Valid Inference in Partially Unstable GMM Models

Transcription

1 Valid Inference in Partially Unstable GMM Models Hong Li Deartment of Economics Brandeis University Waltham, MA 02454, USA Ulrich K. Müller Deartment of Economics Princeton University Princeton, NJ 08544, USA Abstract The aer considers time series GMM models where a subset of the arameters are time varying. We focus on an emirically relevant case with moderately large instabilities, which are well aroximated by a local asymtotic embedding that does not allow the instability to be detected with certainty, even in the limit. We show that for many forms of the instability and a large class of GMM models, usual GMM inference on the subset of stable arameters is asymtotically unaffected by the artial instability. In the emirical analysis of resumably stable arameters such as structural arameters in Euler conditions one can thus ignore moderate instabilities in other arts of the model and still obtain aroximately correct inference. JEL Classification: C32 Keywords: Structural Breaks, Parameter Stability Test, Contiguity, Euler Condition We thank two anonymous referees and the editor, Bernard Salanié, for very helful comments and suggestions, as well as Graham Elliott and Mark Watson and seminar articiants at Houston University, U.C. Davis, Brandeis University, the Federal Reserve Board, thenyfederalreservebank, theatlantafederal Reserve Bank, Iowa State University, the Econometric Society World Congress 2005 and the University of Pennsylvania for helful discussions. Müller gratefully acknowledges financial suort by the NSF via grant SES Corresonding author. Tel: ; fax:

2 1 Introduction Instabilities in the arameters of econometric time series models are a lausible and emirically widesread henomenon. Time varying market conditions, rules and regulations and technological innovations change the economic environment. As ointed out by Lucas (1976), these environmental changes induce behavioral changes of rational economic agents, which results in time varying arameters in many econometric relationshis. In addition, missecifications of econometric models can also manifest themselves in the form of time varying arameters. Emirically, Ghysels (1998), Stock and Watson (1996), Boivin (1999) and Cogley and Sargent (2005), for instance, find instabilities in macroeconomic and finance relationshis. Econometric theory has focussed to a large extent on the roblem of testing the null hyothesis that a time series model is stable over time against the alternative of arameter variation whose exact form is unknown: See, for instance, Nyblom (1989), Andrews (1993), Andrews and Ploberger (1994), Sowell (1996), Bai and Perron (1998), Hansen (2000), Andrews (2003) and Elliott and Müller (2006) for some recent contributions. Much less work is concerned with the next ste: What is one to do once instabilities are susected? One useful result, established in Bai (1994) and generalized in Bai and Perron (1998), concerns inference in linear regressions with a discrete number of arameter shifts at unknown times. If the arameter shifts are large in the sense that reasonable tests detect the instability with robability one in the limit, then standard inference on the coefficients in the various regimes remains asymtotically valid when the regime dates are based on least-squares break date estimators. In many alications, however, instabilities are arguably not sufficiently large for these asymtotics to generate reliable aroximations: Linde (2001) for instance argues that economically imortant changes in monetary olicy lead to arameter instabilities that are small in the sense of being difficult to detect emirically. More generally, the instabilities in bivariate relationshis between macroeconomic data series documented in Stock and Watson (1996) are often only borderline significant. In such instances, more accurate aroximations are generated by asymtotics where reasonable tests detect the instabilities with robability strictly smaller than one in the limit see Elliott and Müller (2007) for some quantitative results on break date confidence sets. 1

3 Inthisaer, weanalyzemodelswhereonlya subset of arameters are unstable, and focus on such small instabilities. We ask the question how to conduct valid inference on the stable subset of arameters. The answer turns out to be more straightforward than it might seem: For a very wide range of unstable arameter aths, and for a large class of Hansen s (1982) Generalized Method of Moments (GMM) models, standard GMM inference (ignoring the artial instability) remains asymtotically valid for the subset of stable arameters. To develo some intuition for this result, consider the linear model Y t = t θ 1,t + Z t θ 2 + ε t,ε t i.i.d.(0,σ 2 ) (1) where t and Z t are two (ossibly correlated) scalar random variables. By standard OLS algebra (the Frisch-Waugh Theorem), the t-statistic on θ 2 in a regression of Y t on ( t,z t ) is numerically identical to the t-statistic in a regression of Y t on ( t, Z t ),where Z t = Z t β t is the residual of a regression of Z t on t.intermsof Z t, (1) becomes Y t = t (θ 1,t + β)+ Z t θ 2 + ε t = t (θ 1,0 + β)+ Z t θ 2 + t (θ 1,t θ 1,0 )+ε t. (2) If ( t,z t ) is aroximately stationary in the sense that T P 1 T Z t t =0also imlies T P 1 [st ] Z t t 0 for all s [0, 1], andθ1,t θ 1,0 is a smooth function of t, thenthe additional error term t (θ 1,t θ 1,0 ) is aroximately orthogonal to Z t. If in addition, the amount of arameter instability is small in the sense that su t T θ 1,t θ 1,0 = O(T 1/2 ), then the error term ε t dominates the randomness of the OLS estimator of θ 2, and inference on θ 2 remains largely unaffected by the instability of θ 1. In contrast, if ( t,z t ) is a ersistent series, lack of correlation between Z t and t does not imly lack of correlation between t (θ 1,t θ 1,0 ) and Z t, and the resence of t (θ 1,t θ 1,0 ) invalidates standard inference for θ 2, even for small arameter instabilities. The same result also holds for more general GMM inference, using the usual linearization arguments. The aroriate analogy for the constraint that ( t,z t ) are not too ersistent is the assumtion that samle averages of the derivative of the moment condition are aroximately the same in all arts of the samle. This holds for most linear or non-linear globally stationary models, such as stationary Vector Autoregressive models. It tyically fails to hold, though, for models that generate deterministically or stochastically trending data. A leading economic examle of a artially stable GMM model are Euler moment conditions of otimizing agents under a time varying olicy environment. Rational economic 2

