedsemi-parametric methods to evaluate whether two fractional stochastic processes sharecommon trends. Analogously, Hualde and Velasco (2008) propose to check for the absenceof cointegration by comparing the estimates of the cointegration vector obtained with OLSand those obtained with a GLS type of estimator. Breitung and Hassler (2002) proposea multivariate score test statistic to determine the cointegration rank that is obtainedby solving a generalized eigenvalue problem of the type proposed by Johansen (1988).Alternatively, Robinson and Yajima (2002) and Nielsen and Shimotsu (2007) suggest atesting procedure to evaluate the cointegration rank of the multivariate coherence matrixof two, or more, fractionally differenced series. Chen and Hurvich (2003, 2006) estimatecointegrated spaces and subspaces by the eigenvectors corresponding to the r smallesteigenvalues of an averaged periodogram matrix of tapered and differenced observations.

Despite the effort spent in defining testing procedures for the presence of fractionalcointegration, for a long time the literature in this area lacked a fully parametric multivariatemodel explicitly characterizing the joint behaviour of fractionally cointegrated processes.Interestingly, Granger (1986, p.222) already introduced the idea of common trends betweenI(d) processes, but the subsequent theoretical works, see among many others Johansen(1988), have mostly been dedicated to cases with integer orders of integration. Onlyrecently, Johansen (2008) and Johansen and Nielsen (2012) have proposed the FCVARd,b

model, an extension of the well-known VECM to fractional processes, which is a toolfor a direct modeling and testing of fractional cointegration. Johansen (2008) studies theproperties of the model while Łasak (2010) suggests a profile likelihood approach toestimate the parameters and to test the hypothesis of absence of cointegration relationsin the Granger (1986) model under the assumption that d = 1. Recently, Johansen andNielsen (2012) have extended the estimation method of Łasak (2010) to the FCVARd,b

model, deriving the asymptotic properties of the profile maximum likelihood estimatorwhen 0 ≤ d− b < 1/2 and b 6= 1/2. Other contributions in the parametric framework forfractional cointegration are in Avarucci and Velasco (2009), Franchi (2010) and Łasak and

2.1. INTRODUCTION 31

Velasco (2015).This paper shows that the FCVARd,b model is not globally identified when the

number of lags, k, is unknown. For a given number of lags, several sub-models with thesame conditional densities but different values of the parameters may exist. Hence theparameters of the FCVARd,b model cannot be uniquely identified. The multiplicity ofnot-identified sub-models can be determined for any FCVARd,b model with k lags. Ananalogous identification problem, for the FIVARb model is discussed in Tschernig, Weber,and Weigand (2013a,b). This paper provides a detailed illustration of the identificationproblem in the FCVARd,b framework. It is proved that the I(1) condition in the VECMof Johansen (1988) can be generalized to the fractional context. In analogy with theI(1) condition for integer orders of integration, this condition is named F(d), and it is anecessary and sufficient condition for the identification of the parameters of the system.If the F(d) condition is not satisfied, the FCVARd,b parameters, including fractional andco-fractional parameters, d and b, cannot be uniquely determined.

The problems of identification in the FCVARd,b model is studied along the followinglines. First, Proposition 2.2.2 extends the results in Theorem 3 of Johansen and Nielsen(2012), highlighting the close relationship between the lag structure and the lack ofidentification, and deriving a necessary and sufficient condition for identification associatedto any lag-length. Proposition 2.2.2 also highlights the consequence of the indeterminacyof the lag-length on the fractional parameters d and b. Second, the paper shows theconsequence of the lack of identification on the likelihood function, both asymptoticallyand in finite samples. Differently from the standard case, where the integration ordersare fixed to integer values, the estimation of the FCVARd,b involves the maximizationof the profile log-likelihood with respect to d and b, but the latter is affected by theindeterminacy generated by the over-specification of the lag-length. Moreover, we discussa further identification issue, that emerges when the cointegration rank is unknown. Undercertain restrictions on d and b, the FCVARd,b with full rank and k lags is equivalent to theFCVARd,b with rank 0 and k + 1 lags. This last finding precludes the possibility to testfor the absence of cointegration when the true number of lags is unknown based on theunrestricted FCVARd,b model. Finally, we prove that the FCVARd,b is identified for anylag k > 1, both in the known and unknown rank cases, if the fractional parameter d isrestricted to be equal to the true fractional order, such that the F(d) condition is satisfiedby construction. Building on this result, we show that to solve the identification problemit is sufficient to restrict the parameter set of d to belong to the sub-interval of R+ thatincludes the true fractional order, d0, but excludes other values of d < d0 associated toequivalent models. The information about the true fractional order can be obtained by theexact local Whittle estimator of Shimotsu and Phillips (2005).

This paper is organized as follows. Section 2.2 discusses the identification problemfrom a theoretical point of view. Section 2.3 discusses the consequences of the lack of iden-tification on the inference on the parameters of the FCVARd,b model both asymptotically


and in finite samples. Section 2.4 discusses the problems when the cointegration rank andthe lag-length are both unknown. Section 2.5 concludes the paper.

2.2 The Identification ProblemThis section provides a discussion of the identification problem related to the FCVARd,b

model

Hk : ∆dXt = αβ′∆d−bLbXt +k∑i=1

Γi∆dLibXt + εt εt ∼ iidN(0,Ω), (2.1)

where Xt is a p-dimensional vector, α and β are p× r matrices, and r defines the cointegra-tion rank. Ω is the positive definite covariance matrix of the errors, and Γj , j = 1, . . . , k,are p × p matrices loading the short-run dynamics. The operator Lb := 1 − ∆b is theso called fractional lag operator, which, as noted by Johansen (2008), is necessary forcharacterizing the solutions of the system in (2.1) and obtaining the associated Grangerrepresentation in the fractional context. Following Definition 1 in Johansen and Nielsen(2012, p.2672), if Xt follows (2.1), then Xt is a fractional process of order d, denoted asF(d), and co-fractional of order d− b. The symbol Hk defines the model with k lags andθ = vec(d, b, α, β,Γ1, ...,Γk,Ω) is the parameter vector. The parameter space of modelHk is

ΘHk= α ∈ Rp×r0 , β ∈ Rp×r0 ,Γj ∈ Rp×p, j = 1, . . . , k, d ∈ R+, b ∈ R+, d ≥ b > 0,Ω > 0,

where r0 is the true cointegration rank and it is assumed known. The results of this Sectionare obtained under the maintained assumption that the true cointegration rank is knownand such that 0 < r0 < p. An extension to the case of unknown rank and number of lags ispresented in Section 2.4.

Similarly to Johansen (2010), the concept of identification and equivalence betweentwo models is formally introduced by the following definition.

Definition 2.2.1. Let P = Pθ, θ ∈ Θ be a family of probability measures, that is, astatistical model. We say that a parameter function g(θ) is identified if g(θ1) 6= g(θ2)

implies that Pθ1 6= Pθ2 . On the other hand, if Pθ1 = Pθ2 and g(θ1) 6= g(θ2), the parameterfunction g(θ) is not identified. In this case, the statistical models Pθ1 and Pθ2 are equivalent.

As noted in Johansen (1995a, p.177), the product αβ′ is identified but not the matricesα and β because if there was an r × r matrix ξ, the product αβ′ would be equal to αξβ′ξwhere αξ = αξ and βξ = β(ξ′)−1. In the following we do not discuss the identificationof the α and β, that is generally solved by a proper normalization of β. It can be showninstead that the other parameters of the FCVARd,b model in (2.1) might not be identified,i.e. several equivalent sub-models associated with different values θ, can be found when k

2.2. THE IDENTIFICATION PROBLEM 33

is overspecified. An illustration of the identification problem is provided by the followingexample.

Example 1: Consider the FCVARd,b model with one lag,

H1 : ∆dXt = αβ′∆d−bLbXt + Γ1∆dLbXt + εt, (2.2)

which can be written as∆d[Ip + αβ′ − Γ1

]+ ∆d−b [−αβ′]+ ∆d+bΓ1

Xt = εt.

First, examine the restriction, H(0)1 : Γ0

1 = 0. Under H(0)1 , the model in equation (2.2)

can be rewritten as ∆d0 [Ip + αβ′] + ∆d0−b0 [−αβ′]

Xt = εt.

Second, consider instead the restriction H(1)1 : Ip + αβ′ − Γ1

1 = 0. It follows that∆d1−b1

[−αβ′

]+ ∆d1+b1 [Ip + αβ′]

Xt = εt.

Given that the condition αβ′∆d0−b0 = αβ′∆d1−b1 must hold in both sub-models, hencemodel (2.2) under H(0)

1 is equivalent to the model (2.2) under H(1)1 if and only if

[Ip + αβ′]∆d0 = [Ip + αβ

′]∆d1+b1 .

This leads to the system of two equations in d0, b0, d1 and b1d0 − b0 = d1 − b1

d0 = d1 + b1

(2.3)

which has a unique solution when d1 = d0 − b0/2 and b1 = b0/2. Since the restrictionsH

(0)1 and H

(1)1 lead to equivalent descriptions of the data, it follows that the fractional order

of Xt implied by both models must be the same. However, in H(0)1 the fractional order

is represented by the parameter d0, i.e. Xt ∼ F(d0) since ∆d0Xt ∼ F(0), while in H(1)1

the fractional order is given by the sum d1 + b1, i.e. Xt ∼ F(d1 + b1). The identificationcondition defined in 2.2.1 is clearly violated, as the conditional densities of H(0)

1 and H(1)1

are such that

pH

(0)1

(X1, ..., XT , θ0|X0, X−1, . . .) = pH

(1)1

(X1, ..., XT , θ1|X0, X−1, . . .), (2.4)

where θ0 = vec(d0, b0, α, β,Ω) and θ1 = vec(d1, b1, α, β,Γ11,Ω) with Γ1

1 = Ip + αβ′.Example 1 can be extended to a generic lag-length k0 ≥ 0. Consider the model Hk0

Hk0 : ∆d0Xt = α0β′0∆d0−b0Lb0Xt +

k0∑i=1

Γ0i∆

d0Lib0Xt + εt εt ∼ N(0,Ω0), (2.5)


with k0 ≥ 0 lags, and |α′0,⊥Γ0β0,⊥| 6= 0 with Γ0 = Ip −∑k0

i=1 Γ0i . When a model Hk

with k > k0 is considered, then Hk0 is associated with the set of restrictions H(0)k :

Γk0+1 = Γk0+2 = ... = Γk = 0 imposed on Hk. However, there may be several alternativerestrictions on Γk0+1,Γk0+2, ...,Γk leading to an equivalent sub-model as the one obtainedunder H(0)

k .The following Proposition states the necessary and sufficient condition, called the

F(d) condition, for identification of the parameters of the model Hk.

Proposition 2.2.2. Consider a FCVARd,b model with k lags,

i) Given k > k0 ≥ 0, the F(d) condition, defined as |α′⊥Γβ⊥| 6= 0 with Γ = Ip −∑ki=1 Γi, is a necessary and sufficient condition for the identification of the set of

parameters of Hk in equation (2.5).

ii) Given k0 and k, with k ≥ k0, the number of equivalent sub-models that can beobtained from Hk is m = b k+1

k0+1c, where bxc denotes the greatest integer less or

equal to x.

iii) For any k ≥ k0, all the equivalent sub-models are found for parameter valuesdj = d0 − j

j+1b0 and bj = b0/(j + 1) for j = 0, 1, ...,m− 1.

Proposition 2.2.2 has several important consequences that are worth being discussedin detail. First of all, the F(d) condition only holds for the sub-model of Hk for whichd = d0 and b = b0, i.e. for the sub-model of Hk associated to the restriction H

(0)k : Γk0+1 =

Γk0+2 = ... = Γk = 0. In the Example 1, the F(d) condition is only verified for H(0)1 ,

while for H(1)1 we have that |α′⊥Γ1β⊥| = 0, since Γ1 = Ip − (Ip + αβ′) = −αβ′. Note

that the assumption |α′0,⊥Γ0β0,⊥| 6= 0 imposed on model (2.5) guarantees that it is notpossible to find restrictions on Hk0 for which two or more sub-models are equivalent.In this sense Proposition 2.2.2 generalizes Theorem 3 in Johansen and Nielsen (2012).Indeed, while in Johansen and Nielsen (2012) the F(d) condition is only imposed onthe Hk0 model with k = k0 by assumption, Proposition 2.2.2.i) shows that a necessaryand sufficient condition for the identification of the parameters of any Hk model, withk > k0, is the validity of the F(d) condition. This has important consequences in practicalapplications when the true number of lags is unknown and it is potentially over-specified.Note that if the number of lags is under-specified there is no identification problem, but thismisspecification model leads to the invalidity of the results in Johansen and Nielsen (2012).In addition, Proposition 2.2.2.ii) characterizes the number of equivalent sub-models of Hk

for a given k0, showing that their multiplicity depends on k and k0. Table 2.1 summarizesthe number of equivalent sub-models for different values of k0 and k. Interestingly, as aconsequence of Proposition 2.2.2.ii), there are cases in which k > k0 does not necessarilyimply a lack of identification. For example, when k = 2 and k0 = 1 there are no setsof restrictions on H2 leading to a sub-model equivalent to the one obtained under the

2.2. THE IDENTIFICATION PROBLEM 35

k0 ↓ k → 0 1 2 3 4 5 6 7 8 9 10 11 12

0 1 2 3 4 5 6 7 8 9 10 11 12 131 – 1 1 2 2 3 3 4 4 5 5 6 62 – – 1 1 1 2 2 2 3 3 3 4 43 – – – 1 1 1 1 2 2 2 2 3 34 – – – – 1 1 1 1 1 2 2 2 25 – – – – – 1 1 1 1 1 1 2 2

Table 2.1: Table reports the number of equivalent models (m) for different combinations of k andk0. When k0 > k the Hk is under-specified.

restriction d = d0, b = b0, Γ1 = Γ01 and Γ2 = 0. Hence, in this case, the multiplicity, m,

of equivalent sub-models is 1. When k0 is small there are several equivalent sub-modelsfor small choices of k. As k0 increases, multiple equivalent sub-models are only foundfor large values of k. For example, when k0 = 5, then two equivalent sub-models canonly be found for suitable restrictions of the H11 model. Moreover, Proposition 2.2.2.iii)shows that each sub-model of Hk equivalent to Hk0 with |α′⊥Γβ⊥| = 0 has values of dand b that are fractions of d0 and b0. Interestingly, when k is very large compared to k0,the (m− 1)-th sub-model is associated with dm−1 ≈ d0 − b0 and bm−1 ≈ 0, i.e. locatedclosely to the boundary of the parameter space.

Interestingly, the lack of identification of b can be interpreted as a sort of aliasingproblem, in which the unit by which time is measured can be changed arbitrarily: forexample, setting k0 = 0 and d0 = b0 = 1, and letting Xt be the value at year t, one has:

Π(0)(L)Xt = Π(0)0 Xt + Π

(0)1 ∆Xt time measured in years

Π(1)(L)Xt = Π(0)0 Xt + 0∆1/2Xt + Π

(0)1 ∆Xt time measured in semesters

......

Π(364)(L)Xt = Π(0)0 Xt + 0∆

1365Xt + · · ·+ 0∆

364365Xt + Π

(0)1 ∆Xt time measured in days

where the FCVAR process is represented as ∆dj−bjΠ(j)(L)Xt = εt and Π(j)(L) =∑j+1i=0 Π

(j)i (1− L)ibj , dj := 1− j

j+1, bj := 1

j+1.

