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Economic Modelling
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Taylor Rule or optimal timeless policy? Reconsidering the Fed's behavior since 1982☆
Patrick Minford a,b,1, Zhirong Ou a,⁎a Cardiff University, UKb CEPR, UK
☆ Weare grateful for helpful comments fromVo PhuongNelson, Ricardo Reis, participants at the RES (2010) annualreferees. We also thank Zhongjun Qu and Pierre Perron forstructural break.⁎ Corresponding author at: B14, Aberconway building
We compare three standard New Keynesian models differing only in their representations of monetary policy—the Optimal Timeless Rule, the original Taylor Rule and another with ‘interest rate smoothing’—with the aim oftesting which if any can match the data according to the method of indirect inference. We find that the OptimalTimeless Rule performs the best, eitherwith calibrated parameters orwith estimatedparameters. Thismodel canalso account for the widespread finding of apparent ‘Taylor Rules’ and smoothed interest rates in the data, eventhough neither of these represents the true policy.
In this paper our aim is to uncover the principles according towhich the Board of Governors of the US Federal Reserve System(the Fed) has conducted monetary policy since the early 1980s. We doso in a novel way by asking which such principles can, when combinedwith a widely-acceptedmacromodel, replicate the dynamic behavior ofthe US economy during the sample period. By ‘principles’ we meaneither an explicit rule the Fed follows (such as an interest-rate settingrule) or some other economic relationship that it aims to ensure occurs(such as a fixed exchange rate or as here an optimality condition).
The main context for this work is the influential paper by Taylor(1993), who—building on earlier work by Henderson and McKibbin(1993a,b) which argued for the efficacy of interest rate rules—suggested that the Fed actually had been for some time systematicallypursuing a particular interest rate rule, reacting directly to two ‘gaps’,
Mai Le, DavidMeenagh, Edwardconference and our anonymoussharing their code for testing of
one between inflation and its target rate, the other between outputand its natural rate. Such a ‘Taylor Rule’ was subsequently adoptedwidely inNewKeynesianmodels to represent the behavior ofmonetarypolicy (Rotemberg and Woodford, 1997, 1998; Clarida et al., 1999,2000; Rudebusch, 2002; English et al., 2003).
However, Minford et al. (2002) and Cochrane (2011) have shownthat a Taylor Rule is not identifiedwhen considered as a single equationrelationship. Estimates of such a ‘rule’may emerge from the data whenthe Fed is following quite other monetary policies; this is because avariety of relationshipswithin the economycan imply a relationship be-tween interest rate, inflation and the output gap whichmimics a TaylorRule. In the presence of such an identification problem, direct estima-tion of Taylor Rules on the data does not establish whether the Fedwas actually pursuing them or not. Some other way of testing hypothe-ses about monetary policy must be found which makes use of identify-ing restrictions from a fully specifiedmodel. The one proposed here is toset up competing structural models which differ solely according to themonetary policies being followed, and to distinguish between thesemodels according to their ability to replicate the dynamic behavior ofthe data. Thus for example if one were to fail to reject just one ofthese models and reject the rest, it would be reasonable to argue thatthis model succeeds because in it not only the rest of the economy butalsomonetary policy is well-specified. Of course other less decisive em-pirical outcomes of the tests are entirely possible.
The rest of this paper is organized as follows: Section 2 reviews thework estimating monetary policy rules and makes a critique of it interms of identification; Section 3 outlines the model and the rules wepropose to test; Section 4 explains our methodology and sets out our
3 While one may argue that various announcements, proposals and reports pub-lished by the central bank directly reveal to econometricians the bank's reaction func-tion, it is worth noting that what the Fed actually does is not necessarily the same thingas what its officials and governors say it does. So these documents, while illuminating,can complement but cannot substitute for econometric evidence.
4 From the money demand and money supply equations, ψ2ΔRt=πt−m+ψ1ΔEt−1
yt+1+Δεt−μt. Substitute for Et−1yt+1 from the IS curve and then inside that the real
interest rate from the Fisher identity giving ψ2ΔRt ¼ πt−mþ ψ11γ
� �φ ΔRt−ΔEt−1ðf
πtþ1Þ þ Δyt−Δυtg þ Δεt−μt ; then, rearrange this as ψ2− ψ1φγ
� �Δ Rt−R�ð Þ ¼ πt−mð Þ−
ψ1φγ ΔEt−1πtþ1 þ ψ1
γ Δ yt−y�ð Þ− ψ1γ Δυt þ Δεt−μ t , where the constants R* and y* have
been subtracted from Rt and yt respectively, exploiting the fact that when differencedthey disappear. Finally, Rt=r*+π*+γχ−1(πt−π*)+ψ1χ−1(yt−y*)+{(Rt−1−R*)−ψ1φχ−1ΔEt−1πt+1−ψ1χ−1(yt−1−y*)−ψ1χ−1Δυt+γχ−1Δεt−γχ−1μt} where wehave used the steady state property that R*=r*+π* and m=π*. In effect we have founda linear combination of the equations of this model that mimics the Taylor Rule. In generalwe can write the model as A(L)xt=Bx∗+Dwt, where x, x∗,w are respectively the vectors ofendogenousvariables, constants anderrors.A(L) is thematrix of coefficients including thoseon lag and (expected) lead values. The general solution of this model will be a VARMA. Onecould also find linear combinations of the solution equations that would yield Rt=R∗+q1(yt−y∗)+q2(πt−π∗)+ηt, where ηt is an error term which includes lagged endogenousvariables (as deviations from equilibrium) and current and lagged errors. Our purpose hereis to illustrate that a widely-used model exists with a different monetary policy rule whichcould be confused with the Taylor Rule being estimated. It is possible that more complexmodels could generate sufficient identifying restrictions in single equation estimation;
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finding that the Fed pursued an optimal timeless policy; Section 5 re-veals how this can explain the apparent ‘Taylor Rules’ the data woulddisplay; in Section 6 we extend these discussions to allow for full evalu-ation of the models basing on their best numerical versions; Section 7concludes.
