Testing for Multiple Structural Changes in Cointegrated Regression Models

Transcription

1 Testing for Multiple Structural Changes in Cointegrated Regression Models Mohitosh KEJRIWAL Krannert School of Management, Purdue University, 43 West State Street, West Lafayette, IN 4797 Pierre PERRON Economics Department, Boston University, 27 Bay State Rd., Boston, MA 2215 Downloaded by [Purdue University] at 9:23 24 August 212 We consider testing for multiple structural changes in cointegrated systems and derive the limiting distribution of the sup-wald test under mild conditions on the errors and regressors for a variety of testing problems. We show that even if the coefficients of the integrated regressors are held fixed but the intercept is allowed to change, the limit distributions are not the same as would prevail in a stationary framework. We also propose a sequential procedure that permits consistent estimation of the number of breaks present. We show via simulations that our tests maintain the correct sie in finite samples and are much more powerful than the commonly used LM tests, which suffer from important problems of nonmonotonic power in the presence of serial correlation in the errors. KEY WORDS: 1. INTRODUCTION Issues related to structural change have received considerable attention in the statistics and econometrics literature. Andrews 1993 and Andrews and Ploberger 1994 provided a comprehensive treatment of the problem of testing for structural change assuming that the change point is unknown. Bai 1997 studied the least squares estimation of a single change point in regressions involving stationary and/or trending regressors, and derived the consistency, rate of convergence, and limiting distribution of the change point estimator under general conditions on the regressors and the errors. Perron and Zhu 25 analyed the properties of parameter estimates in models in which the trend function exhibits a slope change at an unknown date and the errors can either be stationary, I, or have a unit root, I1, where here and throughout the article we refer to an I process as one whose partial sums satisfies a functional central limit theorem with a Brownian motion as the limit random variable, and I1 is the partial sum of an I series. With integrated variables, the case of interest is when the variables are cointegrated. Accounting for parameter shifts is crucial in cointegration analysis, which normally involves long spans of data, which are more likely to be affected by structural breaks. Bai, Lumsdaine, and Stock 1998 considered a single break in a multiequation system. They showed consistency of the maximum likelihood estimates and obtained a limit distribution of the break date estimate under a shrinking shifts scenario. Kejriwal and Perron 28b studied the properties of the estimates of the break dates and parameters in a linear regression with multiple structural changes involving I1, I, and trending regressors. With respect to testing, Hansen 1992b developed tests of the null hypothesis of no change in cointegrated models in which all coefficients are allowed to change. An extension to partial changes was analyed by Kuo 1998, who considered the Sup and Mean LM tests directed against an alternative of a one-time change in parameters. Hao 1996 also Change point; Cointegration; Sequential procedure; Unit root; Wald test. suggested using the exponential LM test. Seo 1998 considered the sup, mean, and exp versions of the LM test within a cointegrated VAR setup; however, these test procedures are based on the fully modified estimation method Phillips and Hansen 199, which has been shown to lead to tests with very poor finite-sample properties Carrion-i-Silvestre and Sansó-i- Rosselló 26. The results of Quintos and Phillips 1993 also suggest that the LM tests are likely to suffer from the problem of low power in finite samples. Moreover, simulation experiments of Hansen 2 have shown that the LM test behaves quite poorly in the presence of structural changes in the marginal distribution of the regressors, whereas the sup-wald test is reasonably robust to such shifts. Hansen 23 considered multiple structural changes in a cointegrated system, although his analysis was restricted to the case of known break dates. Finally, Qu 27 proposed a procedure to