A final remark on the relevance of the F(d) concerns the possibility of polynomialfractional cointegration, that arises when |α′⊥Γβ⊥| = 0. This means that models withpolynomial fractional cointegration up to order m = b k+1

k0+1c can be obtained from the

FCVARd,b model for some combinations of k and k0. Consider again model H(1)1 in

Example 1. After simple algebraical manipulations, model H(1)1 can be formulated as

∆d2Xt = ∆d2−2b1(αβ′Lb1Xt − Γ1∆b1Lb1Xt) + εt, (2.6)

where d2 = d1 + b1 and Γ1 = −αβ′. Equation (2.6) defines a model for polynomial frac-tional cointegration as studied in Johansen (2008, p.667) and further extended in Franchi


(2010). Polynomial fractional cointegration has analogies in the context of modeling I(2)variables in the standard VECM context, see Johansen (1995b), and the number of polyno-mial fractional trends depends on the rank of the matrix α′⊥Γβ⊥. In particular, imposing theF(d) condition on the FCVARd,b model does not only guarantee that the parameters d, band Γ1, ...,Γk are identified, but also rules out cases of polynomial fractional cointegration.

2.3 Identification and InferenceThis section illustrates, by means of numerical examples, the problems in the estimation ofthe parameters of the FCVARd,b that are induced by the lack of identification outlined inSection 2.2. As shown in Johansen and Nielsen (2012), the parameters of the FCVARd,b canbe estimated following a profile likelihood approach, where the estimates of the fractionalparameters, d and b, are obtained first by maximizing the profile log-likelihood

ψ = arg maxψ∈K

`T (ψ), (2.7)

where ψ = (d, b)′, K =η ≤ b ≤ d ≤ d

where d > η > 0 and

`T (ψ) = − log |S00(ψ)| −r∑l=1

log(1− λh(ψ)). (2.8)

The quantities λ(ψ) and S00(ψ) are obtained from the residuals, Rit(ψ) of the reducedrank regression of ∆dXt (i = 0) and ∆d−bLbXt (i = 1) on ∆dLbXt, . . . ,∆

dLkbXt. Theproduct moment matrices Sij(ψ) for i, j = 0, 1 are Sij(ψ) = T−1

∑Tt=1Rit(ψ)R′jt(ψ) and

λh(ψ) for h = 1, . . . , p are the solutions, sorted in decreasing order, of the generalizedeigenvalue problem

|λ(ψ)S11(ψ)− S10(ψ)S−100 (ψ)S01(ψ)| = 0. (2.9)

Given d and b, the estimates α, β, Γj , j = 1, . . . , k, and Ω are found by reduced rankregression as in Johansen (1988). Although the the statistical model (2.5) is defined for all0 < b0 ≤ d0, the asymptotic properties of the ML estimator are derived in Johansen andNielsen (2012) when the true values satisfy 0 ≤ d0 − b0 < 1/2 and b0 6= 1/2, for whichβ′0Xt is (asymptotically) a stationary process. Therefore, the following analysis is carriedout for combinations of d0 and b0, which satisfy such constraint. Unfortunately, imposingthe constraint d− b < 1/2 in the estimation of ψ does not provide sufficient informationto achieve identification since, by construction, d0 − b0 = di − bi ∀i = 1, . . . ,m withm = b k+1

k0+1c, by Propostion 2.2.2.iii).

Since the values of ψ that maximize `T (ψ) must be found numerically, the con-sequences of the lack of identification of the FCVARd,b model on the expected profilelog-likelihood when k > k0 are explored by means of Monte Carlo simulations. In particu-lar, since the asymptotic value of `T (ψ) is not available in closed-form as a function of

2.3. IDENTIFICATION AND INFERENCE 37

the FCVARd,b parameters, the asymptotic values of `T (ψ) are approximated by averagingover M simulations the value of `T (ψ) computed for different values of ψ and a large T .This provides a precise numerical approximation of the expected profile log-likelihood,E[`T (ψ)]. Therefore, M = 100 simulated paths are generated from model (2.5) withT = 50, 000 observations and p = 2. The fractional parameters of the system are d0 = 0.8

and b0 = d0. The restriction b0 = d0 simplifies the readability of the results withoutloss of generality, since the plots display ¯

T (d) = 1M

∑Mi=1 `i,T (d) as a function of d in a

two dimensional Cartesian system. The cointegration vector is β0 = [1,−1]′, the vectorof adjustment coefficients is α0 = [0.5,−0.5]′, and the matrices Γ0

i , i = 1, ..., k0, fordifferent values of k0 are chosen such that the roots of the characteristic polynomial areoutside the fractional circle, see Johansen (2008). The average profile log-likelihood, ¯

T (d),and the average of the function f(d) = |α′⊥(d)Γ(d)β⊥(d)| are computed with respect toa grid of alternative values for d = [dmin, . . . , dmax]. The average of f(d) over the Msimulations is a an estimate of the value of the F(d) condition for different values of d.Hence F(d) = 1

M

∑Mi=1 fi(d) for d = [dmin, . . . , dmax] is plotted together with ¯

T (d). Dueto space constraints, the results of the simulations cannot be shown for a large number ofparameter combinations. However, the results obtained with other parameters confirm thereported evidence.

Figure 2.1 reports the values of ¯T (d) and F(d) when k = 1 lags are chosen but

k0 = 0. It clearly emerges that the two global maxima of ¯T (d) are associated to the pair

of values d0 = 0.8 and d1 = 0.4, but in d1 = 0.4 the F(d) line is equal to zero. Similarly,

0.4 0.5 0.6 0.7 0.8 0.9 1−5.69

−5.68

−5.67x 10

−5 Expected Likelihood and F(d) condition fod different values of d

0.4 0.5 0.6 0.7 0.8 0.9 1−2

0

2

Expected LogL

F(d) condition

d=d*=0.8

d=d*/2=0.4

Zero Line

Figure 2.1: Figure reports simulated values of l(d) and F(d) for different values of d ∈ [0.3, 0.9]on the x-axis. The observations from the DGP are generated with k0 = 0 lags and model Hk

with k = 1 lags is estimated. The parameters of the DGP are d0 = b0 = 0.8, β0 = [1,−1]′,α0 = [−0.5, 0.5]′.

as reported in Figure 2.5.12 in the Supplementary material, the expected log-likelihood


function has three humps around d0 = 0.8, d1 = 0.4 and d2 = d0 − 2/3d0 = 0.2667

when k = 2 and k0 = 0. As in the previous case, the F(d) line is approximately equalto zero around d1 and d2. Consistently with the theoretical results presented in Section 2,the F(d) line is instead far from zero in d0 = 0.8 also in this case. Figure 2.2 reports thecontour plot of the expected profile log-likelihood function in the 2-dimensional spaceof (d, b) ∈ R2, with d ≥ b. The plot clearly highlights the presence of two equivalentpeaks located inside the isolines with level -14.1928 that, as expected, are associatedwith the vectors ψ0 = [0.8, 0.8]′ and ψ1 = [0.4, 0.4]′. Notably, the function l(ψ) quicklydecreases at the extremes of the parameter space, i.e. when d > d0 and b > b0 or whend < d0 − b0/2 and b < b0/2. Instead, the function remains rather high and flat in theinterval b0/2 < b ≤ d < d0. This may induce further identification problems in finitesamples as discussed in Section 2.3.1.

-14.299

-14.2862

-14.282

-14.2777

-14.2735 -14.2

692

-14.2692 -14.2

65

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608

-14.2608

-14.2

565

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523

-14.2523

-14.2

48

-14.2

48

-14.248

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438

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438

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395

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395

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353

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31

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268

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268

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225

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225

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183

-14.

2183

-14.2183

-14.2183

-14.2

14

-14.2

14

-14.2

14

-14.2

14

-14.214

-14.2

14

-14.2

098

-14.2

098

-14.2

098

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-14.2

098

-14.2098

-14.2

055

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-14.2

055

-14.2055

-14.2

055

-14.2

055

-1

4.2

01

3

-14.2013

-14.2013

-14.2

013

-14.2013

-14.2013

-14.1971

-14.1971

-14.1971

-14.1

971

-14.1

971

-14.1

971

-14.1928

-14.1

928

-14.1928

-14.1

928

0.3 0.4 0.5 0.6 0.7 0.8

0.3

0.4

0.5

0.6

0.7

0.8

Figure 2.2: Figure reports the contour plot of the values (rescaled by a 10,000) of the function l(ψ)for different combinations of d ∈ [0.3, 0.9] (x-axis) and b ∈ [0.3, 0.9] (y-axis). The observationsfrom the DGP are generated with k0 = 0 lags and model Hk with k = 1 lag is estimated. Theparameters of the DGP are d0 = b0 = 0.8, β0 = [1,−1]′, α0 = [−0.5, 0.5]′. The empty area isassociated to values of b > d for which the log-likelihood is not defined.

A slightly more complex evidence arises when k0 > 0. Figure 2.3 reports ¯T (d) and

F(d) when k0 = 1 while k = 2 is chosen. The ¯T (d) function is globally maximized in the

region around d0 = 0.8, thus supporting the theoretical results outlined in Propostion 2.2.2,i.e. when k = 2 and k0 = 1 there is no lack of identification. However, another interestingevidence emerges. The lT (d) function is flat and high in the region around d = 0.5, possiblyinducing identification problems in finite samples. This issue will be further discussedin Section 2.3.1. When k = 3, we expect m = 4

2= 2 equivalent sub-models associated

with d0 = 0.8 and d1 = d0 − 1/2d0 = 0.4. Indeed, by looking at Figure 2.5.13 in theSupplementary material, it emerges that the ¯

T (d) line has two global maxima around the


values d0 = 0.8 and d1 = 0.4. As expected, in the region around d1 = 0.4 the F(d) line isclose to zero.

0.4 0.5 0.6 0.7 0.8 0.9 1−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Expected Likelihood and F(d) condition for different values of d

0.4 0.5 0.6 0.7 0.8 0.9 1−160

−140

−120

−100

−80

−60

−40

−20

0

20

F(d) condition

Expected profilelikelihood

Figure 2.3: Figure reports simulated values of l(d) and F(d) for different values of d ∈ [0.4, 1]on the x-axis. The observations from the DGP are generated with k0 = 1 lags and model Hk

with k = 2 lags is estimated. The parameters of the DGP are d0 = b0 = 0.8, β0 = [1,−1]′,α0 = [−0.5, 0.5]′, and Γ1 =

[0.3 −0.20.4 −0.5

].

2.3.1 Identification in Finite Samples

The purpose of this Section is to shed light on how the lack of mathematical identificationaffects the estimates of the FCVARd,b in finite samples. Figure 2.4 reports the finite sampleprofile log-likelihood function, `T (d), against a fine grid of values of d. Each plot reportsthe function `T (d) obtained by fitting model H1 on a distinct simulated path of lengthT = 1, 000, generated under model H0. The plot clearly highlights the consequences ofthe lack of identification in finite samples. In Panel a), the global maximum of `T (d) isfound around d = 0.4, while in Panel b) it is around 0.8. As expected in Panel a), the f(d)

line is near 0 when d = 0.4, while it is far from zero in Panel b) when d = 0.8.Proposition 2.2.2 and Table 2.1 show that there are many combinations of k and

k0 associated to identified FCVARd,b models, for example the case k0 = 1 and k = 2.In these cases, the expected profile log-likelihood should not display multiple equivalentmaxima associated with fractions of d0. However, poor finite sample identification, namelyweak identification, might arise also in this cases. For example, Figure 2.5 reports thefinite sample profile log-likelihood function relative to the estimation of the H2 modelon two simulated paths of H1 with T = 1, 000. In Panel a), the global maximum is in aneighborhood of d = 0.4, and the function f(d) is close to zero in d = 0.4. Hence, theestimated matrices Γ1 and Γ2 are such that |α′⊥Γβ⊥| ≈ 0. On the other hand, with another


0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2860

-2850

-2840

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-1

0

1

l(d)

F(d) condition

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2850

-2840

-2830

-2820

-2

-1

0

1

l(d)

F(d) condition

Figure 2.4: Figure reports the values of the profile log-likelihood l(d) and F(d) for different valuesof d ∈ [0.3, 1] (x-axis) for two different simulated paths with T = 1, 000 of the FCVARd,d whenk0 = 0 and model H1 is estimated. The parameters of the DGP are d0 = b0 = 0.8, β0 = [1,−1]′,α0 = [−0.5, 0.5]′.

simulated path, the global maximum is found around d = 0.8, where the function f(d) isfar from zero, Panel b). As it emerges from this example, for any choice of k > k0 thereis the risk of obtaining estimates of the fractional parameters, d and b, that are far fromthe true ones. Tschernig et al. (2013a) discuss an analogous identification problem in

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2770

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−1

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0

l(d)

F(d)

zero−line

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−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

l(d)

F(d)

zero−line

Figure 2.5: Figure reports the values of the profile log-likelihood l(d) and F(d) for different valuesof d ∈ [0.3, 1] (x-axis) for two different simulated paths with T = 1, 000 of the FCVARd,d whenk0 = 1 and model H2 is estimated in the data. The parameters of the DGP are d0 = b0 = 0.8,β0 = [1,−1]′, α0 = [−0.5, 0.5]′, and Γ1 =

[0.3 −0.20.4 −0.5

].

the FIVARb model. The FIVARb extends the FIVAR model allowing the autoregressivestructure to depend on the fractional lag operator, Lb, hence inducing more flexibility inthe short-run terms. Tschernig et al. (2013a) show that an identification problem arises


when the eigenvalues of the characteristic polynomial in the Lb operator are either closeto 0 or to 1. Similarly to the FCVARd,b, the lack of identification leads to an high andflat log-likelihood function for a wide range of combinations of d and b. However, inthe FCVARd,b model, the F(d) condition provides a necessary and sufficient conditionfor the identification. It is therefore crucial to develop a robust estimation procedure thatguarantees that the estimated FCVARd,b parameters are correctly identified and satisfy theF(d) condition also when the lag-length is potentially overspecified.

2.3.2 Constrained Likelihood

In the previous sections, we have proved that the FCVARd,b model suffers from identifica-tion problems when k is over-specified. In particular, a number of equivalent parametriza-tion associated to fractions of the true d0 and b0 can be found for several choices of k > k0.As illustrated above, this identification problem has clear consequences from a statisticalpoint of view since an unique ML estimator of d and b does not exist. We therefore pro-pose a new approach that is based on the idea of transforming the unrestricted maximumlikelihood problem, whose properties have been studied in Johansen and Nielsen (2012)only for the case k = k0, into a constrained maximum likelihood problem by imposinga very mild restriction on the parameter space of d. In particular, we suggest that d and bmust be the solutions of the following constrained maximum likelihood problem

ψ = arg maxψ

`T (ψ), (2.10)

s.t. δmin ≤ d ≤ d

s.t. η ≤ b ≤ d

where `T (ψ) is defined in (2.8), d > η > 0 and δmin determines the lower bound on theparameter d. Restricting the parameter space of d is supported by the following lemma,which is a direct derivation of Proposition 2.2.2.

Lemma 2.3.1. Let ΘHk= d = d0, b ∈ (0, d0], α ∈ Rp×r, β ∈ Rp×r,Γj ∈ Rp×p, j =

1, . . . , k; Ω > 0 be the restricted parameter space of model ΘHkwith d = d0 ∈ R+, then

the statistical model P = Pθ : θ ∈ ΘHk is identified, i.e. Pθ1 = Pθ2 implies θ1 = θ2 for

all θ1, θ2 ∈ ΘHk, and |α′⊥Γβ⊥| 6= 0 ∀θ ∈ ΘHk

.