2. Taylor Rules, estimation and identification
Taylor (1993) suggested that a good rule for monetary policywould set the Federal Funds rate according to the following equation:
iAt ¼ πAt þ 0:5xt þ 0:5 πA
t −π�� �þ g ð1Þ
where xt is for the percentage deviation of real GDP from trend, itA
is the annual rate of inflation averaged over the past four quar-ters, with inflation target π∗ and real GDP growth rate g both setat 2%.
Known as the original ‘Taylor Rule’, Eq. (1) was found to havepredicted the movement of actual Fed rates well for much of theperiod from 1987 until the banking crisis of the late 2000s. Thissuccess convinced many economists that the Fed's policy at thetime could be conveniently described by this equation. Thus anumber of variants have also been proposed; for example, Claridaet al. (1999) suggested a rule that allows for policy inertia cantake the form:
iAt ¼ 1−ρð Þ α þ γπ πAt −π�� �
þ γxxth i
þ ρiAt−1 ð2Þ
with ρ showing the degree of ‘interest rate smoothing’. Others haveincluded lags or leads of the inflation and output gap terms toaccount for backward-looking or forward-looking behaviors—thisincludes Rotemberg and Woodford (1997, 1998) and Clarida et al.(2000).
Thusmanyhave attempted to estimate these rules in order to uncoverthe underlying policy rule. A common practice of this is to estimate it as asingle-equation regression (as in Rotemberg and Woodford, and Clarida,Gali and Gertler just cited).2 Others (such as Smets and Wouters, 2007and Ireland, 2007) have taken the alternative approach of including andestimating it in a full DSGE model; we consider this alternative below.Many have claimed that a Taylor Rule fitted the data well. However,while econometricians have to deal with the usual difficulties encoun-tered in estimating a Taylor-type equation (e.g., Carare and Tchaidze,2005 and Castelnuovo, 2003), Minford et al. (2002) and Cochrane(2011) have pointed out that single equation estimates face an identifi-cation problem—see also Minford (2008) whichwe use in what follows.
Lack of identification occurs when an equation could be confusedwith a linear combination of other equations in the model. In thecase of the Taylor Rule, DSGE models give rise to the same correla-tions between interest rate and inflation as the Taylor Rule, even ifthe Fed is doing something quite different, such as targeting themoney supply. For example, Minford et al. (2002) show this in aDSGE model with Fischer wage contracts (see also Gillman et al.,2007).
In effect, unless the econometrician knows from other sources ofinformation that the central bank is pursuing a Taylor Rule, he cannotbe sure he is estimating a Taylor Rule when he specifies a Taylor Rule
2 We include in ‘single equation estimation’ methods such as Instrumental Variableswhich deal with endogeneity but do not make use of the identifying restrictions from afull model. For example, the exogenous variables used as instruments would not be dif-ferent across the same model with different monetary policy rules.
equation because under other possible monetary policy rules a simi-lar relationship to the Taylor Rule is implied.3
To illustrate the point, we may consider a popular DSGEmodel butwith a money supply rule instead of a Taylor Rule:
This model implies a Taylor-type relationship that looks like: Rt=r∗+π∗+γχ−1(πt−π∗)+ψ1χ−1(yt−y∗)+wt, where χ=ψ2γ−ψ1ϕ,and the error term, wt, is both correlated with inflation and outputand autocorrelated; it contains the current money supply/demandand aggregate demand shocks and also various lagged values (thechange in lagged expected future inflation, interest rate, the outputgap, the money demand shock, and the aggregate demand shock).This particular Taylor-type relation was created with a combinationof equations—the solution of the money demand and supply curvesfor interest rate, the Fisher identity and the IS curve for expected fu-ture output.4 But other Taylor-type relations could be created withcombinations of other equations, including the solution equations,generated by the model. They will all exhibit autocorrelation and con-temporaneous correlation with output and inflation, clearly of differ-ent sorts depending on the combination used.
All the above apply to identifying a single equation being estimated;thus one cannot distinguish a Taylor Rule equation from the equationsimplied by the model and alternative rules when one just estimatesthat equation. One could attempt to apply further restrictions but suchrestrictions are hard to find. For example, one might restrict the errorprocess of a Taylor Rule in some distinct way, say to being seriallyuncorrelated. But the error in a Taylor Rule, which represents ‘monetary
note however that large DSGE models with nominal rigidity can generally be reduced tothe IS/Phillips-Curve/Monetary-policy form used here, with the error terms sweeping upexpressions that do not fit the structure. While of course the errors in the linear combi-nation of equations resembling the Taylor Rule will differ in principle from the errorterm in the Taylor Rule which represents ad hoc ‘other’ policy reactions to events, thereis no obvious way of distinguishing them, since both contain endogenous variables andboth are persistent.
6 They also comment that such a rule may produce indeterminacy; however thisdoes not occur in the model here.
7 We have allowed for an AR(1) process for serial correlation in all equation errors. We
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judgment’ based on factors other than the two gaps, may well be auto-correlated because those factors are persistent.
However, when a ‘monetary rule’ is chosen for inclusion in a com-plete DSGEmodel with rational expectations, then the model imposesover-identifying restrictions through the expectation terms which in-volve in principle all the model's parameters. Thus a model with aparticular rule is in general over-identified so that estimation by fullinformation methods of that particular model as specified is possible.One way of putting this is that there are more reduced-form parame-ters than structural parameters. Another is to note that the reducedform will change if the structural description of monetary policychanges—a point first made by Lucas (1976) in his ‘critique’ of con-ventional optimal policy optimization at that time (for some illustra-tions of how reduced forms will change for a model like the one inthis paper, see Meenagh et al., 2009a, 2009b). So if the econometri-cian posits a Taylor Rule then he will retrieve its coefficients andthose of the rest of the model under the assumption that it is thetrue structural monetary rule. He could then compare the coefficientsfor a model where he assumes some other rule. He can distinguish be-tween the two models via their different reduced forms and hencetheir different fits to the data. Thus it is possible to identify the differ-ent rules of monetary policy behavior via full information estimation.
