Some Perils of Policy Rule Regression: The Taylor Rule Revisited Julio A. Carrillo∗ University of Toulouse (GREMAQ)

Patrick F`eve University of Toulouse (GREMAQ and IDEI) and Banque de France (Research Division)

April 2006 This version: February 2008

Abstract This paper analyzes the potential misidentification resulting from estimating policy rules trough single equations. In specific, we focus in the Taylor rule’s regression popularized in applied monetary economics. Using a simple dynamic, stochastic general equilibrium model, we show how easy is to find reduced forms relations among the interest rate and inflation that resemble the Taylor rule, even though the reduced form parameters are not policy invariant (thus illustrating the Lucas critique). Further, we show how limited–information estimation techniques (as GMM) can be subjected to structural bias whenever the econometrician misidentifies the actual policy. This turns out to be revealing for screening strong policy misspecifications, for which we propose to execute a series of regressions with specific set of instruments in order to rule out this possibility. We finally performed this robustness exercise for the Taylor rule as given by Clarida et al. (2000) using latest U.S. data. Our results point out that a strong misidentification is present in their estimations. Notably, weak evidence is found in terms of consistency favoring the monetary policy inertia hypothesis. Keywords: exogenous and endogenous policy rules, policy rule regression, nearly observational equivalent models, indeterminacy, policy misspecification JEL Class.: C22, E40, E52 ∗

Corresponding author: GREMAQ–Universit´e de Toulouse I, Manufacture des Tabacs, bˆ at. F, Aile Jean– Jacques Laffont, 21 all´ee de Brienne, 31000 Toulouse, France. Tel: + (33) 5-6112-8765. Fax: + (33) 5-6112-8637. Email: [email protected] and [email protected]. We would like to thank S. Auray, P. Beaudry, L. Christiano, F. Collard, B. Djembissi, M. Dupaigne, T. Kehoe, O. Licandro, J. Motis, F. Portier and J. V. Rios–Rull. All the assistants to the Toulouse Lunch Seminar, the 9th International Conference on Theory and Methods of Macroeconomics in Lyon, the X Workshop on Dynamic Macroeconomics in Vigo, the Banque de France’s Small Monetary Macro–Models workshop in Paris, the 2006 Southern Workshop in Macroeconomics in Auckland City, the 2006 Spring Meeting of Young Economists in Sevilla, the 21st Annual Congress of the European Economic Association in Vienna, and the 2006 Latin American Meeting of the Econometric Society in Mexico City for helpful discussions and comments. All errors are of course our own. J. A. Carrillo gratefully acknowledges the financial support granted by CONACyT (The National Council for Sciences and Technology of Mexico). The views expressed herein are those of the authors and not necessary those of the Banque de France.

1

Introduction

This paper analyzes the potential misidentification resulting from estimating policy rules trough single equations. We start by analyzing policy misspecifications from the point of view of a dynamic, stochastic general equilibrium (DSGE) model. We later jump to empirical grounds to test the implications of misidentifying the reaction function of the Federal Reserve in the context of the Taylor rule’s regression.

In recent years there has been an increasing interest about the identification and estimation of the policy rules that some economic authorities are assumed to follow. Notably, applied monetary economics estimates single equations using instrumental variables in order to compute the reaction function of the central bank, which is specified generally as a Taylor type form, i.e. with the nominal interest rate responding to deviations of inflation and output away their targets. The most notable example is guided by Clarida, Gal´ı and Gertler (2000) who estimate a Taylor rule with interest rate inertia using the Generalized Method of Moments (GMM). They document how changes in the coefficients’ value across their estimations could explain episodes for economic (and monetary) stability and volatility for different time periods.

The work of Clarida et al. (2000) has been criticized by recent contributions as it disregards the fact that their single equation could be more precisely estimated if it is included into a more elaborated system (Linde, 2002; Lubik and Schorfheide, 2004). Moreover, the single equation approach suffers from a lack of identification about structural parameters (especially when weak instruments are used), contrary to full–information environments (Mavroeidis, 2004a, 2004b; Nason and Smith, 2003). Another and more problematic issue is presented in Beyer and Farmer (2004, 2005), where the existence of observationally equivalent models makes impossible to relate a single class of policy with aggregate fluctuations that are driven by either fundamental shocks or sunspot shocks. Thus, knowing only the coefficients’ value of the central bank reaction function turns out to be insufficient to explain episodes of economic stability or volatility. All these critics, in general, call for a broader view when analyzing policy rules than the one settled by single equations.

The objectives of this paper are twofold. First, we ask what exactly is the econometrician 2

estimating when looking at the economic authorities’ reaction function: is it indeed the policy rule or just simply a reduced form? Second, we ask how useful is the information elicited from the estimation. The first peril concerns thus the misidentification of policy behavior and its possible causes. The second peril cares about the consistency properties of the instrumental variable–GMM (IV-GMM) estimator used in single equation analysis under the presence and absence of misspecifications.

Concerning the first peril, we use a simple monetary DSGE model to show rather easily that policy misidentifications may happen more often then expected. Due to the nature of general equilibrium dynamics, one can find reduced forms relations among the interest rate and inflation that resemble the Taylor rule, even though other policy was in place. In that case, the econometrician would undergo the Lucas critique since the estimated coefficients of the wrongly assumed policy rule are not policy invariant. This result follows closely the contributions of Beyer and Farmer (2004, 2005) about the existence of multiple observational equivalences among DSGE models.1

About the second peril, we show that the estimators used (generally, the IV–GMM) in the policy rule regressions can be subjected to a strong bias in the presence of misspecifications. In the later case, the error term of the policy rule equation could not fulfil the orthogonality conditions, resulting in bias in the estimated coefficients. In contrast, in the absence of misspecifications, the IV-GMM estimators are consistent to any choice of instruments. This latter result goes beyond the observational equivalence of Beyer and Farmer, since we focus in the econometric properties of the DSGE models. The test is simple: if the econometrician makes no mistakes in specifying the shape of the policy reaction function, then any set of instrument will deliver the same coefficients; in contrast, important misspecifications would make the error term in the regression equation to break the orthogonality conditions and different biases would result from different set of instruments. Thus, in order to screen important misspecifications, a series of regressions with very different instrument set can be performed.

To illustrate these findings, we consider the Taylor rule regression performed in Clarida et al. 1

They argue that several model structures share the same reduced form in terms of observed variables; the analyst is thus constrained by the limited information about the true structure of the economy and will be unable to retrieve the deep parameters of the model, if she is not willing to impose strong restrictions.

3

(2000). Our results suggest that their estimations about the Federal Reserve reaction function may suffer from important misspecifications, remarkably concerning the output gap and the monetary policy inertia terms. The latter are related with the willingness of the monetary authorities to smooth the interest rate. These results provide evidence that single equation analysis might not provide significant information about the structural preferences of the central bank.

The paper is organized as follows. Section 1 introduces the idea of policy misidentification and the principle of policy rule regression using a simple rational expectations model. The asymptotic properties of the IV–GMM estimators are also covered. In section 2, we build a simple DSGE model to account for the misidentification in the context of monetary models. In section 3 we run the Taylor rule as specified in Clarida et al. (2000) using U.S. data and report the results. Finally, the last section presents some concluding remarks.

1

An Introductory Example to Policy Rule Regression

What we call a policy rule regression is an estimation procedure that relies on limited information estimation techniques, as Instrumental Variables (IV) or in a broad sense, the Generalized Method of Moments (GMM), in the form of a single equation. These methods have gain in popularity because of the agreed upon clich´e that “all models are false” and because they put few restrictions on the data. Although specifying the whole model is not necessary for estimation purposes, a single equation approach is not prevented of policy rule misspecification and could undergo the Lucas critique. The obvious consequence is to produce spurious policy recommendations. In this section, we are interested in showing two issues: (i) the presence of observational equivalent models make policy misidentification a common and problematic subject, and (ii) radical robustness exercises within limited information estimation techniques may elicit the presence of misspecification.