It follows from Lemma 2.3.1 that once the parameter d is fixed to d0, then all theFCVARd,b parameters are uniquely identified for any lag k > k0. Under the constraintd = d0, the profile log-likelihood function `T (ψ) only varies with respect to b and it hasan unique maximum around b0. Interestingly, Lemma 2.3.1 provides theoretical supportto the procedure, adopted in Bollerslev, Osterrieder, Sizova, and Tauchen (2013) andCaporin, Ranaldo, and Santucci de Magistris (2013), of estimating the FCVARd,b model byrestricting the fractional parameter d to a constant value and by maximizing the profile log-likelihood function with respect to b only. Figure 2.6 reports the value of the sliced profile


log-likelihood with respect to different values of b, when the parameter d is fixed at thetrue value d0 = 1. It clearly emerges that, irrespectively of the choice of k > k0, the profile

0.6 0.7 0.8 0.9 1 1.1

×104

-5.72

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-5.7

-5.69

-5.68

-5.67

k=1

k=2

k=3

k=4

Figure 2.6: Figure reports the values of the expected profile log-likelihood, l(ψ), fordifferent values of b ∈ [0.51, 1.2] (x-axis) when d = d0 = 1. The sample size is T =20, 000 and k0 = 0, while Hk with k = 1, 2, 3, 4 is estimated. The parameters of the DGPare d0 = b0 = 1, β0 = [1,−1]′, α0 = [−0.5, 0.5]′.

log-likelihood function is uniquely maximized around b0. This is a direct consequenceof Lemma 2.3.1. Figure 2.5.14 in the Supplementary material confirms this result alsowhen b0 < d0. As expected the value of the sliced profile log-likelihood at the optimumis the highest for the model with k = 4 lags in both figures, since the model H4 nestsall the other models with k < 4. However, the profile log-likelihood function becomesvery flat when k increases. This is due to the efficiency loss caused by the inclusion in themodel Hk of matrices of parameters, Γj , j > k0, that should be theoretically excluded. Insmall samples, this may generate a problem of weak identification analogous to the onediscussed in Section 2.3.1.

Since there exists an unique maximum of the profile log-likelihood function when dis restricted to d0, then the asymptotic properties found in Johansen and Nielsen (2012)still hold. However, since d0 is unknown in practice, we rely on a constrained optimizationmethod which sets to zero the probability of selecting a maximum outside a given intervalfor the parameter d. This means that a lower bound for d, namely δmin, must be determinedsuch that the optimization of the profile log-likelihood is performed in an region whichshould only contain one maximum. In the following, we illustrate a simple and direct wayto select δmin in a data-driven fashion. In principle, any semi-parametric estimator of thefractional order of the series, e.g. the exact local Whittle estimator of Shimotsu and Phillips(2005), could be adopted to determine the fractional order of the system and a value for


δmin could be easily determined by setting a lower bound based on the point estimate.Unfortunately, a multivariate version of the exact local Whittle in which all the processesshare the same degree of fractional integration is not yet available in the literature. Indeed,under the assumption of fractional cointegration the multivariate log-likelihood of themodel cannot be determined due to the singularity of the coherence matrix at the origin,see the discussion in Nielsen and Shimotsu (2007) among others. Similarly to Nielsen andShimotsu (2007), we therefore recommend to obtain a semi-parametric estimate of d as

d =1

p

p∑i=1

di (2.11)

where di is the univariate exact local Whittle estimate of the parameter d on the i-th series.Doing this choice, we assume that the fractional orders of all the variables of the systemXt is the same.1 The exact local Whittle is defined as

di = arg mind∈D

R(di, Xt,i) i = 1, ..., p (2.12)

with

R(di, Xt,i) =1

m

m∑j=1

log(λ−2dij

)+ log

1

m

m∑j=1

I∆diXt,i(λj)

, (2.13)

where I∆diXt,i(λj) is the periodogram of the fractional difference of the series Xt,i evalu-

ated at the Fourier frequency λj , where the number of frequencies used in the estimation isqT and D is the admissible set of values of d, which according to Shimotsu and Phillips(2005) has length no larger than 9

2. Under Assumptions 1-5 of Shimotsu and Phillips

(2005), di is a consistent estimator of d and asymptotically Gaussian with

√qT (di − d0)

d→ N

(0,

1

4

). (2.14)

where the asymptotic variance does not depend on any nuisance parameter and the rate ofconvergence depends on qT . Therefore, once d is estimated, then δmin can be determined as

δmin = d− c · d, (2.15)

where c ∈ [0, 1]. Given this choice for δmin, it is possible to evaluate the probability thatthe restriction d ≥ δmin is associated to an identified system. Under identification, theML estimator of Johansen and Nielsen (2012) can consistently estimate d0 and b0 for anyk ≥ k0. Let’s first define the set D∗ = [δmin,+∞). If d0 ∈ D∗ and d0 − b0/2 /∈ D∗, thenall the parameters of the FCVAR model can be identified for all choices of k. Since δmin

1The order of fractionality of a p−dimensional vector Xt is defined to be the maximum of the fractionalorders of the series contained in Xt, see Johansen (2008). Our results are based on the hypothesis that theseries in Xt share the same fractional order d.


depends on the ELW estimator, which has the Gaussian distribution in (2.14), we cancompute the probability of the event δmin ∈ (d0 − b0/2, d0), denoted as Pd0−b0/2,d0 :=

Pr(d0 − b0/2 < δmin < d0), that is as

Pd0−b0/2,d0 = Pr

(Z ≤ 2c

1− cd0√qTp

)− Pr

(Z ≤ 2cd0 − b0

1− c√qTp

), (2.16)

where Z follows a standard normal distribution. Note that the first probability in (2.16)tends to 1 as T → ∞, while the second one tends to 0 only if 2cd0−b0

1−c < 0, that is ifb0 > 2cd0. Figure 2.7 displays the probability Pd0−b0/2,d0 for different combinations of T ,b0 and c when qT = T 0.5 and d0 = 0.8. First, it emerges that when b0 is very small, theprobability of δmin being in the interval (d0 − b0/2, d0) approaches zero, since the sizeof the interval (d0 − b0/2, d0) shrinks to zero. It should be noted that the case b0 ≈ 0 isnot empirically relevant. Indeed, when b0 ≈ 0 the strength of the cointegration relationis minimal, thus excluding the practical implementation of the FCVARd,b. Moreover,Johansen and Nielsen (2012) have proved the asymptotic distribution of the ML estimatorfor values of b0 > d0− 1/2, thus ruling out a number of cases for low values of b0. Second,

0.30.25

0.2

c

0.150.1

0.050.3

0.4

0.5

0.6

b0

0.7

0

0.2

0.4

0.6

0.8

1

Pd

0-b

0/2

,d0

0.30.25

0.2

c

0.150.1

0.050.3

0.4

0.5

0.6

b0

0.7

1

0.8

0.6

0.4

0.2

0

Pd

0-b

0/2

,d0

Figure 2.7: Figure reports the probability that δmin is in the interval (d0 − b0/2, d0), for differentvalues of b0 ∈ (0.3, 0.8) and c ∈ [0.01, 0.3], with qT = T 0.5 and d0 = 0.8. In the left panelthe series generated have sample size T = 1, 000 while in the right panel the sample size isT = 50, 000.

the probability Pd0−b0/2,d0 increases with the strength of the cointegration relation (b0) andwith T . For large values of b0, the probability of δmin being in the interval (d0 − b0/2, d0)

approaches one. When T = 50, 000, the probability Pd0−b0/2,d0 is close to 1 already forintermediate values of b0 and for almost all choices of c. In particular, choosing a small cin the range between 5% and 15%, that is setting the lower bound close to d, leads to thebest asymptotic results since the condition b0 > 2cd0 is more easily verified. On the otherhand, in small samples, e.g. T = 1, 000, the probability of d0 < δmin might not be zero if


c is too small, thus leading to a reduction in the joint probability Pd0−b0/2,d0 . This clearlydefines a trade-off between asymptotic precision and finite sample robustness in the choiceof c. Figure 2.8 shows that the probability of Pd0−b0/2,d0 tends to 1 as T →∞ for differentcombinations of d0 and b0 when c = 15%. In particular, the probability is already close to85% for the case d0 = 0.6 and b0 = 0.4 when T = 500 and it is almost equal to 1 whenT = 5, 000. In the next paragraph, we show how imposing the lower bound constraint in

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

d0=1, b

0=0.8

d0=0.8, b

0=0.6

d0=0.6, b

0=0.4

Figure 2.8: Figure reports the probability that δ is in the interval (d0 − b0/2, d0), for differentvalues combinations of d0 and b0 and for different sample sizes. The probability curves are relatedto a choice of c = 15%. The x-axis reports the sample size T and the y-axis the probabilityPd0−b0/2,d0 .

(2.10) is generally sufficient to solve the identification problem with a very mild restrictionon the parameter space.

Monte Carlo simulations

In this paragraph, we discuss the results of a number of Monte Carlo simulations to supportthe need for the approach based on the constrained log-likelihood outlined in (2.10) asopposed to the unconstrained2 one when the number of lags is unknown. Figure 2.9 reportsthe contour plot of the Monte Carlo estimates of the parameters d and b when a sample ofT = 2, 500 observations is generated by the following bivariate FCVARd,b model

∆d0Xt = α0β′0∆d0−b0Lb0Xt + εt t = 1, . . . , T (2.17)

where d0 = 1 and b0 = 0.8. For each generated sample, the model H2 is estimated on thedata. According to Proposition 2.2.2, three equivalent models can be found associated to

2The "unconstrained" profile maximum likelihood estimator is defined to be in a compact set K =η ≤ b ≤ d ≤ d

.


different combinations of d and b, i.e. ψ0 = [1, 0.8], ψ1 = [0.6, 0.4] and ψ2 = [0.47, 0.27].From Panel a) of Figure 2.9 it clearly emerges that maximizing the constrained log-likelihood function (2.10) solves the identification problem discussed above. Indeed,almost the entire probability mass of ψ, based on M = 1, 000 Monte Carlo estimates, isconcentrated around ψ0. Only in a very limited number of cases the estimates are locatedaround [0.8,0.5], and this could be attributed to the variability of the estimates in finitesamples. Instead, when the optimal parameters d and b are found by maximizing theunrestricted likelihood function, see Panel b), a large portion of the probability mass islocated away from ψ0 = [1, 0.8]. In particular, when the profile log-likelihood function isnot constrained, the bivariate distribution of ψ is clearly multi-modal, as a consequenceof the lack of identification as outlined in Proposition 2.2.2. For comparison, Figure 2.10

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Figure 2.9: Figure reports the contour plot ofM = 1, 000 Monte Carlo estimates of the parametersd (x-axis) and b (y-axis) when a sample of T = 2, 500 observations is generated by a FCVARd,bmodel with k0 = 0, d0 = 1, b0 = 0.8 and the cointegration vectors given by β0 = [1,−1]′ andα0 = [−0.5, 0.5]′. Model H2 is estimated on the data. The left panel is relative to the estimatesbased on the constrained log-likelihood (2.10) where c = 15% and qT = T 0.5. The right panelreports the contour plot for the unrestricted estimates.

reports the distribution of ψ when the number of lags is correctly specified, i.e. k = 0.Not surprisingly, the distribution of ψ is well centered around ψ0, and the estimates aremore efficient than those obtained with k > 0 since fewer FCVARd,b parameters mustbe estimated under correct lag specification. However, k0 is unknown in practice and istypically determined by a general-to-specific sequence of LR tests. In Section 2.4.1 wediscuss the nesting structure of the FCVARd,b model under unknown cointegration rankand lag-length and the optimal sequence of LR tests when the parameter space of d isproperly restricted.

Figures 2.5.16-2.5.20 in the Supplementary material highlight the robustness of theconstrained likelihood approach for different sample sizes and different combinations ofk0 and k. As T increases, the estimates based on the unconstrained likelihood still display

2.4. UNKNOWN COINTEGRATION RANK 47

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Figure 2.10: Figure reports the contour plot of M = 1, 000 Monte Carlo estimates of theparameters d (x-axis) and b (y-axis) when a sample of T = 2, 500 observations is generatedby a FCVARd,b model with k0 = 0, d0 = 1, b0 = 0.8 and the cointegration vectors given byβ0 = [1,−1]′ and α0 = [−0.5, 0.5]′. Model H0 is estimated on the data.

the identification problem, while the constrained estimates are all centered around d0 andb0, see Figure 2.5.15 which is based on T = 10, 000. This confirms that the probabilityPd0−b0/2,d0 converges to 1 as T →∞ as illustrated in Figure 2.8. If T = 1, 000 most of theprobability mass is still concentrated around d0 and b0, although the density is much moredispersed than the case with T = 10, 000, see Figures A.5 and A.6. This is not surprisinggiven the flatness of the log-likelihood function in small samples. Finally, the resultsdo not qualitatively change when data are generated under H1 with Γ1 =

[0.3 −0.2−0.4 0.5

]and model H3 is estimated, see Figure 2.5.18 As expected, the estimates based on theunconstrained likelihood are bimodal, since two equivalent sub-models of H3 can befound associated to k0 = 1, see Table 2.1. Instead, the estimates based on the constrainedlikelihood are unimodal and centered around the true values d0 and b0. Finally, whend0−b0 ≈ 0.5, see Figure 2.5.19, the slow convergence rate makes the profile log-likelihoodfunction extremely flat, although the sample size is moderately large, thus leading to ratherdispersed estimates of ψ. However, compared to the unrestricted estimates which are foundeverywhere in the interval 0.3 < b < d < 1, the constrained estimates are much moreoften located in the region around d0 and b0.

2.4 Unknown cointegration rank

We now extend the previous results to allow both the cointegration rank and the lag-lengthto be unknown. This is the relevant case in empirical applications, when testing for the


presence of a cointegration relationship between two (or more) fractional processes butthere is no preliminary information on the optimal choice of k. The unrestricted FCVARd,b

model is formulated as:

Hr,k : ∆dXt = Π∆d−bLbXt +k∑i=1

Γi∆dLibXt + εt, (2.18)

where 0 ≤ r ≤ p is the rank of the p× p matrix Π. The parameter space of model Hr,k is

ΘHr,k= α ∈ Rp×r, β ∈ Rp×r,Γj ∈ Rp×p, j = 1, . . . , k, d ∈ R+, b ∈ R+, d ≥ b > 0,Ω > 0.

Compared to the parameter space of Hk in Section 2.2, the set ΘHr,kalso contains the

cointegration rank, r, among the unknown parameters. Model Hr,k exhibits further identi-fication issues than those illustrated in Section 2.2.

Example 2: Consider the model with k = 1 lags and rank 0 ≤ r ≤ p, given by

Hr,1 : ∆dXt = Π∆d−bLbXt + Γ1∆dLbXt + εt,

where the set of parameters is θ = vec(d, b,Π,Γ1).Examine now the following two sub-models of Hr,1. First, model Hp,0 is

Hp,0 : ∆dXt = Π∆d−bLbXt + εt,

with θ = vec(d, b, Π) is the set of parameters. Second, model H0,1 is

H0,1 : ∆d∗Xt = Γ∗1∆d∗Lb∗Xt + εt.

where θ∗ = vec(d∗, b∗,Γ∗1) is the set of parameters. Both Hp,0 and H0,1 can be written as[∆d−b(−Π) + ∆d(Ip + Π)

]Xt = εt, (2.19)

and [∆d∗(I − Γ∗1) + ∆d∗+b∗(Γ∗1)

]Xt = εt. (2.20)

Imposing the restrictions d = d∗ + b∗, b = b∗ and −Π = Ip − Γ∗1 on model Hp,0 in (2.19)leads to Hp,0 and H0,1 being equivalent. Indeed, the probability densities are

pHp,0(X1, . . . , XT ; θ|X0, X−1 . . .) = pH0,1(X1, . . . , XT ; θ∗|X0, X−1, . . .), (2.21)

when θ = vec(d∗ + b∗, b∗,Γ∗1 − Ip, 0) and θ∗ = vec(d∗, b∗, 0,Γ∗1).However, the sub-model H0,1 is not always a re-parametrization of Hp,0. Indeed,

applying the restrictions d∗ = d− b, b∗ = b and Γ∗1 = Ip + Π on model H0,1 in (2.20), itfollows that

pHp,0(X1, . . . , XT ; θ|X0, X−1, . . .) = pH0,1(X1, . . . , XT ; θ∗|X0, X−1, . . .), (2.22)

where θ = vec(d, b, Π, 0) and θ∗ = vec(d− b, b, 0, Ip + Π). However, the equality (2.22)

holds if and only if d− b ≥ b > 0, i.e. d ≥ 2b. This implies that H0,1 = Hp,0 ∩d ≥ 2b

.