However, the identification problem does not go away, even whena model is over-identified in this way. The problem is that the deci-sion to include the Taylor Rule in such a model is justified by thefact that it fits the data in single equation estimation; but as wehave seen such a choice could be the victim of identification failureas the rule could be mimicking the joint behavior of the rest of themodel and some other (true) monetary rule. If so, including it in themodel will produce a mis-specified model whose behavior will notfit the data as well as the properly-specified model with the truemonetary policy equation. To detect this and also to find the truemodel we need not only to test this model but also to test possiblewell-specified alternatives. Thus we need to check whether there isa better model which with its over-identifying restrictions may fitthe data more precisely.
This points the way forward. One may specify a complete DSGEmodel with alternative monetary rules and use the over-identifyingrestrictions to determine their differing behavior. One may then testwhich of them is accepted by the data. This is the approach takenhere.
3. A simple New Keynesian model for interest rate, output gap andinflation determination
We follow a common practice among New Keynesian authors ofsetting up a full DSGE model with representative agents and reducingit to a three-equation framework consisting of an ‘IS’ curve, a Phillipscurve and a monetary policy rule. Under rational expectations the IScurve derived from the household's optimization problem and thePhillips curve derived from the firm's optimal price-setting problemunder Calvo (1983) contract can be shown to be:
xt ¼ Etxtþ1−1σ
� �ı̃t−Etπtþ1
� �þ vt ð3Þ
πt ¼ βEtπtþ1 þ γxt þ κuwt ð4Þ
where xt is the output gap, ı̃t is the deviation of interest rate from itssteady-state value, πt is the price inflation, and vt and ut
w are the de-mand and supply shocks, respectively.5
5 Note γ and κ are functions of other structural parameters and some steady-staterelations (see calibration Table 2 in the next section). For a full derivation of Eqs. (3)and (4), see Walsh (2003).
This model can be closed by adding to it a monetary policy equation;this is normally a Taylor Rule in the New Keynesian literature, but otherpolicy alternatives can also be considered. Thus, for our comparison herewe consider three popular monetary regime versions widely suggestedfor the US; these are the Optimal Timeless Rule when the Fed commitsto minimizing the social welfare loss from volatile inflation and output,the original Taylor Rule and finally its interest-rate-smoothed version,Eq. (2).
Many normative monetary policy studies in the literature focusingon Taylor Rules have argued that policies of this sort are roughly op-timal. Here, the Optimal Timeless Rule is introduced and compared tothese, as by pursuing this rule the Fed would have acted optimallywith full precision. This optimal rule can be derived by minimizingthe social welfare loss function
SWLt ¼ψ2
αx2t þ π2t
h ið5Þ
with respect to the Phillips curve, under full commitment. It suggeststhat if the Fed was a strict, consistent optimizer, it would have keptinflation equal to a fixed fraction of the first difference in the outputgap, thus
πt ¼ −αγ
xt−xt−1ð Þ ð6Þ
where α is the weight it puts on output variation loss relative to infla-tion, γ is the Phillips curve's (degree of rigidity) constraint. For a fullderivation, see Woodford (1999) and McCallum and Nelson (2004).
Note that unlike the typical Taylor Rules that specify an explicitpolicy instrument response to the economy's condition, in the Opti-mal Timeless Rule the instrument response is implicit. The rule setsan optimal trade-off between the economic outcomes; since in thismodel interest rates are the monetary instrument, these are implicitlyset to achieve this trade-off. Svensson and Woodford (2004) catego-rized this kind of targeting rule as ‘high-level monetary policy’. Theyargued that by connecting the central bank's monetary principle toits ultimate policy objectives this rule is more transparent and robustcompared to other instrument rules such as Taylor Rules.6
Thus, implementing such an optimal policy would require the Fedto fully understand the model (including the shocks hitting the econ-omy) and to set the policy instrument (here, the funds rate) to what-ever that supports the optimal trade-off. Of course the Fed may makeerrors of implementation that cause the rule not to be met exactly—‘trembling hand’ errors, ξt. Here, since Eq. (6) is a strict optimality con-dition, we think of such policy mistakes as due either to an imperfectunderstanding of the model or to an inability to identify and react tothe demand and supply shocks correctly. This differs from the error itwouldmake in pursuing Eqs. (1) and (2), which consist of the Fed's dis-cretionary departures from these rules.
Thus the models being compared are summarized in Table 1.7
These are different only in the monetary policies being followed.
also transformed the Taylor Rules to quarterly versions so that the frequency of interestrate and inflation in these rules is consistent with what the rest of the model has defined.All constants are dropped as demeaned, detrended data will be used (see ‘data’ inSection 4.2).
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Hence by comparing their capacity to fit the data, one shouldbe able to tell which rule, when included in this simple NewKeynesian framework, can best explain the facts and therefore bestdescribe the underlying policy. We go on to investigate this inSection 4.
9 For equations involving expectations terms we use the method of robust Instru-mental Variables estimation suggested by McCallum (1976) and Wickens (1982). Inthat case we set the lagged endogenous data as instruments and calculate the fittedvalues from a VAR(1)—this also being the auxiliary model we choose in what follows.10 We have drawn the innovations in our tests by time vectors to preserve possiblecontemporaneous correlations.11 Specifically, they found that the bias due to bootstrapping was just over 2% at the95% confidence level and 0.6% at the 99% level. They suggested possible further refine-ments in the bootstrapping procedure which could increase the accuracy further; how-ever, we do not feel that it is necessary to pursue these here.12 Other time series models (including VARs of higher orders) may also be used. Butwe choose a VAR(1) here because it provides us with a parsimonious description of thedata. VARs of higher order describe the data more precisely, thus considerablystrengthening the test's power of rejection. In Appendix E we redo the exercise usinga VAR(2) and VAR(3) instead; it turns out that it fails to change our conclusion of mod-
4. Evaluation of the models' performance
4.1. The method of indirect inference
We evaluate the models' capacity in fitting the data using themethod of indirect inference first proposed by Minford et al.(2009) with substantial refinements by Meenagh et al. (2009a,2009b) and Le et al. (2010, 2011).8 The approach employs an auxil-iary model that is completely independent of the theoretical modelto produce a description of the data against which the performanceof the theoretical model is evaluated indirectly. Such a descriptioncan be summarized either by the estimated parameters of the auxil-iary model or by functions of these; we will call these the des-criptors of the data. While these are treated as the ‘reality’, thetheoretical model being evaluated is simulated to find its impliedvalues for them.