detect whether cointegration is present when the cointegrating vector changes at some unknown date, possibly multiple dates. The literature on testing for multiple structural changes is relatively sparse but is practically important, because single-break tests can suffer from nonmonotonic power when the alternative involves more than one break. As stressed by Perron 26, most tests may exhibit nonmonotonic power functions if the number of breaks present is greater than the number explicitly accounted for in the construction of the tests. The aim of the present work is to provide a comprehensive treatment of issues related to testing for multiple structural changes occurring on unknown dates in cointegrated regression models. Our work builds on that of Bai and Perron 1998, who studied a similar treatment in a stationary context. Our framework is sufficiently general to allow both I and I1 variables in the regression. The assumptions about the distribution of the error processes 21 American Statistical Association Journal of Business & Economic Statistics October 21, Vol. 28, No. 4 DOI: /jbes

2 54 Journal of Business & Economic Statistics, October 21 Downloaded by [Purdue University] at 9:23 24 August 212 are sufficiently mild to allow for general forms of serial correlation. Moreover, we analye both pure and partial structural change models. A partial change model is useful in allowing potential savings in degrees of freedom, a particularly relevant issue for multiple changes. It is also important in empirical work, because it helps isolate the variables responsible for the failure of the null hypothesis. We derive the limiting distribution of the sup-wald test under the null hypothesis of no structural change against the alternative hypothesis of a given number of cointegrating regimes. We also consider the double-maximum tests proposed by Bai and Perron We provide critical values for a wide variety of models that are relevant in practice. Finally, our simulation experiments show that with serially correlated errors, the commonly used sup, mean, and exp- LM tests suffer from nonmonotonic power problems. This is true for cases with a single break as well as those with multiple breaks. We propose a modified sup Wald test that exhibits a power function that is monotonic with respect to the magnitude of the breaks while maintaining reasonable sie properties. The article is organied as follows. Section 2 presents the model and assumptions. Section 3 describes the testing problems and the test statistics used. Section 4 presents our theoretical results regarding the limit distributions of the tests for a wide variety of cases. This is first done for models involving nontrending regressors, no serial correlation in the errors, and exogenous regressors. These restrictions are relaxed in Sections 4.2, 5.1, and 5.2, respectively. Asymptotic critical values are presented in Section 4.3. Section 6 presents simulation experiments that address issues related to the sie and power of the tests, including a comparison with the often used LM tests. Section 7 provides some concluding remarks. The Appendix presents technical derivations. 2. THE MODEL AND ASSUMPTIONS Consider the following linear regression model with m breaks m 1 regimes: y t = c j ft δ f bt δ bj x ft β f x bt β bj u t t = T j 1 1,...,T j 1 for j = 1,...,m1, where T =, T m1 = T, and T is the sample sie. In this model, y t is a scalar-dependent I1 variable, x ft p f 1 and x bt p b 1 are vectors of I variables, whereas ft q f 1 and bt q b 1 are vectors of I1 variables defined by ft = f,t 1 u f t, bt = b,t 1 u b t, x ft = μ f u f xt, and x bt = μ b u b xt, where for simplicity, f and b are assumed to be either O p 1 random variables or fixed finite constants. For ease of reference, the subscript b on the error term represents break and the subscript f represents fixed across regimes. The break points T 1,...,T m are treated as unknown. This is a partial structural change model in which the coefficients of only a subset of the regressors are subject to change. When p f = q f =, we have a pure structural change model with all coefficients allowed to change across regimes. It will be useful to express 1 in matrix form as Y = Gα Wγ U, where Y = y 1,...,y T, G = Z f, X f, Z f = f 1,..., ft, X f = x f 1,...,x