Fixed-b Inference for Testing Structural Change in a Time Series Regression

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1 Fixed- Inference for esting Structural Change in a ime Series Regression Cheol-Keun Cho Michigan State University imothy J. Vogelsang Michigan State University August 29, 24 Astract his paper addresses tests for structural change in a weakly dependent time series regressions. he cases of full structural change and partial structural change are considered. Heteroskedasticity-autocorrelation (HAC) roust Wald tests ased on nonparametric covariance matrix estimators are explored. wo di erent ways to construct the HAC estimator and accordingly the Wald statistic are analyzed. Fixed- theory is developed for the HAC estimators which allows xed- approximations for the test statistics. For the case of the reak date eing known, the xed- limits of the statistics depend on the reak fraction and the andwidth tuning parameter as well as on the kernel. he analysis is further extended to the case of partial structural change. A simulation study compares the nite sample properties and local power across di erent tests. When the reak date is unknown, supremum, mean and exponential Wald statistics are commonly used for testing the presence of the structural reak. Fixed- limits of these statistics are otained and critical values are taulated. Keywords: HAC estimator, kernel, andwidth, partial structural change, reak point Introduction his paper focuses on xed- inference of heteroskedasticity and autocorrelation (HAC) roust Wald statistics for testing for a structural reak in a time series regression. We focus on kernel-ased nonparametric HAC estimators which are commonly used to estimate the asymptotic variance. HAC estimators allow for aritrary structure of the serial correlation and heteroskedasticity of weakly dependent time series and are consistent estimators of the long run variance under the assumption that the andwidth (M) is growing at a certain rate slower than the sample size ( ). Under consistency assumptions, the Wald statistics converge to the usual chi-square distriutions. However, ecause the critical values from the chi-square distriution are ased on a consistency approximation for the HAC estimator, the chi-square limit does not re ect the often sustantial nite sample randomness of the HAC estimator. Furthermore, the chi-square approximation does not capture the impact of the choice of the kernel or the andwidth on the Wald statistics. he

2 sensitivity of the statistics to the nite sample ias and variaility of the HAC estimator is well known in the literature; Kiefer and Vogelsang (25) among others have illustrated y simulation that the traditional inference with a HAC estimator can have poor nite sample properties. Departing from the traditional approach, Kiefer and Vogelsang (22a), Kiefer and Vogelsang (22), and Kiefer and Vogelsang (25) otain an alternative asymptotic approximation y assuming that the ratio of the andwidth to the sample size, = M=, is held constant as the sample size increases. Under this alternative nesting of the andwidth, they otain pivotal asymptotic distriutions for the test statistics which depend on the choice of kernel and andwidth tuning parameter. Simulation results indicate that the resulting xed- approximation has less size distortions in nite samples than the traditional approach especially when the andwidth is not small. heoretical explanations for the nite sample properties of the xed- approach include the studies y Hashimzade and Vogelsang (28), Jansson (24), Sun, Phillips, and Jin (28, hereafter SPJ), Gonçalves and Vogelsang (2) and Sun (24). Hashimzade and Vogelsang (28) provide an explanation for the etter performance of the xed- asymptotics y analyzing the ias and variance of the HAC estimator. Gonçalves and Vogelsang (2) provide a theoretical treatment of the asymptotic equivalence etween the naive ootstrap distriution and the xed- limit. Higher order theory is used y Jansson (24), SPJ (28) and Sun (24) to show that the error in rejection proaility using the xed- approximation is more accurate than the traditional approximation. In a Gaussian location model Jansson (24) proves that for the Bartlett kernel with andwidth equal to sample size (i.e. = ), the error in rejection proaility of xed- inference is O( log ) which is smaller than the usual rate of O( =2 ). he results in SPJ (28) complement Jansson s result y extending the analysis for a larger class of kernels and focusing on smaller values of andwidth ratio. In particular they nd that the error in rejection proaility of the xed- approximation is O( ) around =. hey also show that for positively autocorrelated series, which is typical for economic time series, the xed- approximation has smaller error than the chi-square or standard normal approximation even when is assumed to decrease to zero although the stochastic orders are same. In this paper xed- asymptotics is applied to testing for structural change in a weakly dependent time series regression. he structural change literature is now enormous and no attempt will e made here to summarize the relevant literature. Some key references include Andrews (993), Andrews and Ploerger (994), and Bai and Perron (998). Andrews (993) treats the issue of testing for a structural reak in the GMM framework when the one-time reak date is unknown and Andrews and Ploerger (994) derive asymptotically optimal tests. Bai and Perron (998) consider multiple structural change occurring at unknown dates and cover the issues of estimation of reak dates, testing for the presence of structural change and testing for the numer of reaks. 2

3 For a comprehensive survey of the recent structural reak literature see Perron (26). Because a structural change in a regression relationship e ectively divides the sample into two regimes, we consider two di erent HAC estimators when constructing Wald statistics. Asymptotically, estimators of the regression parameters are uncorrelated across the two regimes ecause of weak dependence. One HAC estimator imposes this zero covariance restriction while the other estimator does not and this leads to two possile Wald statistics, W ald (S) and W ald (F ), that can e used in practice. When the reak date is known and coe cients of all regressors are suject to structural change (i.e. full structural change), oth Wald statistics have pivotal xed- limits ut these limits are di erent. he xed- critical values increase as the andwidth gets igger and as the hypothesized reak date is closer to the oundary of the sample. When some of the regressors are not suject to structural change (i.e. partial structural change), the Wald statistic ased on the unrestricted HAC estimator (W ald (F ) ) still has the same pivotal xed- limit as in the case of full structural change. We carry out a local power analysis and nd several patterns. he local power of oth Wald statistics is lower with igger especially for the QS kernel. Power also improves as the reak date gets closer to the middle of the sample regardless of andwidth or kernel. With the within-regime e ective andwidths matched across the two statistics, the power di erence is more evident when a ig value of is used with the QS kernel. A simulation study on nite sample performance reveals that the Wald statistic ased on the unrestricted HAC estimator has etter overall size performance especially when the quadratic spectral (QS) kernel and large andwidths are used in the case of persistent data. When we compare the performance of xed- inference with traditional inference, we nd that the latter is suject to sustantially more size distortions. Similar to the known patterns in models without structural change, we nd that size distortions decrease as one uses igger andwidth or as the hypothesized reak date is closer to the middle of the sample. When there is strong persistence in the data, rejections using xed- critical values can e aove nominal levels although these distortions are much smaller compared to traditional inference. he remainder of this paper is organized as follows. Section 2 reviews xed- asymptotic theory in a regression with no structural change. In Section 3 the asic set up of the full structuralchange model. Some preliminary results are laid out and the two HAC estimators are introduced. Section 4 derives the xed- limits of the two Wald statistics. Section 5 explores patterns in the xed- critical values when the reak date is treated as a priori. Section 6 compares empirical null rejection proailities across di erent approaches of inferences, di erent choices of kernel and HAC estimators under various data generating process (DGP) speci cations. Section 7 examines local power. Section 8 generalizes the results in Section 4 to a model with partial structural change. It is shown that the xed- limit of W ald (F ) in the partial structural change model is same as in the 3