1.1

One shock economy: Two observationally equivalent models

In what follows, we present two model economies that have the same reduced forms. Both are deliberately stylized in order to deliver clear results about policy rule regression. Further references on observational equivalent representations of determinate and indeterminate models can be found in Beyer and Farmer (2004, 2005).

4

1.1.1

The exogenous rule economy

We consider an economy that expresses a single endogenous variable yt as a linear function of the conditional (on the available information at time t) expectation of this same variable in period t + 1 and a policy instrument xt governed by an exogenous stationary process. This economy takes the form: yt = aEt yt+1 + bxt

(1)

xt = ρxt−1 + σεt

with εt ∼ iid(0, 1)

(2)

where b 6= 0, |a| < 1, |ρ| < 1, σ > 0 and Et is the conditional expectation operator with respect to the current and past values of {xt , xt−1 , ..., yt , yt−1 , ....}. The later assumptions imply that the equilibrium will be determinate. The AR(1) specification of xt is widely used in macroeconomic dynamics. It allows to simply represent the time series behavior of the forcing variable. Moreover, one must remark that xt can be correlated with yt . In this case, we have just to modify (1), taking into account the feedback effect of xt on yt . The AR(1) specification then represents the pure exogenous component in xt . A stationary solution would be thus obtained iterating (1) forward. Without loss of generality, we omit a constant term both in (1) and (2) since the endogenous and exogenous variables can be considered as deviations from their long– term average value. Equation (1) can represent either a linear or a log–linear approximation to the equilibrium conditions of a non–linear model. Despite its simplicity, this simple linear representation embodies several model economies.2 Using forward substitutions, the endogenous variable is expressed by yt = bEt

T X

ai xt+i + aT +1 Et yt+T +1

i=0

Excluding explosive paths, the forward solution takes the simple form: yt = bEt

∞ X

ai xt+i

i=0

Using the exogenous policy rule (2), the solution is Ã∞ ! X yt = b ai ρi xt i=0

= 2

b xt 1 − aρ

See section 2 for a typical illustration.

5

∀ t

(3)

This reduced form expresses the endogenous variable yt as a linear function of the exogenous variable xt . This solution shows that when the parameter ρ of the exogenous policy rule changes, the reduced form parameter b/(1 − aρ) will be evidently affected. The lack of policy invariance of the reduced form parameter is a basic illustration of the Lucas critique. Note as well that we can rewrite the reduced form of model (1)-(2) as 1 − aρ yt b yt = ρyt−1 + σy εt

xt =

(4) ∀ t

(5)

where σy = bσ/(1 − aρ). We now move to the second economy. 1.1.2

The endogenous (forward–looking) rule economy

We assume that the endogenous variable yt is still represented by equation (1). However, in this economy the policy instrument is stated to answer the future values of the endogenous variable. This model is represented by yt = aEt yt+1 + bxt xt = ηEt yt+1

(6)

where η 6= 0. The later forward–looking policy rule is said to be endogenous since it reacts to expected changes of the endogenous variable, receiving feedback from the dynamics of the economy. Moreover, the variable xt is not fixed by extraneous process. It is also common in the literature to suggest rules as (6), been one famous example the Taylor rule involving the interest rate reacting to forward inflation and output gap in monetary economics. The aggregate dynamics of the above model are obtained by replacing in the endogenous equation the policy rule, which leaves yt = (a + bη)Et yt+1 For the purpose of our analysis, we assume that the absolute value of a + bη is greater than one. Thus, the equilibrium is indeterminate and can be driven by the sunspot shock νy,t = yt −Et−1 yt .3 Stating the above equation in a backward perspective, and replacing Et yt+1 in the policy rule 3 For further details about indeterminate models or irregular equilibria see Farmer (1999) or Benhabib and Farmer (1999).

6

equation, delivers the following reduced form η yt a + bη 1 yt = yt−1 + νy,t a + bη

xt =

(7) ∀ t

(8)

It is clear that the two reduced forms, (2)–(3) or equivalently (4)–(5) for the determinate model and (7)–(8) for the indeterminate model, are observationally equivalent. Using actual data, it will be impossible to decide whether {yt , xt }Tt=1 is generated by a determinate or an indeterminate process without any extra information about the way the economic policy (namely, the variable xt in our simple example) is conducted. This introductory example with exogenous and endogenous policy rule is another illustration of the lack of identification from the reduced forms in linear rational expectations models (Beyer and Farmer, 2004, 2005). We now investigate the issue of policy rule regression. 1.1.3

Policy rule regression

Assume an econometrician is asked to estimate the policy rule behind the behavior of xt . If the only information the econometrician has is a scatter plot between xt and yt , as is shown in the hypothetical data set of figure 1, a natural question would be how to discriminate between the exogenous rule - eq. (2) - and the endogenous rule - eq. (6) ? The answer is that in the single shock economy this is not possible. It is straightforward to see that if ρ=

1 a + bη

and V ar(νy,t ) = σy ,

both economies are observationally equivalent, since their reduced forms are just the same. Let us assume that the econometrician is about to estimate a forward–looking policy rule, of the form xt = ηprr Et yt+1 + εprr,t

(9)

where εprr,t is the policy rule estimation error. The estimation properties of the above equation are conditional on the model economy that describes the real world. If the true world is depicted as the exogenous rule economy states, then the reduced form parameter ηprr ≡ (1 − aρ)/bρ and εprr,t ≡ 0. In this case, the lack of policy invariance of the reduced form parameter provides an illustration of the Lucas critique. In the contrary, if the forward–looking policy rule economy is the true world, then ηprr ≡ η and εprr,t ≡ 0. We claim that it is precisely because the policy rule error term, εprr,t , is similar in both models (in this case, equivalent to zero) we cannot discriminate using standard econometric techniques between the two competing policy rules. 7

Figure 1: Policy rule regression for both models economies in the one shock world x

t

ε

prr,t

x =plim η t

prr

E y

t t+1

=η

prr

E y

t t+1

yt+1 Note: The solid line represents the estimation equation of the “assumed” policy rule, determined by the reduce form of each model economy. The two dotted lines represent upper and lower bounds of the policy rule estimation error. The estimated policy rule coincides with both models’ reduced forms, since in the one shock world no bias is present.

The following is trivial, but it will help to fix ideas on the relevant message to be presented in the next section. As the usual practice, we consider an Instrumental Variables technique (hereafter, IV–GMM) in order to estimate the policy parameter of equation (9). To avoid endogeneity issues, empirical studies use a set of predetermined – or weekly exogenous – instrumental variables. We follow here exactly the same empirical strategy. For simplicity and tractability, we assume that the econometrician uses a single instrument, which is enough to fulfilled the necessary condition for identification of the policy rule parameter ηprr . In this case, our set of possible instruments is defined by Zt = {xt−1 , yt−1 }. In terms of observable variables, equation (9) can be rewritten as: xt = ηprr yt+1 − ηprr νy,t+1 + εprr,t

(10)

Let zt ∈ Zt denote a single instrument known in period t. The later implies that the probabilistic limit of the IV estimator using instrument zt is given by plim ηprr (zt ) ≡

E(xt zt ) = ηprr E(yt+1 zt )

(11)

for any zt ∈ Zt .4 Using IV estimation is completely uninformative about the true nature of the 4

This follows from the fact that, according with the two models presented, E[(εprr,t − ηprr νy,t+1 )zt ] = 0 for

8

policy rule, since still we ignore if ηprr ≡ (1 − aρ)/bρ, according to the first model, or ηprr ≡ η according to the second.