ft, U = u 1,...,u T, W = w 1,...,w T, w t = 1, bt, x bt, γ = δ b1, β b1,...,δ b,m1, β b,m1, α = δ f, β f and W is the matrix which diagonally partitions W at the m-partition T 1,...,T m, that is, W = diagw 1,...,W m1 with W i = w Ti 1 1,...,w Ti for i = 1,...,m 1. Kejriwal and Perron 28b analyed the properties of the estimates of the break dates and the other parameters of the model under general conditions on the regressors and the errors. In this article, our interest lies in testing the null hypothesis of no structural change versus the alternative hypothesis of m changes as specified by the model 1. Thus the data-generating process is assumed to be given by 1 with p b = q b =. As a matter of notation, p denotes convergence in probability, d denotes convergence in distribution, and denotes weak convergence in the space D[, 1] under the Skorohod metric. In addition, x t = x ft, x bt, u xt = u f xt, u b xt, t = ft, bt, μ = μ f, μ b, and λ ={λ 1,...,λ m } is the vector of break fractions defined by λ i = T i /T for i = 1,...,m.Wemake the following assumptions on ξ t = u t, u f t, u b t, uf xt, u b xt, a vector of dimension n = q f p f q b p b 1. Assumption A1. The vector ξ t satisfies the following multivariate functional central limit theorem FCLT: T 1/2 [Tr] t=1 ξ t Br, with Br = B 1 r, B f r, B b r, B f xr, B b x r is a n vector Brownian motion with symmetric covariance matrix σ 2 f 1 b 1 f 1x b 1x 1 = f 1 b 1 f x1 b x1 ff bf ff x bf x fb bb fb x bb x fb x bf x ff xx bf xx ff x bb x fb xx bb xx = lim T T 1 ES T S T =, where S T = T t=1 ξ t, = lim T T 1 T t=1 Eξ t ξ t, and = lim T T 1 T 1 T j j=1 t=1 Eξ tξ tj. We also assume σ 2 > q f q b p f p b and p lim T T 1 T t=1 u 2 t = lim T T 1 T t=1 E[u 2 t ] σ 2 uȧssumption A2. The vector {xtut} satisfies assumption A4 of Qu and Perron 27, so that T 1/2 [Tr] t=1 uf xt, u b xt u t σ Q 1/2 W x r, where W x r = W xf r, W xb r is a p f p b vector of independent Wiener processes, and Q = [ Q ff x Q bf x Q fb x Q bb x Assumption A3. For all t and s, a Eu xt u t s =, b Eu xt u t u s =, and c Eu xt u t u xs =. Assumption A4. The matrix ff is positive definite. bf fb bb ].

3 Kejriwal and Perron: Testing for Multiple Structural Changes 55 Downloaded by [Purdue University] at 9:23 24 August 212 Assumption A5. T 1 [Ts] t=1 x tx p t sq and T 1 [Ts] t=1 u xt u p xt sq, uniformly in s [, 1], for some positive definite matrices Q and Q. Assumption A1 requires that the errors satisfy a multivariate FCLT. The conditions required for this to hold are very general see, e.g., Davidson It can be shown to apply to a large class of linear processes, including those generated by all stationary and invertible ARMA models. Assumption A2 guarantees that an FCLT also holds for the sequence {u xt u t }.Assumption A3 restricts somewhat the class of models applicable but is quite mild. Sufficient conditions for it to hold are for a, that the I regressors are uncorrelated with the errors contemporaneously even conditional on the I1 variables; for b, that the autocovariance structure of the I regressors be independent of the errors; and similarly for c, that the autocovariance structure of the errors be independent of the I regressors. This assumption is needed to guarantee that W x and B are uncorrelated and, being Gaussian, are thus independent. Without this condition, the analysis would be much more complex. Assumption A4 rules out cointegration among the I1 regressors. Assumption A5 is standard for I regressors but rules out trending regressors, which we relax in Section 4.2. Under the alternative hypothesis, the estimates of the parameters are obtained by minimiing the global sum of squared residuals. For each m-partition T 1,...,T m, denoted by {T j }, the associated least squares estimates of α and γ are obtained by minimiing m1 SSR T T 1,...,T m = T i t=t i 1 1 [ yt c i ft δ f x ft β f bt δ bi x bt β bi] 2. 