Indirect inference has been widely used in the estimation ofstructural models. Early examples of this include Smith (1993),Gregory and Smith (1991, 1993), Gourieroux et al. (1993),Gourieroux and Monfort (1996) and Canova (2005). Here wemake a different use of indirect inference as our aim is to evaluatean already estimated or calibrated structural model. The commonelement is the use of an auxiliary time series model. In estimationthe parameters of the structural model are chosen such that whenthis model is simulated it generates estimates of the auxiliarymodel similar to those obtained from the actual data. The optimalchoices of parameters for the structural model are those that min-imize the distance between a given function of the two sets of esti-mated coefficients of the auxiliary model. Common choices of thisfunction are the actual coefficients, the scores or the impulseresponse functions. In model evaluation the parameters of thestructural model are taken as given. The aim is to compare the per-formance of the auxiliary model estimated on simulated data de-rived from the given estimates of a structural model—which istaken as a ‘true’ model of the economy, the null hypothesis—withthe performance of the auxiliary model when estimated from theactual data. If the structural model is correct then its predictionsabout the impulse responses, moments and time series propertiesof the data should statistically match those based on the actualdata. The comparison is based on the distributions of the two sets
8 Le et al. (2011) deadwith a wide range of practical issues raised by this methodology.
of parameter estimates of the auxiliary model, or of functions ofthese estimates.
The testing procedure thus involves first constructing the errorsimplied by the previously estimated/calibrated structural modeland the data. These are called the structural errors and are backedout directly from the equations and the data.9 These errors are thenbootstrapped and used to generate for each bootstrap new data basedon the structural model.10 An auxiliary time series model is thenfitted to each set of data and the sampling distribution of the coeffi-cients of the auxiliary time series model is obtained from these esti-mates of the auxiliary model. A Wald statistic is then computed todetermine whether functions of the parameters of the time seriesmodel estimated on the actual data lie in some confidence intervalimplied by the sampling distribution of these. This bootstrap estimateof the small sample distribution is generally more accurate for smallsamples than the asymptotic distribution; it is also shown to be consis-tent given that the Wald statistic is ‘asymptotically pivotal’ by Le et al.(2011), who also showed that it had quite good accuracy in small sam-ple Monte Carlo experiments.11
We take a VAR(1) that includes interest rate, output gap and infla-tion as the auxiliary model, and treat as the descriptors of the data theVAR coefficients and the variances of these variables.12 The Wald sta-tistic is computed from these. Thus, by testing the models against thedata we ask if any of these models is able to explain the observed dy-namics and volatility of the chosen variables using the simulated jointdistribution of these at a given confidence level.13 The Wald statisticis given by:
Φ−Φ� �′X−1
ΦΦð ÞΦ−Φ� � ð7Þ
el comparison qualitatively, though increasing the test's power in such a way tends toreject all these models.13 Note that by assessing the VAR coefficients one would have examined the VAR'simpulse responses, the co-variances and the auto/cross correlations of the data implic-itly, since the latter are functions of the former.
A
B
0.25
0.2
0.15
0.1
0.05
04 3 2 1 0 -1 -2
-3-2
-10
1
Fig. 1. The principle of testing using indirect inference.
117P. Minford, Z. Ou / Economic Modelling 32 (2013) 113–123
whereΦ is the vector of estimated descriptors yielded in each simulation,with Φ and Σ representing, respectively, the means and variance–covariancematrix of those calculated fromall simulations. The full testingprocedure can be illustrated diagrammatically as in Fig. 1.
While the first panel in Fig. 1 summarizes the main steps of themethodology just described, the ‘mountain-shaped’ diagram in panelB gives an example of how the ‘reality’ is compared to the model's pre-diction using theWald testwhen two parameters of the auxiliarymodelare considered. Suppose the real-data estimates of these parameters aregiven at R and there are two competing models, each implies a jointdistribution of these parameters shown by the ‘mountains’ (α and β,respectively). Since R lies outside the 95% contour of α, this modelwould be rejected at 95% confidence level; the other model that gener-ated β would not be rejected, however, since R lies inside. In practicethere are usually more than two descriptors to be considered andhenceforth to deal with the extra dimensions the test is carried out bycalculating the Wald statistic (Eq. (7)).
15 Note that by defining the output gap as the HP-filtered log output we have effec-tively assumed that the HP trend approximates the flexible-price output in line withthe bulk of many others. To estimate the flexible-price output from the full DSGE mod-el that underlies our three-equation representation, we would need to specify thatmodel in detail, estimate the structural shocks within it and fit the model to the unfil-tered data, in order to estimate the output that would have resulted from these shocksunder flexible prices. This is a substantial undertaking well beyond the scope of this pa-per, though something worth pursuing in future work. Le et al. (2011) test the Smetsand Wouters (2007) US model by the same methods as we use here. This has a TaylorRule that responds to flexible-price output. It is also close to the Timeless OptimumRule since, besides inflation, it responds mainly not to the level of the output gap butto its rate of change and also has strong persistence so that these responses cumulatestrongly. They find that the best empirical representation of the output gap treats the
4.2. Data and calibration
4.2.1. DataWe use the quarterly data published by the Federal Reserve Bank of
St. Louis from 1982 to 2007.14 Most discussions of the Fed's behavior(especially those based on Taylor Rules) are concerned with periodsthat begin sometime in the mid- or late 1980s but we chose 1982 asour starting point because many (including Bernanke and Mihov,1998; Clarida et al., 2000) have argued that it was around then thatthe Fed switched from using non-borrowed reserves to setting theFunds rate as the instrument of monetary policy. Taylor (1993) originallysuggested a later starting point for his specification and plainly one couldchoose a variety of different sample periods and test for that. In theAppendix we show that our choice of 1982 is in line with the Qu andPerron (2007) test that suggests a break in the data between 1980 and1984; we also show that our benchmark results are robust to this choice.
14 http://research.stlouisfed.org/fred2/.