1.2

Multiple shocks world: Two nearly observational equivalent models

Dynamic macroeconomic models include several shocks – productivity, government spending, tastes, money supply – in order to improve the specification of the endogenous variable. Moreover, a typical exercise in the business cycle literature is to identify the various sources of aggregate fluctuations and thus to evaluate their relative contribution. We thus extend the previous models to the case of multiple exogenous variables. 1.2.1

Exogenous rule economy

Assume there exists m different exogenous stationary processes, among economic policy decisions and fundamental shocks, affecting the dynamics of the economy. Specifically, the model economy is described by yt = aEt yt+1 +

m X

bj xj,t

(12)

j=1

where as before |a| < 1 and each exogenous variable xj,t follows an AR(1) process xj,t = ρj xj,t−1 + σj εj,t

with εj,t ∼ iid(0, 1)

(13)

where |ρj | < 1 and σj > 0. We assume that the innovations verify E(εj,t εj 0 ,t ) = 0

∀j 6= j 0 ,

so, the exogenous variables are uncorrelated. For exposition purposes, we denote x1,t as the policy instrument of interest. Iterating (12) forward and using the exogenous policy rules (13), the solution is given by yt =

m X j=1

bj xj,t 1 − aρj

(14)

or, its equivalent representation: x1,t

m 1 − aρ1 1 − aρ1 X bj ρj = Et yt+1 − xj,t b1 ρ1 b1 ρ1 1 − aρj

(15)

j=2

Now, suppose that the econometrician wants to estimate the same policy rule as before. Our set of possible instruments changes to Z = {x1,t−1 , yt−1 }.5 In terms of the policy instrument of any zt ∈ Zt . 5 Other instruments, like xj,t are of not concern, since they are uncorrelated with x1,t .

9

interest, the estimation equation rewrites: x1,t = ηprr Et yt+1 + εprr 1 We see immediately that the solution of the model implies that ηprr ≡ 1−aρ b1 ρ1 and εprr ≡ P bj ρj m 1 − 1−aρ j=2 1−aρj xj,t . The later tell us that the plim ηprr (zt ) will not be free from bias for b1 ρ1

all zt ∈ Z, i.e. plim ηprr (zt ) ≡

m E(xt zt ) 1 − aρ1 X bj ρj E(xj,t zt ) = ηprr − E(yt+1 zt ) b1 ρ1 1 − aρj E(yt+1 zt )

(16)

j=2

In particular, we have that only the plim ηprr (x1,t−1 ) will be a consistent estimate of ηprr , whereas plim ηprr (yt−1 ) can be stated as

1 − aρ1 µ b1 ρ1

where µ=

V (y /x ) Pm t 1,t 2 V (yt /x1,t ) + j=2 (ρj /ρ1 ) V (yt /xj,t )

Since V (yt /xj,t ) denotes the variance of yt conditional on xj,t , µ can be interpreted as the weighted contribution of x1,t to the variance of yt . This is clear when all the weights are the same, i.e. when ρ1 = . . . = ρm and µ simply reduces to µ=

V (yt /x1,t ) V (yt )

If the contribution of x1,t to the variance of yt is large, we have µ ' 1 and the IV–GMM estimator is the same as the one obtained in the single exogenous variable case. Conversely, if the contribution of x1,t to the variance of yt is very small, we have µ ' 0 and plim ηprr (yt−1 ) ' 0. Figure 2 illustrates precisely the latter case for the policy rule regression.

This first insight about the consistency properties of the IV–GMM estimator can be summarized as follows: changing the instrument set challenges parameter robustness and, thus, may point out the presence of policy misspecification. 1.2.2

The endogenous rule economy

We now study the implied dynamics of the forward–looking rule economy with multiple shocks. Let us assume that the only difference with the model in the preceding subsection is the determination of the x1,t policy instrument, which follows the rule:6 x1,t = ηEt yt+1 6

Our results are left unaffected when this rule includes an iid policy shock.

10

Figure 2: Policy rule regression with multiple variables for the exogenous rule economy x1,t ε

prr,t

x = plim η 1,t

prr

E y

t t+1

x1,t= ηprr Et yt+1

yt+1 Note: The solid line is the estimation equation of the reduced form of the model with multiple exogenous variables and exogenous rule. The two thin dotted lines represent upper and lower bounds of the regression equation error term, determined by the exogenous variables distribution supports, given in the reduced form. The gross dotted line is the estimated policy rule when the exogenous variable x1,t explains a tiny portion of the variance of yt .

11

After substituting the later into (12), the dynamics of the economy is given by: yt = (a + b1 η)yt+1 +

m X

bj xj,t − (a + bη)νy,t+1

j=2

As before, we assume that |a + b1 η| > 1, so the economy displays an irregular equilibrium and is given by:

m

X bj 1 yt = yt−1 − xj,t−1 + νy,t a + b1 η a + b1 η j=2

where νy,t is, again, a sunspot shock that verifies νy,t = yt − Et−1 yt . If the econometrician does the same job as before, the policy rule parameter ηprr ≡ η and εprr,t ≡ 0. Unambiguously, plim ηprr (zt ) is consistent for any instrument, since plim ηprr (zt ) = ηprr given that E(νt+1 zt ) = 0 for zt ∈ {x1,t−1 , yt−1 }. These two sections have insisted in two important results: i) Policy misspecification may happen more often than expected, due to the existence of -at least nearly- observationally equivalent models, and ii) in the presence of multiple shocks, one should challenge the consistency of the IV–GMM estimators in order to elicit the presence of misidentification in the means of using different instrument sets. The rest of the paper applies this methodology to the Taylor rule of applied monetary economics.

2

The Taylor rule revisited

Since Taylor (1993), abundant empirical evidence and a quite number of monetary models have defended the use of the Taylor rule as an effective interest rate policy rule to control prices and stabilize the economy. The surprising fit of this equation in terms of inflation and output gap for the Volcker–Greenspan era could perfectly induce the hypothesis that the Federal Reserve Board preferences can be rationalized trough a linear equation linking these three variables. Figure 3 illustrates a part of this rationale, as it suggests the existence of a linear relationship between the federal funds rate and the one–quarter forward annualized inflation rate of the U.S. for 1982:4 trough 2004:3. Such relationship shows that the estimated coefficient in the figure would be grater than one for the sample period considered. The estimating equation of Clarida 12

Figure 3: Nominal interest rate vs inflation, the Volscker–Greenspan era 12

Federal funds rate

10

8

6

4

45° line 2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Annualized one−period forward inflation Note: Period 1982:4 trough 2004:3, with quarterly frequency data. Source: the Federal Reserve System and the Bureau of Economic Analysis.

et al. (2000) would become the common wisdom when referring to the Taylor rule for the U.S. case, where the nominal interest rate reacts to changes in inflation and output away their targets, plus a term reflecting policy inertia. The stabilization of some macroeconomic variables, specially the price index, during the tenures of Paul Volcker and Alan Greenspan would make economists to think that an aggressive reaction of the funds rate (responding by more than a one-by-one basis) would fight inflation instead of accommodating it. Furthermore, the usual monetary models of sticky–prices7 accept the so called Taylor principle, in what an aggressive reaction of the nominal interest rate to inflation gap brings about determinacy to nominal and real variables.8

However, some sceptics have argued that the estimation of interest rate rules trough single equations might not provide sufficient information about the true behavior of policymakers (e.g. Beyer and Farmer, 2004, 2005; Lubik and Schorfheide, 2004; Mavroeidis 2004a, 2004b). Notably, Hetzel (2000) remarks that the estimated rules could describe rather the final interest rate equilibrium than the central bank’s performance. This is, they could entail, instead, the 7

See Walsh (2003), ch. 5, for a survey. Although this is not a general result, since it heavily depends on special arrangements for price–setting and expectations formation. 8

13

reaction of agents given their expectations.