Hence, H0,1 ⊆ Hp,0. The next proposition extends this example for any combination of kand r.


Proposition 2.4.1. Consider an unrestricted FCVARd,b model

Hr,k : ∆dXt = Π∆d−bLbXt +k∑j=1

Γj∆d−bLbXt + εt (2.23)

where 0 ≤ r ≤ p is the rank of the matrix Π and k is the number of lags. Consider thefollowing sub-models of Hr,k: Hp,k−1 with parameter set θ = vec(d, b, Π, Γ1, ..., Γk−1, Ω),and H0,k with parameter set θ∗ = vec(d∗, b∗,Γ∗1, ...,Γ

∗k,Ω

∗).

i) For any k > 0, model H0,k is equivalent to Hp,k−1 if the condition d ≥ 2b imposed

on model Hp,k−1 is satisfied. Hence H0,k=Hp,k−1 ∩d ≥ 2b

.

ii) The nesting structure of the FCVARd,b model is represented by the following scheme:

H0,0 ⊂ H0,1 ⊂ H0,2 ⊂ · · · ⊂ H0,k

∩ ∩ ∩ ∩H1,0 ⊂ H1,1 ⊂ H1,2 ⊂ · · · ⊂ H1,k

∩ ∩ ∩ ∩...

...... . . . ...

∩ ∩ ∩ ∩Hp,0 ⊂ Hp,1 ⊂ Hp,2 ⊂ · · · ⊂ Hp,k

with

H0,1 ⊆ Hp,0

H0,2 ⊆ Hp,1

...

...H0,k ⊆ Hp,k−1

It follows from Proposition 2.4.1i) that model H0,k can always be re-parametrizedas model Hp,k−1. On the other hand, model Hp,k−1 can be formulated as H0,k only whenthe condition d ≥ 2b on model Hp,k−1 holds. This leads to the peculiar nesting structuredisplayed in Proposition 2.4.1.ii). Notably the interpretation of the two models Hp,k−1 andH0,k is slightly different, although they are equivalent descriptions of the data. In modelHp,k−1, the process Xt has p non-common stochastic trends fractional order d− b. Instead,in model H0,k, then the process Xt has p non-common stochastic trends with fractionalorder d∗. A similar identification problem, due to indeterminacy between d, b and k, arisesalso in the univariate FAR(k) model studied in Johansen and Nielsen (2010)

∆dYt = π∆d−bLbYt +k∑i=1

γi∆dLibYt + εt,

where Yt is an univariate process and π is a scalar. Following the same procedure of theproof of Proposition 2.4.1, it follows that M0,k = M1,k−1 ∩

d ≥ 2b

, where M0,k defines

the FAR model with π = 0 and k lags, while M1,k−1 defines the FAR model with π 6= 0

and k − 1 lags. Therefore, the FAR(k) model has a similar circular nesting structure asin Proposition 2.23.ii). In Johansen and Nielsen (2010), the theoretical results are indeedobtained under the maintained assumption that the true number of lags k0 is known.

The following Corollary shows that indeterminacy between cointegration rank andlag-length is not limited to Hp,k−1 and H0,k, but it can be extended to any cointegrationrank 0 < s < p.


Corollary 2.4.2. For any k > 0, model Hs,k−1 with 0 < s < p and d ≥ 2b is equivalentto H0,k , if and only if the matrix Γ∗ = Ip −

∑kj=1 Γ∗j in model H0,k has rank equal to s.

In other words, if the matrix Γ∗ = Ip −∑k

j=1 Γ∗j in H0,k has reduced rank of order0 < s < p, the models Hs,k−1 and H0,k are equivalent under d ≥ 2b in Hs,k−1. This meansthat H0,k ⊆ Hs,k−1 for any 0 < s ≤ p, if rank(Γ) = s.

2.4.1 Model selection under unknown rank and lag-length

The peculiar nesting structure of the FCVARd,b obviously impacts on the joint selectionof the number of lags and the cointegration rank. Indeed, the likelihood ratio statisticfor cointegration rank r, denoted as LRr,k := −2 logLR(Hr,k|Hp,k), see Johansen andNielsen (2012, p.2698), is given by

− 2 logLR(Hr,k|Hp,k) = T (`(r,k)T (dr,k, br,k)− `(p,k)

T (dp,k, bp,k)), (2.24)

where `(r,k)T is the profile log-likelihood of the FCVARd,b model with rank r and k lags.

Analogously, dr,k and br,k are the arguments that maximize `(r,k)T . The asymptotic properties

of the LRr,k test, under the maintained assumption of correct specification of the lag-length,i.e. k = k0, are provided in Johansen and Nielsen (2012). Unfortunately, the values of theprofile log-likelihoods `(0,k)

T (d0,k, b0,k) and `(p,k−1)T (dp,k−1, bp,k−1) are equal when d ≥ 2b

in model Hp,k−1, and the number of the parameters of the model Hp,k−1 is the same asin H0,k. Hence, the equality of `(0,k)

T (d0,k, b0,k) and `(p,k−1)T (dp,k−1, bp,k−1) influences the

general-to-specific sequence of tests for the joint selection of the cointegration rank andthe lag-length. Indeed, assuming that the general-to-specific procedure for the optimal lagselection terminates in Hp,k−1, then it would be impossible to know whether the optimalmodel is Hp,k−1 or H0,k if the estimates dp,k−1 and bp,k−1 are such that dp,k−1 ≥ 2bp,k−1.

Therefore, a problem of joint selection of k and r arises in the FCVARd,b whenthe cointegration rank is unknown and potentially equal to 0 or p. Moreover, under H0,k

with k > 0, the parameter b is defined but it does not have the usual interpretation ascointegration gap, since it only appears in the generalized lag operator. A test for the nullhypothesis that r = 0 has been proposed by Łasak (2010) and extended in Łasak andVelasco (2015) to allow for multiple degrees of fractional cointegration. Łasak (2010)derives the asymptotic distribution of the maximum eigenvalue and trace tests for the nullhypothesis of absence of cointegration relation in the Granger (1986) system, which is notaffected by the problem of joint indeterminacy between cointegration rank and number oflags. Alternatively, a solution to the indeterminacy in the FCVARd,b framework is to rely ona preliminary estimate of the cointegration rank based on a frequency domain procedure,following for example the testing procedure of Nielsen and Shimotsu (2007). Instead,in the section below, we show that it is sufficient to impose a constraint the fractionalparameter d to solve in the problem of indeterminacy of cointegration rank and lag-length.


Model selection with an identification restriction

Unfortunately, a solution to the joint indeterminacy of cointegration rank and lag-length isnot available within the unrestricted FCVARd,b framework. However, a simple solutionto the identification problem caused by the indeterminacy of cointegration rank and lag-length can be achieved by a suitable restriction of the parameter space of d. Consider themodel with unknown rank and unknown lag structure. The model can be expressed bythe parameter set Θr,k = d0 ∈ R+, b ∈ (0, d0],Γj ∈ Rp×p, j = 1, . . . , k, α ∈ Rp×r, β ∈Rp×r,Ω > 0 where 0 ≤ r ≤ p and k ≥ 0 are unknown. The following lemma holds.

Lemma 2.4.3. Let ΘHr,k= d = d0, b ∈ [0, d0], α ∈ Rp×r, β ∈ Rp×r,Γj ∈ Rp×p, j =

1, . . . , k; Ω > 0 be the restricted parameter space of model ΘHr,kwith d = d0 ∈ R+ for

0 ≤ r ≤ p and k ≥ 0, then the nesting structure for the statistical models P = Pθ : θ ∈Θr,kr=0,...,p

k=0,1,... can be written as

H0,0 ⊂ H0,1 ⊂ · · · ⊂ H0,k

∩ ∩ ∩H1,0 ⊂ H1,1 ⊂ · · · ⊂ H1,k

......

...∩ ∩ ∩

Hp,0 ⊂ Hp,1 ⊂ · · · ⊂ Hp,k

When d = d0 is fixed, Lemma 2.4.3 proves that the FCVARd,b has a nesting structurethat does not exhibit the problem outlined in Proposition 2.4.1, since Hp,k−1 and H0,k aretwo distinct models. Therefore, under the assumption that d = d0, one could develop ageneral-to-specific sequence of LR tests which consists of iterating the tests LRp,k−1 :=

−2 logLR(Hp,k−1|Hp,k) over k with fixed p (full rank) until the null hypothesis is rejectedin k∗. Under the restriction d = d0, the distribution of these tests is a classical χ2(p2),since , under full rank, the test statistic reduces to the one in Nielsen (2006, eq. 3.1).Therefore, the results in Nielsen (2006) can be applied in the fractional context also andthe likelihood ratio test for Γk = 0 is asymptotically χ2(p2) even if a fractional unit root ispresent in the DGP. Therefore, the asymptotic power of each LRp,k−1 test of rejecting anunderspecified model should reach the maximum, although the true cointegration rank issmaller than p, and the asymptotic size of each test is a. Once the lag is selected, then thesequence of cointegration rank tests LRr,k∗ = −2 logLR(Hr,k∗|Hp,k∗) can be performedover r ∈ [0, p−1] with k fixed to k∗. Following the results of Johansen and Nielsen (2012),the asymptotic distribution of LRr,k∗ is a functional of the type II fractional Brownianmotion for r = 0, . . . , p− 1 if br,k∗ > 0.5 and it is a χ2(q2) if br,k∗ < 0.5 where q = p− r.The asymptotic distribution for the case br,k∗ > 0.5 is tabulated in MacKinnon and Nielsen(2014).

In practice, since d0 is unknown, we suggest to set a mild constraint on the parameterset of d to enforce identification at each step in the selection procedure. Analogously to the


discussion in Section 3.2, the estimates of dr,k and br,k, for any 0 ≤ r ≤ p and k ≥ 0, mustbe the solutions of the following constrained maximum likelihood problem

ψr,k = arg maxψ

`(r,k)T (ψr,k), (2.25)

s.t. δmin ≤ dr,k ≤ d

s.t. η ≤ br,k ≤ dr,k (2.26)

where the lower bound on the parameter dr,k, δmin, can be determined by a preliminaryestimate of the fractional order of the process as in (2.15) and d > η > 0. Therefore, underthe constraint dr,k ≥ δmin, we can test Hp,k against Hp,k−1, without the risk of having anequivalent parametrization in H0,k under the null hypothesis. It is important to stress thatδmin does not depend on r and k so that it can be determined a priori. If the probabilityP (d0 − b0/2 < δmin < d0) = 1, then the standard asymptotic results discussed above areunchanged since P (LRp,k−1 > Cp2(a)|d0 − b0/2 < δmin < d0) = P (LRp,k−1 > Cp2(a))

where Cp2(a) is the critical value of a χ2(p2) distribution at the a significance level.Otherwise if P (d0 − b0/2 < δmin < d0) < 1 we could experience loss of power and/orsize distortions in the sequence of LR tests that are most likely proportional to the extentin which this probability is smaller than 1.

H0,0 H0,1 H0,2 H0,3 H0,4

0 0 0 0 0H∗1,0 H1,1 H1,2 H1,3 H1,4

92.5 1.6 1.3 1.5 1.3H2,0 H2,1 H2,2 H2,3 H2,4

1.4 0 0.1 0.1 0.3

H0,0 H0,1 H0,2 H0,3 H0,4

0 0 0 0 0H1,0 H∗1,1 H1,2 H1,3 H1,4

0 93.8 1.2 1.9 1.6H2,0 H2,1 H2,2 H2,3 H2,4

0 1.4 0 0.1 0

Table 2.2: The two tables report the percentage of selected models when the DGP is aFCVARd,b model with k0 = 0 (left) and k0 = 1 (right) lags, r0 = 1 cointegration rank andfractional parameters d0 = b0 = 0.8. The results are based on M = 1000 Monte Carlosimulations and T = 2500 observations. The sequence of LR test is performed with atheoretical significance level of a = 1% for both the selection of the cointegration rankand the optimal number of lags. The estimations of d and b for each model are carried outunder the restriction d > δmin, with δmin = d− c · d, with c = 0.15.

The results reported in Table 2.2 confirm the reliability of the constrained sequentialselection procedure which selects the true model in almost 93% of cases when k0 = 0

and 94% when k0 = 1. Interestingly, a model smaller than the true one is never selectedsuggesting a power close to 100% for both the lag and rank tests. Concerning the empiricalsize of the tests for the optimal number of lags, it emerges that the probability of selectingeach of the models with k > k0 is close to the theoretical significance level that is set to 1%,thus signaling a good finite-sample approximation of the asymptotic χ2(p2) distribution.

2.5. CONCLUSION 53

Moreover, the probability associated to the selection of model Hp,k0 , when the true modelis H1,k0 , is 1.4% in both cases, that is again close to the theoretical significance level ofa = 1%. The results for different setups for the DGP are reported in the Supplementarymaterial and they all confirm this evidence. Notably, the contour plots in Figure 2.11 showthat the distribution of dr∗,k∗ and br∗,k∗ is unimodal and centered around the true values inall cases. Finally, Table 2.3 compares the selection frequencies based on the constrainedlog-likelihood approach (left panel) and those obtained when estimating unrestrictedly(right panel). Again, imposing the constraint d > δmin leads to the a selection frequency ofthe true model that is in line with the size of each individual sets (again we set a = 1%).Instead, the true model (with r0 = 0 and k0 = 1) is never selected when relying onthe unrestricted estimator, but, in line with Theorem 2.4.1, the model with full rank andk0 − 1 = 0 lags is selected most of the times.

d0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

b

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

d0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

b

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 2.11: Figure reports the contour plot of M = 1, 000 Monte Carlo estimates of theparameters d (x-axis) and b (y-axis) when a sample of T = 2, 500 observations is generated by aFCVARd,b model with k0 = 0 (left) and k0 = 1 (right), d0 = 1, b0 = 0.8 and the cointegrationvectors given by β0 = [1,−1]′ and α0 = [−0.5, 0.5]′. The estimates are associated to the optimalselected model, Hr∗,k∗ . In the left panel k0 = 0. In the right panel k0 = 1.