We measure ı̃t as the deviation of current Fed rate from thesteady-state value which is interpreted here as a linear trend (at aquarterly rate for compatibility with the quarterly inflation rate), out-put gap, xt, as the percentage deviation of real GDP from its HP-trendvalue,15 and quarterly inflation, πt, as the log difference in CPI be-tween the current and the last quarter. We use demeaned data forsimplicity. These are plotted in the Appendix where we also reportthe Unit Root test results.
4.2.2. CalibrationWe use values of model parameters that are commonly calibrated
and accepted for the US economy in the literature, as listed in Table 2:the time discount rate is set to 0.99, implying a 1% quarterly (4% an-nual) interest rate in the steady state. σ and η are set to 2 and 3 re-spectively as in Carlstrom and Fuerst (2008), who emphasized onthe values' consistency with the inelastic intertemporal consumptionand labor supply in the US data. The Calvo contract (non-adjusting)probability (ω) of 0.53 and the price elasticity of demand (θ) of 6are both taken from Kuester et al. (2009). This Calvo probability islower than most; but we chose it because Le et al. (2011) found
output trend as a linear or HP trend instead of the flexible-price output—this TaylorRule is used in the best-fitting ‘weighted’ models for both the full sample and the sam-ple from 1984. Thus while in principle the output trend should be the flexible-priceoutput solution, it may be that in practice these models capture this rather badly sothat it performs less well than the linear or HP trends.
β Time discount factor 0.99σ Inverse of intertemporal consumption
elasticity2
η Inverse of labor elasticity 3ω Calvo contract price non-adjusting
probability0.53
GY Steady-state gov. expenditure to
output ratio0.23
YC Steady-state output to consumption ratio 1
0:77 (implied value)
κ κ ¼ 1−ωð Þ 1−ωβð Þω 0.42 (implied value)
γ γ ¼ κ ηþ σ YCð Þ 2.36 (implied value)
θ Price elasticity of demand 6αγ ≡ 1
θ Optimal trade-off rate on the Timeless Rulea 16 (implied value)
Parameters on the interest-rate-smoothed Taylor Ruleρ Interest rate smoothness 0.76γπ Inflation response 1.44γ′
x Output gap response 0.14ρv Demand shock persistence 0.93 (sample estimate)ρuw Supply shock persistence 0.80 (sample estimate)ρξ Policy shock persistence
-Model one (Optimal Timeless) 0.38 (sample estimate)-Model two (Original Taylor) 0.39 (sample estimate)-Model three (IRS Taylor) 0.39 (sample estimate)
a Nistico (2007) found that the relative weight α could be shown as the ratio of theslope of the Phillips curve to the price elasticity of demand, and so α=γ/θ.
Table 4Performance of model with the Taylor Rules.
Tests for chosen data features Baseline model with
Original Taylor Rule(model two)
‘IRS’ Taylor Rule(model three)
Directed Wald for dynamics 100 99.8Directed Wald for volatilities 99.2 99Full Wald for dyn. & vol. 100 99.7
118 P. Minford, Z. Ou / Economic Modelling 32 (2013) 113–123
price rigidity to be less than Smets and Wouters did; it turns out (seeTable 7) that when we reestimate the model indirectly we find avalue close to this.
The steady-state output–consumption ratio of 1/0.77 is calculatedbased on the steady-state government-expenditure-to-output ratio of
Table 3Performance of model with the Optimal Timeless Rule.
Note: Estimates reported in panel B are magnified by 1000 times as their original values.
0.23 calibrated by Foley and Taylor (2004).16 For the interest-rate-smoothed Taylor Rule, we follow Carlstrom and Fuerst (2008) and setthe inflation and output responses, respectively, to 1.44 and 0.14, andthe smoothing parameter to 0.76. The second part of Table 2 then re-ports the sample estimates of the autoregressive coefficients of theequation errors (which are all significant at 5% level); it shows that inall model versions the demand and supply shocks are highly persistentcompared to the policy shock.
4.3. The test results
We now present the test results for the models being compared.Recall that this is based on the model's ability to fit the data's dynam-ics and volatility, as captured by a VAR(1). Since there are three en-dogenous variables, a VAR(1) representation of these generatestwelve data descriptors of interest; these are the nine VAR coeffi-cients and the three data variances. Our VAR(1) representationtakes the form:
ı̃txtπt
24
35 ¼
β11 β12 β13β21 β22 β23β31 β32 β33
24
35 ı̃t−1
xt−1πt−1
24
35þ errors;
with the variables' order being interest rate (1), output gap (2) andinflation (3). We calculate Wald statistics for two subsets of thedata descriptors: one (the VAR coefficients) relating to the dynamicsin the data and one (the variances) relating to its volatility. We callthese ‘directedWald’ statistics as they are directed at these narroweraspects of the data. We also calculate what we term the ‘full’ Waldstatistic where the whole set of descriptors is considered simulta-neously. In both cases we report the Wald statistic as a percentile,i.e. the percentage point where the data value comes in the bootstrapdistribution; thus 1 minus the percentile /100 is the p-value of themodel.
Table 3 reveals the performance of the Optimal Timeless Rulemodel. Panel A shows that while at 95% confidence level the modelhas slightly underpredicted the interest rate's response to the laggedoutput and the output autocorrelation, other VAR coefficients esti-mated with the real data all lie well within their individual 95%bounds as the model suggests.17 Overall, the test returns a directed
16 Here have assumed Y=C+G and used the steady-state G/Y ratio to calculate Y/C.17 Although theWald statistics provide uswith our tests, we also report for this first caseonly the calculated 95% bounds for each individual estimate of the data descriptors. Theseshow where the data estimate for each descriptor lies within the model distribution forthat descriptor alone. Theymay give clues about sources ofmodelmis-specification. Thesecomparisons are similar to the widespread comparison of moments (including cross-moments) in the data with those simulated from themodel. However, these comparisonsdo not take account of thesemoments' joint distributionwhich is relevant to whether thedata is compatible with themodel on all these features simultaneously. Unfortunately theindividual data moment comparisons taken as a group are not a reliable guide to whetherthe data moments will lie inside the model's joint distribution for them. See Le et al.(2011). For this the Wald must be used.
Table 5Summary of model performance with standard calibration.
Model versions Directed Wald fordynamics
Directed Wald forvolatilities
Full Wald fordyn. & vol.