A minor interest has been left to discussion about the problems that may arise when there is a monetary rule misidentification, specially for policy recommendations. We claim, as Beyer and Farmer (2004, 2005), that this misspecification cannot be ruled out, since there exists - nearly - observational equivalent models with similar reduced forms. In addition, we also provide a characterization of the IV–GMM estimators (commonly used in single equation analysis) that can help the econometrician to elicit the presence of a misspecified rule. We now present two model economies with two rival policy rules.

2.1

The money growth rule economy

Consider an economy composed by a unit mass of infinite–lived expected utility maximizer agents, indexed by i ∈ [0, 1]. There is also an infinite number of firms, mass one, producing differentiated final goods, indexed by j ∈ [0, 1], with labor as the only input. For the ease of the analysis, let us assume that there is no capital accumulation. Additionally, there exist a government who collects lump–sum taxes and provides money, bonds and lump–sum transfers to the agents in the economy.

Households The representative household carries from period t − 1 into t the nominal balances Mt , the total revenues from bond holdings Rt−1 Bt of a riskless one–period bond issued by the government, where R is the gross nominal interest rate. In period t, the household has a disposable nominal income Wt ht − Tt per ht worked hours, where Wt is the nominal wage and the Tt denotes lump– sump taxes. Additionally, the household receives from the government the lump–sum transfer R1 Nt and the profits payed out from firms Ψt = 0 Ψj,t dj at the end of the period. All the revenues are used to buy the consumption bundle, money balances and bond holdings for the next period. Consequently, its budget constraint is Bt+1 + Mt+1 + Pt Ct + Tt 6 Wt ht + Rt−1 Bt + Mt + Nt + Ψt

(17)

where Pt is the general price index. We assume, as Cooley and Hansen (1995), that the household needs cash in order to buy Ct and Bt+1 (financial markets open first), so its CIA constraint takes the form Pt Ct + Tt + Bt+1 6 Mt + Rt−1 Bt + Nt . 14

(18)

Total profits, Ψt , are not included since they are not available till the end of the period.9 Let us assume that the instantaneous utility function over consumption and labor is described by U (Ct , ht ) = log Ct −ht . Therefore, the representative household maximization problem is simply max

Cτ ,hτ ,Mτ +1 ,Bτ +1

Et

∞ X

β τ −t (log Cτ − hτ )

τ =t

subject to (17) and (18). The subjective discount factor β lies in the (0, 1) interval, and Et denotes the conditional expectations operator given the available information at period t. The optimal behavior in consumption, labor, money and bonds are resumed by (taking for granted £ ¤ £ ¤ the existence of a positive interest rate) 1 = βWt Et (Pt+1 Ct+1 )−1 and Rt βEt (Pt+1 Ct+1 )−1 = (Pt Ct )−1 . Parallel to the lifetime utility maximization, the household must decide how much of each good j consume. If we assume that the consumption bundle is formed trough the Dixit–Stiglitz aggregator,

·Z Ct =

1

0

ξ−1 ξ

ξ ¸ ξ−1

Cj,t dj

(19)

where ξ > 1 is the elasticity of substitution among the consumption goods, then the minimizaR tion of consumption expenditures, i.e. minCj,t j∈[0,1] Pj,t Cj,t w.r.t. equation (19), leads to the following household demand for good j: ·

Cj,t

Pj,t = Pt

¸−ξ Ct

This implies that the general price index is given by Pt =

(20) hR 1 0

1−ξ Pj,t dj

i

1 1−ξ

.

Firms We consider a representative firm whose output, Yj,t , is produced by the linear technology Yj,t = At hj,t

(21)

where At is a common technology shock for all firms. Prices are flexible, in the sense that they can be adjusted once per period, but as Gal´ı (1999) we assume that entrepreneurs set their prices at the end of period t − 1, i.e. before observing any shock of period t.10 This is the source 9

If the nominal interest rate is positive, (17) and (18) imply that Mt+1 = Wt ht + Ψt . This can be saw also as a variation of the Mankiw and Reis (2002)’s sticky–information model, in which price–sluggishness comes from the lags in the available information of firms; here this lag is reduced to a single period. It is worth noting that the sticky–information assumption allows for some inflation inertia. We avoid the usual Calvo pricing to allow for a discussion later on about the choice of a “suitable” monetary policy rule. 10

15

of stickiness in the model. Firms choose prices according to µ ¶ Pj,t ∈ argmax Et−1 δt Pj,t Yj,t − Wt hj,t

w.r.t equations

(19),

(20) and

(22)

(21).

where δt = β Ct1Pt is the appropriate discounting rate for the firm, given that it is own by households. All firms are essentially equal and at equilibrium they all choose the same price, given by Pt =

t ξEt−1 YCttW At

Yt (ξ − 1)Et−1 C t

(23)

Government and central bank The budget constraint of the government is given by Pt Gt − Tt = Mt+1 − Mt − Nt + Bt+1 − Rt−1 Bt where Gt is real government spending, which without loss of generality can be normalized to zero, and M0 and B0 are given. Money supply is described by Mt+1 = γt Mt . The gross growth rate of money, γt , follows an exogenous rule that will be given bellow.

16

Monetary policy rule and shocks The gross growth rate of money and the technical progress are stochastically constant or stationary around a long–term value. We consider two fundamental uncorrelated disturbances for each of them. Both are assumed to follow AR(1) processes log γt = ργ log γt−1 + (1 − ργ ) log γ¯ + σε,γ εγ,t

(24)

log At = ρa log At−1 + (1 − ρa ) log A¯ + σε,a εa,t

(25)

where εk,t ∼ iid(0, 1), with E(εk,t εk0 ,t ) = 0 ∀ k 6= k 0 . Further, σε,k > 0 and |ρk | ∈ (0, 1) for all k ∈ {a, γ}.

© Definition 1 (Equilibrium) An equilibrium is a sequence of prices {Pj,t : j ∈ [0, 1]},Wt , ª∞ ª∞ © Rt t=0 and a sequence of allocations {Cj,t , Yj,t , hj,t : j ∈ [0, 1]}, Mt+1 , Bt+1 t=0 , such that given prices, these sequences maximize profits, households’ utility, and clears all markets every t. A log–linear approximation around the deterministic steady state characterizes the solution of the model in terms of the forcing variables denoted by the vector [ρa ργ σε,a σε,γ ]0 . The following equations characterize the aggregate dynamics at equilibrium (a hat denotes the percentage deviations from the deterministic steady state): yˆt = cˆt

(26)

w ˆt = Et [ˆ pt+1 + cˆt+1 ]

(27)

ˆ t = Et (ˆ R πt+1 + cˆt+1 ) − cˆt

(28)

pˆt = Et−1 (w ˆt − a ˆt )

(29)

π ˆt+1 = γˆt+1 − yˆt+1 + yˆt

(30)

with (24) and (25) to complete the system. Equations (27) and (28) come from the decision rules of the household, (29) from the price choice of firms, and (30), where πt = pt − pt−1 is the inflation rate, is an implication from the CIA constraint and the government budget constraint. The aggregate dynamics at equilibrium of the model economy are presented in the next proposition.

17

Proposition 2.1 (Equilibrium for the money growth rule economy) The aggregate dynamics of the economy denoted by equations (24)-(30) present a regular or determinate equilibrium, giving the following solutions for output, inflation, and the interest rate: yˆt = γˆt − ργ (1 + ργ )ˆ γt−1 + ρa a ˆt−1 µ ¶ π ˆt = γˆt−1 + (1 − L) ργ (1 + ργ )ˆ γt−1 − ρa a ˆt−1

(31)

ˆ t = ργ γˆt R

(33)

(32)

where L is the lag operator, such that Lxt = xt−1 . Proof: See appendix.