2.5 ConclusionThis paper discussed in detail some identification problems that affect the FCVARd,b

model of Johansen (2008). The main finding is that the fractional parameters of the systemcannot be uniquely determined when the lag structure is over-specified. In particular,the multiplicity of equivalent sub-models is provided in closed form given k and k0. Itis also shown that a necessary and sufficient condition for the identification is that theF(d) condition, i.e. |α′⊥Γβ⊥| 6= 0, is fulfilled. A simulation study highlights the practicalproblem of multiple humps in the expected profile log-likelihood function as a consequence


H0,0 H∗0,1 H0,2 H0,3 H0,4

0 91.9 1.1 1.8 0.8H1,0 H1,1 H1,2 H1,3 H1,4

0 0.3 0.2 0 0.2H2,0 H2,1 H2,2 H2,3 H2,4

0 0.2 0.1 0 0

H0,0 H∗0,1 H0,2 H0,3 H0,4

0 0 0.7 0.9 1.9H1,0 H1,1 H1,2 H1,3 H1,4

23 1.1 1.6 2.0 1.8H2,0 H2,1 H2,2 H2,3 H2,4

62.9 0.4 1.0 1.4 1.3

Table 2.3: The two tables report the percentage of selected models when the DGP is aFCVARd,b0 model with k0 = 1 lags, r0 = 0 cointegrating rank and fractional parametersd0 = 0.8 b0 = 0.4. The results are based on M = 1000 Monte Carlo simulationsandT = 2500 observations. The sequence of LR test is performed with a theoreticalsignificance level of a = 1% for both the selection of the cointegration rank and theoptimal number of lags. The estimations of d and b are carried out under the restrictiond > δmin with δmin = d− c · d, with c = 0.15 on the left panel and unrestrictedly on theright panel.

of the identification problem and the over-specification of the lag structure. Furthermore,the simulations reveal a problem of weak identification, characterized by the presence oflocal and global maxima of the profile likelihood function in finite samples. We also provethat it is sufficient to restrict d to d0 to solve the identification problem. However, since d0 isunknown, we impose a lower-bound restriction on d, where the lower bound is determinedon the basis of a preliminary semiparametric estimate of d0. This imposes the mildestrestriction on the parameter space of the FCVARd,b model. The Monte Carlo simulationsshow that the estimates of the model parameters are unimodal and centered around thetrue values in most cases. It is also proved that model H0,k is equivalent to model Hp,k−1

under certain conditions on d and b. Unfortunately, the F(d) condition does not provideany information for the identification in this case, but it is again sufficient to impose asuitable lower bound restriction on the parameter space of d to solve this identificationproblem and retrieve a nesting structure of FCVARd,b model that allows testing for theunknown lag-length and cointegration rank in the standard general-to-specific fashion.

AcknowledgementsThe authors are grateful to Niels Haldrup, Søren Johansen, Katarzyna Łasak, Bent Nielsenand Morten Ørregaard Nielsen for their suggestions that improved the quality of this work.The authors are also grateful to an anonymous referee for providing insightful comments.The authors would like to thank also James MacKinnon, Rocco Mosconi, Paolo Paruolo,the participants to the Third Long Memory Symposium (Aarhus 2013), the participantsto the CFE’2013 conference (London 2013), and the seminar participants at Queen’sUniversity and at Bologna University for helpful comments.

2.5. CONCLUSION 55

Proofs

2.5.1 Proof of Proposition 2.2.2

Let us define the model Hk0 under k0 ≥ 0 as

k0∑i=−1

Ψi,0∆d0+ib0Xt = εt, (2.27)

and the model Hk with k > k0 as

k∑i=−1

Ψi∆d+ibXt = εt. (2.28)

It is possible to show, that, for a given k0, m sub-models equivalent to the model in(2.27) can be obtained imposing suitable restrictions on the matrices Ψi, i = −1, ..., k ofthe model Hk. The equivalent sub-models, H(j)

k , j = 1, . . . ,m− 1, are found for

Ψ−1 = Ψ−1,0 corresponding to d− b = d0 − b0 (2.29)

Ψ(`+1)(j+1)−1 = Ψ`,0 corresponding to d+ [(`+ 1)(j + 1)− 1]b = d0 + `b0,

for ` = 0, . . . , k0 j = 0, 1, . . . ,m− 1

Ψs = 0 for s 6= (`+ 1)(j + 1)− 1,

and ` = 0, . . . , k0 j = 0, 1, . . . ,m− 1.

The matrices Ψ−1,0 = −α0β′0 and Ψ−1 = −αβ′ load the terms ∆d0−b0Xt and

∆d−bXt respectively. This implies that d0 − b0 = d− b in all equivalent sub-models. For agiven j > 0, a system of k0+2 equations (2.29) in d and b is derived from the restrictions onthe matrices Ψi. The solution of this system is found for b = b0/(j+1) and d = d0− j

j+1b0.

All sub-models H(j)k , j = 1, . . . , k are such that Ψ−1 = −αβ′ = −α0β

′0 = Ψ−1,0 and

Ψ0 = 0, This implies that αβ′+Γ = Ψ0 = 0. It follows that the sub-models for j = 1, ..., k

are such that |α′⊥Γβ⊥| = 0. Only for j = 0, the condition |α′⊥Γβ⊥| 6= 0 is satisfied.For a given k > k0, the number of restrictions to be imposed on Ψi that satisfies the

system in (2.29) is b k+1k0+1c. Hence, the number of equivalent sub-models is m = b k+1

k0+1c.

2.5.2 Proof of Lemma 2.3.1

Consider two models H(1)k and H

(2)k defined in ΘHk

, given by

k∑j=−1

∆d0+jb1Ψ(1)j Xt = εt and

k∑j=−1

∆d0+jb2Ψ(2)j Xt = εt


with d0 ≥ b1 > 0 and d0 ≥ b2 > 0. We want to prove that H1k and H2

k are equal if only ifb1 = b2 and Ψ

(1)j = Ψ

(2)j , j = 1, . . . , k and Ω1 = Ω2.

Given that Pθ is Gaussian for all θ ∈ ΘHkwe should check that the characteristic

polynomials

Πi(z) =k∑

j=−1

(1− z)d0+jbiΨ(i)j , i = 1, 2

are equal. They are equal if

(1−z)d0+jb1 = (1−z)d0+jb2 ⇐⇒ (1−z)b1 = (1−z)b2 ⇐⇒ b1 = b2, ∀j = −1, . . . , k

andΨ

(1)j = Ψ

(2)j , ∀j = −1, . . . , k

Finally, the variance of the innovations are Ω1 = Ω2 by construction since the error termsεt is the same in H

(1)k and H

(2)k . Therefore, the statistical model P = Pθ : θ ∈ ΘHk

isidentified.

2.5.3 Proof of Proposition 2.4.1

The unrestricted FCVARd,b model is given by

Hr,k : ∆dXt = Π∆d−bLbXt +k∑j=1

Γj∆d−bLbXt + εt, (2.30)

where 0 ≤ r ≤ p is the rank of the matrix Π and k is the number of lags. The model inequation (2.23) can be written as

k∑j=−1

Ψj∆d+ibXt = εt,

where Ψ−1 = −Π, Ψ0 = Ip + Π−∑k

i=1 Γi and Ψk = −(1)k+1Γk.Now consider the following sets of restrictions on model (2.23):

Hp,k−1 : Π is a p× p matrix and Γk = 0

H0,k : Π=0.

The model Hp,k−1 can be written in compact form as:

k−1∑i=−1

Ψi∆d+ibXt = εt (2.31)

where Ψ−1 = Π, Ψ0 = Ip + Π −∑k−1

i=1 Γi and Ψk−1 = (−1)kΓk−1. The matrices Π andΨi, i = −1, ..., k − 1 define the model under the restriction Hp,k−1.

2.5. CONCLUSION 57

Similarly, the model H0,k can be written as:

k∑i=0

Ψ∗i∆d∗+ib∗Xt = εt, (2.32)

with Ψ∗−1 = 0, Ψ∗0 = Ip + 0 −∑k

i=1 Γ∗i and Ψ∗k = (−1)k+1Γ∗k. The matrices Ψ∗i , i =

−1, ..., k, define the model under the restriction H0,k.Imposing the following set of restrictions on the matrices Ψi and Ψ∗i :

Ψ−1 = Ψ∗0

Ψ0 = Ψ∗1...

Ψk−1 = Ψ∗k,

(2.33)

it follows that the two models Hp,k−1 and H0,k are equivalent when the systemd− b = d∗

d = d∗ + b∗

...

d+ (k − 1)b = d∗ + kb∗

(2.34)

has an unique solution. Suppose that the system (2.34) is solved for d and b. The uniquesolution in this case is d = d∗ + b∗ and b = b∗, which satisfies the condition d ≥ b > 0.Now suppose that the system (2.34) is solved for d∗ and b∗. The unique solution in thiscase is d∗ = d − b and b∗ = b, which satisfies the condition d∗ ≥ b∗ > 0 if and only ifd ≥ 2b. Therefore, if d ≥ 2b it follows that H0,k ≡ Hp,k−1. Hence, H0,k ⊂ Hp,k−1.

2.5.4 Proof of Corollary 2.4.2

Using a procedure similar to that adopted in the proof of Proposition 2.4.1, it is straight-forward to show that, when d ≥ 2b, the model Hs,k−1 with 0 < s < p and model H0,k

are equivalent if Γ∗ = Ip −∑k

i=1 Γ∗i = Ψ∗0 is a matrix with rank s in model (2.32) and therestriction r = s is imposed on model (2.31), so that Π = αβ′ where α and β are p × smatrices.

2.5.5 Proof of Lemma 2.4.3

Consider the models Hp,k−1 and H0,k for k = 0, 1, . . . in equations (2.31)-(2.32) andimpose the constraint d = d0. Then,

Hp,k−1 :k−1∑i=−1

Ψi∆d0+ib = εt


H0,k :k∑i=0

Ψ∗i∆d0+ib∗Xt = εt.

It follows that Hp,k−1 ∩H0,k = ∅ because there is no solution to the system of equations(2.34) when d = d0 is fixed. Therefore, the nesting structure in 2.4.3 follows.

Supplementary Material

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2.84

−2.839

−2.838

−2.837

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

Expected Profile Likelihood and F(d) condition for different values of d

F(d) condition

Expected logL

d=d*−2b

*/3=0.2667

d=d*−b

*/2=0.4

d=d*=0.8

Zero Line

Figure 2.5.12: Figure reports simulated values of l(d) and F(d) for different values of d ∈[0.25, 0.9] (x-axis). The observations from the DGP are generated with k0 = 0 lags and modelHk with k = 2 lags is estimated. The parameters of the DGP are d0 = b0 = 0.8, β0 = [1,−1]′,α0 = [−0.5, 0.5]′.

2.5. CONCLUSION 59

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.01

−0.005

0

Expected Likelihood Function and F(d) condition for different values of d

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

0

2

Expected Likelihood

F(d) condition

Figure 2.5.13: Figure reports simulated values of l(d) and F(d) for different values of d ∈ [0.3, 1](x-axis). The observations from the DGP are generated with k0 = 1 lags and model Hk withk = 3 lags is estimated. The parameters of the DGP are d0 = b0 = 0.8, β0 = [1,−1]′, α0 =[−0.5, 0.5]′,and Γ1 =

[0.3 −0.20.4 −0.5

].

0.6 0.7 0.8 0.9 1 1.1

×104

-5.76

-5.74

-5.72

-5.7

-5.68

k=1

k=2

k=3

k=4

Figure 2.5.14: Figure reports the values of the expected profile likelihood, l(ψ), for differentvalues of b ∈ [0.51, 1.1] (x-axis) when d = d0 = 1. The sample size is T = 20, 000 andk0 = 0, while Hk with k = 1, 2, 3, 4 is estimated. The parameters of the DGP are d0 = 1and b0 = 0.8, β0 = [1,−1]′, α0 = [−0.5, 0.5]′.


0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

Figure 2.5.15: Figure reports the Monte Carlo distribution (M = 1000) of the estimated dwhen the DGP is a FCVARd,b with d0 = 0.8, b0 = d0 and k0 = 1 lags, while a FCVARd,b

model with k = 2 lags is fit on the data. The sample size is T = 1000.

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Figure 2.5.16: Figure reports the contour plot of M = 1, 000 Monte Carlo estimates of theparameters d (x-axis) and b (y-axis) when a sample of T = 10, 000 observations is generated by abivariate FCVARd,b model with k0 = 0, d0 = 1, b0 = 0.8 and the cointegration vectors given byβ0 = [1,−1]′ and α0 = [−0.5, 0.5]′. Model H2 is estimated on the data. The left panel is relativeto the estimates based on the constrained log-likelihood in (10) where c = 15% and qT = T 0.5.The right panel reports the contour plot for the unrestricted estimates.

2.5. CONCLUSION 61

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2



d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Figure 2.5.19: Figure reports the contour plot of M = 1, 000 Monte Carlo estimates of theparameters d (x-axis) and b (y-axis) when a sample of T = 2, 500 observations is generated by abivariate FCVARd,b model with k0 = 1, d0 = 1, b0 = 0.8 and the cointegration vectors given byβ0 = [1,−1]′, α0 = [−0.5, 0.5]′ and Γ1 =

[0.3 −0.2−0.4 0.5

]. Model H3 is estimated on the data. The

left panel is relative to the estimates based on the constrained log-likelihood in (10) where c = 15%and qT = T 0.5. The right reports the contour plot for the unrestricted estimates.

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

d

b

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


2.5. CONCLUSION 63

H0,0 H0,1 H0,2 H0,3 H0,4

0 0 0 0 0H1,0 H1,1 H1,2 H1,3 H1,4

93.6 1.5 1.3 1.4 1H2,0 H2,1 H2,2 H2,3 H2,4

1 0 0 0.1 0.1

H0,0 H0,1 H0,2 H0,3 H0,4

0 0 0 0 0H1,0 H1,1 H1,2 H1,3 H1,4

91.7 1 1.8 1.1 1.5H2,0 H2,1 H2,2 H2,3 H2,4

1.4 0.1 0.2 0.7 0.5

Table 2.4: The two tables report the percentage of selected models when the DGP is aFCVARd,b0 model with k0 = 0 lags, r0 = 1 cointegrating rank and fractional parametersd0 = 0.8 b0 = 0.6 (left) and d0 = 0.4 and b0 = 0.4 (right). The results are based onM = 1000 Monte Carlo simulations andT = 2500 observations. The sequence of LR testis performed with a theoretical significance level of a = 1% for both the selection of thecointegration rank and the optimal number of lags. The estimations of d and b for eachmodel are carried out under the restriction d > δmin with δmin = d− c · d, with c = 0.15.


2.6 ReferencesAvarucci, M., Velasco, C., 2009. A Wald test for the cointegration rank in nonstationary

fractional systems. Journal of Econometrics 151 (2), 178–189.

Bollerslev, T., Osterrieder, D., Sizova, N., Tauchen, G., 2013. Risk and return: Long-run relations, fractional cointegration, and return predictability. Journal of FinancialEconomics 108 (2), 409–424.

Breitung, J., Hassler, U., 2002. Inference on the cointegration rank in fractionally integratedprocesses. Journal of Econometrics 110 (2), 167–185.

Caporin, M., Ranaldo, A., Santucci de Magistris, P., 2013. On the predictability of stockprices: A case for high and low prices. Journal of Banking & Finance 37 (12), 5132–5146.

Chen, W., Hurvich, C., 2003. Semiparametric estimation of multivariate fractional cointe-gration. Journal of the American Statistical Association 98, 629–642.

Chen, W., Hurvich, C., 2006. Semiparametric estimation of Fractional CointegratingSubspaces. Annals of Statistics 34, 2939–2979.

Christensen, B. J., Nielsen, M. Ø., 2006. Asymptotic normality of narrow-band leastsquares in the stationary fractional cointegration model and volatility forecasting. Journalof Econometrics 133 (1), 343–371.

Franchi, M., 2010. A representation theory for polynomial cofractionality in vector autore-gressive models. Econometric Theory 26 (04), 1201–1217.

Granger, C. W. J., August 1986. Developments in the study of cointegrated economicvariables. Oxford Bulletin of Economics and Statistics 48 (3), 213–28.

Hualde, J., Velasco, C., 2008. Distribution-free tests of fractional cointegration. Economet-ric Theory 24, 216–255.

Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of EconomicDynamics and Control 12, 231–254.

Johansen, S., 1995a. Likelihood-based inference in cointegrated vector autoregressivemodels. Oxford University Press, Oxford.

Johansen, S., 1995b. A stastistical analysis of cointegration for I(2) variables. EconometricTheory 11 (01), 25–59.


2.6. REFERENCES 65

Johansen, S., October 2010. Some identification problems in the cointegrated vectorautoregressive model. Journal of Econometrics 158 (2), 262–273.

Johansen, S., Nielsen, M. Ø., 2010. Likelihood inference for a nonstationary fractionalautoregressive model. Journal of Econometrics 158 (1), 51–66.


Łasak, K., September 2010. Likelihood based testing for no fractional cointegration.Journal of Econometrics 158 (1), 67–77.