Optimal Timeless Rule(model one)
86.4 89.6 77.1
Original Taylor Rule(model two)
100 99.2 100
‘IRS’ Taylor Rule(model three)
99.8 99 99.7
Table 7SA estimates of model parameters.
Parameters Definitions Calibrations SA estimates
TimelessRulemodel
TaylorRulemodel
β Time discount factor 0.99 Fixed at 0.99σ Inverse of intertemporal
consumption elasticity2 1.46 1.16
η Inverse of labor elasticity 3 3.23 3.85ω Calvo contract price non-adjusting
probability0.53 0.54 0.61
GY Steady-state gov. expenditure to
output ratio0.23 Fixed at 0.23
YC Steady-state output to consumption
ratio
10:77 Fixed at 1/0.77
κ κ ¼ 1−ωð Þ 1−ωβð Þω 0.42 0.40 0.25
γ γ ¼ κ ηþ σ YCð Þ 2.36 2.06 1.33
α Relative weight of loss assigned tooutput variations against inflation
0.39 0.58 –
αγ ≡1
θ Optimal trade-off rate on theTimeless Rule
16
13:6 –
θ Price elasticity of demand 6 3.6 –
γπ Interest-rate response to inflation 1.44 – 2.06γ′
x Interest-rate response to output gap 0.14 – 0.06ρ Interest rate smoothness 0.76 – 0.89ρv Demand shock persistence 0.93 0.94 0.95ρuw Supply shock persistence 0.80 0.79 0.77ρξ Policy shock persistence
-With Optimal Timeless Rule 0.38 0.42 –
-With Taylor Rule 0.39 – 0.40
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Wald percentile of 86.4. This means at 95% (or even at 90%) confi-dence level the dynamic behavior of the data fails to reject themodel according to the joint distribution of this implied by themodel. This indicates that the model has in general captured thedata dynamics pretty precisely.
Panel B then shows that themodel is also correctly sized, as the datavariances not only lie individually within the 95% bounds, but are alsojointly explained by the model at the 95% level (and marginally at the90% level) since 89.6 is the directed Wald percentile.
The model's overall fit to data is finally assessed in panel C, wherethe full Wald jointly evaluates both the subsets of descriptors justassessed. This is reported at 77.1, implying a p-value for the model of0.229. Such a lowWald percentile indicates that themodel has generat-ed a joint distribution of the data descriptors whose mean is not too fardistant fromwhatwe observe in reality tomerit rejection of themodel;thus even at the 90% confidence level the data fails to reject the modeljointly on both dynamics and size. We therefore conclude that the USfacts do not reject the Timeless Rule model as the data-generating pro-cess in this post-1982 period.
We now consider the model versions with the Taylor Rules. Table 4shows that both these alternatives are rejected not merely at the 95%but even at the 99% level by both the dynamics and the volatility of thedata. Thus from the statistical viewpoint these standard Taylor Rulemodels could not have generated the post-1982 US as many haveclaimed.
We now summarize our results in Table 5. Comparison by columnsshows that, with the Optimal Timeless Rule, the model stands out inexplaining all our concerned data features by consistently yielding thelowest Walds. This version is, too, the only model version capable ofexplaining the dynamics and volatility of the data both separately andjointly. The other twomodel versions with Taylor Rules are both unableto capture either of these, and are both completely rejected overall.Thus on the assumption that the IS and Phillips Curve parameters arecorrect and that the chosen parameters of monetary policy rules arealso the best available, we can confidently reject the Taylor Rule formu-lations of monetary policy, whereas we cannot reject the OptimalTimeless Rule formulation.
Table 8Performance of models under calibration and estimation.
5. The underlying policy in the light of the test results
5.1. What roles could the standard Taylor Rules have played?
Our analysis thus suggests that the widespread success reported insingle-equation Taylor Rule estimation could simply be a statistical illu-sion. These ‘rules’, instead of being structural, may well be representing
Table 6‘Taylor Rule’ in the data and the accountability of the Optimal Timeless Rule.
Least squared estimates of Eq. (8) Wald percentile of the
γπ γx ρ Adj. R2 Timeless Rule model
0.08 0.05 0.89 0.92 21.7
some other data relationships implied by the true model embeddedwith the true policy rule (e.g., the Optimal Timeless Rule). We illustratethis possibility using indirect inference in what follows. Thus we treatthe Optimal Timeless Rule model again as the true model (the nullhypothesis) and ask whether the existence of empirical Taylor Ruleswould be consistent with it.
Suppose we estimate an arbitrarily specified ‘Taylor Rule’ of theform:
ı̃t¼ γππtþγxxtþρ ı̃t−1þξt ð8Þ
where variables have their usual meaning. The Least Squared esti-mates of Eq. (8) are shown in columns one to four in Table 6.
Now, to use the method of indirect inference we treat this TaylorRule regression as the auxiliary model, the benchmark descriptor ofthe data and ask whether the Timeless Rule model can generatepseudo-data such that it would replicate this benchmark result.The test returns a directed Wald percentile of 21.7 (the last columnin Table 6), indicating that it is statistically possible for the TimelessRule model to imply that such a ‘Taylor Rule’ would be observed—such a low Wald percentile suggests that the model would generatea joint distribution of these Taylor Rule parameterswhosemean is fairlyclose to the Least Squared estimates. Thus the probability is fairly high.
Tests for chosen features Timeless Rule modelunder
Taylor Rule modelunder
Calibration Estimation Calibration Estimation
Directed Wald fordynamics (and p-value)
86.4 77.7 (0.223) 99.8 89.6 (0.104)
Directed Wald forvolatilities (and p-value)
89.6 90.3 (0.097) 99 94.9 (0.051)
Full Wald for dyn. & vol.(and p-value)
77.1 68.6 (0.314) 99.7 87.6 (0.124)
Table 9‘Taylor Rules’ in the data and the accountability of the estimated Timeless Rule model.
Note: a. Time series model: VAR(1) (without constant). b. H0: no structural break; H1:one structural break. c. Observation sample (adjusted): 1972Q2–2007Q4.