2

The money growth rule economy does not present any indeterminacy as long as the growth rate of money follow an exogenous stationary process. Despite its simple resolution, the preset–price assumption or lagged information arrival, is able to make inflation persistent at least for one period, since the inflation rate does not answer immediately to a monetary policy shock.11 We now present the second economy, where the monetary policy is represented by a Taylor rule.

2.2

The Taylor rule economy

The behavior of households, firms and the fiscal authority is completely analogous as the money growth rule economy. The central bank, however, is assumed to follow a Taylor rule of the form ˆ t = ηEt π R ˆt+1 + σu ut

(34)

where ut is an iid(0,1) policy shock and σu > 0. We consider this rule, that relates the nominal interest rate only with inflation, as the implied dynamic properties will depend on a single parameter. Moreover, previous empirical results suggest that the estimates of η are unambiguously significant and positive for the Volcker–Greenspan era, while the output gap has shown a less important contribution in predicting the target interest rate (Clarida et al., 2000).12

Since the growth rate of money is now endogenous, the Taylor rule economy is summarized by the system of equations (25)–(30) plus (34). As the recurrent assumption, we set η > 1 to represent an aggressive interest rate rule. The dynamic properties of the economy are described by the proposition below. 11

The same qualitative results can be found in Mankiw and Reis (2002), although in a larger magnitude. We have investigated also rules that include policy inertia terms, or lagged interest rate terms, and all the results are qualitative unaltered, though unnecessary cumbersome. 12

18

Proposition 2.2 (Equilibrium for the Taylor rule economy) If η > 1, the aggregate dynamics of the model economy described by (25)–(30) and (34) present an irregular or indeterminate equilibrium, with two different sunspot shocks, giving the following solutions for output and inflation: · ¸ 1 yˆt = yˆt−1 + (η − 1)ρa a ˆt−2 + σu ut−1 + (η − 1)νw,t−1 + νy,t η

(35)

· ¸ 1 π ˆt = πt−1 + ρa (1 − L)at−2 − σu ut−1 + (1 − L)νw,t−1 − νy,t−1 η

(36)

where νw,t = (w ˆt − Et−1 w ˆt ) and νy,t = (ˆ yt − Et−1 yˆt ) are two sunspot shocks. Proof: See appendix.

2

The first shock in beliefs, νw,t from nominal wages, is a nominal sunspot that is carried into the real variables; this is an example of a nominal indeterminacy that is translated into real indeterminacy (Carlstrom and Fuerst, 2000, for another illustration). The second sunspot, νy,t from the output gap, results of having an aggressive coefficient in the interest rate rule; indeed, in a preset–price economy, an active Taylor rule leads to real indeterminacy. This contrasts the Taylor principle statement of achieving determinacy trough an aggressive rule found in some sticky–prices models (as Calvo pricing).

Indeed, the so called Taylor principle breaks when we consider environments where prices are assumed to be more flexible. Recall that the sluggishness in the price index in the preset–price model comes from the lagged available information of firms, that prevent them to change their prices instantaneously to current innovations in the shocks of the economy. A result of the sticky–information models, from which the preset–price is the simplest representation, is that they can explain some inflation inertia without imposing exogenous restrictions in price–setting, such as Calvo pricing (Walsh, 2003, ch. 5) or rule-of-thumb indexation (Gal´ı and Gertler, 1999). Whereas one model specification is better than the other is a discussion that lies beyond this paper. Here, we just want to point out that the Taylor principle is not as general as it is usually perceived, and that its efficiency to stabilize inflation depends on specific assumptions about price–setting.

19

3

Policy rule regression: The Taylor rule

In this section, we study the properties of the IV–GMM estimators embodied by the structure of both model economies. In addition, we proposed two quantitative exercises in which we exemplify the relevance of policy misidentification. The first is about a calibration experience within the money growth rule economy while the second is an empirical exercise concerning the estimation of the Taylor rule for the U.S.

3.1

The econometrics of the money growth rule economy

Assume that an econometrician is asked to retrieve the preferences of the central bank into a policy rule equation. In this subsection, we assume that the “true” world is the one described by the money growth rule economy. Our interest is on the risk and consequences of misspecification, so we let the econometrician to perform a Taylor rule regression, i.e., to estimate the following equation: ˆ t = ηprr Et π R ˆt+1 + εprr,t .

(37)

Note, however, that if the true world is given by system (24)–(30), which imply the solution of proposition 2.1, one can find a linear relationship between the nominal interest rate and inflation gap as:13 1 + ργ ρa ˆ t = 1 Et π ˆt+1 − σ²,γ εγ,t + (1 − L)ˆ at R ργ ργ ργ

(38)

Some worthy observations are the following. First, the relation above is an equilibrium condition, not a policy rule. Second, in the preset–price model, the nominal interest rate will react by more than a one-by-one basis to inflation gap as long as the growth rate of money be a stationary process, taken as granted that no other shock is present. Third, if the above were to be estimated by IV–GMM, as commonly done in the literature, not all of the predetermined instruments would fulfil the orthogonality conditions.

This latter remark is next formally presented. For the ease of the exposition, assume that the econometrician uses a single instrument, denoted by zt , to estimate the policy rule parameter trough IV–GMM. Since in this world ηprr ≡

1 ργ

13

and εprr,t ≡

1+ργ ργ σ²,γ εγ,t

−

ρa ργ (1

− L)ˆ at , the

Note that solution for inflation in proposition 2.1 can be rewritten as π ˆ t+1 = γˆt + (1 + ργ )(ργ γt − γt + εγˆ ,t ) − ˆ t = ργ γˆt , solving the later for the interest rate gives the expression below. ρa (ˆ at − a ˆt−1 ), since R

20

IV–GMM estimator of ηprr would be the ratio between two covariances, as14 plim ηprr (zt ) ≡

ˆ t zt ) E(εprr,t zt ) E(R = ηprr + E(ˆ πt+1 zt ) E(ˆ πt+1 zt )

(39)

The last term constitutes the potential bias of the policy rule regression. The classic assumption is that this term is equal to zero for the consistency of the IV–GMM estimator. In the presence of misidentification, this is not necessarily the case.

As in previous empirical works, let us consider the following set of weakly exogenous or pren o ˆ t−1 . We are particularly interested in the special determined instruments Z = π ˆt−1 , yˆt−1 , R effect of changing the estimating instrument. Proposition 3.1 shows the probabilistic limits for the IV–GMM estimator of η, conditional on each of the above instruments and given that the world is the money growth rule economy.15 Proposition 3.1 (IV–GMM estimators) The probabilistic limit of the GMM estimators ˆ t−1 } of ηprr are given by: plim ηprr (zt ) for zt ∈ Z = {ˆ πt−1 , yˆt−1 , R plim η(ˆ πt−1 ) =

1 + ργ

ρ3a V ar(ˆ at )(1 − ρa )2 µ ¶ 4 2 3 2 ργ ργ V ar(ˆ γt )(1 + ργ (1 − ργ )) − ρa V ar(ˆ at )(1 − ρa )

plim η(ˆ yt−1 ) =

1 − ργ

ρ3a V ar(ˆ at )(1 − ρa ) µ ¶ ργ ρ3γ V (ˆ γt )(1 − ρ2γ (1 + ργ )) + ρ3a V (ˆ at )(1 − ρa )

ˆ t−1 ) = 1 plim η(R ργ where V ar(kt ) =

2 σε,k 1−ρ2k

for k ∈ {γ, a}.