Łasak, K., Velasco, C., 2015. Fractional cointegration rank estimation. Journal of Business& Economic Statistics 33 (2), 241–254.

MacKinnon, J. G., Nielsen, M. Ø., 2014. Numerical distribution functions of fractionalunit root and cointegration tests. Journal of Applied Econometrics 29 (1), 161–171.

Nielsen, B., 2006. Order determination in general vector autoregressions. Vol. 52 of LectureNotes–Monograph Series. Institute of Mathematical Statistics, pp. 93–112.

Nielsen, M. Ø., Shimotsu, K., 2007. Determining the cointegration rank in nonstationaryfractional system by the exact local Whittle approach. Journal of Econometrics 141,574–596.

Robinson, P. M., Marinucci, D., 2003. Semiparametric frequency domain analysis offractional cointegration. In: Robinson, P. M. (Ed.), Time Series with Long Memory.Oxford University Press, pp. 334–373.

Robinson, P. M., Yajima, Y., 2002. Determination of cointegrating rank in fractionalsystems. Journal of Econometrics 106, 217–241.

Shimotsu, K., Phillips, P. C., 2005. Exact local Whittle estimation of fractional integration.Annals of Statistics 33(4), 1890–1933.

Tschernig, R., Weber, E., Weigand, R., 2013a. Fractionally integrated VAR models with afractional lag operator and deterministic trends: Finite sample identification and two-stepestimation. Tech. Rep. 471, University of Regensburg, Department of Economics.

Tschernig, R., Weber, E., Weigand, R., 2013b. Long-run identification in a fractionallyintegrated system. Journal of Business & Economic Statistics 31 (4), 438–450.

CH

AP

TE

R

3A NEW ESTIMATION METHOD OF A

FRACTIONAL COINTEGRATED MODEL

Federico CarliniAarhus University and CREATES

Katarzyna ŁasakVU University and CREATES

Abstract

We consider the Fractional Vector Error Correction model proposed in Avarucci(2007), which is characterized by a richer lag structure than the models proposed inGranger (1986) and Johansen (2008, 2009). In particular, we discuss the propertiesof the model of Avarucci (2007) in comparison to the model of Johansen (2008,2009). Both models generate the same class of processes, but the properties of thetwo models are still different. Opposed to the model of Johansen (2008), the modelof Avarucci has a convenient nesting structure, which allows for testing the numberof lags and the cointegration rank as in the standard I(1) cointegration frameworkof Johansen (1995) and hence might be attractive for econometric practice. We findthat some identification problems arise in the model of Avarucci (2007) but theyhave a different nature compared with those of Johansen and Nielsen (2012) that arediscussed in Carlini and Santucci de Magistris (2017). Due to a larger number ofparameters, the estimation turns out to be more complicated in the model proposed byAvarucci. We propose a 4-step estimation procedure for this model that is based onthe switching algorithm employed in Carlini and Mosconi (2014) together with theGLS procedure of Mosconi and Paruolo (2014). We check the performance of ourestimation procedure in finite samples by means of a Monte Carlo experiment.

67

68 CHAPTER 3. A NEW ESTIMATION METHOD OF A FRACTIONAL COINTEGRATED MODEL

3.1 Introduction

The econometrics literature on fractional co-integration has developed rapidly in recentyears. An empirically attractive modelling strategy is to use parametric inference, basedon an econometric model that fully describes the system under consideration. It allowsidentification of the long-run and short-run structure of the model, as well as of the commonstochastic trends and the impulse response functions summarizing the system dynamics.Three different Fractional Vector Error Correction Models (FVECM) have been proposedin the literature due to Granger (1986), Johansen (2008, 2009) and Avarucci (2007). Thesemodels turn out to be almost identical in the simplest case without short run dynamics, butmore generally they are characterized by different lag structure specifications.

The model proposed in Johansen (2008, 2009) has a convenient algebraic structure.The inference for this model has been developed in Johansen and Nielsen (2012). However,there exist potential identification problems in this model, as mentioned in Johansen andNielsen (2012) and further discussed in Carlini and Santucci de Magistris (2017).

In this paper we demonstrate that an identification problem also arises in the modelproposed by Avarucci (2007). The estimation problem is slightly more complicated in thismodel due to the multiplicative structure of the parameters involved. However, designingtesting procedures for the lag length and the cointegration rank is less complicated due tothe fact that the nesting structure follows the usual structure known for the I(1) CointegratedVector AutoRegressive (CVAR) model. We propose a 4 step algorithm to estimate themodel parameters. The algorithm is based on the approach of Carlini and Mosconi (2014)that maximizes the profile likelihood function using a switching algorithm and implementsthe GLS procedure proposed in Mosconi and Paruolo (2014).

In a Monte Carlo experiment we investigate the small sample properties of theestimates of all the parameters of this model obtained from the proposed estimationprocedure.

The remainder of the paper is organized as follows. Section 2 presents the FCVARmodel proposed in Johansen (2008,2009) and the FECM model suggested in Avarucci(2007). In Section 3 we discuss the identification issues and the nesting structure of theFECM model. Section 4 introduces the profile likelihood and a 4 step switching algorithmto estimate the parameters of the FECM model. Section 5 describes the small sampleproperties of our estimation procedure by means of a Monte Carlo experiment. Section 6concludes.

3.2. MODEL COMPARISON 69

3.2 Model comparison

3.2.1 Johansen’s FCVAR model

The model of Johansen (2008, 2009), which we denote in this paper as FCVAR, is givenby the following dynamics

∆dXt = αβ′∆d−bLbXt +k∑j=1

Γj∆dLjbXt + εt, εt ∼ iid(0,Ω), (3.1)

where the vector of variables Xt is p−dimensional, the loadings α and the cointegratingrelations β are p × r matrices with 0 ≤ r ≤ p, Γj are p × p matrices of the shortrun dynamics and the fractional difference operator is given by the binomial expansion∆d := (1−L)d =

∑∞j=0(−1)j

(dj

)Lj and the fractional lag operator is defined Lb = 1−∆b.

As shown in Johansen (2008, 2009), this model could be derived from the standardVAR model ∆Yt = αβ′LYt +

∑kj=1 Γj∆L

jYt + εt analysed in Johansen(1995), where thelag operator L is such that LXt = Xt−1, and the difference operator ∆ = 1 − L, in thefollowing way. First replace the difference operator ∆ and the lag operator L = 1−∆ byfractional difference operator ∆b and the fractional lag operator Lb = 1−∆b, respectively,to obtain ∆bYt = αβ′(1−∆b)Yt +

∑kj=1 Γj∆

bLjbYt + εt. Next we define Yt = ∆d−bXt toget the model (3.1).

The Granger representation of the model (3.1) is given in Johansen (2008, 2009):

Xt = C∆−d+ εt + ∆−(d−b)+ Y +

t + µt, (3.2)

where µt is a deterministic component generated by initial values, C = β⊥(α′⊥Γβ⊥)−1α′⊥

and Y +t =

t−1∑n=0

τnεt−n, so Y +t is fractional of order zero. Thus the solution of model

(3.1) implies that Xt is an integrated of order d (I(d)) process, while ∆bXt and β′Xt areI(d− b).

In general the cointegration rank r and number of lagged differences k is not knownand needs to be determined. However, the nesting structure, as described in Carlini andSantucci di Magistris (2017) turns out to be of the following form:

H0,0 ⊂ H0,1 ⊂ H0,2 ⊂ . . . ⊂ H0,k

∩ ∩ ∩ ∩H1,0 ⊂ H1,1 ⊂ H1,2 ⊂ . . . ⊂ H1,k

∩ ∩ ∩ ∩...

...... . . . ...

∩ ∩ ∩ ∩Hp,0 ⊂ Hp,1 ⊂ Hp,2 ⊂ . . . ⊂ Hp,k


with

H0,1 ⊂ Hp,0

H0,2 ⊂ Hp,1

...

...H0,k ⊂ Hp,k−1

,

where Hp,k denotes the hypothesis that the model (3.1) has cointegration rank r = p andk lagged differences. Therefore, the joint identification of r and k, if both are unknown,becomes tricky.

It is also shown that there exists a number of equivalent FCVAR models, whichcauses problems with identification of fractional parameters d, b and lag length whenthe cointegration rank r is known, see Carlini and Santucci di Magistris (2017). Theydemonstrate that for any k ≥ k0, where k0 denotes the number of lagged differences in thetrue DGP, the following holds:

• Given k0 and k, with k ≥ k0, the number of equivalent sub-models that can beobtained is m = [ k + 1

k0 + 1], where [x] denotes the greatest integer less than or equal to

x.

• For any k ≥ k0, all the equivalent sub-models are found for parameter valuesdj = d0 − j

j + 1b0 and bj = b0/(j + 1) for j = 0, 1, ...,m− 1.

• α, β are the same in these models.

Further, they give the number of equivalent sub-models in the following table

k0 ↓ k → 0 1 2 3 4 5 6 7 8 9 10 11 120 1 2 3 4 5 6 7 8 9 10 11 12 131 - 1 1 2 2 3 3 4 4 5 5 6 62 - - 1 1 1 2 2 2 3 3 3 4 43 - - - 1 1 1 1 2 2 2 2 3 34 - - - - 1 1 1 1 1 2 2 2 25 - - - - - 1 1 1 1 1 1 2 2

Thus there are identification problems regarding the parameters d, b,Γj, j = 1, . . . , k.

3.2. MODEL COMPARISON 71

3.2.2 Avarucci’s FECM model

The model of Avarucci (2007), which we denote in this paper as FECM, is given by thefollowing dynamics

∆dXt = αβ′∆d−bLbXt+k∑j=1

BjLj∆d−bLbXt+

k∑j=1

AjLj∆dXt+εt εt ∼ iid(0,Ω)

(3.3)where Bj = −Aj(αβ′). Avarucci (2007) imposes the restriction Xt = 0 for t < 1. It issimilar to the model of Lobato and Velasco (2006) for testing for fractional unit root in theunivariate framework.

The model has been derived using a standard assumption in a parametric framework(see Robinson and Hualde (2003), Dueker and Startz (1998)) that the dynamics of thestationary process can be given by an autoregressive representation. Consider a fractionallycointegrated system in a triangular form, i.e.

ξ′∆dXt = u1t,

β′∆d−bXt = u2t,with d− b ≥ 0. (3.4)

The triangular representation (3.4) can be shown to be equivalent to the FVECMwithout short run dynamics, i.e.

∆dXt = αβ′∆d−bLbXt + ξt, (3.5)

where α = −ξ⊥(β′ξ⊥)−1 and β′α = −Ir, and r is the cointegration rank. The processut has the V AR(k) representation A(L)ut = vt. Then ξt is also a V AR(k) process, i.e.:ξt =

∑kj=1Ajξt−j + εt. Consider the model (3.5), then

ξt = ∆dXt − αβ′∆d−bLbXt, (3.6)

can be written as ∆dXt = αβ′∆d−bLbXt +∑k

j=1 Ajξt−j + εt and further using (3.6)∆dXt = αβ′∆d−bLbXt+

∑kj=1Aj[∆

dXt−j−αβ′∆d−bLbXt−j]+εt to give finally ∆dXt =

αβ′∆d−bLbXt +∑k

j=1Aj∆dXt−j +

∑kj=1Bj∆

d−bLbXt−j] + εt, where Bj = −Aj(αβ′).

The model (3.3) can also be written in another form. The representation proposedbelow is coherent with the representation in Johansen (2008) . The model in (3.3) and canbe reformulated as:

∆d−b

Ip − k∑j=1

AjLj

(∆bIp − αβ′Lb)Xt = εt. (3.7)

This representation emphasizes the nature of the process. In fact, the FECM model is aseries of two systems: a VAR process identified by the lag polynomial (I −

∑kj=1AjL

j)


and a FCVAR process identified by the lag polynomial ∆d−b(∆bIp−αβ′Lb). The followingscheme represents the FECM process:

εt → A(L)−1 → Vt → Πd,b(Lb)−1 → Xt

The input of the system is the Gaussian error term εt transformed in a VAR process Vtthrough the transfer function A(L)−1. Finally the VAR process Vt is transformed intoa Fractionally Cointegrated process by means of the transfer function Πd,b(Lb)

−1 :=

∆b−d(∆bIp − αβ′Lb)−1.In linear system theory, the dynamics of two systems connected in series can be

analysed by checking the zeros and poles of their transfer functions contemporaneously.Hence, the dynamics of the FECM can be found by checking the characteristic roots ofthe polynomial A(z) and Πd,b(y), where y = 1− (1− z)b. This means that we generatefractional cointegration if det(Πd,b(y)) = 0 has some of the characteristic fractional rootsequal to one and β′α is a full rank matrix.

The FECM model is characterized by a different (and more complicated) lag structurethan the model proposed in Granger (1986)

∆dXt = αβ′∆d−bLbXt−1 +k∑j=1

ΓjLj∆dXt + εt, (3.8)

and the model (3.1) discussed in the previous section. In fact, model (3.3) contains boththe usual lags based on a standard lag operator present in Granger’s model (3.8) and lagsusing the fractional lag operator. The latter are different than those present in the model(3.1) of Johansen. However, in the very particular case of d = b = 1 with k = 0 all threemodels reduce to the standard ECM. Besides, when k = 0, i.e. the short run dynamicscomponents are not present, then models (3.3) and (3.1) are equal apart for the initialvalues. The solution of Johansen’s model depends on the initial values, for t < 0, whilethe model of Avarucci implicitly has the restriction for which the process starts in t = 0.

The moving average representation (MA) of the model (3.3) is given in Avarucci(2007). Following his Theorem 2.2, Xt has the representation

Xt = C∆−d+ Vt + C∗∆−d+b+ Vt + ∆−d+2b

+

t−1∑j=1

ΦjVt−j, (3.9)

where∑∞

j=0 ||Φj||2 <∞, and C = β⊥(α′⊥β⊥)−1α′⊥, and C∗ = −[βα′ + Cβα′ + βα′C +

Cβα′C], where Φj, j = 1, . . . , t−1 are p×pmatrices, Vt = A(L)−1εt and if c is a genericp× r matrix then c := c(c′c)−1 and c⊥ is a p× (p− r) matrix such that c′⊥c = c′c⊥ = 0.Thus, Xt and β′Xt are Type II I(d) and I(d− b) processes respectively.

The proof of Theorem 2.2 is largely based on Theorem 8 in Johansen (2008) and theMA representation (3.9) is based on the solution (3.2) given in Johansen (2008, 2009).

3.3. STATISTICAL IDENTIFICATION OF THE FCVAR MODEL 73

Therefore both models generate the same class of processes. However, in the modelproposed by Avarucci (2007), cointegration always occurs if b > 0 unlike in the model ofJohansen (2008, 2009) where the system can not be cointegrated for b > 0 if αβ′ is a fullrank matrix.

3.3 Statistical identification of the FCVAR modelIn this section we illustrate the proof of the identification of the FECM model when theorder k of the VAR is known. The proof of identification follows the same steps as inJohansen and Nielsen (2012).

Proof

1. A parametric model is identified when fλ0(xt|It−1) = fλ(xt|It−1) implies λ0 =

λ, where f(xt|It−1) is the conditional density function. In the FECM model theparameter vector is given by λ = vec(d, b, α, β, A1, . . . , Ak,Ω).

2. In the statistical model εt is assumed to be Gaussian. We are interested in thefirst and the second moment of f . Hence we have to show that the conditionsEλ0(xt|It−1) = Eλ(xt|It−1) and varλ0(xt|It−1) = varλ(xt|It−1) imply λ0 = λ. Theequality for conditional variances requires that Ω0 = Ω.