4 full structural change model. Section 9 covers the case where the reak date is unknown providing the xed- critical values. Section summarizes and concludes. Proofs and supplemental results are collected in the Appendix. 2 Review of the Fixed- Asymptotics We consider a weakly dependent time series regression with p regressors given y y t = x t + u t : () Model () is estimated y ordinary least squares (OLS) giving = x t x t! t= t=! x t y t ; and u t = y t x t are the OLS residuals. he centered OLS estimator is given y = x t x t! t= t=! v t ; where v t = x t u t. We make the following two assumptions where [x] denote the integer part of the real numer, x. Assumption. P [r ] t= x tx t p! rq; uniformly in r 2 [; ]; and Q exists. Assumption 2. =2 P [r ] t= x tu t = =2 P [r ] t= v t ) W p (r); r 2 [; ]; where = ; and W p (r) is a p standard Wiener process. For a more detailed discussion aout the regularity conditions under which Assumptions and 2 hold, refer to Kiefer and Vogelsang (22). See Davidson (994), Phillips and Durlauf (986), Phillips and Solo (992), and Wooldridge and White (988) for more details. he matrix Q is the second moment matrix of x t and is typically estimated using the quantity Q = P t= x tx t. he matrix is the asymptotic variance of =2 P t= v t; which is, for a stationary series, given y X = + ( j + j) with j = E(v t vt j): j= Being a long run variance, is commonly estimated y the kernel-ased nonparametric HAC estimator = X t= s= jt K sj M v t v s = + j= K j M j + j ; 4

5 where j function. = P t=j+ v tv t j ; v t = x t u t, M is a andwidth, and K() is a kernel weighting Under some regularity conditions (see Andrews (99), De Jong and Davidson (2), Hansen (992), Jansson (22) or Newey and West (987)), is a consistent estimator of, i.e. p!. hese regularity conditions include the assumption that M=! as M;!. his asymptotics is called traditional asymptotics throughout this paper. In contrast to the traditional approach, xed- asymptotics assumes M = where is held constant as increases. Assumptions and 2 are the only regularity conditions required to otain a xed- limit for. Under the xed- approach, for 2 (; ], Kiefer and Vogelsang (25) show that where f W p (r) = W p (r) ) P(; f W p ) ; (2) rw p () is a p-vector of standard Brownian ridges and the form of the random matrix P(; f W p ) depends on the kernel. Following Kiefer and Vogelsang (25) we consider three classes of kernels which gives three forms of P. Let H p (r) denote a generic vector of stochastic processes. H p (r) denotes its transpose. hen P(; H p ) is de ned as follows: Case Suppose K(x) is twice continuously di erentiale everywhere (Class ) such as the Quadratic Spectral kernel (QS), then P (; H p ) Z Z r 2 K where K () is the second derivative of the kernel K(): s Case 2 Suppose K(x) is the Bartlett kernel (Class 2), then H p (r)h p (s) drds; (3) P (; H p ) 2 Z H p (r)h p (r) dr Z H p (r)h p (r + ) + H p (r + )H p (r) dr: (4) Case 3 Suppose K(x) is continuous, K(x) = for jxj, and K(x) is twice continuously di erentiale everywhere except for jxj = (Class 3) (e.g. Parzen kernel), then Z Z jr sj P (; H p ) 2 K H p (r)h p (s) drds (5) jr + K _() sj< Z H p (r + )H p (r) + H p (r)h p (r + ) dr; where K _() = lim h# [(K() K( h)) =h] ; i.e., K _() is the derivative of K(x) from the left at x = : 5

6 result Inference regarding is ased on the asymptotic normality of the OLS estimator given y the p ( ) = x t x t t=! =2 X t=! d x t u t! Q W p () N(; Q Q ); which follows directly from Assumptions and 2. Suppose we are interested in testing the null hypothesis, H : R lp = r, where l is the numer of the restrictions. De ne the Wald statistic as W ald = R r R Q Q R R Under traditional asymptotics we otain the well known result whereas under xed- asymptotics we have W ald d! 2 l ; r : W ald ) W l () P(; f W l ) Wl (): (6) he limit of the Wald statistic depends on the sequence of andwidths y which we index. It is important to note that we are not viewing the choice of sequence as a andwidth rule for the choice of M in practice. Rather, we are simply pointing out that di erent asymptotic approximations are otained for the two assumptions regarding M. Under the xed- approach, the random matrix P(; W f l ) approximates the randomness in and its dependence on M (through ) and the kernel K(): In contrast the traditional approach approximates y a constant that does not depend on M or the kernel. Once structural change is allowed in the model, existing results in the xed- literature no longer apply and new results are required. 3 Model of Structural Change and Preliminary Results Consider a weakly dependent time series regression model with a structural reak given y y t = wt + u t ; (7) wt = x t; x 2t ; = ; 2 ; x t = x t (t ); x 2t = x t (t > ); where x t is p regressor vector, 2 (; ) is a reak point, and ( ) is the indicator function. Recalling that [ ] denotes the integer part of, note that x 2t = for t = ; 2; :::; [ ] and x t = for t = [ ] + ; :::; : For the time eing the potential reak point is assumed to e 6