4 agents adat their otimal behavior to olicy changes. Econometrically, this leads to reduced form equations that exhibit time varying arameters. At the same time, structural arameters describing references and technology might very well remain constant, and their values are crucial for conducting roer olicy analysis. For concreteness, consider a stylized two equation system involving (i) a New Keynesian Phillis Curve (NKPC), which is a rational exectations Euler condition in inflation and unemloyment ga, and (ii) a reduced-form rocess for the unemloyment ga, the driving variable of the NKPC (see Blanchard and Gali (2007) for the theoretical derivation of this secification). Exressing the NKPC in first differences and with an AR(2) secification for the unemloyment ga, we obtain π t = φe t π t+1 + κs t + ε t (3) s t = ρ 1,t s t 1 + ρ 2,t s t 2 + ξ t (4) where π t and s t are the inflation rate and unemloyment ga, resectively, E t is the conditional exectation at date t, and the disturbance terms ε t and ξ t are i.i.d. mean zero. In this examle, economic theory has direct imlications for the stability of the various arameters: the coefficients ρ 1 and ρ 2 are functions of current monetary olicy. With a time varying monetary olicy, ρ 1 and ρ 2 therefore become unstable. The Euler equation (3), in contrast, is derived from the economic agents otimization roblem. As long as references and technology remain constant through time, economic theory imlies φ and κ to be stable, even in the face of a time varying monetary olicy. Our results imly that one might estimate the system (3) and (4) by standard GMM ignoring the time variability in ρ 1 and ρ 2, and one still obtains aroximately valid inference about the structural arameters φ and κ, at least as long as the time variability in ρ 1 and ρ 2 is not too ronounced. We return to this examle in Section 4 below and find that this is a reasonable aroximation in a Monte Carlo exercise calibrated to U.S. data. Also see Li (2008) for an emirical alication of our results to an investment model. We also find that oular tests of stability of a subset of arameters are tyically affected by instabilities in the non-tested arameters. An additional contribution of this aer is the derivation of a class of modified tests whose rejection robability is unaffected by local instabilities in the non-tested arameters. Our results allow for arameter instabilities of a magnitude that corresonds to local alternatives of efficient stability tests. Formally, in such asymtotics the magnitude of the 3

5 instability is of the order T 1/2 in a samle of size T. As noted above, this does not mean that our results only aly to economically insignificant instabilities. In many alications, there is substantial uncertainty about the resence and form of instabilities. Accurate aroximations are then generated by a modelling strategy in which there is only limited information about the instability asymtotically, as in the T 1/2 neighborhood. What is more, from a more theoretical ersective, it makes sense to focus on local deviations from standard model assumtions in a robustness analysis. After all, when arameter instabilities are large, the roblem can be detected consistently with an aroriate test and, at least for a finite number of discrete shifts, the inclusion of the aroriate dummies leads to valid inference, as demonstrated by Bai and Perron (1998). In contrast, when arameter instabilities are of the order T 1/2, there is no way of knowing for sure whether the arameters are unstable, and there is no obvious remedy if one believes they are. Our results recisely cover this latter case, where it is challenging to derive more immediate aroaches to time varying nuisance arameters. In alications of our results, one must decide whether these asymtotic considerations yield accurate aroximations. This is, of course, true of all asymtotically justified inference. But the roblem is more severe here, since it is not hard to see that in general, our results do not hold for asymtotics with a fixed magnitude of the time variation, 1 where, for instance, θ 1,t θ 1,0 = 1[t T/2] rather than θ 1,t θ 1,0 = T 1/2 1[t T/2] in (1). 2 A formal solution is to reject the null hyothesis for the value of a arameter of interest using the usual GMM Wald test only if the observed -value of a good arameter stability test is larger than some threshold >0, and to never reject otherwise. Since the -value of a good arameter stability test converges to zero in robability for instabilities that are larger than the T 1/2 neighborhood, this aroach yields uniformly valid inference over all 1 As noted above, inference based on Bai and Perron s (1998) results face the same roblem in reverse their aroach yields oor aroximations for local arameter instabilities. See Elliott and Müller (2007) for details. 2 For such a fixed arameter instability θ 1,t θ 1,0 = 1[t T/2], the outcome deends strongly on roerties of {( t,z t )} T :For{( t,z t )} T i.i.d. and indeendent of {ε t} T, the heteroskedasticity robust t-statistic on θ 2 remains unconditionally asymtotically valid, but not conditionally on {( t,z t )} T. Stationary but autocorrelated {( t,z t )} T indeendent of {ε t} T yield that unconditionally, T 1/2 (ˆθ 2 θ 2 ) N (0,V 2 ), with V 2 in general different from the limit of White s estimator. Finally, under weak exogeneity where, for instance, t = Y t 1 and Z t = Y t 2, ˆθ 2 is inconsistent. 4

6 magnitudes of the artial instability (at the cost of being otentially very conservative for large instabilities). But in ractice, icking too small relative to the actual samle size might still yield oor aroximations, so that a judgement must still be made. In Section 3 below, we rovide small samle results for two data generating rocesses to shed some light on this issue, and we find that the local asymtotics considered in this aer tend to generate accurate aroximations for samle sizes and amounts of instability that are lausibly encountered in macroeconomic and financial alications. On a technical level, the analysis of time series models with time varying arameters faces the difficulty that these models tend to generate nonstationary data. This comlicates the justification of asymtotic aroximations, such as those generated from Laws of Large Numbers. We address these difficulties by roviding sufficient conditions for the unstable model to be contiguous to the corresonding stable model. In the analysis of arameter stability tests for fully secified arametric models, the concet of contiguity has been emloyed before in Andrews and Ploberger (1994) and Elliott and Müller (2006), although these aers address more secific forms of arameter instability than considered here. Contiguity ensures that aroximation errors that are o (1) in the stable model remain o (1) in the corresonding unstable model. It therefore suffices to make aroriate assumtions on the stable model, and derive the corresonding roerties of the unstable model via contiguity. The results we establish with this indirect reasoning might be of indeendent interest; they might be used, for instance, to justify the efficient tests of arameter instability in Sowell (1996), the median unbiased estimators of the amount of local instability in Stock and Watson (1998) or the inference for the date of a single break in linear regression in Elliott and Müller (2007) for a wide class of data generating rocesses, and results of this aer have been used in the derivation of efficient arameter ath estimators in Müller and Petalas (2007). The next section contains the asymtotic results of this aer: Subsection 2.1 introduces the model and discusses a high-level condition on the artially unstable GMM model. The high-level condition is sufficient for the main asymtotic result resented in subsection 2.2, followed by a discussion of its imlications for econometric ractice. In subsection 2.4 we derive contiguity based arguments to justify the high-level condition on the unstable model by aroriate assumtions about the roerties of the corresonding stable model. Section 3 contains a Monte Carlo study that investigates the small samle relevance of the asymtotic 5