Proof: See appendix.

2

In this example, two out of three instruments do not deliver consistent estimators. This comes from the fact that the policy rule error εprr is not an uncorrelated innovation, given its intrinsic relation with the persistency of technology. Just the interest rate not imply any bias, since in the preset–price model as stated in the money growth rule economy this variable is purely 14

Since all variables are zero-mean, the covariance between two variables is given by the crossed expectation. In ˆ t zt ) = ηprr E(ˆ this sense, multiplying instrument zt into (37) and taking total expectations yields: E(R πt+1 zt ) + E(εprr,t zt ). The rest is just solving the later for plim η˜prr . 15 Clarida, Gal´ı and Gertler (2000) include up to four lags in the inflation rate. To keep tractable results, we do not introduce over–identifying conditions.

21

determined by money growth rate.

It is clear that the instruments considered are qualitatively different, and that these differences surely influence the absolute size of the bias. For example, if the output gap is mostly driven by technology shocks than money growth disturbances, we should expect that the size of the bias will be greater for this instrument than for any other else. 3.1.1

A calibrated exercise

The aim of this subsection is to illustrate numerically, within some standard values of the literature for the U.S., the probabilistic limits provided in proposition 3.1. Quite surprisingly, two out of three calibrated estimators are not so different from their empirical counterparts of the estimated Taylor rule for the U.S. Table 1 presents the parametrization that we consider for this quantitative exercise.

Table 1: Parameter values for calibration Parameters (ρa , σε,a ) (ργ , σε,γ )

Values (0.98, 0.0072) (0.49, 0.0089)

Source King and Rebelo (1999) Cooley and Hansen (1995)

Note : The technology coefficients are updated by King and Rebelo (1999) over the sample period 1947:1–1996:4, quarterly observations for the U.S. Cooley and Hansen (1995) reports the parameters of an AR(1) process for the M1 growth over the period 1954:1–1991:2.

Table 2: The calibrated and estimated policy rule parameter

Money growth rule economy Clarida et al. (2000)

plim ηprr (ˆ πt−1 ) plim ηprr (ˆ yt−1 ) ˆ t−1 ) plim ηprr (R

2.17 0.50 2.04 2.15

Note: The estimation of Clarida et al. (2000) corresponds to the tenures of Paul Volcker and Alan Greenspan over the sample period 1979:3–1996:4, with quarterly frequency data.

The first three lines of Table 2 contains the calibrated values for the different plim ηprr conditional to the instrument used; the bottom of the table shows the estimation of Clarida et al. (2000) of the inflation coefficient for the Taylor rule in the Volcker–Greenspan era. As noted before, only the nominal interest rate yields a consistent IV–GMM estimator, whereas 22

the others are not free from biases. In fact, the greatest bias is obtained with the output gap, which reaches −1.54 estimate points. This should not be very surprising since this variable is mostly driven by technology shocks, as it is illustrated by its money growth rule model-based variance decomposition in table 3, where after 100 periods the technology innovations accounts for 93.2 per cent of the total variance of output. Notice as well that output reacts immediately to a money shock, but the effects of the later are not persistent and decay at increasing rates over time, been extensively dominated by the technology disturbance. Table 3 also shows the Table 3: Variance decomposition (in %) ˆt R

π ˆt

yˆt

Horizon 1 2 3 4 5 .. .

a ˆt – 17.3 17.3 17.3 17.3

γˆt – 82.7 82.7 82.7 82.7

a ˆt 0 0 0 0 0

γˆt 100 100 100 100 100

a ˆt 0 37.2 53.4 62.7 68.6

γˆt 100 62.8 46.6 37.3 31.4

100

17.4

82.6

0

100

93.2

06.8

Note: The variance decomposition was performed given the calibrated values of Table 1.

variance decomposition for inflation and the nominal interest rate, which are mostly affected by money shocks; more than 80 per cent of their variation is explained by money disturbances at every period. This is the reason for which their biases are rather small and null, respectively. It is also remarkable the similarity between the magnitudes of these IV–GMM plim estimators and the baseline point estimate of Clarida et al. (2000), shown at the bottom of Table 2. This observation naturally rises the question of a possible misidentification on the current estimations of Clarida et al., since even a simple preset–price model with a money growth rule is able to match the empirical value of the inflation coefficient in the Taylor rule. Let us now explore the properties of the IV–GMM estimators if the true world were given by the Taylor rule economy described in subsection 2.2.

3.2

The econometrics of the Taylor rule economy

If the true world is the Taylor rule economy, consistency for any predetermined instrument is trivially achieved following the assumptions detailed in subsection 2.2. The following Proposition resumes these results. 23

Proposition 3.2 (Consistency) If the true monetary policy is given by the Taylor rule, then, ˆ t−1 } and given that the policy shock ut for any predetermined instrument zt ∈ Z = {ˆ πt−1 , yˆt−1 , R is an iid process, the plim ηprr (zt ) = η. Proof: See appendix.

2

The statement of Proposition 3.2 is general for any specification of the Taylor rule, given that the monetary policy shock, ut , be an uncorrelated process. And is also general to the character of the equilibrium of the model economy, i.e. determinate or indeterminate. Thus, if the policy rule regression is to be performed, the lack of robustness when varying from one set of instruments to another could embody three different causes: a) The policy rule is misspecified; b) The policy shock presents some degree of persistency; or c) Both, a) and b) happen.

We emphasize the fact that any predetermined instrument will yield a consistent estimator if the assumptions of Proposition 3.2 are verified, no matter the quantitatively information contained in each of them, i.e., no matter by which fundamental or sunspot shock the fluctuations of the instrument are mostly driven. It is straightforward to recommend, then, that when performing a robustness test in a policy rule regression, a dramatically change in the set of instruments must be preferred in order to clearly ruling out misspecification. 3.2.1

An empirical exercise

This subsection is intended to explore the threat of policy rule misidentification with real data. We analyze the case for the U.S. using different set of instruments: the Clarida et al. (2000)’s original instrument set,16 and the rest composed separately by the lags of the federal funds rate, inflation rate of the GDP deflator, the Congressional Budget Office’s output gap, the unemployment rate, and the labor income share, the last three as different measures of real activity;17 the period considered corresponds to the Volcker–Greenspan era, i.e. 1979:3–2004:3, with quarterly frequency data.

At first stage, we estimate equation (34), hereafter called the reduced Taylor rule, and only afterwards we look to reproduce the estimations of Clarida et al. (2000), CGG hereafter, to 16

Conformed by four lags in the federal funds rate, annualized inflation, CBO’s output gap, commodities price inflation, the growth rate of money stock (M2), and the short-run–long-run interest rate spread. 17 Unemployment is considered as an alternative measure of the output gap in Clarida et al. (2000) estimations, whereas the labor income share is a proxy to labor costs, which is strongly correlated with the output gap (see Gal´ı and Gertler 1999 for a discussion).

24

study the effects of changing the instrument set on their extended version of the Taylor rule. The reduced Taylor rule is estimated using IV techniques with a single instrument within five different predetermined variables: interest rate, inflation, output gap, unemployment and labor income share lagged one period. Table 4: Estimations of the reduced Taylor rule, variation in the instrument ηbprr Inflation lag

1.63 (0.28)

Interest rate lag

2.41 (0.50)

Output gap lag

1.44 (1.19)

Unemployment lag

2.18 (0.70)

Labor income share lag

1.86 (0.46)

Note: Sample period for estimation: 1979:3–2004:3, quarterly data U.S. data.