Appendix A. The Qu–Perron breakpoint test
120 P. Minford, Z. Ou / Economic Modelling 32 (2013) 113–123
This illustrates the identification problem with which we beganthis paper: a Taylor-type relation in the data may well be generatedby a model where there is no structural Taylor Rule at all. Hence, anysingly estimated/calibrated Taylor Rule, no matter how well it mightpredict the actual movement of the Funds rate, is not by itself evi-dence that it is what the Fed's policy being followed. In our case, asit shows, it is merely a statistical relation implied by the OptimalTimeless Rule.
5.2. An implication for the ‘interest rate smoothing’ illusion
This also sheds light on the debate over the observation of ‘interestrate smoothing’—a ‘puzzle’ raised by Clarida et al. (1999), who claimedthat as optimal response the Timeless Rulewould require one-off interestrate adjustments, whereas evidence fromTaylor Rule regressions usual-ly implied highly smoothed interest rates. They argued that this slug-gishness in interest rate adjustment was hard to rationalize as optimalbehavior.
Whilemanyhave attempted to explain this discrepancy fromdifferentangles (e.g., Rotemberg andWoodford, 1997, 1998 andWoodford, 1999,2003a,b theoretically, and Sack andWieland, 2000 and Rudebusch, 2002empirically), our tests above support the Optimal Timeless Rule but rejectthe interest-rate-smoothed Taylor Rule, implying that the Fed has beenresponding optimally without deliberately smoothing the interest rates.It is the persistence in the shocks themselves that induces the appearanceof inertia in the Funds rate setting. Furthermore we show that regressionof interest-rate-smoothed Taylor Rule can successfully fit the data eventhough this data was being produced by the Optimal Timeless Rule.Hence we would argue that the Fed's optimal responses have beenmisinterpreted as ‘policy inertia’ due to this sort of misleading regression.
6. Simulated annealing and model evaluation with finalparameter selection
We have just evaluated three competing models differing solely inthe setting of the monetary policy rule. All of them share the same ISand Phillips Curve parameters, and so if these parameters could be as-sumed to be the true ones we would be testing these three policyrules solely. Of course we cannot strictly make this assumption, sincethese parameters could in principle be calibrated anywhere within arange permitted by the models' theoretical structure. A full test ofthese models would examine all these possible parameter values anduse only the best the model could deliver. We now examine whetherour results are robust to re-estimating eachmodelwith the best param-eters that can be found for it.
To test the model itself one is compelled to search over the fullrange of potential values permitted by the model's theory, to findthe best set of values from the model's viewpoint: if rejected onthat set then the model itself is rejected. Such a search is the purposeof indirect estimation. We perform this search by using the powerfulsearching algorithm of Simulated Annealing (SA) due to Ingber(1996). This mimics the behavior of the steel cooling process inwhich steel is cooled, with a degree of reheating at randomly chosen
moments in the cooling process to ensure that the defects are mini-mized globally. Thus the algorithm searches in a chosen range of pa-rameter values acceptable by the theory until it finds a set of valuesthat minimizes the distance between the model and the data andhence minimizes the Wald statistic (Eq. (7)) (here the full Wald sta-tistic). We then re-evaluate the models using our standard testingprocedure with these best-fitting parameter values.
Table 7 compares the SA estimates of these models to our bench-mark calibration. These are not substantially different from the calibrat-ed values. However, in terms of the Wald percentiles they improve themodels' performance substantially, as summarized in Table 8. This isparticularly true for the Taylor Rule model: the new estimates shiftit from being strongly rejected to being unrejected at the 90% level.Nevertheless it is still greatly dominated by the Optimal TimelessRule model which, with newly estimated parameters, also improvesto being unrejected at the 70% level. Its p-value is 0.314 against theTaylor Rule's 0.124. Hence, although it is possible to fit the post-1982US with a Taylor Rule model, policy in this period is still better under-stood as the Optimal Timeless Rule, which comes closest to the data inprobability terms.
Finally, we should see that this estimated Timeless Rule wouldalso explain other ‘Taylor Rules’ in the data, besides Eq. (8) that we il-lustrated using calibrated parameters in the last section: the Waldpercentiles in Table 9 suggest all the rules here are explained at95%, including case one and case three exceptionally at under 20%.Hence, from this last exercise we also know that the Timeless Rulemodel is robust in generating essentially the whole range of empiricalTaylor Rules that have been estimated for the US post-1982.
7. Conclusion
In this paper we have attempted to identify the principlesgoverning the US monetary policy since the early 1980s. TheTaylor Rule is widely regarded as a good description of these prin-ciples. Yet there is an identification problem plaguing efforts toestimate it as a single equation: other relationships implied bythe DSGE model in which it is embedded could imply a relation-ship that mimicked a Taylor Rule. To get around this problem wehave set up three complete models, each fully identified, withthe same New Keynesian structure but differing only in their mon-etary policy rules. These rules are the Optimal Timeless Rule, theoriginal Taylor Rule and another with interest rate smoothing.We show, using the method of indirect inference, that the OptimalTimeless Rule significantly outperforms the others in replicatingboth the dynamics and the volatility of the data. We also showthat if the Optimal Timeless Rule model was operating it wouldhave produced data in which regressions of typical Taylor Ruleswould have been found. In short, the policy of the Fed in thisperiod appears to have been approximately optimal and the factthat its behavior looks like a Taylor Rule with interest ratesmoothing is a statistical artifact.
Fig. A.1. US data between 1982q3 and 2007q4.
Appendix B. Plots of time series
Table A.2Unit Root test on the demeaned, detrended data (1982Q4–2007Q4).
Time series 5% critical value 10% critical value ADF test statistics p-Values *
Note: ‘*’ denotes the Mackinnon (1996) one-sided p-values.
Appendix C. Unit Root test for stationarity
Appendix D. Choosing a different starting date
Table A.3Estimates of the models when starting in 1984.