7 results. Section 4 concludes. Proofs are collected in an Aendix. 2 Asymtotic Results 2.1 Model and High-Level Condition Consider a GMM model with the unknown m 1 arameter vector θ, anelementofthe arameter sace Θ R m. The observed data in a samle of size T is given by a triangular array of random q 1 vectors {y T,t } T, defined on a robability sace (Ω, S,P), onwhich also all following random elements are defined. A triangular array construction for the data is necessary to accommodate the artial instability in the arameter θ. The GMM oulation moment condition is embodied in the known, integrable function g : R q Θ 7 R for m, such that in the stable GMM model, the true arameter θ 0 satisfies E[g(y T,t,θ 0 )] = 0 for all t T. Let {θ T,t } T Θ T be the arameter ath in the corresonding unstable model such that E[g(y T,t,θ T,t )] = 0 for all t T, T 1. (5) For notational convenience, we will dro the deendence of y T,t and θ T,t on T if no confusion arises. Also, let g t (θ) be g(y t,θ). All limits are taken as T. We write for convergence in robability (in P ), for weak convergence of the underlying robability measures, [ ] denotes the greatest lesser integer function and is the sectral matrix norm. The delimiters of integrals are zero and one, if not indicated otherwise. We analyze the asymtotic roerties of the usual GMM estimator ˆθ, defined as " # 0 " # " # 0 " # T 1 g t (ˆθ) Q T T 1 g t (ˆθ) =inf T 1 g t (θ) Q T T 1 g t (θ), (6) θ Θ where Q T is a sequence of (ossibly random) ositive definite matrices. Denote by G t (θ) =G T,t (y T,t,θ) the m matrix of the artial derivatives g(y T,t,θ)/ θ 0 (if it exists). We imose the following high-level condition. Condition 1 TheunstableGMMmodelsatisfies (i) T 1/2 (θ t θ 0 )=f(t/t ) t T,T 1 for some nonstochastic, bounded and iece-wise continuous function f :[0, 1] 7 R m with at most a finite number of discontinuities, and left and right limits everywhere. 6

8 (ii) In some neighborhood Θ 0 of θ 0, g t (θ) is differentiable in θ a.s. for t T,T 1. (iii) T P 1/2 T g t(θ t ) N (0,V) for some ositive definite matrix V. (iv) ˆθ θ 0. (v) Q T Q0 for some ositive definite matrix Q 0, and there exist ositive definite matrices ˆV T such that ˆV T V. (vi) T P 1 T G t(θ 0 ) = O (1), T 1 su t T G t (θ 0 ) 0 and for any decreasing neighborhood Θ T of θ 0 contained in Θ 0,i.e.Θ T = {θ : θ θ 0 <c T } Θ 0 for some sequence of real numbers c T 0, T P 1 T su θ Θ T G t (θ) G t (θ 0 ) 0. (vii) For all 0 λ 1, T P 1 [λt ] G t(θ 0 ) λγ for some full column rank m matrix Γ. Part (i) of Condition 1 assumes the instability in the arameters to be of order T 1/2. This is the neighborhood in which efficient tests of arameter stability have nontrivial local asymtotic ower. The form of the instability is described by the function f. Byletting some elements of f to be zero, the GMM model becomes only artially unstable. The main interest of the aer is how to conduct asymtotically valid inference about the stable subset of arameters. The restrictions on the non-zero arts of the function f are quite weak; in articular, note that we do not assume differentiability of f. The conditions on f are sufficient to ensure that f can be uniformly aroximated by a sequence of ste functions. The arameter instability is assumed to be nonstochastic, in contrast to, say, Stock and Watson (1996, 1998), Primiceri (2005) and Cogley and Sargent (2005). But under an alternative assumtion of stochastic arameter aths, the following results continue to hold as long as Condition 1 holds conditional on almost all realizations of the ath. Such an assumtion, of course, restricts the ossible deendence between the disturbances of the model and the stochastic arameter ath, but it covers the models of exogenous time varying arameters models oular in alied work, including those cited above. Almost all realizations of a Wiener rocess on the unit interval are bounded and continuous, and hence may serve as functions f as secified in art (i). The key assumtion for the result in this aer is the aroximate linearity of T P 1 [λt ] G t(θ 0 ) in λ as imosed in art (vii) (which, given the condition in art (vi), is equivalent to the aroximate linearity of T P 1 [λt ] G t(θ t )). This assumtion entails that averages of G t (θ 0 ) are aroximately equal to Γ in all arts of the samle. It is tyically justified for globally stationary models, such as stationary Vector Autoregressive models. Even 7