Table 4 summarizes the results (standard deviations are stated in parenthesis). All estimates look significant, except when using the output gap as instrument. Indeed, they could all, precluding ηbprr (ˆ yt−1 ), be considered as statistically similar.18 To explain why the estimation using the output gap is not statistically different from zero, we could argue that this may be due to a weak covariance between π ˆt+1 and yˆt−1 , which entails a correlation coefficient between these two variables near to zero. In fact, this is exactly the case, since the correlation coefficient reaches -0.13 and is not statistically different from zero at 5 per cent of significance.

The stability of the coefficient of inflation holds even when one changes to instruments that are not so correlated with the inflation rate, as unemployment and the labor share (the contemporaneous correlation coefficients are 0.33 and 0.64, although they are significant). The relative stability of the reduced Taylor rule imply that the true policy may be a reaction function of the interest rate to expected inflation. This could be challenged by using a larger set of instruments.

We now turn to the CGG’s extended Taylor rule to apply the same methodology as before. The 18 Denoting unemployment and the labor income share as un and ls, respectively, the 95 per cent confidence ˆ t−1 ) ∈ interval of all these estimates have un interception at some regions: ηbprr (ˆ πt−1 ) ∈ [1.08, 2.18], ηbprr (R [1.42, 3.40], ηbprr (unt−1 ) ∈ [0.79, 3.56] and ηbprr (lst−1 ) ∈ [0.95, 2.77].

25

extended Taylor rule includes the output gap and lagged interest rate terms as to reflect policy inertia into the policy rule regression equation, and it writes as following (constant terms are omitted): ¤ £ ˆ t = (1 − ρR ) ηπ Et π ˆ t−1 + εprr,t , R ˆt+1 + ηy Et yˆt+1 + ρR (L)R

(40)

where ρR (L) = ρR,1 + ρR,2 L is the lag–polynomial to describe the inertia of monetary policy and ρR ≡ ρR (1).19 These terms are assumed to represent the “Federal Reserve’s tendency to smooth changes in the interest rate” (CGG, pp. 152). The period considered is the same as before, 1979:3–2004:3. The first set of instruments used is the one considered by CGG described earlier. We then change to 5 different instrument sets, each one within 8 lags including separately: 1) the inflation rate, 2) the interest rate, 3) the CBO’s output gap, 4) the unemployment rate and, 5) the labor income share, to test for parameter stability.20 The estimation method used is IV–GMM using 12 lags to estimate the optimal weighting matrix corrected for serial correlation trough the Newey–West procedure. Table 5 resumes the results, with standard deviations in parenthesis and the p-value of the J-statistic under it.

Table 5: Estimations of the extended Taylor rule, different set of instruments CGG instruments Inflation lags Interest rate lags Output gap lags Unemployment lags Labor income share lags

ρbR 0.86

ηbπ 2.21

ηby 0.46

J–stat 7.81

(0.01)

(0.19)

(0.10)

0.99

0.65

1.84

0.0018

3.59

(0.10)

(0.35)

(0.35)

0.46

1.36

-0.22

2.21

-0.67

( 0.36)

(0.43)

(0.36)

0.85

1.02

32.25

-0.18

2.38

(0.11)

(199.13)

(3.19)

0.67

0.57

1.71

0.79

0.98

(0.22)

(0.07)

(0.56)

0.91

0.64

2.01

0.29

1.37

(0.16)

(0.46)

(0.62)

0.85

Note: Estimation period: 1979:3–2004:3, quarterly U.S. data.

Despite of the estimation using the output gap instrument set, which is completely uninformative since it may be considered as weak instruments, all the other estimations provide interesting 19

See Clarida et al. (2000) for further details. We choose 8 lags in the alternative instrument sets as to control for excessive moments conditions. Nevertheless, the results are qualitatively invariable to the choice of larger horizons. 20

26

results. The first is that the coefficient of inflation is quite stable and significant for all instrument sets, except for the output gap’s. The second observation is that, in fact, the later is the only stable coefficient in the whole experience. Notably, the rejection of the policy inertia parameters when using the interest rate instrument set raises serious doubts about the good specification of the interest rate partial adjustment hypothesis and the existence of any desire of smoothing the funds rate by the central bank, as referred in CGG (this result is not new, see Rudebusch, 2002, for a discussion). Also, the irrelevance of the output gap coefficient for all instruments sets may provide evidence that the weight of this variable in the extended Taylor rule is rather small or null.

More experiences like the one presented might be enlighten to elucidate any misspecification in policy rule regression analysis. Here, we have some evidence that the extended Taylor may be misspecified, since only the inflation coefficient seems to be stable across different instrument sets.

4

Concluding Remarks

This paper provides a general overview of the policy rule regression analysis. Our main finding is that, within the framework of a rational expectations model, policy rule regression that uses a limited information estimation method tell us very little about the real structural behavior of policy–making. In the presence of nearly observational equivalent models, policy misidentification may happen more often then expected. Therefore, a series of robustness experiences may be useful to elicit any misspecification.

Indeed, when the policy rule is misidentified, the consistency of the IV–GMM estimators may be challenged. Using very different instruments, such as their variance decomposition be qualitatively different, could be a powerful tool to rule out misspecification.

In the case of the Taylor rule regression, we show that even a preset–price model with exogenous money growth rule can match the value of the inflation coefficient in the estimated Taylor rule for the U.S. Therefore, misidentification cannot be ruled out in advantage in this case. Some empirical robustness test to the CGG extended Taylor rule show that the stability of the estimated parameters fails for all variables and all instrument sets, except for the inflation

27

coefficient. Notably, there could be evidence against the monetary policy inertia hypothesis or the desire of the FED to smooth at quarterly level the funds rate (result also found in Rudebusch, 2002), and the relevance of the CBO’s output gap measure into the estimated monetary policy rule for the 1979:3–2004:3 sample period, with quarterly data.

28

References Benhabib, J., Farmer, R., 1999. Indeterminacy and Sunspots in Macroeconomics. In: Taylor JB, Woodford M (Eds), Handbook of Macroeconomics, vol.1. North-Holland: Amsterdam. p. 387-448. Beyer, A., Farmer, R., 2004. On the Indeterminacy of New-Keynesian Economics. European Central Bank; working paper 323. Beyer, A., Farmer, R., 2005. Testing for Indeterminacy: An Application to U.S. Monetary Policy: Comment on paper by Thomas Lubik and F. Schorfheide. American Economic Review, forthcoming. Carlstrom, C.T., Fuerst, T.S., 2000. Forward–Looking Versus Backward–Looking Taylor Rules. Federal Reserve Bank of Cleveland; working paper 00–09. Clarida, R., Gal´ı, J., Gertler, M., 2000. Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory. Quarterly Journal of Economics; CXV (1); 147-180. Cooley, T.F., Hansen, G.D., 1995. Money and the Business Cycle. In: Cooley TF (Ed), Frontiers of Business Cycle Research. Princeton University Press, Princeton, NJ. p. 175-216. Farmer, R., 1999. The Economics of Self-Fulfilling Prophecies. MIT Press: Cambridge, Massachusetts. Second edition. Gal´ı , J., Gertler, M., 1999. Inflation Dynamics: A Structural Econometric Analysis. Journal of Monetary Economics; 44; 195-222. Hetzel, R.L., 2000. The Taylor Rule: Is it a Useful Guide to Understanding Monetary Policy? Federal Reserve Bank of Richmond; Economic Quarterly; 86 (2); 1-33. King, R.G., Rebelo, S.T., 1999. Resuscitating Real Business Cycles. In: Taylor JB, Woodford M (Eds), Handbook of Macroeconomics, vol.1. North-Holland: Amsterdam. p. 927-1007. Linde, J., 2002. Estimating New–Keynesian Phillips Curves: A Full Information Maximum Likelihood Approach. Sveriges Riksbank; Working Paper Series; 129. Lubik, T., Schorfheide, F., 2004. Testing for Indeterminacy: An Application to US Monetary Policy. American Economic Review; 94(1); 190-217. Mankiw, G., Reis, R., 2002). Sticky Information versus Sticky Prices: a Proposal to Replace the New Keynesian Phillips Curve. Quarterly Journal of Economics; 117; 1295-1328. Mavroeidis, S., 2004a. Weak Identification of Forward–looking Models in Monetary Economics. Oxford Bulletin of Economics and Statistics; forthcoming.