Parameters Definitions Calibrations SA estimates
TimelessRulemodel
TaylorRulemodel
β Time discount factor 0.99 Fixed at 0.99σ Inverse of intertemporal
consumption elasticity2 2.67 2.83
η Inverse of labor elasticity 3 2.53 3.40ω Calvo contract price
non-adjusting probability0.53 0.48 0.64
GY Steady-state gov. expenditure to
output ratio0.23 Fixed at 0.23
YC Steady-state output to
consumption ratio
10:77 Fixed at 1/0.77
κ κ ¼ 1−ωð Þ 1−ωβð Þω 0.42 0.57 0.21
γ γ ¼ κ ηþ σ YCð Þ 2.36 3.41 1.46
α Relative weight of loss assignedto output variations againstinflation
0.39 0.58 –
αγ ≡ 1
θ Optimal trade-off rate on theTimeless Rule
16
15:9 –
θ price elasticity of demand 6 5.9 –
γπ Interest-rate response toinflation
1.44 – 1.34
γ′x 0.14 – 0.09
(continued on next page)
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Our main text has suggested that the Optimal Timeless Rule modelhas a better fit to the data between 1982 and 2007 compared to theTaylor Rule model. While this choice of starting date is supported bythe Qu–Perron test, the data suggests the best date (of break) wasin 1984. Here we report both the estimates (Table A.3) and the per-formance (Table A.4) of the models when 1984 is chosen as thestarting date; it turns out that our benchmark results are robust tothis choice, i.e., a) it fails to change the model estimates significantly,and b) it fails to change the models' ranking.
Table A.4Performance of the models when starting in 1984.
Tests for chosen features Optimal TimelessRule model
Taylor Rulemodel
Directed Wald for dynamics 96.4 88.2Directed Wald for volatilities 2.3 99.3Full Wald for dynamics & volatilities 92.3 96.9
122 P. Minford, Z. Ou / Economic Modelling 32 (2013) 113–123
Appendix E. Choosing a higher order VAR representation
We also compare the models' performance in fitting the datadescriptors implied by VAR(2) and VAR(3) representations. While theVAR(1) we used in the main text has parsimoniously described howthe data behave, a VAR of higher orders would describe such behaviorin more detail. Clearly this would increase the rejection power of thetest since more (detailed data behavior) of the models are now askedto fit. Hence in practice this also points a way of further discriminatingbetween models whose performance is hardly distinguishable withparsimonious auxiliaries though checking robustness is our main pur-pose here.
The result of this exercise is reported in Table A.5. It shows when aVAR(2) or VAR(3) is substituted for the test it rejects both modelsstrongly, with failure mostly from fitting the dynamics part albeit theTimeless Rule model is fairly well sized still. Nevertheless the modelranking we established in the main text is unchanged: in this case thenormalized t statistics (in parentheses) indicate that—although beingrejected—the Timeless Rule model remains considerably closer to thedata compared to the Taylor Rule, which means that our ranking isrobust to the choice of among these VARs.
Table A.5Performance of models with different auxiliaries (1982–2007).
Tests for chosen features VAR(2) VAR(3)
TimelessRule
TaylorRule
TimelessRule
TaylorRule
Directed Wald for dynamics(normalized t-stat)
99.9 (4.33) 100 (9.38) 100 (10.1) 100 (13.7)
Directed Wald for volatilities(normalized t-stat)
93.7 (1.41) 99.9 (6.59) 90.7 (1.06) 100 (6.37)
Full Wald for dyn. & vol.(normalized t-stat)
100 (4.87) 100 (12.1) 100 (9.80) 100 (15.0)
Non-technical summary
For decades it has been ‘conventionalwisdom’ tomodel themonetarypolicy in the US using a Taylor Rule. Such a rule, named after the econo-mist John Taylor who first suggested it was not only likely to have goodmacroeconomic results but also resembled the behavior of the Fed
since themid-1980s, relates interest rates to the level of inflation relativeto the Fed's implicit target and to the gap between output and potentialoutput or its ‘natural rate’; it also may include lagged interest rateswhich have the effect of slowing down the responses to inflation andoutput—known as ‘interest rate smoothing’. While most research onthis field has focused on different specifications and statistical methodsof estimation for such a rule, few have squared up to the risk that esti-mates of it may be representing something else rather than the true be-havior of the Fed. This is known as the ‘identification problem’ of TaylorRules. In this paper, our purpose is to reconsider the principle accordingto which the Fed has conducted its policy since the early 80s. We do soin a novel way to get around the ‘identification problem’, and we findthat the Fed's behavior over the past decades is better understood asthe effort to optimize social welfare. We also find that this behaviormay well be misinterpreted as a Taylor Rule, since it produces interestrate reactions in this period that somewhat resemble that rule, including‘interest rate smoothing’.
Estimation of Taylor Rules has been a popular exercise for researchersaiming atmodeling the prevailingmonetary policy. However, whilemostmodelers believe that they could summarize the Fed's behavior by run-ning a Taylor-type regression, not all of them are aware of the fact thata statistical relationship between nominal interest rate and the two‘gaps’—one between inflation and its target, and the other between out-put and its natural rate—does not by itself provide any evidence for thetrue policy. This is because a variety of relationships within the economycan imply such a relationship, in a way that mimics a Taylor Rule. Hence,direct estimation of Taylor Rules on the data does not establish whetherthe Fed was actually pursuing them or not, as we simply do not knowwhat these rules are standing for.
One way to solve this problem is to set up competing models thatdiffer solely according to the monetary policies being followed, and todistinguish between these models according to the ability to replicatethe dynamic behavior of the data. This allows us to pick up the ‘true’model (as well as the ‘true’ policy) out of those that are merely ap-proximating it.
In this paperwe consider threepopular variants of hypothetical policy.These are: an optimality condition required by social welfare maximiza-tion, a standard Taylor Rule, and its modified version with ‘interest ratesmoothing’. We find the only model version that fails to be stronglyrejected by the data is the one where the Fed is assumed to pursuethe optimality condition. Interestingly, the implementation of such apolicy would have produced data in which regressions of an interest-rate-smoothed Taylor Rule would have been found, for which the‘true’ policy may be misunderstood as Taylor-type and the sluggishmovements of nominal interest rate required as optimal responsemay bemisinterpreted as deliberate ‘interest rate smoothing’ as a resultof ‘policy inertia’.
In short, our study here suggests the policy of the Fed since theearly 80s appears to have been approximately optimal and the factthat its behavior looks like a Taylor Rule with interest-rate smoothingis a statistical artifact.
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