9 certain globally nonstationary models, such as a linear regression with stationary regressors but trending disturbance variance, can satisfy this requirement. On the other hand, most models that generate (stochastically or deterministically) trending data fail to satisfy (vii) of Condition 1, even after scale normalizations that ensure T P 1 T G t(θ 0 )=O (1). Part (iii) assumes a multivariate Central Limit Theorem to hold for the scaled samle averageofthemomentcondition,evaluatedatthetruetimevaryingarameter. Giventhe GMM oulation moment condition (5), this is a natural condition. At the same time, in order to invoke such a Central Limit Theorem, a suitable set of moment and deendence conditions on the random variables {g t (θ t )} T need to checked in the unstable model, a comlication to which we return in Section 2.4 below. Parts (iv) (vii) imose high-level conditions on the asymtotic roerties of the unstable GMM model, which would be fairly standard for a stable model, i.e. if f was equal to zero. Part (iv) can usually be justified by the uniform convergence of h T P i 0 h 1 T g t(θ) QT T P i 1 T g t(θ) over θ Θ to a nonstochastic function whose unique minimizer is θ 0. A suitable estimator ˆV T of V, the asymtotic variance of T P 1/2 T g t(θ t ), is tyically given by the non-arametric long-run variance estimators of Newey and West (1987) and Andrews (1991). The third assumtion in art (vi) controls the average variability of G t (θ) as a function of the arameters. It is imlied by the more rimitive conditions A.2 and A.3 of Andrews (1987). See Gallant and White (1988) and Andrews (1992) for further discussion. Again, for unstable models with nonzero f, these convergences in robability are less standard, and we rovide a suitable argument in Section 2.4 below. 2.2 Validity of Standard GMM Inference on Stable Parameters The following main result establishes the asymtotic roerties of standard GMM inference that ignores the arameter instability. Theorem 1 Under Condition 1, (i) T 1/2 ˆΣ 1/2 θ (ˆθ T P 1 T θ t) N (0,I m ), (ii) T P 1/2 T g t(ˆθ) N (0, (I Γ(Γ 0 Q 0 Γ) 1 Γ 0 Q 0 )V (I Γ(Γ 0 Q 0 Γ) 1 Γ 0 Q 0 ) 0 ), where ˆΣ θ =(ˆΓ 0 Q T ˆΓ) 1ˆΓ0 Q T ˆVT Q ˆΓ(ˆΓ0 T Q T ˆΓ) 1 and ˆΓ = T P 1 T G t(ˆθ) Γ. Furthermore, if in addition, su λ [0,1] T P 1 [λt ] G t(θ 0 ) λγ 0 and T P 1/2 [ T ] g t(θ t ) V 1/2 W ( ) with W a 1 standard Wiener rocess, then 8

10 (iii) T P 1/2 [ T ] g t(ˆθ) ζ( ), whereζ(λ) =V 1/2 W (λ) λγ(γ 0 Q 0 Γ) 1 Γ 0 Q 0 V 1/2 W (1) + ³ R λ Γ f(l)dl λr 1 f(l)dl. 0 0 Part (i) of Theorem 1 shows that standard asymtotically Gaussian inference based on ˆθ and ˆΣ θ remains valid for the stable subset of the arameters (where θ t isthesameforall t and equal to θ 0 in the corresonding row): for the stable subset, the conventional GMM estimator is asymtotically unbiased and Gaussian. Wald statistics involving only stable arameters are asymtotically chi-squared under the null hyothesis, and have the same noncentrality arameter under local alternatives as the corresonding fully stable model. It is immediate from Condition 1 (i) and (iv) that the GMM estimator ˆθ is consistent for the average arameter value, ˆθ T P 1 T θ t 0. Part (i) of Theorem 1 shows how to conduct asymtotically valid inference about this average. In most alications, however, the average of a time varying arameter does not have a structural interretation. To see why the artial instability does not sill over to the estimators of the stable subset of arameters, consider the following first order Taylor exansion of the first order condition for (6) 0=ˆΓ 0 Q T T P 1/2 T g t(ˆθ) = ˆΓ 0 Q T T P 1/2 T g t(θ t )+ˆΓ 0 Q T (T P 1 T G t )T 1/2 (ˆθ θ 0 ) ˆΓ 0 Q T T P 1 T G t T 1/2 (θ t θ 0 ) (7) where the jth row of G t is the jth row of G t evaluated at some θ t,j that lies on the line segment between θ t and ˆθ. Standard arguments imly that under Condition 1, T P 1 T G t Γ. The main insight concerns the term T P 1 T G t T 1/2 (θ t θ 0 )=T P 1 T G t f(t/t ). Thisis a weighted average of the columns of { G t } T, withweights{f(t/t )} T. If the averages of G t (θ 0 ) (and hence G t ) are aroximately equal to Γ in all arts of the samle, as assumed in Condition 1 (vii), then the weighted average is aroximately the simle average times the average weight: T 1 P T G t f(t/t ) lim T ΓT P 1 T f(t/t ). In the context of deriving the asymtotic local ower of stability tests, similar results were established in Ploberger, Krämer, and Kontrus (1989), Andrews (1993) and Sowell (1996); also see Stock and Watson (1998). Theorem 1 (i) now follows from rearranging (7) and taking limits, revealing the relevance of this result for conducting asymtotically valid inference in artially stable models. As a consequence of art (ii) of Theorem 1, Hansen s (1982) overidentification test remains asymtotically chi-squared with m degrees of freedom, even in the unstable model. 9

11 The overidentification test has no ower against the alternative of (locally) time varying arameters this result was obtained by Ghysels and Hall (1990) for a single break and is imlied by Sowell s (1996) asymtotic decomosition of the samle moment condition; also see Newey (1985) and Hall and Sen (1999). Therefore, when conducting inference about stable arameters in a artially unstable model as described in Condition 1, rejection by the overidentification test cannot be exlained by the artial instability. As usual, it still indicates incorrect moment conditions. 2.3 Stability Tests in Partially Unstable Models Part (iii) of Theorem 1 requires the strengthening of Condition 1 (iii) to a Functional Central Limit Theorem to hold for the artial sums of the samle moment conditions evaluated at the true time-varying arameter, and the convergence in Condition 1 (vii) to be uniform. The result serves as a basis for understanding the asymtotic local ower of a wide range of arameter stability tests. The statistics analyzed in Nyblom (1989), Sowell (1996) and Elliott and Müller (2006), as well as the LM versions of the tests derived in Andrews (1993) and Andrews and Ploberger (1994) can be written as functions of T P 1/2 [ T ] g t(ˆθ). Of secial interest here are the roerties of stability tests in artially stable models. Suose one is interested in the first m 0 m elements of θ. Let C be the m m 0 selection matrix C =[I m0, 0 m0 (m m 0 )] 0, and consider the case of efficient GMM estimation, so that Q T = ˆV 1 T. One might invoke the analysis of Sowell (1996), who derives tests of H 0 : θ t is constant in t against H 1 : θ t deends on t (8) that efficiently discriminate between the limiting distribution of T P 1/2 [ T ] g t(ˆθ) under the null and local alternatives of (8). Secifically, Sowell s Corollary 2 shows that the limit random rocess J(λ) =Σ 1/2 θ (W m (λ) λw m (1)) + Σ 1 θ ³ R λ 0 f(l)dl λr 1 0 f(l)dl of ˆΓ 0 ˆV 1 T T P 1/2 [ T ] g t(ˆθ) J( ) is asymtotically sufficient for the local alternative f,where Σ θ =(Γ 0 V 1 Γ) 1 and W m is a m 1 standard Wiener rocess, so that W m (λ) λw m (1) is a m 1 Brownian Bridge. Secializing his Corollary 2 further, one obtains that tests of (8) that maximize ower against alternatives where only the first m 0 elements of θ are time 10