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Mavroeidis, S., 2004b. Identification Issues in Forward-Looking Models Estimated by GMM, with an Application to the Phillips Curve. University of Amsterdam; Department of Quantitative Economics; working paper. Nason, J., Smith, G., 2003. Identifying the New Keynesian Phillips Curve. Queen’s University, Kingston, Ontario; Department of Economics; manuscript. Rudebusch, G.D., 2002. Term structure evidence on interest rate smoothing and monetary policy inertia. Journal of Monetary Economics; 49; 1161-1187. Taylor, J.B., 1993. Discretion versus Policy Rule in Practice. Canergie Rochester Conference on Public Policy; 39; 195-214. Walsh, C.E., 2003. Monetary Theory and Policy. The MIT Press: Cambridge, Massachusetts. Second Edition.

30

Appendix

A

Solution for the money growth rule economy

Since the preset-price model presented in section 2 does not allow for any predetermined endogenous variable, i.e. there is no capital accumulation, the aggregate dynamics of the constantmoney-growth economy collapse into a single equation. As a solution strategy, we make use of ¯ t ) and the following auxiliary variables: p∗t = pˆt − m ˆ t and wt∗ = w ˆt − m ˆ t , where m ˆ t = log(Mt /M ¯ ¯ Mt = Mt−1 γ¯ is the stock of money if there is no shocks in the money growth rate. The later implies that m ˆ t+1 − m ˆ t = γˆt and Et m ˆ t+1 = m ˆ t+1 . Accordingly, equations (27), (29) and (30) can be rewritten as: ¡ ¢ wt∗ = Et p∗t+1 + yˆt+1 + γˆt (41) p∗t = Et−1 (wt∗ − a ˆt )

(42)

p∗t

(43)

= γˆt − yˆt

The last two expressions imply yˆt = γˆt + ρa a ˆt−1 − Et−1 wt∗ .

(44)

Placing the later and (43), forwarded one period, into (41) yields wt∗ = (1 + ργ )ˆ γt ,

(45)

which in (44) give us the solution for output, expressed in proposition 2.1. The inflation and nominal interest rate solutions are straightforward retrieved from equations (28) and (30).

B

Solution for the Taylor rule economy

Since (42) implies wt∗ = p∗t + ρa a ˆt−1 + νw,t ,

(46)

where νw,t = wt − Et−1 wt , equation (41) can be stated as Et (p∗t+1 − p∗t ) = ρa a ˆt−1 − Et yt+1 − γˆt + νw,t .

(47)

Substituting p∗t on the later, according to (43), yields: Et γˆt+1 = ρa a ˆt−1 − yˆt + νw,t .

(48)

ˆ t = Et γˆt+1 . Since the nominal interest rate is Equations (26), (28) and (30) imply that R set according to the rule (34), this means that output is governed by the following dynamic equations: yˆt = η yˆt+1 − (η − 1) (ρa a ˆt−1 + νw,t ) − σu ut − ηνy,t+1 , (49) where νy,t+1 = yt+1 −Et yt+1 . To characterize the determinacy conditions, setting (29) one period ahead and taking it into (27) yields: w ˆt = w ˆt+1 + yˆt+1 − ρa a ˆt − νw,t+1 − νy,t+1 . 31

(50)

The two later expressions conform the system Xt = AXt+1 + BFt + CSt where

µ Xt =

w ˆt yˆt

(51)

¶

is the vector of independent endogenous variables, the first nominal and the second real ¶ µ a ˆt Ft = ut contains the fundamental disturbances and µ St =

νw,t+1 νy,t+1

¶

is the sunspot shocks. A, B and C are matrices of parameters, equal to µ ¶ 1 1 A= , 0 η µ B=− and

ρa 0 (η − 1)ρa L σu µ

C=−

1 1 (η − 1)L η

¶ ,

¶

respectively, where L is the lag operator. The dynamic properties of the above system depend on the eigenvalues of A, which are equal to 1 and η. Since one eigenvalue lies within the unit circle, we have at least one degree of indeterminacy, denoted by νw,t+1 . In other words, regardless the dynamic environment of the real variables, a nominal sunspot shock will be carried into the real economy; this comes from the indetermination of the growth rate of money.21 If the second eigenvalue η is outside the unit circle, then a second degree of indeterminacy will appear in the solution, denoted by νy,t+1 . Eventually, this is the assumption we made in section 2.2 when we chose an aggressive interest rate rule. Since the output gap will be driven not only by the fundamental disturbances, as technology or the policy shock, but also by the sunspot (or beliefs) innovations, solution (35) in proposition 2.2 comes simply from stating (49) in a backward perspective. For the inflation rate, first note that (50) can be rewritten as (one period backward) Et−1 w ˆt − w ˆt−1 = ρa a ˆt−1 − Et−1 yˆt .

(52)

Substituting the later and the solution for output into the first–order difference of equation (29) yields the solution for the inflation rate as given in proposition 2.2. 21

Notably, we can determine the solution for Et γˆt+1 , but not for γˆt+1 itself, which leaves free the innovation νγ,t+1 , that is translated into prices and nominal wages.

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C

Proof of Proposition 3.1

The moments are deduced from the solution equations (31)–(33). Since the fundamental shocks are independent from each other, E(xt x0t+s ) = Cov(xt , x0t+s ) = 0 for any integer s and x 6= x0 , we have only to worry about the autocovariances of money growth and technology. Thus, according to equation (53), the relevant moments can be rewritten as E[(ˆ at − a ˆt−1 )ˆ πt−1 ] = ρ2a V ar(ˆ at )(1 − ρa )2 E[(ˆ at − a ˆt−1 )ˆ yt−1 ] = −ρ2a V ar(ˆ at )(1 − ρa ) ˆ t−1 ] = 0 E[(ˆ at − a ˆt−1 )R E(ˆ πt+1 π ˆt−1 ) = ρ4γ V ar(ˆ γt )(1 + ργ (1 − ρ2γ )) − ρ3a V ar(ˆ at )(1 − ρa )2 E(ˆ πt+1 yt−1 ) = ρ3γ V (ˆ γt )(1 − ρ2γ (1 + ργ )) + ρ3a V (ˆ at )(1 − ρa ) The plim of the IV–GMM estimators is just the sum between the true reduced form parameter ηprr and the specific instrument bias.

D

Proof of Proposition 3.2

The policy rule regression equation (37) in terms of observable variables is ˆ t = ηprr π R ˆt+1 + ηprr νπ,t+1 + εprr,t . Equation (34) means εprr,t = σu ut ; solution (36) implies that νπ,t+1 = (ˆ πt+1 − Et π ˆt+1 ) = 0. Thus, the plim of the IV–GMM estimator of ηprr given zt is written as plim η˜prr (zt ) ≡

ˆ t zt ) E(R E(ut zt ) = η + σu E(ˆ πt+1 zt ) E(ˆ πt+1 zt )

(53)

Given the solutions (35) and (36), we see immediately that E(ut zt ) = 0, for any choice of zt ∈ Z and E(ηprr ) = η. There is no bias caused by the instruments and all the three estimators are consistent, provided that E(ˆ πt+1 zt ) 6= 0.

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