7 known. We discuss the case of eing unknown in Section 9. he regression model (7) implies that coe cients of all explanatory variales are suject to potential structural change and this model is laeled the full structural change model. We are interested in testing the presence of a structural change in the regression parameters. Consider the null hypothesis of the form where H : R = ; (8) R = (R ; R ) ; (l2p) and R is an l p matrix with l p: Under the null hypothesis, we are testing that one or more linear relationships on the regression parameter(s) do not experience structural change efore and after the reak point. ests of the null hypothesis of no structural change aout a suset of the slope parameters are special cases. For example we can test the null hypothesis that the slope parameter on the rst regressor did not change y setting R = (; ; : : : ; ). We can test the null hypothesis that none of the regression parameters have structural change y setting R = I p. In order to estalish the asymptotic limits of the HAC estimators and the Wald statistics, Assumptions and 2 given in the previous section are su cient. hose Assumptions imply that there is no heterogeneity in the regressors across the segments and the covariance structure of the errors is assumed to e same across segments as well. Notation For later use, we de ne a l l nonsingular matrix A such that R Q Q R = AA ; and R Q W p (r) d = AW l (r); (9) where W l (r) is l standard Wiener process: We focus on the OLS estimator of given y = w t wt t=!! w t y t ; t= which can e written for each segment as = x t x t!! [ ] X x t y t x t x A t t= t= t= x t y t A ; () [ ] t= 7

8 2 = x 2t x 2t t=!! x 2t y t t= t=[ ]+ x t x A t=[ ]+ Fixed- results depend on the limiting ehavior of the following partial sum process S t = tx w j u j = j= tx j= w j y j x j x 2j 2 x t y t A : () tx = w j u j x j j= x 2j 2 2 : (2) Under Assumptions and 2 the limiting ehavior of and the partial sum process S t are given as follows. Proposition Let 2 (; ) e given. Suppose the data generation process is given y (7) and let [r ] denote the integer part of r where r 2 [; ]. hen under Assumptions and 2 as! ; p p ( ) p A d (Q) W! p () 2 2 (( )Q) ; (W p () W p ()) and where and =2 S[r ] ) F p (2) (r; ) = W p (r) F p (r; ) F p () (r; ) = W p (r) W p () F () p (r; ) F (2) p (r; ) r W p() (r );! r (W p() W p ()) (r > ): ; See the Appendix for the proof. It is easily seen that the asymptotic distriutions of and 2 are Gaussian and are independent of each other. Hence the asymptotic covariance of and 2 is zero. he asymptotic variance of p ( ) is given y Q Q ; where Q Q and : ( )Q ( ) In order to test the null hypothesis (8), HAC roust Wald statistics are considered. hese statistics are roust to heteroskedasticity and autocorrelation in the vector process, v t = x t u t : he generic form of the roust Wald statistic is given y W ald = R RQ Q R R ; 8

9 where Q = P! [ ] t= x tx t P t=[ ]+ x tx ; t and is an HAC roust estimator of. We analyze two estimators for : he rst one, denoted y (F ) ; is newly introduced in this paper and it is constructed using the residuals directly from the dummy regression (7): (F ) = X t= s= jt K sj M v t v s; (3) where v t = w t u t = x t u t; x 2t u t 2p. We denote the components of v t as v () t = x t u t = x t u t (t ) and v (2) t = x 2t u t = x t u t (t > ). Notice that (F ) is the variance estimator one would e using if the usual "Newey-West-Andrews" approach is applied directly to the dummy regression (7). he other estimator of is the HAC estimator appearing in the existing literature (e.g. Bai and Perron (26)) which is given y! (S) = () ( ) (2) ; (4) where (2) = () = [ ] [ ] X X jt K [ ] [ ] t= s= t=[ ]+ s=[ ]+ M sj jt K v () t v () s ; (5) M 2 sj v (2) t v (2) s : (6) Note that () is constructed using v () t (data from the pre-reak regime) and uses the andwidth M and pre-reak sample size [ ]. Likewise (2) is constructed using v (2) t (data from the postreak regime) and uses the andwidth M 2 and post-reak sample size [ ]. In Andrews (993) and Bai and Perron (26), the estimator (4) is used to allow for a potential structural change in the long run variance itself. In this paper we maintain the assumption that does not have structural change ecause allowing for heterogeneity in results in a non-pivotal xed- limit of the Wald statistic. Finding an estimator of that has a pivotal xed- limit when has structural change is a topic of ongoing research. At a super cial level, the estimators (F ) and (S) look di erent ut they are directly related. 9

10 Using v t = (v () t ; v (2) t ) we can write (F ) as! = (F ) P [ ] t= P (F ) (F ) 2 (F ) 2 (F ) 22 P P t= s= K jt P P t= s= K jt t=[ ]+ P t=[ ]+ s= K jt sj M P [ ] s= K jt sj M P [ ] sj M sj M v () t v () v (2) t v s () v () t v s () v (2) t v s () P t= P t= P s= K jt P s= K jt sj M sj M s P [ ] P t= s=[ ]+ K jt sj M P P t=[ ]+ s=[ ]+ K jt [ ] () P [ ] P t= s=[ ]+ K jt sj M P [ ] s= K jt sj M v (2) t v s () ( [ ]) (2) v () t v s (2) v (2) t v s (2) A v () t v s (2) sj M v (2) t v s (2) v () t v s (2) A A (7) We see that the diagonal locks of (F ) are nearly the same HAC estimators used y (S) (exactly the same when is an integer) except that the same andwidth, M, is used for each diagonal lock of (F ) whereas the diagonal locks of (S) can have di erent andwidths. In terms of the o -diagonal components, (S) is a restricted version of (F ) with the o -diagonal locks set to, i.e. (S) imposes a zero covariance etween and 2 matching the zero asymptotic covariance etween and 2. In contrast (F ) does not impose this zero covariance which is consistent with the possiility that the nite sample covariance etween and 2 is not equal to zero (which is true in general). Note, however, that (F ) uses the same andwidth for oth regimes whereas (S) allows di erent andwidths in the two regimes. hus, from the andwidth perspective, (F ) is restrictive relative to (S). he next section provides asymptotic results for the two HAC roust Wald statistics under the traditional asymptotics and under the xed- asymptotics. 4 Asymptotic Results 4. Asymptotic limits under the traditional approach he goal of the traditional approach is to nd conditions under which the HAC estimator is consistent. One requirement for consistency is that M grows with the sample ut at a slower rate. hen under additional regularity conditions, () and (2) are consistent estimators of and the limit of (S) is straightforwardly given y (S) = () ( ) (2) (2p2p)! p! : ( ) However estalishing consistency of (F ) requires some additional calculation eyond existing results in the literature.