12 varying may be based on functionals of (C 0 ˆΣ 1 θ C) 1/2 C 0ˆΓ [ T ] 0 ˆV 1 T T 1/2 g t (ˆθ) W m0 ( ) W m0 (1) (9) ³ R +(C 0 Σ 1 θ C) 1/2 C 0 Σ 1 θ f(l)dl R f(l)dl where W m0 =(C 0 Σ 1 θ C) 1/2 C 0 Σ 1/2 θ W m. In general, as long as Σ θ is not block diagonal, the asymtotic distribution (9) deends on whether or not the last m m 0 elements in f are zero. The asymtotic null distribution of the usual tests for instability in the first m 0 elements of θ are thus tyically affected by instabilities in other arameters, as long as the arameter estimators are not asymtotically uncorrelated. In other words, these tests are not in general valid tests of H 0 : C 0 θ t is constant in t against H 1 : C 0 θ t deends on t (10) which allows for local instabilities of the last m m 0 arameters in θ under the null hyothesis. As a solution to this roblem, consider the class of modified test statistics that are functions of [ T ] (C 0 ˆΣ θ C) 1/2 C 0 ˆΣ θˆγ 0 ˆV 1 T T 1/2 g t (ˆθ) (C 0 Σ θ C) 1/2 C 0 Σ θ J( ) (11) = W m0 ( ) W ³ R m0 (1) + (C 0 Σ θ C) 1/2 C 0 f(l)dl R f(l)dl 1, 0 0 where W m0 =(C 0 Σ θ C) 1/2 C 0 Σ 1/2 θ W m. The asymtotic null distribution of stability tests based on (11) does not deend on the local instabilities in the last m m 0 elements of θ because C 0 f is equal to zero whenever the first m 0 elements of f are. More formally, the right-hand side of (11) is a maximal invariant of J for the grou of transformations J 7 J + Σ 1 θ (0 1 m 0, F 0 ) 0 for continuous functions F :[0, 1] 7 R m m 0 satisfying F (0) = F (1) = 0. This grou corresonds to the transformations induced by local instabilities in the last m m 0 elements of θ, sothatefficient tests based on (11) yield best invariant tests of (8) based on J. 3 3 The results of Müller (2007) rovide a sense in which this rocedure is asymtotically efficient for the original GMM roblem, rather than only for the limiting rocess J. 11

13 When one alies the same tye of test statistic to (9) and (11), such as the functional corresonding to Nyblom s (1989) statistic N(ψ( )) = R 1 0 ψ(λ)0 ψ(λ)dλ, whereψ( ) is the lefthand side of (9) and (11), one obtains the same asymtotic distribution when all arameters are stable, and thus the same critical value. But in contrast to (9), under the null hyothesis (10) of stability of the first m 0 elements of θ, the second summand in (11) is equal to zero, indeendent of the last m m 0 elements of f. Therefore, as long as all otential instabilities are local, one might only test the stability of those arameters that one is actually interested in, and the result of tests based on (11) will not be affected by instabilities in the non-tested arameters. 4 If a stability test based on a functional of (11) rejects, it indicates that the resumably stable subset of arameters is not stable after all. If it is known for sure that the last m m 0 arameters are stable, however, tests based on (11) tyically have lower ower than tests based on (9): By the formula for the inverse of a artitioned matrix, C 0 Σ 1 θ C (C0 Σ θ C) 1 is ositive semi-definite, and zero only if Σ θ is block diagonal. The signal-to-noise ratio against alternatives of the form f = Cg for some function g :[0, 1] 7 R m 0 in (9) is (C 0 Σ 1 θ C) 1/2 C 0 Σ 1 θ C =(C0 Σ 1 θ C)1/2, which is larger than the corresonding ratio (C 0 Σ θ C) 1/2 C 0 C =(C 0 Σ θ C) 1/2 in (11). For tests that seek to detect otential instabilities in all arameters, i.e. C = I m, (11) reduces to (9). In summary, for a artially stable GMM model under Condition 1, standard asymtotically Gaussian GMM inference about the stable subset of arameters remains valid. Also, rejection of the overidentification test continues to indicate mistaken moment conditions. The rejection robability of usual stability tests for a subset of arameters, in contrast, is tyically affected by instabilities in the non-tested arameters. As a solution, we suggest basing inference on a class of modified statistics that are functions of (11), whose asymtotic rejection robabilities are a function of the stability of the arameters under consideration only. 4 One might worry about the local nature of the robustness against instabilities in the last m m 0 elements of θ of this aroach. Formally, one might resolve this issue by never rejecting H 0 in (10) whenever the -value of a arameter stability test for the last m m 0 elements in θ is smaller than some threshold >0, similar to the discussion in the introduction. 12