3. We use the decomposition Ip = ββ′ + β⊥β′⊥ where β′ = (β′β)−1β′ and β′⊥ =

(β′⊥β⊥)−1β′⊥ to identify the parameter α and β defining ˜α = αβ′β0 and ˜β =

β(β′0β)−1 so that αβ′ = ˜α ˜β′ and we assume that this identification strategy isassumed in the following.

4. The equality for conditional means require that

Πλ0(z) = (Ip−A01z−· · ·−A0

kzk)(1− z)d0−b0((1− z)b0Ip−α0β

′0(1− (1− z)b0) =

= (Ip − A1z − · · · − Akzk)(1− z)d−b((1− z)bIp − αβ′(1− (1− z)b) = Πλ(z)

If k > 0 and r > 0 then it is implied that A0j = Aj, j = 1, . . . , k, d0 = d, b0 =

b, α0 = α and β0 = β when b0 = b 6= 1.

5. If b0 = b = 1 then (Ip − A01z − · · · − A0

kzk)((1 − z)Ip − α0β

′0(1 − (1 − z)) =

(Ip−A1z−· · ·−Akzk)((1−z)Ip− αβ′(1−(1−z)) is not generally solved byA0j =

Aj, j = 1, . . . , k, α0 = α and β0 = β. In fact, by the theory of matrix polynomialsin Dennis, Traub, and Weber (1976), this problem can be easily explained. Supposethat you have a given matrix polynomial

B(z) = Ip −B1z − . . .−Bkzk


and Bj, j = 1, . . . , k are p× p fixed square matrices and we want to decompose it as

(Ip −D1z − . . . Dk−1zk−1)(Ip − C1z)

where C1 is called the right solvent of the matrix polynomial B(z). We define thelatent values as the values z1, . . . , zpk such that |B(zk)| = 0 and the right latentvectors as the vectors v1, . . . , vpk such that B(zj)vj = 0 where zj is a latent value.If B(z) has p linearly independent right latent vectors v1,, . . . , vp corresponding tolatent roots z1, . . . , zp, then C1 := QΛQ−1 is a right solvent, where Q = [v1, . . . , vp]

and Λ = diag(z1, . . . , zp), see Dennis et al. (1976). Therefore, in general the rightsolvent is not unique (because we can find many z1, . . . , zp that satisfy the require-ment of the theorem) and there exist different matrices D(l)

j , j = 1, . . . , k − 1, C(l)1

for l = 1, . . . , p that satisfy the decomposition of B(z). For this reason, whenb0 = b = 1 the matrices Aj, j = 1, . . . , k (the Dj, j = 1, . . . , k − 1 matrices in theexample) and α and β (the C1 matrix in the example) are not identified.

6. Suppose now r = 0, then the parameters (d,A1, . . . , Ak) are just identified and itfollows the same argument as in Johansen and Nielsen (2012).

This proof shows that an identification problem occurs when the DGP value of the coin-tegration gap parameter b0 is equal to 1. This identification issue can naturally affect theasymptotic distributions of the parameters of the model. We will discuss in Section 4.2 theconsequences of the identification issue on the estimation method proposed.

3.3.1 Identification issues when the lag length is unknown

Recall from Section 2 that in the FCVAR model there exists a number of equivalent modelswith overspecified lag length. In order to verify whether this also happens in the FECMmodel let us consider the model with just 2 lags:

H2 : ∆d−b

Ip − 2∑j=1

AjLj

(∆bIp − αβ′Lb)Xt = εt. (3.10)

where H2 indicates the model with k = 2 in (3.7)Let us demonstrate under which restrictions the two sub-models of H2

H(0)2 : ∆d0−b0

(Ip − (Ip + A1)L+ A1L

2)(

∆b0Ip − αβ′Lb0)Xt = εt. (3.11)

H(1)2 : ∆d1−b1

(Ip − A1L

) (∆b1Ip − αβ′Lb1

)Xt = εt. (3.12)

can be reparameterized as in Carlini and Santucci de Magistris (2017).First note that the sub-model H(0)

2 in equation (3.11) can be written as:

∆d0−b0(Ip − A1L

)(∆b0Ip − αβ′Lb0

) (Ip − IpL

)Xt = εt.

3.3. STATISTICAL IDENTIFICATION OF THE FCVAR MODEL 75

or equivalently as

∆d0−b0(Ip − A1L


)∆Xt = εt

Therefore,H

(0)2 : ∆d0−b0+1

(Ip − A1L


)Xt = εt (3.13)

Now, let us compare the sub-models (3.12) and (3.13). It is clear that the equations(3.13) and (3.12) reparametrize when αβ′ = αβ′, A1 = A1, b1 = b0 and d0 − b0 + 1 =

d1 − b1. Hence, H(0)2 = H

(1)2 if and only if d0 + 1 = d1.

Furthermore, note that the model H(0)2 is a sub-model of the model H2, when we

impose the restriction A2 + A1 − Ip = 0. Instead, the sub-model H(1)2 is the sub-model of

the model H2 when we impose the restriction A2 = 0.Therefore, the parameter b is always identified. In order to rule out the identification

problem for the parameter d, we only need to assume that the characteristic polynomial

Π(z) =

Ip − k∑j=1

Ajzj

has roots outside the unit circle, which is already assumed in Avarucci (2007). Thereforethe identification problem for the parameter d is not present.

3.3.2 Identification issue when αβ′ = 0

However, the problem of identification can arise when αβ′ = 0. In this situation, (3.7) isgiven by

∆d

Ip − k∑j=1

AjLj

Xt = εt

and the parameter b is not identified. This particular feature of the model has been used inŁasak (2010) to propose a sup-test for no cointegration and is common for all fractionallycointegrated Vector Error Correction models. This identification issue is also relevant inthe FCVAR model when the number of lags is k = 0.

3.3.3 The nesting structure of the FECM model

The nesting structure of the Avarucci model also differs from the nesting structure of theJohansen (2008, 2009) model and it follows the VAR structure. If we define the model

Hr,k : ∆d−b(Ip −k∑i=1

AiLi)(∆bIp − αβ′Lb)Xt = εt, r = 0, . . . , p


then, the nesting structure of the Avarucci’s model is given by

H0,0 ⊂ H0,1 ⊂ H0,2 ⊂ . . . ⊂ H0,k

∩ ∩ ∩ ∩H1,0 ⊂ H1,1 ⊂ H1,2 ⊂ . . . ⊂ H1,k

∩ ∩ ∩ ∩...

...... . . . ...

∩ ∩ ∩ ∩Hp,0 ⊂ Hp,1 ⊂ Hp,2 ⊂ . . . ⊂ Hp,k

,

which makes testing the cointegration rank r and the number of lagged differences k to bestraightforward. For example, the inclusion H2,1 ⊂ H2,2 can be tested by A2 = 0 and theinclusion H1,1 ⊂ H2,1 can be tested on a rank restriction on the matrix αβ′. Moreover, itis simple to prove that the model H0,1 is not nested in H2,0 because the term αβ′Lb is zeroin H0,1.

The nesting structure of the Johansen model is more complicated as discussed inSection 2, see Carlini and Santucci de Magistris (2017) for the details. Therefore, theinference for these two models should be different, despite the fact that they both generatethe same class of processes.

3.4 Estimation of the FECM model

In this section we propose a profile likelihood approach to estimate the parameters ofthe FECM model. We propose to concentrate the likelihood function on the parametersψ = (d, b)′ as in Johansen and Nielsen (2012). The profile maximum likelihood estimatoris defined to be

arg maxψ∈K

`T (ψ) (3.14)

where K is a compact set defined to be K =η ≤ b ≤ d ≤ d

for some values η > 0 and

d > 0 and

`T (ψ) = − 1

2Tlog det(Ω) +

1

pTlog(2π)

Hence, the parameters α, β and Aj, j = 1, . . . , k are considered as nuisance parameters.To maximize the likelihood function we use the same idea developed in Johansen and

Nielsen (2012). For any combination of ψ chosen, we maximize the likelihood functionwith respect to the nuisance parameters α, β and Aj, j = 1, . . . , k with a numerical routinebased on a switching algorithm. Finally, we optimize the likelihood function with respectto ψ to calculate the ML estimator ψ.

3.4. ESTIMATION OF THE FECM MODEL 77

3.4.1 The switching algorithm

For any given values of ψ = (d, b)′, we estimate

∆dXt − αβ′∆d−bLbXt + A(Ik ⊗ α)(Ik ⊗ β′)∆d−bLbZt − A∆dZt = εt.

Now we can use the switching algorithm to maximize the likelihood with respect to ψ.Note that

Ap×pk

=[A1 A2 · · · Ak

]Ztpk×1

=

Xt−1

Xt−2

...Xt−k

We run the following switching algorithm:Step 1. For given values of α, β,Ω, we estimate A in the following equation

∆dXt − αβ′∆d−bLbXt = A[∆dZt − (Ik ⊗ α)(Ik ⊗ β′)∆d−bLbZt] + εt

The parameters A are estimated with ordinary least squares.

Step 2. Given values of A, β,Ω, we estimate α in the following equation:

∆dXt − A∆dZt = [Ip : −A](Ik+1 ⊗ α)(Ik+1 ⊗ β′)∆d−bLb

[Xt

Zt

]+ εt

Using the Mosconi and Paruolo algorithm explained in the Appendix, we estimate α with

generalized least squares by imposing Hα = [Ip : −A], Wt = (Ik+1 ⊗ β′)∆d−bLb

[Xt

Zt

]and K is the matrix such that (Ik+1 ⊗ α) = Kvec(α).

Step 3. For given values of A,α,Ω, we estimate β in the following equation

∆dXt − A∆dZt = [Ip : −A](Ik+1 ⊗ α)(Ik+1 ⊗ β′)∆d−bLb

[Xt

Zt

]+ εt

Again, the Mosconi and Paruolo algorithm is needed to estimate β with generalized

least squares after imposing Hβ = [Ip : −A](Ik+1 ⊗ α), Wt = ∆d−bLb

[Xt

Zt

]where K

is the matrix such that (Ik+1 ⊗ β′) = Kvec(β).

Step 4. For given values of A,α, β we estimate Ω as

Ω =1

T

T∑t=1

εtε′t

and then we evaluate the likelihood.

We iterate Step 1 - Step 2 - Step 3 and Step 4 until convergence.


3.4.2 Estimation and Identification issues

As explained in Section 3, the identification issue in the FECM model arises when the DGPvalue b0 = 1. In particular, the identification issue is relevant for the matrices α, β andAj, j = 1, . . . , k, because they are shown not to be unique when b0 = 1. In particular, wemaximize the profile likelihood function with respect to ψ = (d, b)′. Hence, the maximumlikelihood estimator ψ is always identified, but the estimated nuisance parameters α, β andAj, j = 1, . . . , k are not identified.

By simple algebra, we note that if b0 = 1 then the FECM model with k lags is areparameterization of the FCVAR model where b = 1, d ≥ 1 and k lags. In fact, thecharacteristic polynomial of the FECM model when b = 1 is given by the followingexpression

∆d−1(Ip −k∑j=1

Ajzj)((1− z)Ip − αβ′z) = (1− z)d−1

k+1∑j=0

Ψjzj

while the characteristic polynomial of the FCVAR model when b = 1 is given by thefollowing expression:

(1− z)dIp = αβ′(1− z)d−1z −k∑j=1

Γj(1− z)dzj = (1− z)d−1

k+1∑j=0

Ψjzj.

Furthermore, the FCVAR model with b = 1 is an identified model. Hence, wecould test in the FECM framework the hypothesis H0 : b = 1. If the hypothesis is notrejected, then we study the FCVAR in which d ≥ 1. The asymptotic distribution ofthis hypothesis has to be derived because this is a case of hypothesis testing in whichthe nuisance parameters are not identified. A reference that describe in more detail thisproblem is Hansen (1994).

Finally, we have generated a FECM model with b = 1, b = 0.99, b = 1.01 andT = 100, 000 observations. When we estimate the model setting the initial values of α, βand A to their true values, the switching algorithm converge very slowly and the numberof iterations are approximately of an order of 109. These non-identified and almost-non-identified FECM models make the proposed estimation procedure very difficult to manage.Ideas to improve our switching algorithm is a research problem that is currently beingaddressed.

3.4.3 Initial values

The problem of the switching algorithm is that it is important to have a good initial guessfor parameters α, β and the matrix A. Hence we use as initial guess some of the parametersestimated in the equation

∆dXt − αβ′∆d−bLbXt + Ξ∆d−bLbZt − A∆dZt = εt

3.5. SIMULATION EXPERIMENT 79

with a CSS (Conditional Sum of Squares) profile likelihood method. We get d, b, α, β, A, Ξmaximizing the profile likelihood function (or profile CSS) with a reduced rank regression.In fact, this model can be estimated with a profile likelihood depending on the parametersd, b. Hence, we maximize

`(ψ) = − log det

1

T

T∑i=1

εt(ψ)ε′t(ψ)

− 1

pTlog(2π)

over a compact parameter set K =η ≤ b ≤ d ≤ d

for some values d > η > 0.

The estimates A, α and β are used as initial guess to start up the switching algorithm.Instead the estimates d and b are used as initial values in the optimization routine.

3.5 Simulation experimentThe Monte Carlo exercise is conducted with a simulation of the FECM model using theJensen (2014) algorithm for the generation of the FCVAR model two times. The FECMmodel can be generated as a first step by inverting a FCVARd,b model with 0 lags, obtainingYt given by

Yt = (Ip − A1,0L− . . .− Ak,0Lk)−1εt

The second step is to generate the process

Xt = (∆d0Ip − α0β′0∆d0−b0Lb0)

−1Yt

again with the Jensen (2014) algorithm. We developed a new routine that transforms theparameters of the VAR model in the parameters of the FCVAR model when d = b = 1.With these transformed parameters, we generate the process Xt.

We generate N = 1500 Monte Carlo replications of the following data generatingprocess

(Ip − A1,0L− A2,0L2)(∆d0Ip − α0β

′0∆d0−b0Lb0)Xt = εt

where εt ∼ i.i.d.N(0,Ω0) and t = 1, . . . , T . The numerical parameters of the DGP are

A0 = [A1,0 : A2,0] =

[−0.2 0.2 0.2 0

0 0.3 −0.3 −0.3

]

α0 =

[−0.3

0.3

]β0 =

[1

−0.4

]Ω0 =

[1 0

0 1

]First, we have chosen values of d0 and b0 such that the inverse roots y of the determinant|(1 − y)Ip − α0β

′0y| = 0 are outside the fractional circle Cb0 and fractional unit roots

described in Johansen (2008). Secondly, we have chosen the parameters in A0 such that theinverse roots z of the lag polynomial calculated as |Ip − A1,0z − A2,0z

2| = 0 are outside


the unit circle. Furthermore, we have chosen parameters for which no identification issuesoccur.

For each run of the Monte Carlo simulation we fit the FECM model with two lags,given by

(Ip − A1L− A2L2)(∆dIp − αβ′∆d−bLb)Xt = εt (3.15)

using the switching algorithm discussed in Section 4.1.When we maximize the likelihood function, the first experiment is to use as initial

values for the parameters d and b the true data generating process values d0 and b0, whilethe initial values in the switching algorithm for α, β and Ω are imposed to be α0, β0 and Ω0.The maximization routine climbs the likelihood function within the values d ∈ [0.01, 2]

and b ∈ [0.01, 2] in order to avoid negative - or close to zero - d and b estimates. Further,we impose in the maximization routine the restriction d ≥ b because in the FECM thisinequality must be satisfied.

We introduce two parameters to control the convergence of the switching algorithm.The first parameter is the maximum number of iterations of the switching algorithm N iter.The program stops when a number of N iter chosen is reached. The second parameter is atolerance number Tol. The program stops when the absolute value of the likelihood at stepk + 1 minus the likelihood value at step k is less than Tol. In the Monte Carlo we have setthese two parameters to be N iter = 20, 000 and Tol = 10−8.