11 Proposition 2 Under regularity conditions for the consistency of the HAC estimators () and (2), as! ; (F ) p! : ( ) 2p2p See the Appendix for the proof. Let W ald (S) denote the Wald statistic ased on (S) and let W ald (F ) denote the Wald statistic ased on (F ). hen the results given in this susection, comined with Proposition, give us the limits of the test statistics under the traditional approach: ( ) W l() W ald (S), W ald (F ) ) (W l() W l ()) W l() (W l() W l ()) ; where W l is a l standard Wiener process. Note that for any given value of the limit follows a chi-square distriution with degrees of freedom l: While convenient, the chi-square limit does not capture the impact of the randomness of (S) and (F ) on the Wald statistics in nite samples. 4.2 Asymptotic results under the xed- approach We now provide xed- limits for the HAC estimators and the test statistics. he xed- limits presented in the next Lemma and Corollary approximate the diagonal locks of (F ) y random matrices. Also, it is shown that xed- approach gives a nonzero limit for the o -diagonal lock, which further distinguishes xed- asymptotics from the traditional asymptotics. Lemma Let 2 (; ) and 2 (; ] e given. Suppose M = : hen under Assumptions and 2, as!, where F (2) p (r; ) = (F ) ) P (; F p (r; )) F () p (r; ) = W p (r) F p (r; ) = F p () (r; ) F p (2) (r; ) W p (r) W p ()! ; (8) ; (9) r W p() ( r ) ; (2) r (W p() W p ()) ( < r ) ; (2) and P p (; F p (r; )) is de ned y (3), (4), and (5) with H p (r) = F p (r; ). See the Appendix for the proof. Additional algera leads to an alternative representation for P (; F p (r; )). he proof for this Corollary is omitted.

12 Corollary where 8 >< = >: P (; F p (r; )) C R R jr sj< K jr 2 sj P ; F p () (r; ) C ; F () p (r; ) ; F (2) p (r; ) P R R C ; F p () (r; ) ; F p (2) (r; ) F () ; F p () (r; ) ; F p (2) (r; ) A ; F p (2) ; (22) (r; ) K jr sj 2 p (r; ) F p (2) (s; ) drds, R F p () (r; ) F p (2) (r + ; ) dr, p (r; ) F p (2) (s; ) drds + K _() R p (r; ) F p (2) (r + ; ) dr; F () for Class,2 and 3 kernels respectively. F () he expression for P (; F p (r; )) in this Corollary makes it easier to compare the xed- limit of (F ) with the standard xed- limit (see (2)) appearing in a non-structural change settings. Since each diagonal lock of (F ) is asically a HAC estimator (up to a scale factor; see (7)) ased on one of the pre- or post- reak data, its limit should take the same form as (2), which is veri ed in this Corollary. So each diagonal component of P (; F p (r; )) serves to re ect the randomness and andwidth/kernel-dependence of the associated HAC estimator. Second, unlike the traditional approach, the xed- limit of the o -diagonal component is nonzero. his implies the xed- inference is ale to take account of the covariance etween and 2 which is generally nonzero in nite samples. he next Lemma provides results for (S). Lemma 2 Let 2 (; ) and ; 2 2 (; ] e given. Suppose M = ( ) and M 2 = 2 ( ). hen under Assumptions and 2, as!, () ) P ; F p () (r; ) ; (2) ) P 2 ( ) ; F p (2) (r; ) ; and (S) P ; F p () (r; ) P 2 ( ) ; F p (2) (r; ) A ; See the Appendix for the proof. he limits of the Wald statistics can e derived y using Lemmas and 2 respectively, heorem gives the result for W ald (F ). 2

13 heorem Let 2 (; ) and 2 (; ] e given. Suppose M = : hen under Assumptions and 2, as!, P ; F () l (r; ) W ald (F ) ) See the Appendix for the proof. W l() F (2) l (r; ) (W l() W l ()) W l() (W l() W l ()) he next Corollary provides an alternative representation for the limit given in (23). he proof for this Corollary is given in Cho (24). Corollary 2 For a given value of 2 (; ), the xed- limit of W ald (F ) has the same distriution as ( ) W l() where P ; W f l (r) + P 8 >< CP (; ) = >: (23) ; f W l (r) + CP (; ) + CP (; ) W l (); p p R R jt ( )s j K fwl (t) W f l (s) dtds R 2 fw l ( r )f Wl ( r+ ) ( <r)dr p p for Class-2 kernels, p p R R jt ( )s j K fwl (t) W f l (s) (jt ( )s j<)dtds R 2 K_ ()f W l ( r )f Wl ( r+ ) ( <r)dr p p for Class- kernels, for Class-3 kernels, and W f l (r) and W f l (r) are l Brownian Bridge processes which are independent of each other and of W l (): he limit in (24) shows how the components of (F ) a ect the distriution of W ald (F ) : As mentioned earlier, the random matrix P ; W f l (r) re ects the random nature of (F ) which is part of the estimator of the asymptotic variance of. Notice that the (e ective) andwidth for (F ) turns out to e not : hus we implicitly use the andwidth ratio for (F ) when we use a full sample andwidth ratio for constructing (F ) : he second component, P ; W f l, (r) is related to (F ) 22 (and 2 ) in exactly the same fashion: Finally, the third component, CP (; ), captures the impact of nite sample covariance etween and 2 on structural change inference: heorem 2 gives the results for W ald (S). heorem 2 Let 2 (; ) and ; 2 2 (; ] e given. Suppose M = ( ) and M 2 = 2 ( ). hen under Assumptions and 2, as!, W ald (S) ) W l() (W l() W l ()) 3 (24)