14 2.4 Justifying Condition 1 via Contiguity As noted in Section 2.1 above, the assumtions in arts (iii) (vii) of Condition 1 are fairly standard for stable GMM models. The analysis of unstable models is comlicated by the fact that arameter instability tyically leads to nonstationary data, and otentially comlicated interactions between the time varying arameters and the data generating rocess (think of regression models with lagged deendent variables with time varying coefficients). One way to address these comlications is to restrict the ossible interactions: Ploberger, Krämer, and Kontrus (1989) only consider regression models with strictly exogenous regressors. Sowell (1996) assumes that both the stable and unstable model generate stationary data. In the context of an unstable regression, Stock and Watson (1998) rule out lagged deendent variables. It might be ossible to justify Condition 1 directly by imosing rimitive conditions on the unstable model similar to those in Andrews (1993) (see Ghysels, Guay, and Hall (1997) and Hall and Sen (1999) for additional results based on these assumtions). In Andrews (1993) analysis of the local asymtotic ower of stability tests, {g t (θ 0 )} T is assumed to be near-eoch deendent with time varying mean and finite higher moments. Such conditions allow for a rich set of unstable models, including regression models with only weakly exogenous regressors. At the same time, given the highly technical nature of these rimitive assumtions, for any given model it might not be much harder to establish the high-level Condition 1 from first rinciles. Also, Andrews (1993) does not rovide a discussion of the consistency of the long-run variance estimator ˆV T in the unstable model. We hence refrain from further discussing rimitive conditions on the data and the function g that imly Condition 1 directly. Rather, we now discuss conditions on the likelihood of stable models that imly Condition 1 (iii) (vii) to hold in the unstable model whenever they hold in the corresonding stable model. This indirect reasoning circumvents much of the difficulty of establishing Condition 1 in (locally) unstable models. The difference between the unstable model and the corresonding stable model is the resence of time varying arameters, whose time variation is only big enough to be detectable with some (ossibly high) robability. Even efficient GMM based tests for arameter stability cannot discriminate between the stable and unstable model consistently. But this suggests that no statistic can be of a different robabilistic order in the unstable model than in the stable model. This in turn imlies Condition 1 (iv) (vii) to be true in the unstable GMM 13

15 model whenever they hold in the corresonding stable GMM model (i.e. when f =0). The formal argument relies on the concet of contiguity. Definition 2 Let PT 1 and PT 2 be two sequences of robability measures on a measurable sace (Ω T, F T ).ThesequencePT 1 is called contiguous to P T 2 if T 0 under PT 2 imlies T 0 under PT 1 for all random variables T : Ω T 7 R. See, for instance, Chater 6 of van der Vaart (1998) for further discussion and references. To make the heuristic reasoning rigorous, we thus need to imose some regularity conditions on the generating rocess of the data {y T,t } T to ensure that the unstable model is contiguous to the stable model. Assume that the difference between the density of the stable and unstable model can be described by the evolution of the k 1 arameter β, k, such that for all s T, the density of {y T,t } s (withresecttosomesigmafinite measure)isgivenby Q s f T,t(y T,t,y T,t 1,,y T,1 ; β T,t ) when β takes on the value β T,t at date t. Withk>, this allows the instability in the likelihood to go beyond the instability in the GMM arameter θ. Denote by l T,t (β) =lnf T,t (y T,t,y T,t 1,,y T,1 ; β) the contribution to the log-likelihood of the density at date t, the scores s T,t (β) = l T,t (β)/ β and the Hessians h T,t (β) = s T,t (β)/ β 0.LetF T,t be the σ field generated by {y T,s } t s=1, andf T,0 be the trivial σ field. We again omit the deendence on T of β T,t, s T,t, h T,t and F T,t for simlicity. Also, we refer to the model with density Q T f T,t(y T,t,y T,t 1,,y T,1 ; β 0 ) as the stable model. Condition 2 (i) The unstable arameter vector β t satisfies T 1/2 (β t β 0 )=B(t/T ) for some bounded and iecewise continuous vector function B :[0, 1] 7 R k with at most a finite number of discontinuities, and left and right limits everywhere. (ii) In some neighborhood B 0 of β 0, l t (β) is twice differentiable a.s. with resect to β for t =1,,T. Furthermore, in the stable model, (iii) {s t (β 0 ), F t } is a square-integrable martingale difference array with T P 1 [λt ] E[s t(β 0 )s t (β 0 ) 0 F t 1 ] R λ Υ(l)dl for all 0 λ 1 and some nonstochastic bounded Riemann integrable matrix function Υ : [0, 1] 7 R k k 0, T 1 su t T E[s t (β 0 )s t (β 0 ) 0 F t 1 ] 0 and there exists ν > 0 such that T P 1 T E[ s t(β 0 ) 2+ν F t 1 ]=O (1). (iv) T 1 P T h t(β 0 ) = O (1), T 1 su t T h t (β 0 ) 0 and for any decreasing neighborhood B T of β 0 contained in B 0, T 1 P T su β B T h t (β) h t (β 0 ) 14 0.

16 (v) For all 0 λ 1, T 1 P [λt ] h t(β 0 ) R λ 0 Υ(l)dl. Part (i) makes the same assumtion on the form of the instability in β as Condition 1 (i) does on θ. Parts (iii) (v) are weak regularity conditions on the likelihood of the stable model, see, for instance, Phillis and Ploberger (1996) for a similar set of assumtions. When integration and differentiation can be exchanged and the relevant conditional moments exist, {s t (β 0 ), F t } and {s t (β 0 )s t (β 0 ) 0 +h t (β 0 ), F t } are martingale difference arrays by construction see Hall and Heyde (1980), Chater 6.2. The matrix function Υ reresents the average rate of (conditional) information accrual on the time scale of the the samle fraction. For stationary stable models, Υ is constant and equal to the robability limit of ( T P 1 T h t(β 0 )) and T P 1 T E[s t(β 0 )s t (β 0 ) 0 F t 1 ]. The oint-wise convergences in λ in arts (iii) and (v) are then fulfilled automatically. Lemma 1 Under Condition 2, the unstable model is contiguous to the stable model. In articular, if a stable GMM model satisfies Conditions 1 (iv) (vii) and 2, then Condition 1 (iv) (vii) also holds under the unstable model. Lemma 1 formally states the ossibility of obtaining Condition 1 (iv) (vii) by making assumtions only on the stable GMM model. As argued above, Condition 1 (iv) (vii) is quite standard under stability. Note that one does not need to know the likelihood structure of the data to take advantage of this reasoning, as long as one is willing to assume Condition 2. In a general GMM set-u, Condition 2 lays the role of a regularity condition, akin to more familiar mixing or moment conditions. Under and alternative assumtion of a stochastic arameter ath, as discussed in Section 2.1, the argument based on contiguity may still be alied, as long as Condition 2 holds conditional on almost all realizations of B. See the aendix for details. While contiguity imlies that all o (1) aroximations of the stable model remain asymtotically accurate in the unstable model, it does not in itself justify Condition 1 (iii), the weak convergence of the average samle moment condition to a multivariate normal. At the same time, some rimitive conditions of (Functional) Central Limit Theorems take the form of convergences in robability. To establish those in the unstable model, it suffices to show that they hold in the stable model and to then invoke contiguity. As an examle, consider the case where the moment condition evaluated at the truth g T,t (θ T,t ) is a martingale 15