To simplify the exposition, we introduce new notation for the elements of the matricesin model. The elements of the matrices are

α =

[α1

α2

]β =

[1

β1

]Ω =

[ω11 ω12

ω21 ω22

]A =

[a

(1)11 a

(1)12 a

(2)11 a

(2)12

a(1)21 a

(1)22 a

(2)21 a

(2)22

]

where ω12 = ω21. When we estimate the FECM model the vectors β∗ = [β∗1 : β∗2 ]′ andα∗ = [α∗1 : α∗2]′ are normalized by calculating α = β∗1 α

∗ and β = 1β∗1· β∗.

Furthermore, we initialize the switching algorithm with values αIn, βIn,ΩIn andthe maximization routine with dIn and bIn. The first experiments are conducted choosingαIn = α0, βIn = β0, ΩIn = Ω0, d

In = d0 and bIn = b0. We have chosen different valuesof (d0, b0) such that the inverse roots are outside the circle Cb. We have chosen DGP valuesof b0 ≥ 0.15 because we are not close to the boundary b = 0 and the maximization routineconverged without any problems.

We present the Monte Carlo results when we simulate the process with d0 = 0.8,b0 = 0.6 and T = 100, 000. The results of the sample statistics of the distributions ofthe estimated parameters are reported in Table 3.1. Figure 3.1 displays the plots of thedensities of the estimates of the parameters in Eq. 3.15. These densities are calculated witha non-parametric method and they are smoothed by a Gaussian Kernel.

The biases of the Monte Carlo estimates are smaller than an order of magnitudeof 10−3 and the standard deviations are smaller than 0.1. We analyzed if the Monte

3.6. CONCLUSIONS 81

Carlo sample distributions were normal with a Jarque-Bera test and we do not reject thehypothesis for some of the elements in the matrix A and the fractional parameters d and b.

We tried to check if the switching algorithm is robust with respect to different initialvalues and the results are still the same when the sample size is T = 100, 000. If the samplesize is T = 10, 000 then the initial values αIn, βIn and ΩIn are crucial if we want to findthe global maximum of the likelihood function.

The algorithm behaves differently depending on the choice of d0 and b0. In fact, ifwe simulate a model where d0 = b0 = 0.9, we need over 20,000 iterations of the switchingalgorithm. Further research has to be done to improve the convergence properties of theswitching algorithm when the data generating process is generated with d0 ≥ b0 > 0.9.

In Figure 3.2 and Table 3.2 the results are shown when the DGP takes valuesd0 = b0 = 0.6 and α0, β0, A0 as before and T = 100, 000. In this experiment we fix d = d0

and b = b0 and we leave the switching algorithm to find the maximum of the likelihoodwhen the initial values are at their true values. We notice that in this set up we could notreject the null hypothesis of normality for all the parameters but β.

In Figure 3.3 and Table 3.3 the results are shown Monte Carlo when the DGP hasd0 = b0 = 0.6 and α0, β0, A0 as before and T = 100, 000. In this experiment we fit themodel

(I − A1L− A2L2)(∆dIp − Π∆d−bLb)Xt = εt

where

Π =

[Π11 Π12

Π21 Π22

]and we run the switching algorithm fixing d = d0 and b = b0. We do not reject the nullhypothesis of normality for all the estimated parameters with a Jarque-Bera test.

3.6 Conclusions

In this paper we discuss two fractionally cointegrated models: a FCVAR model proposedin Johansen (2008, 2009) and the FECM model proposed in Avarucci (2007) and Avarucciand Velasco (2010). They both generate the same class of processes, but due to differentlag structures their properties differ significantly. The FECM model turns out to be char-acterised by a more convenient nesting structure, that allows a straightforward way fortesting the cointegration rank and the number of lagged differences to be included as ashort run parameters. Further, the identification problems are far less severe than in theFCVAR model. On the other hand, the estimation of FECM is more complicated due tothe presence of two different parts that model the short run dynamics and the restrictionthat relates their parameters. We propose an estimation procedure, which is based on thesuggestion of Carlini and Mosconi (2014) that maximizes the likelihood function usinga switching algorithm and the GLS procedure of Mosconi and Paruolo (2014) and we


demonstrate by means of Monte Carlo experiment the performance of our procedure infinite samples. Finally, we find that close to the DGPs chosen in the Monte Carlo simula-tions, the estimated parameters d, b, α, Aj, j = 1, 2 are normally distributed, whilst β has afat-tailed distribution.

AcknowledgementsWe would like to thank Charles Bos, Søren Johansen, Niels Haldrup, Rocco Mosconiand Morten Nielsen for great suggestions that helped improving this paper. The financialsupport from CREATES - Center for Research in Econometric Analysis of Time Seriesfunded by the Danish National Research Foundation (DNRF78), VU Amsterdam and theTinbergen Institute is gratefully acknowledged.

AppendixThe following algorithm describes how to estimate a bilinear form with a GLS model.Further details can be found in Mosconi and Paruolo (2014).

Consider the following equation

Yt = Hθ′Wt + εt t = 1, . . . , T

εt ∼ iidN(0,Ω), vec(θ) = Kψ

where Yt and Wt are respectively py × 1 and pw × 1 vectors, H and θ are respectivelypy × r and pw × r matrices. Then, we can estimate ψ as

ψ = (K ′(H ′Ω−1H ⊗ Sww)K)−1K ′vec(SwyΩ−1H)

where Sww =∑T

t=1WtW′t and Swy =

∑Tt=1WtY

′t .

3.6. CONCLUSIONS 83

0.78 0.785 0.79 0.795 0.8 0.805 0.81 0.815 0.820

20

40

60

80

100

120

d

0.54 0.56 0.58 0.6 0.62 0.64 0.660

5

10

15

20

25

30

b

−0.4 −0.38 −0.36 −0.34 −0.32 −0.3 −0.28 −0.26 −0.24 −0.220

2

4

6

8

10

12

14

16

18

20

α1

0.2 0.25 0.3 0.35 0.40

2

4

6

8

10

12

14

16

18

20

α2

−0.41 −0.405 −0.4 −0.395 −0.39 −0.385 −0.380

20

40

60

80

100

120

140

160

180

β

−0.25 −0.2 −0.150

5

10

15

20

25

30

35

40

a11

(1)

0.17 0.18 0.19 0.2 0.21 0.22 0.230

10

20

30

40

50

60

70

80

90

a12

(1)

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.040

5

10

15

20

25

30

35

40

45

a21

(1)

0.285 0.29 0.295 0.3 0.305 0.31 0.315 0.32 0.325 0.330

10

20

30

40

50

60

70

80

a22

(1)

0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.240

5

10

15

20

25

30

35

40

45

50

a11

(2)

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.020

10

20

30

40

50

60

70

80

90

a12

(2)

−0.315 −0.31 −0.305 −0.3 −0.295 −0.29 −0.2850

20

40

60

80

100

120

a21

(2)

Figure 3.1: Distributions of the parameters when the DGP parameters are d0 = 0.8,b0 = 0.6. N = 1, 500 Monte Carlo replications and Yt, t = 1, . . . , 100000.


d b α1 α2 βBias 0.0000 0.0012 0.0001 -0.0005 0.0000

Std.Dev 0.0038 0.0135 0.0214 0.0206 0.0024Skew. -0.1404 0.0117 -0.3912 0.3018 0.0546

Kurtosis 3.1239 3.0231 3.3295 3.2352 3.3915p-value JB test 0.0514 >0.5000 0.0010 0.0010 0.0084

a(1)11 a

(1)12 a

(1)21 a

(1)22

Bias 0.0001 0.0000 0.0002 0.0000Std.Dev 0.0103 0.0049 0.0096 0.0054

Skew 0.2826 -0.2777 -0.1961 0.1228Kurtosis 3.1502 3.1095 3.1846 3.0136

p-value JB test 0.0010 0.0010 0.0052 0.1434a

(2)11 a

(2)12 a

(2)21 a

(2)22

Bias 0.0000 0.0000 0.0000 -0.0002Std.Dev 0.0081 0.0046 0.0033 0.0030

Skew 0.2360 -0.0388 0.0532 0.0482Kurtosis 3.3815 3.0897 2.9097 2.8717

p-value JB test 0.0010 >0.5000 >0.5000 0.4295

Table 3.1: Sample statistics of the Monte Carlo distributions when d0 = 0.8 b0 = 0.6,T = 100000 observations and N = 1500 Monte Carlo replications.

3.6. CONCLUSIONS 85

−0.36 −0.34 −0.32 −0.3 −0.28 −0.260

5

10

15

20

25

30

35

40

α1

0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.350

5

10

15

20

25

30

35

40

45

50

α2

−0.415 −0.41 −0.405 −0.4 −0.395 −0.39 −0.385 −0.380

20

40

60

80

100

120

140

160

180

200

β

−0.23 −0.22 −0.21 −0.2 −0.19 −0.18 −0.17 −0.160

10

20

30

40

50

60

a11

(1)

0.18 0.185 0.19 0.195 0.2 0.205 0.21 0.2150

20

40

60

80

100

120

a12

(1)

−0.03 −0.02 −0.01 0 0.01 0.02 0.030

10

20

30

40

50

60

70

a21

(1)

0.285 0.29 0.295 0.3 0.305 0.31 0.3150

20

40

60

80

100

120

a22

(1)

0.17 0.18 0.19 0.2 0.21 0.22 0.230

10

20

30

40

50

60

70

80

a11

(2)

−0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

a12

(2)

−0.315 −0.31 −0.305 −0.3 −0.295 −0.29 −0.2850

20

40

60

80

100

120

a21

(2)

−0.315 −0.31 −0.305 −0.3 −0.295 −0.29 −0.2850

20

40

60

80

100

120

140

a22

(2)

Figure 3.2: Distributions of the Monte Carlo simulations when d0 = b0 = 0.6 are fixed inthe switching algorithm. The generated FECM paths have T = 100000 observations.


d α1 α2 β1

Bias - -0.0005 0.0002 0.0000Std.Dev - 0.0097 0.0081 0.0023Skew. - -0.1066 0.0750 0.0655

Kurtosis - 3.0495 2.9494 3.4033p-value JB test - 0.1671 0.3867 0.0031

a(1)11 a

(1)12 a

(1)21 a

(1)22

Bias 0.0004 -0.0002 -0.0002 0.0001Std.Dev 0.0068 0.0038 0.0060 0.0035

Skew 0.0432 -0.0834 -0.0087 0.0745Kurtosis 2.8909 2.9927 3.0914 3.0576

p-value JB test 0.4818 0.3513 >0.5000 0.3810a

(2)11 a

(2)12 a

(2)21 a

(2)22

Bias 0.0002 -0.0001 -0.0000 -0.0001Std.Dev 0.0054 0.0038 0.0032 0.0030

Skew 0.0200 0.0748 0.0591 0.0374Kurtosis 2.9412 2.9225 2.8213 2.9423

p-value JB test >0.5000 0.3455 0.1802 >0.5000

Table 3.2: Sample statistics of the Monte Carlo distributions d0 = b0 = 0.6 kept fixed inthe switching algorithm with T = 100000 and N = 1750 Monte Carlo replications.

3.6. CONCLUSIONS 87

−0.36 −0.34 −0.32 −0.3 −0.28 −0.260

5

10

15

20

25

30

35

40

Π11

0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0.1450

10

20

30

40

50

60

70

80

90

100

Π12

0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.350

5

10

15

20

25

30

35

40

45

50

Π21

−0.135 −0.13 −0.125 −0.12 −0.115 −0.11 −0.1050

20

40

60

80

100

120

Π22

−0.23 −0.22 −0.21 −0.2 −0.19 −0.18 −0.17 −0.160

10

20

30

40

50

60

a11

(1)

0.18 0.185 0.19 0.195 0.2 0.205 0.21 0.2150

20

40

60

80

100

120

a12

(1)

−0.03 −0.02 −0.01 0 0.01 0.02 0.030

10

20

30

40

50

60

70

a21

(1)

0.285 0.29 0.295 0.3 0.305 0.31 0.3150

20

40

60

80

100

120

a22

(1)

0.17 0.18 0.19 0.2 0.21 0.22 0.230

10

20

30

40

50

60

70

80

a11

(2)

−0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.020

20

40

60

80

100

120

a12

(2)

−0.315 −0.31 −0.305 −0.3 −0.295 −0.29 −0.2850

20

40

60

80

100

120

a21

(2)

−0.315 −0.31 −0.305 −0.3 −0.295 −0.29 −0.2850

20

40

60

80

100

120

140

a22

(2)

Figure 3.3: Distributions of the Monte Carlo simulations when d0 = b0 = 0.6 are fixed inthe switching algorithm. The generated FECM paths have T = 100, 000 observations.


π11 π12 π21 π22

Bias - 0.0005 0.0000 0.0002 0.0003Std.Dev 0.0098 0.0041 0.0081 0.0034Skew. -0.1072 0.0678 0.0755 -0.0877

Kurtosis 3.0452 3.0158 2.9507 2.9101p-value JB test 0.1663 0.4993 0.3846 0.2338

a(1)11 a

(1)12 a

(1)21 a

(1)22

Bias 0.0004 0.0000 -0.0001 0.0002Std.Dev 0.0068 0.0038 0.0060 0.0035

Skew 0.0428 -0.0699 -0.0070 0.0829Kurtosis 2.8821 3.0308 3.0953 3.0458

p-value JB test 0.4444 0.4578 >0.5000 0.3315a

(2)11 a

(2)12 a

(2)21 a

(2)22

Bias 0.0002 0.0000 0.0000 -0.0001Std.Dev 0.0054 0.0038 0.0033 0.0030

Skew 0.0202 0.0642 0.0589 0.0379Kurtosis 2.9403 2.9068 2.8339 2.9455

p-value JB test >0.5000 0.3859 0.2123 >0.5000

Table 3.3: Sample statistics of the Monte Carlo distributions d0 = b0 = 0.6 kept fixed andT = 100000, N = 1750 Monte Carlo replications and estimation of the matrix Π.

3.7. REFERENCES 89

3.7 ReferencesAvarucci, M., 2007. Three essays on fractional cointegration. PhD Thesis University of

Rome Tor Vergata.

Carlini, F., Mosconi, R., 2014. Twice integrated models. Tech. rep., Politecnico di Milano.

Carlini, F., Santucci de Magistris, P., 2017. On the identification of fractionally cointegratedVAR models with the F(d) condition. Forthcoming in Journal of Business & EconomicsStatistics.

Dennis, Jr, J. E., Traub, J. F., Weber, R., 1976. The algebraic theory of matrix polynomials.SIAM Journal on Numerical Analysis 13 (6), 831–845.

Dueker, M., Startz, R., 1998. Maximum-likelihood estimation of fractional cointegrationwith an application to US and Canadian bond rates. Review of Economics and Statistics80 (3), 420–426.

Granger, C. W., 1986. Developments in the study of cointegrated economic variables.Oxford Bulletin of economics and statistics 48 (3), 213–228.

Jensen, A. N., 2014. Efficient simulation of the Johansen-Nielsen model. Tech. rep.,Economics Department, University of Copenaghen.

Johansen, S., 1995. Likelihood-Based Inference in Cointegrated Vector AutoregressiveModels. Oxford University Press, Oxford.


Johansen, S., 2009. Representation of cointegrated autoregressive processes with applica-tion to fractional cointegration. Econometric Reviews 28, 121–145.


Łasak, K., September 2010. Likelihood based testing for no fractional cointegration.Journal of Econometrics 158 (1), 67–77.

Mosconi, R., Paruolo, P., 2014. Rank and order conditions for identification in simultaneoussystem of cointegrating equations with integrated variables of order two.

Robinson, P. M., Hualde, J., 2003. Cointegration in fractional systems with unknownintegration orders. Econometrica 71 (6), 1727–1766.

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