2 Let ˆα{T j } and ˆγ {T j } be the resulting estimates. Substituting these into the objective function and denoting the resulting sum of squared residuals as S T T 1,...,T m, the estimate of the break points are ˆT 1,..., ˆT m = arg min T1,...,T m S T T 1,...,T m, where the minimiation is taken over all partitions T 1,...,T m such that T i T i 1 ɛt for some ɛ>. The estimates of the regression coefficients are then ˆα = ˆα{ ˆT j } and ˆγ = ˆγ { ˆT j }. Such estimates can be obtained using the algorithm of Bai and Perron 23. Finally, consistent estimates of the matrixes and and thus are ˆ = T 1 T t=1 ˆξ t ˆξ t and ˆ = T 1 T 1 j=1 wj/l T j ˆξ t=1 t ˆξ tj, where ˆξ t = û t, ft, bt,x ft x f,x bt x b, with û t the ordinary least squares OLS residuals from regression 1, x i = T 1 T t=1 x it i = f, b and wj/l is a kernel function that is continuous and even with w = 1 and w2 x dx <. In addition, l as T and l = ot 1/2. Hansen 1992c has demonstrated the consistency of these covariance matrix estimates. 3. THE TESTING PROBLEM AND THE TEST STATISTICS The data-generating process 1 is the most general, and restricted versions may be used in practice. This gives rise to a variety of possible cases for the testing problems considered. We classify these into two categories: a models with only I1 regressors and bmodels with both I1 and I regressors. This classification into two categories is useful, because researchers often are faced with only I1 variables. For category a, we consider the following testing problems for ease of reference, we list the relevant regression under the alternative hypothesis: Testing problems, category a: models with I1 variables only p f = p b =, for all cases. Let H a denotes the restrictions {c j = c, δ bj = δ b for all j = 1,...,m 1}. 1. H a1 ={Ha, q f = } versus H1 a1 ={q f = } y t = c j bt δ bj u t ; 2. H a2 ={Ha, q b = } versus H1 a2 ={q b = } y t = c j ft δ f u t ; 3. H a3 ={Ha, q f = } versus H1 a3 ={c j = c for all j = 1,...,m 1, q f = } y t = c bt δ bj u t ; 4. H a4 ={Ha } versus Ha 1 4 ={no restriction} y t = c j ft δ f bt δ bj u t ; 5. H a5 ={Ha } versus Ha 1 5 ={c j = c for all j = 1,..., m 1} y t = c ft δ f bt δ bj u t. Testing problems, category b: models with both I1 and I variables. Let H b denotes the restrictions {c j = c, δ bj = δ b, β bj = β b for all j = 1,...,m 1}. 1. H b1 ={Hb, p f = q b = } versus H1 b1 ={c j = c for all j = 1,...,m 1, p f = q b = } y t = c ft δ f x bt β bj u t ; 2. H b2 ={Hb, p b = q f = } versus H1 b2 ={c j = c for all j = 1,...,m 1, p b = q f = } y t = c bt δ bj x ft β f u t ; 3. H b3 ={Hb, p f = q f = } versus H1 b3 ={c j = c for all j = 1,...,m 1, p f = q f = } y t = c bt δ bj x bt β bj u t ; 4. H b4 ={Hb, p f = q f = } versus H1 b4 ={p f = q f = } y t = c j bt δ bj x bt β bj u t ; 5. H b5 ={Hb, p b = q b = } versus H1 b5 ={p b = q b = } y t = c j ft δ f x ft β f u t ; 6. H b6 ={Hb, p b = q f = } versus H1 b6 ={p b = q f = } y t = c j bt δ bj x ft β f u t ; 7. H b7 ={Hb, p f = q b = } versus H1 b7 ={p f = q b = } y t = c j ft δ f x bt β bj u t ; 8. H b8 ={Hb, q f = } versus H1 b8 ={q f = } y t = c j bt δ bj x ft β f x bt β bj u t ; 9. H b9 ={Hb, q b = } versus H1 b9 ={q b = } y t = c j ft δ f x ft β f x bt β bj u t ; 1. H b1 ={Hb } versus Hb 1 1 ={no restriction} y t = c j ft δ f bt δ bj x ft β f x bt β bj u t ; 11. H b11 ={Hb } versus Hb 1 1 ={c j = c for all j = 1,...,m 1} y t = c ft δ f bt δ bj x ft β f x bt β bj u t. We now give a brief description of each of the models in the two categories. In category a, case 1 is a pure structural change model that allows for a change in the intercept as well. Case 2 is a partial change model in which only the intercept is allowed to change. Case 3 also is a partial change model but in which the intercept is not allowed to change. Cases 4 and 5 are

4 56 Journal of Business & Economic Statistics, October 21 Downloaded by [Purdue University] at 9:23 24 August 212 block partial models in which a subset of the I1 coefficients is allowed to change. In category b, cases 1 3 are partial change models in which the intercept is not allowed to change across regimes. Case 4 is a pure change model in which all I1 and I coefficients, as well as the intercept, are allowed to change. Case 5 is a partial change model that involves only an intercept shift. Case 6 is a partial change model in which the I coefficients are not allowed to change. Similarly, case 7 is a partial change model in which the I1 coefficients are not allowed to change. Cases 8 11 are block partial models in which a subset of coefficients of at least one type of regressor is not allowed to change. We consider three types of tests. The first type of test applies when the alternative hypothesis involves a fixed