17 difference array with resect to the sigma fields G T,t,whereg T,s (θ T,s ) is measurable with resect to G T,t for all s<t. Droing again the deendence on T for simlicity, we can verify the conditions given in McLeish (1974) and establish the following Lemma. Lemma 2 If in the unstable model, {g t (θ t ), G t } T is a martingale difference array and there exists ς > 0 such that T P 1 T E[ g t(θ t ) 2+ς G t 1 ] = O (1), and in the stable model, Condition 1 arts (i),(ii),(vi) and (vii) hold, T 1/2 su t T g t (θ 0 ) 0 and T P 1 T g t(θ 0 )g t (θ 0 ) 0 V for some ositive definite matrix V, then under Condition 2, T P 1/2 T g t(θ t ) N (0,V) in the unstable model. Furthermore, if in addition T P 1 [λt ] g t(θ 0 )g t (θ 0 ) 0 P λv for all 0 λ 1 in the stable model, then T 1/2 [ T ] g t(θ t ) V 1/2 W ( ) in the unstable model, where W is a 1 standard Wiener rocess. To aly Lemma 2, the only condition that needs to be verified in the unstable model is that {g t (θ t ), G t } T is a martingale difference array with slightly more than two conditional moments, which are bounded in robability on average. This is often further facilitated by contiguity: Suose g t (θ t ) is of the form x t 1 ε t in the unstable model, with x t measurable with resect to G t and su t E[ ε t 2+ς G t 1 ] M ε a.s under the unstable model. Then T P 1 T E[ g t(θ t ) 2+ς G t 1 ] M ε T P 1 T x t 1 2+ς a.s. in the unstable model, and it suffices to show that T P 1 T x t 1 2+ς = O (1) inthestablemodeltoconcludeby contiguity that it is also O (1) in the unstable model. Interestingly, one can justify Condition 1 art (iii) entirely with assumtions on the stable model when the likelihood can be arametrized in a way such that the moment condition becomes a linear combination of the derivatives of the log-likelihood. The leading case for this is, of course, maximum likelihood estimation, although it also covers instances where only a subset of the likelihood derivatives are exloited as moment conditions. The roof of the following Lemma relies heavily on LeCam s Third Lemma (see van der Vaart (1998),. 90), an asymtotic change of measure from the stable to the unstable model. Lemma 3 If Condition 2 holds and T P 1/2 T g t(θ 0 ) T 1/2 F P 0 T s t(β 0 ) 0 under the stable model for some k matrix F,thenT P 1/2 T g t(θ 0 ) N (0,V) in the stable model and T P 1/2 T g t(θ t ) N (0,V) in the unstable model, where V = F 0R Υ(s)ds F. To sum u, a reasoning via contiguity justifies the high level Condition 1 for the unstable model mostly by reference to the corresonding stable model: Whenever a stable model 16

18 satisfies Conditions 1 and 2, then Condition 1 (iv) (vii) also holds under the unstable model. In general, Condition 1 (iii) under the unstable model requires an additional argument, but contiguity either simlifies the alication of an aroriate central limit theorem (Lemma 2) or, in the secial context of Lemma 3, is also imlied by contiguity whenever Condition 1 (iii) holds in the stable model. Theorem 1 thus holds for a wide range of data generating rocesses, including regression models with lagged endogenous variables and models with additional local time variation in unmodelled arameters. 3 Monte Carlo Results The results of the last section show that usual GMM inference about a stable subset of arameters remains asymtotically valid under certain conditions. The two main assumtions are that (i) the instabilities are local in the sense that even efficient stability tests do not detect them with robability one and (ii) the derivative of the moment samle condition has aroximately equal averages in all arts of the samle. This section exlores the accuracy of this asymtotic result in small samles by two Monte Carlo exeriments. The first exeriment concerns a simle time series OLS regression, and we exlore the quantitative imlications of assumtion (ii) by considering null rejection robabilities as a function of the ersistence in the regressors. The second exeriment concerns GMM inference for the structural arameters in the New Keynesian Phillis Curve examle introduced in the introduction. We calibrate the data generating rocess to quarterly US data and exlore the range of magnitudes of the instability for which our results remain useful aroximations. 3.1 OLS Regression The first exeriment considers the linear regression Y t = t θ 1,t + Z t θ 2 + θ 3 + ε t, t =1,,T (12) where ε t i.i.d. N (0, 1) and ( t,z t ) 0 is a zero-mean stationary Gaussian VAR(1) indeendent of {ε t } with coefficient matrix ri 2, Et 2 = EZt 2 = 1 and E[ t Z t ] = ρ Z. Let R t =( t,z t, 1) 0, and denote by ˆε t the OLS residuals of regression (12). This is an exactly identified GMM roblem, where the derivative of the moment condition equals G t (θ) =R t Rt,sothatˆΓ 0 = T P 1 T R trt 0 and, for heteroskedasticity robust inference, we 17