14 P ; F () l (r; ) + W l() P 2 ( ) ; F (2) l (r; ) (W l() W l ()) : (25) See the Appendix for the proof. Corollary 3 For a given value of 2 (; ), the xed- limit of W ald (S) has the same distriution as ( ) W l() P ; W f l (r) + P 2 ; W f l (r) W l (): (26) he proof is given in Cho (24). he major di erence etween (26) and (24) lies in the term CP (; ) which is the limit associated with the cross product term of (F ) : A second di erence is distinct andwidth ratios used in each regime. In constructing the HAC estimator (F ), one uses a single value of andwidth M and accordingly a single andwidth ratio M. On the one hand this andwidth ratio implicitly determines the e ective andwidth ratios within each regimes: for the pre-reak M ( ) = ( ) = M = regime and ( ) for the post-reak regime. On the other hand, if one uses (S) ; a researcher uses two distinct andwidths for each regime respectively giving two andwidth ratios = M and 2 = M 2 ( ) : In order to sensily compare the performance of W ald(f ) and W ald (S), we should ensure that the e ective within-regime andwidth ratios are the same across the two statistics. Speci cally, if the andwidth ratio used for W ald (F ) is, then should make comparisons to W ald (S) with = and 2 =. Con guring andwidth ratios in this way allows us to isolate the impact of the presence of CP (; ) on inference. 5 Critical Values While the xed- limiting distriutions are nonstandard, asymptotic critical values are easily otained via simulations. We approximate the Wiener processes in the limiting distriutions using scaled partial sums of, i.i.d. N(; ) random variales. Critical values are taulated ased on 5, replications. he 95% critical values for l= and 2 are provided for selected values of and in ales. through ale 2.7. he reak fraction, ; runs from. to.9 with. increments. he andwidths considered are :2; :4; :6; :8; :; :2; : : : ; : ales. through.6 contain the critical values for W ald (F ) and ale 2. through 2.7 provide the critical values for W ald (S) : he critical value tales for W ald (S) only cover the case where = 2 although ale 2.7 provides some critical values with 6= 2 : ales. through.6 display two main patterns of the critical values. First, for each given the critical values increase as the andwidth gets igger. his is expected given the well known = 4

15 downward ias induced into HAC estimators from estimation error. he xed- approximation captures this downward ias and re ects it through larger critical values. Second, for a given value of the andwidth the critical values display a V-shaped pattern as a function of : As the reak point moves closer to zero or one, the critical values increase and the minimum critical values occur at = :5. his V-shaped pattern is present ut is less pronounced in the critical values for W ald (S). 6 Empirical Null Rejection Proailities In this section we report the results of a nite sample simulation study that illustrates the performance of the xed- critical values relative to the traditional critical values under the null hypothesis of no structural change. he DGP used in this simulation study is given y y t = + 2 x t + u t ; x t = x t + t ; u t = u t + t + ' t ; where t and t are independent of each other with t ; t i.i.d. N(; ). We use the following parameter values: ( ; 2 ) = (; ); 2 f:2; :4; :5g; 2 f:5; :8; :9g; (; ') 2 f(; ); (:5; ); (:5; :5); (:9; :5); (:9; :9)g; with various cominations of parameters laeled according to the tale DGP ' x t u t v t = x t u t A :5 AR() IID White Noise B :5 :5 AR() AR() Serially Correlated C :5 :5 :5 AR() ARMA(,) Serially Correlated D :8 :5 :5 AR() ARMA(,) Serially Correlated E :8 :9 :5 AR() ARMA(,) Serially Correlated F :9 :9 :9 AR() ARMA(,) Serially Correlated he value of measures the persistence of the time varying regressor x t : he parameters and ' jointly determine the serial correlation structure of the error term u t. Bigger values of these three parameters leads to higher persistence of the time series v t x t u t except for speci cation A. We report results for sample sizes = 5; ; 5 and the numer of replications is 2,5. he nominal level of all tests is 5%. We compute the W ald (S) and W ald (F ) statistics for testing the joint null hypothesis of no structural change in oth the and 2 parameters. 5

16 6. W ald (S) and W ald (F ) : raditional Inference vs. Fixed- Inference We rst examine the empirical null rejection proailities of the two statistics. We compare the performance of the traditional chi-square critical values with xed- critical values. ales 3. through 3.6 report empirical null rejection proailities for W ald (S) using the two critical values. In practice one can construct (S) using di erent andwidths ratios and 2 in the two regimes. o conserve space, we only provide results for homogeneous andwidths choices (i.e. = 2 ). ales 3. and 3.2 give results for DGP A for = :2; :4, ales 3.3 and 3.4 give results for DGP D for = :2; :4, and ales 3.5 and 3.6 give results for DGP E for = :2; :4. Each tale reports results using the Bartlett and QS kernels for a range of values of. Consider the results for DGP A (no serial correlation in v t ) given in ales 3. and 3.2. When = 5, we see that oth the Bartlett and QS kernel deliver tests that over-reject when the chisquare critical value is used. As the andwidth gets igger, this tendency to over-reject ecomes more and more pronounced. Rejections using xed- critical values are similar when small andwidths are used. As the andwidth increases, rejections using xed- critical values systematically decrease towards the nominal level of.5. Increasing reduces over-rejections for oth critical values when is small. In contrast, when is large, rejections using the chi-square critical value tend to persist as increases ut rejections tends towards.5 when xed- critical values are used. With = 5, the QS kernel has empirical rejection proailities close to.6 for all values of when xed- critical values are used. It is clear that the xed- approximation is often a sustantial improvement over the traditional approximation. ales and show the results for the data with stronger dependence. Patterns are similar to ales except that over-rejection prolems tend to e higher in all cases. Using chi-square critical values leads to severe over-rejection prolems regardless of andwidth or kernel. Using xed- critical values tends to reduce over-rejection prolems especially when larger values of are used. he QS kernel tends to e much less over-sized than the Bartlett kernel. hese nite sample patterns are very similar to patterns found y Kiefer and Vogelsang (25) in non-structural change settings. Comparing ales 3., 3.3, 3.5 ( = :2) with ales 3.2, 3.4, 3.6 ( = :4) we see that overrejection prolems tend to e greater with = :2 than with = :4. his makes sense ecause the = :2 case has a relatively small sample size for regime compared to regime 2 whereas for = :4 the regime sample sizes are similar. We next examine null rejection proailities of the W ald (F ) statistic. In Figure we plot rejection proailities for W ald (F ) for the DGP given y ( ; 2 ; ; ; ') = (; ; :; :5; :5) for the case of = 5: Results are given for two kernels: Bartlett and QS and andwidth ratio = :2. For each kernel, rejections are computed using the chi-square critical value and the xed- critical value. We plot rejections across a grid of reak points given y 2 f:; :2; :3; :4; :5; :6; :7; :8; :9g. 6

17 Figure shows the large size distortions associated with traditional inference. Over-rejections are sustantial especially when is close to the endpoints of the sample. In general we see a V-shaped pattern in over-rejections as a function of with the least over-rejections occurring at the middle of the sample ( = :5). Rejections using xed- critical values are much closer to.5 and rejections are less sensitive to the location of the reak point. Figure also shows that the kernel matters. Rejections using the QS kernel when xed- critical values are used are closer to.5. ales 4. through 4.6 provide additional results for W ald (F ) for most of the DGPs given in ales Again rejections are reported using traditional chi-square critical values and xed- critical values. he tale also reports some rejections for W ald (S) which are discussed in the next susection. he patterns in ales are generally very similar to the patterns seen in ales It is clear that the xed- approximation is a sustantial improvement over the traditional approximation. 6.2 Fixed- Inference: W ald (S) vs. W ald (F ) In this susection the nite sample rejections of the two Wald statistics are compared under the xed- framework. As discussed after Corollary 2, to make comparisons of W ald (S) and W ald (F ) sensile with respect to andwidths, we use andwidth ratios for W ald (S) given y = and 2 = where is the andwidth ratio for W ald (F ). : (27) By using (27) we ensure that the within regime andwidth to (regime) sample size ratios for W ald (S) are the same as the andwidth to (full) sample size ratio used y W ald (F ). We use the following eight andwidth/reakpoint speci cations satisfying (27): A : = :2; = :4; = :2; 2 = :5; B : = :2; = :; = :5; 2 = :25; C : = :2; = :2; = :; 2 = :25; D : = :5; = :5; = :; 2 = :; E : = :5; = :; = 2:; 2 = 2:; F : = :4; = :5; = :25; 2 = :83; G : = :2; = :5; = 2:5; 2 = :625; H : = :2; = :; = 5:; 2 = :25: ales 4. through 4.3 give results for relatively small andwidths (A, B, and C ). With small andwidths W ald (S) and W ald (F ) have very similar size distortions. ales 4.4 through 4.6 report 7

18 results for the relatively large andwidths (D, E, F, G, and H ). Note that compared to the small andwidth cases, there is noticeale di erence in the performance of the two statistics specially when the QS kernel is used. As the DGP ecomes more persistent (such as DGP F or G) the di erences etween W ald (S) and W ald (F ) ecome clear. ale 4.4 shows that under DGP F and andwidth E the empirical null rejection proaility is 9.8% for W ald (S) and 3.6% for W ald (F ) when = 5 with the QS kernel. In contrast rejections are similar for the Bartlett kernel. he di erences when = are smaller (5.96% vs..92%); see ale 4.5. Once reaches 5 (ale 4.6), the di erences etween W ald (S) and W ald (F ) are very small for either kernel. general W ald (F ) is less size distorted than W ald (S) when is relatively small and persistence in the data is not weak. For larger sample sizes the two tests have similar null rejection proailities. From a size perspective, W ald (F ) is preferred over W ald (S). he etter size performance of W ald (F ) can e explained y examining CP (; ) in (24) and the corresponding nite sample counterparts (F ) 2 (or (F ) 2 ) in (7). (F ) 2 is estimating the sample covariance etween and 2 which is driven y nite sample covariance etween v () t and v (2) t across the two regimes. Larger andwidths lead to (F ) 2 and (F ) 2 estimators that etter re ect the nite sample covariance etween and 2 and CP (; ) captures the impact of (F ) 2 and (F ) 2 on the sampling distriution of W ald (F ). As the data ecomes more persistent, the covariance etween and 2 ecomes more pronounced in small samples so there is ene t to including (F ) 2 and (F ) 2 in. As increases, the covariance etween and 2 decreases and it ecomes more reasonale to impose the restriction that the o -diagonal locks of are zero. 7 Local Power Analysis of Fixed- Inference In this section we investigate the local power of W ald (S) and W ald (F ) under the xed- approach. he local alternative is given y with R = (R ; are easily otained as H : R = =2 (l) In (28) R ) : Under the local alternative (28), the xed- limits of the Wald statistics W ald (F ) ) W l() P ; F () l (r; ) W l() (W l() W l ()) A F (2) l (r; ) (W l() W l ()) A ; 8

19 W ald (S) ) W l() P W l() ; F () where the nonsingular matrix A is de ned in equation (9). (W l() W l ()) A l (r; ) + P 2 ( ) ; F (2) l (r; ) (W l() W l ()) A ; 7. Comparison of the Local Asymptotic Power of W ald (S) and W ald (F ) In this susection the local asymptotic power of W ald (S) and W ald (F ) is compared for the case R = (I 2 ; I 2 ). he andwidth speci cations are the same as those used in Section 6.2. Local asymptotic power was computed using the same simulation methods used to compute asymptotic xed- critical values and we set A = (; ) where is a scalar parameter. Figures 2-5 plot local asymptotic power for the Bartlett and QS kernels for a selection of reakpoint and andwidth speci cations. As the gures illustrate, there is no sustantial di erence in the local asymptotic power etween W ald (S) and W ald (F ) for the case of the Bartlett kernel. However, when the QS kernel is used, di erences in power emerge with large andwidths as depicted y Figures 3-5. For example, using = as shown in Figures 4 and 5, we see sustantial power loss associated with W ald (F ) with the QS kernel. Comparing the Bartlett and QS kernels, we see that the Bartlett kernel delivers higher power than the QS kernel for oth Wald statistics. he loss of power associated with W ald (F ) relative to W ald (S) occurs exactly for the kernel and andwidth speci cations that resulted in W ald (F ) having less size distortions than W ald (S). Similarly, the higher power of the Bartlett kernel relative to the QS kernel comes at the cost of greater size distortions. As has een documented repeatedly in the xed- literature, there is an inherent trade-o etween reduction of over-rejection prolems and loss of power. Bandwidth/kernel cominations that reduce over-rejection prolems also tend to reduce power. 7.2 Impact of Breakpoint Location and Bandwidth on Power Next we focus on the impact of the reakpoint location,, and andwidths on local asymptotic power. Figures 6-9 display local asymptotic power for a range of : Each gure depicts power curves for the range of = :; :2; :3; :4; :5 for the andwidth ratio = :. Figures 6 and 7 depict power for the W ald (F ) statistic for the Bartlett and QS kernels respectively. Figures 8 and 9 depict power for W ald (S) for the Bartlett and QS kernels respectively. For a given ; the andwidths and 2 for W ald (S) are chosen so that they match the within-regime e ective andwidths of W ald (F ) ; and. For a given ; smaller values of implies larger values of max ; and the local power for W ald (F ) decreases as max ; increases. First notice that regardless of 9