19 set ˆV T = T P 1 T R trtˆε 0 2 t. The arameter r governs the degree of autoregressive ersistence in the nonconstant regressors, and our asymtotic results formally aly when r < 1. We base tests for the resence of an instability on analogues of Nyblom s (1989) statistic. Let C be a 3 m 0, m 0 3 matrix, which is constructed of those columns of I 3 that corresond to the coefficients whose stability is to be tested. For instance, to test the stability of θ 2, C =(0, 1, 0) 0. With ˆΣ θ =(ˆΓ 0 ˆV 1ˆΓ) T 1, the non-modified Nyblom statistic based on (9) is then given by 5 Ã N = T 2 s=1 s=1 C 0ˆΓ 0 ˆV 1 T! 0 s ³ R tˆε t C 0 ˆΣ 1 θ C 1 Ã C 0ˆΓ 0 ˆV 1 T and the modified Nyblom statistic based on (11) is Ã! 0 Ã s 1 M = T 2 C 0 ˆΣ θˆγ 0 ˆV 1 T R tˆε t ³C 0 ˆΣ θ C C 0 ˆΣ θˆγ 0 ˆV 1 T! s R tˆε t (13)! s R tˆε t. (14) By Theorem 1 (iii), under the null hyothesis of all coefficients being constant, the asymtotic distribution of both N and M is as tabulated in Nyblom (1989). We consider two forms of instability in θ 1 : a break in the middle of the samle, θ 1,t = ht 1/2 1[t >T/2]; and a Gaussian random walk, θ 1,t = ht 1/2 W (t/t ), wherew is a standard Wiener rocess indeendent of {ε t,r t } T. Small instabilities (denoted as sm in the table) corresond to h =5and h =8in the single break case and the random walk case, resectively; large instabilities (denoted as lg in the table) corresond to h =10and h =16. We set the samle size T =100and, as a benchmark, r =0.5. Table 1 reorts emirical rejection robabilities of heteroskedasticity robust two-sided t-tests on θ 1 and θ 2 (t 1 and t 2 ) under the null hyothesis, of the usual Nyblom statistics (13) for the constancy of all three coefficients (N all )andofθ 1 and θ 2 (N 1 and N 2 ), and of the modified Nyblom statistics (14) for the constancy of the coefficients θ 1 and θ 2 (M 1 and M 2 ). The number of relications is 50,000. All tests are based on 5% nominal level asymtotic critical values. When θ 1,t is time varying, the true value of θ 1 is set to T P 1 T θ 1,t in the comutation of t 1.Byinsectionof(13)and(14),notethatM 2 and N 1 are invariant to the transformation 5 This version of the heteroskedasticity robust Nyblom (1989) statistic differs from what is suggested in Hansen (1990) and often emloyed in ractice, that is T 1 P T P ³ s=1 (C0 s R tˆε t ) 0 T 1 P T C0 R t RtCˆε t (C 0 P s R tˆε t ). The otimality result of Sowell (1996) discussed above imlies that in the resence of heteroskedasticity, (13) is the more owerful statistic, at least asymtotically. 18

20 {R t } 7 {AR t } for any invertible lower triangular matrix A. This transformation leaves the regressor with the time varying arameter unchanged u to scale, so that the results for M 2 and N 1 do not deend on the value of ρ Z even in the unstable model, and this also holds for t 2 by the same argument. Similarly, in the stable model with h =0, the distribution of all tests does not deend on ρ Z. << Table 1 about here >> The emirical rejection robability of the t-test on the stable coefficient θ 2 is very little affected by the instability in θ 1, as redicted by Theorem 1 (i). This remains true even for instabilities that are large enough to be detected by tests with high robability for the large instability, the -value of N all is smaller than 0.1% for more than one in four realizations. The magnitudes of the instabilities considered here are very large by emirical standards. Cogley and Sargent (2005) find instabilities in arameters of monetary VARs that they consider substantial from an economic oint of view, but which are detected by 5% nominal level Nyblom statistics less than 25% of the time. Stock and Watson (1996) reject the stability of the seven arameters describing univariate AR(6) models for 40 out of 76 U.S. ostwar macroeconomic time series on the 10% level using Andrews (1993) QLR statistic. But based on Stock and Watson s (1998) method of obtaining median unbiased estimates for the magnitude h of a random walk instability by inverting the QLR test statistic, the largest estimate of h for these 76 models is less than 12. Similarly, in Ghysel s (1998) alication in asset ricing, he mostly rejects the stability of two- and three-arameter versions of a conditional consumtion-based CAPM for 12 industry and 10 size sorted ortfolios, using 7 different instruments, and often on the 1% significance level. But his test statistics imly median unbiased estimates of h that are always smaller than 11. What is more, for T = 250, one needs to double the magnitude of the instabilities to obtain roughly similar size distortions as reorted in Table 1. All this suggests that the results of this aer are of emirical relevance for many arameter instabilities that one might encounter in a financial or macroeconomic alication. Also, as imlied by Theorem 1 (i), the t-test of θ 1 using the seudo true value T P 1 T θ 1,t has a rejection robability close to the nominal level. The usual Nyblom statistic for the stability of θ 2,N 2,isstronglyaffected by the instability in θ 1 when ρ Z 6=0, in contrast to the modified statistic M 2. Comaring the ower of M 1 and N 1,wefind 19

21 substantial losses in ower of the modified statistic only when ρ Z =0.9. << Figure 1 about here >> The results in Table 1 are based on regressors that follow a VAR(1) rocess with coefficient r =0.5, so that ersistence in the regressors is moderate. Figure 1 deicts the null rejection robability of the t-test t 2 of the stable arameter θ 2 as a function of r [0, 1). While technically a stationary VAR, for large autoregressive roots r<1, linearity of T P 1 [λt ] R trt 0 in λ, i.e., Condition 1 (vii), is a oor aroximation, and the test dislays substantial overrejections. As one might exect, unreorted simulations show that r needs to be relatively closer to unity to induce overrejections of a similar degree in larger samles. Nevertheless, our results should be alied cautiously for models with ersistent data, such as macro data in levels. We recommend conducting a model secific Monte Carlo analysis when ersistence is an issue. 3.2 A Stylized New Keynesian Phillis Curve Model The second exeriment is based on the New Keynesian Phillis Curve (NKPC) data generating rocess (3) and (4) of the introduction. With φ<1, the unique reduced form of the two-equation system is where ν t = ε t + γξ t and α 1 = κ(ρ 1 + φρ 2 ) 1 φρ 1 φ 2 ρ 2,α 2 = π t = α 1 s t 1 + α 2 s t 2 + ν t (15) s t = ρ 1 s t 1 + ρ 2 s t 2 + ξ t κρ 2 1 φρ 1 φ 2 ρ 2, and γ = κ 1 φρ 1 φ 2 ρ 2. (16) We adot the anticiated utility assumtion of the learning literature that agents know the true value of the arameters at each eriod, but behave as if the arameters remained constant in the future cf. Kres (1998). Under this assumtion, time varying arameters of the model (4) lead to time varying arameters of the reduced form arameters in (16), with the current values of α 1, α 2 and γ determined by the current values of ρ 1 and ρ 2. Note that due to the interaction via the exected future inflation term, instabilities in ρ 1 and ρ 2 lead to unstable reduced form arameters α 1 and α 2, even when the Euler equation in (3) is assumed stable throughout. 20