20 andwidth or kernel, power is lowest for = : and steadily rises as increases to :5. We should note that power is symmetric in and power for values of larger than :5 can e inferred from the gures. Second, comparing Figures 6 and 8 (Bartlett) with Figures 7 and 9 (QS) we see that the Bartlett kernel gives higher local power than the QS kernel especially for large within-regime e ective andwidths. A similar nding was documented y Kiefer and Vogelsang (25) in models without structural change. o isolate the impact of andwidths on power, Figures through 3 depict power for a range of values for a given kernel and = :5. hese gures illustrate that increasing the andwidth generally reduces power and this is especially true for the QS kernel. For the Bartlett kernel, once = :5, further increases in have little e ect on power. In contrast, increasing from :5 to : signi cantly reduces power when using the QS kernel although the drop in power is even more dramatic when increases from : to :5. hat power of W ald (F ) and W ald (S) is decreasing in the andwidth is not surprising given that this pattern holds in models without structural change. 8 Partial Structural Change Model his section derives the xed- limit of W ald (F ) in the partial structural change model. he main result of this section is that the limit is the same as the limit for the full structural change model. In contrast, the W ald (S) statistic in the partial structural change case has a non-pivotal xed- limit. 8. Setup and Assumptions he regression model with partial structural change is given y y t = zt + x t + x 2t 2 + u t (29) = zt + Xt + u t ; where x t is p and z t is q vector and x t = x t (t ); x 2t = x t (t > ); Xt = (x t x 2t); and = ( 2): he coe cients on the x t regressors are unrestricted in terms of a structural change whereas the coe cients on the z t regressors are assumed to not have structural change. Denote y = (y ; y 2 ; : : : ; y ) ; X = (X ; X 2 ; : : : X ) ; Z = (z ; z 2 ; : : : ; z ) ; u = (u ; u 2 ; : : : ; u ) : 2

21 he parameters (; ) are estimated y OLS and the OLS residual vector can e written as u = ey e X = u e X P Z u; where ey = (I P Z ) y; e X = (I PZ ) X; and P Z = Z(Z Z) Z : he residual for an individual oservation is given y Also, note that u t = u t e X t ex t = X t X Z(Z Z) z t = z t Z Z Z u: (3) he following assumptions replace Assumptions and 2 in Section 2: Assumption. =2 P [r ] xt u t t= ) W z t u p+q (r) W t p+q (r); where is a p (p + q) 2 matrix, 2 is a q (p + q) matrix, and W p+q (r) is a (p + q) vector of independent Wiener process. Assumption 2. p lim P [r ] t= z tzt = rq ZZ ; p lim P [r ] t= x tx t = rq xx ; and p lim P [r ] t= x tzt = rq xz uniformly in r 2 [; ], and Q ZZ and Q xx exist. ex () t p ex (2) t p C A : 8.2 Asymptotic Results We continue to focus on tests of the null hypothesis of no structural change in the x t slope parameters of the form H : R = r with R l2p = R ; R lp lp! and r = : (3) Recall that the OLS estimator, = ; 2 can e rewritten as = t= ex t e X t! t= ex t ey t! : (32) Proposition 3 Under Assumptions and 2, as! =2 d! Q W p+q () Q xz Q ex X e ZZ 2W p+q () (W p+q () W p+q ()) ( )Q xz Q ZZ ; 2W p+q () 2

22 and p R H r ) R Qxx W p+q() where Q ex X e = p lim P e t= X tx e t : (W p+q() W p+q ()) ; (33) See the Appendix for the proof. As seen from the aove proposition, and 2 are not asymptotically independent in the partial structural change regression model. his is true ecause we are projecting out the variation of explanatory variales z t so that and 2 depend on the entire series of x t and z t. he dichotomy that is dependent only on the pre-reak data and 2 depends only on the post-reak data no longer holds in the partial structural change model. he dependence manifests in the common term, Q xz Q ZZ 2W p+q () in the aove Proposition. However, this term cancels out in (33) when the restriction matrix takes the form of (3). As a result, and also as suggested y equation (33), in principle we need to estimate only for testing for partial structural change. Because (F ), extended for the case of partial structural change, does not impose any restrictions on the asymptotic correlation etween and 2, W ald (F ) continues to allow asymptotically pivotal xed tests for partial structural change. While not ovious at rst glance, W ald (F ) has the same xed- limit in the partial structural change case as it does in the full structural change case. In contrast, if we extend (S) to the partial structural change case, (S) incorrectly imposes the restriction that and 2 are asymptotically uncorrelated and W ald (S) no longer allows asymptotically pivotal xed- inference. For that reason we do not pursue an analysis of W ald (S) for the case of partial structural change. he Wald statistic for testing for partial structural change is given y W ald = R RQ ex X e Q ex X er R ; (34) where Q ex X e = P e t= X tx e t. For constructing W ald (F ) we use the HAC estimator (F ) which n o is computed using ext u t t= : (F ) = X t= s= jt K sj M t s ; (35) where t = e X t u t : By the Frisch-Waugh-Lovell heorem this is the straightforward extension of W ald (F ) to the case of partial structural change. he next Lemma provides the limit of the scaled partial sum process of t premultiplied y an appropriate term. Lemma 3 Let S t = P t j= j. Under Assumptions and 2, as! ; RQ ex X e =2 S [r ] ) R Qxx F () p+q (r; ) F (2) p+q (r; ) ; 22

23 where F () p+q (r; ) = W p+q (r) F (2) p+q (r; ) = W p+q (r) W p+q () r W p+q() ( r ) ; ( < r ) : r (W p+q() W p+q ()) See the Appendix for the proof. As Lemma 3 shows, the partial sums of the inputs to (F ) are asymptotically proportional to the same nuisance parameters as p R r. his is the key condition for an asymptotic pivotal xed- limit. he next heorem provides the xed- limit of W ald (F ) : heorem 3 Let 2 (; ) and 2 (; ] e given. Suppose M =. hen under Assumptions and 2, W ald (F ) weakly converges to the same limit in (23), i.e. as!, W ald (F ) ) W l() P ; F () l (r; ) (W l() W l ()) F (2) l (r; ) W l() (W l() W l ()) : See the Appendix for the proof. According to heorem 3, the limit of W ald (F ) in the partial structural change model is the same as in the full structural change model. 9 When the Break Date is Unknown ests for a potential structural reak with an unknown reak date are well studied in Andrews (993), Andrews and Ploerger (994), and Bai and Perron (998). Andrews (993) considers several tests ased on the supremum across reakpoints of Wald and LM statistics and shows they are asymptotically equivalent. Andrews and Ploerger (994) derive tests that maximize average power across potential reakpoints. Here we construct tests using only the W ald (F ) statistic ecause W ald (F ) allows valid testing in oth the full and partial structural change models whereas W ald (S) does not. We denote the W ald (F ) statistic computed using reak date [ ] y W ald (F ) ( ): Denote the limit of W ald (F ) ( ) as W ald (F ) () where the form of W ald (F ) () depends on whether traditional or xed- asymptotic theory is eing used. In the case of xed- theory, W ald (F ) () depends on P (; F p (r; )). As argued y Andrews (993) and Andrews and Ploerger (994), reak dates close to the end points of the sample cannot e used and so some trimming is needed. o that end de ne = [; ] with < < to e the set of admissile reak dates. he tuning parameter,, denotes the amount of trimming of potential reak dates. We consider the three 23

24 statistics following Andrews (993) and Andrews and Ploerger (994) de ned as SupW (F ) sup W ald (F ) ( ); (36) 2 MeanW (F ) X W ald (F ) ( ); (37) 2 ExpW (F ) X exp 2 W ald(f ) ( ) A : (38) 2 he asymptotic limits of these statistics follow from the continuous mapping theorem and are given y SupW (F ) ) sup 2(; ) Z W ald (F ) (); MeanW (F ) ) W ald (F ) ()d; Z ExpW (F ) ) log exp 2 W ald(f ) () d : In ales 5. and 5.2 xed- critical values for SupW (F ) ; MeanW (F ) ; and ExpW (F ) are provided for l = 2, = :5; :; :2 and for 2 f:2; :4; :6; :8; :; :2; :3; :::; :9; g: ales 6. through 8.3 presents the simulation result for some of the DGP speci cations introduced in Section 6 and for = ; 5 and : Fixed- critical values are used for the Bartlett and QS kernels. For the Bartlett kernel, results are also reported using the traditional critical values otained y Andrews (993) and Andrews and Ploerger (994). Several patterns stand out for the null rejection proailities associated with SupW (F ) in ale 6. to 6.3. First, rejections using the traditional critical values are often sustantially aove the 5% nominal level unless persistence is very weak and a small andwidth is used. Rejections can e close to %. he situation is much improved y the use of xed- critical values ut severe over-rejections are still possile. Size distortions are higher with more persistence in the data, with a smaller value of ; and a smaller value of. As was true in the case of a known reak date, the QS kernel gives less size distortion than the Bartlett kernel although the use of large andwidths causes under-rejections. But over-rejections and under-rejections dissipate as grows. he Bartlett kernel can su er from over-rejections that are not easily removed just y using a ig andwidth. A larger value of helps along with more trimming. Similar patterns hold for the MeanW (F ) and ExpW (F ) statistics; see ales and respectively. Summary and Conclusions In this paper xed- asymptotics is applied to the prolem of testing for the presence of a structural reak in a weakly dependent time series regression. wo di erent HAC estimators and accordingly 24

25 two di erent Wald statistics are investigated. he W ald (F ) statistic is the Wald statistic that one otains when structural change is expressed in terms of dummy variales interacted with regressors. he W ald (S) statistic is a restricted version of W ald (F ) where the o -diagonal locks of the HAC estimator are set to zero mimicking the asymptotic zero covariance etween OLS estimators in the two regimes. We derive the xed- limits of the two statistics, and we compare xed- inference with traditional inference. In a model with full structural change, oth Wald statistics have pivotal xed- limits. However, in models with partial structural change, the naturally extended version of W ald (F ) has the same pivotal xed- limit as in the full structural change case whereas the naturally extended version of W ald (S) does not have a pivotal xed- limit. In a simulation study we examined the nite sample size distortions associated with the xed- approach and the traditional approach. he simulation results indicate that traditional inference is suject to more severe size distortions. When small andwidths are used, the gap is not large, ut with igger andwidths the di erence ecomes sustantial. Overall, rejections otained when using xed- critical values are closer to the nominal level compared to using the traditional chi-square critical values. When using xed- critical values, we found that nite sample size distortions ecomes more pronounce as gets smaller or as gets closer to or : Local asymptotic power is decreasing in and power is highest for structural change located near the center of the sample. In a comparison of the W ald (F ) and W ald (S) statistics for full structural change models we found that W ald (F ) tends to e less size distorted than W ald (S) when using xed- critical values especially when serial correlation is strong and a large andwidth is used. he etter size performance of W ald (F ) comes at the cost of lower power re ecting the usual trade-o etween size roustness and power typically found in xed- analyses. he choice etween W ald (F ) and W ald (S) ecomes a choice etween tolerance for over-rejections relative to desire for high power. At a practical level, W ald (F ) is appealing ecause it retains the same asymptotic pivotal xed- limit in models with partial structural change whereas W ald (S) ecomes nonpivotal. Finally, using W ald (F ) we taulated some xed- critical values for SupW; MeanW; and ExpW statistics which are commonly used for testing the presence of a structural reak when the reak date is not known a priori. A simulation study revealed that over-rejections are a igger concern when the reak date is treated as unknown. Critical values ased on traditional asymptotics can lead to very severe over-rejection prolems. Rejections using xed- critical values are less distorted especially when the QS kernel is used with a andwidth that is not too small. When the Bartlett kernel is used, xed- rejections can show sustantial over-rejections. 25