value m = k of changes. We consider the Wald test, scaled by the number of regressors whose coefficient are allowed to change, defined by T k 1qb p b p f q f F T λ, k = k ˆγ R R W M G W 1 R 1 R ˆγ, 3 SSR k where R is the conventional matrix such that Rγ = γ 1 γ 2,...,γ k γ k1 and M G = I GG G 1 G. Here SSR k is the sum of squared residuals under the alternative hypothesis. Following Bai and Perron 1998, we define the following set for some arbitrary small positive number ɛ, k ɛ = {λ : λ i1 λ i ɛ,λ 1 ɛ,λ k 1 ɛ}. The sup-wald test is then defined as sup-f T k = sup λ k ɛ F T λ, k. Because in the current cases the estimates ˆλ ={ˆλ 1,...,ˆλ k } with ˆλ i = ˆT i /T for i = 1,...,k obtained by minimiing the global sum of squared residuals correspond to those that maximie the test F T λ, k, we have sup-f T k = F T ˆλ, k. The second type of test applies when the alternative hypothesis involves an unknown number of changes between 1 and some upper bound M. Again following Bai and Perron 1998, we consider a double-maximum test based on the maximum of the individual tests for the null of no break versus m breaks m = 1,...,M, defined by UD max F T M = max 1 m M sup λ m ɛ F T λ, m. This test is arguably the most useful for determining the presence of structural changes. Simulations of Bai and Perron 26 showed that with multiple changes, the power of tests for a single break can be quite low in finite samples, especially for certain types of multiple changes e.g., two breaks with identical first and third regimes. In addition, tests for a particular number of changes may have nonmonotonic power when the number of changes is greater than specified. Finally, their simulations demonstrated that the power of UD max was nearly as high as that of the sup-f T test based on the true number of changes. The third testing procedure is a sequential one based on the estimates of the break dates obtained from a global minimiation of sum of squared residuals, as described by Bai and Perron Consider a model with k breaks, with estimates denoted by ˆT 1,..., ˆT k, which are obtained through global minimiation of the sum of squared residuals. Testing the null hypothesis of k breaks versus the alternative hypothesis of k 1 breaks involves performing a one-break test for each of the k 1 segments defined by the partition ˆT 1,..., ˆT k, then assessing whether the maximum of the tests is significant. More precisely, the test is defined by SEQ T k 1 k = max 1 j k1 sup τ j,ε T { SSR T ˆT 1,..., ˆT k SSR T ˆT 1,..., ˆT j 1,τ, ˆT j,..., ˆT k } /SSR k1, where j,ε ={τ; ˆT j 1 ˆT j ˆT j 1 ε τ ˆT j ˆT j ˆT j 1 ε}. Note that this differs from a purely sequential procedure, because for each value of k, the break dates are reestimated to obtain those corresponding to the global minimiers of the sum of squared residuals. 4. ASYMPTOTIC DISTRIBUTIONS OF THE TESTS An important issue that arises with integrated regressors is the correlation between the regressors and the errors. We first considerthe case in which all I1 regressors are strictly exogenous. We then examine the case of endogenous regressors and show that if the regression is augmented with leads and lags of the the first differences of the I1 regressors, then the limiting distribution of the tests is the same as that obtained when all I1 regressors are strictly exogenous. Thus, for now we assume f 1 = b 1 =, which we later relax in Section 5.2. We also start with the following assumption, which imposes serially uncorrelated errors in the cointegrating regression, which we relax in Section 5.1: Assumption A6. Let ξ t = u f t, u b t, uf xt, u b xt, the errors {u t } form an array of martingale differences relative to {F t }=σ- field{ξ t s, u t 1 s; s > }. 4.1 Main Theoretical Results As a matter of notation, we define the following functionals, where W 1 = σ 1 B 1 : b b 1 b hg, a, b = G dw 1 GG G dw 1, a a a λi 1 λi f G = G dw 1 GG k1 λi G dw 1, gg, a, b = agb bga agb bga/bab a, and G a,b r = Gr λ b λ a 1 1 λ b λ a 1 G. In addition, by convention, λ = and λ k1 = 1. The limit distributions of the tests when only I1 variables are involved are stated in the following theorem. Theorem 1. Assume that Assumptions A1 A6 hold and that f 1 = b 1 =. For the testing problems in category a, the limit distribution of sup λ k ɛ F T λ, k is sup λ k ɛ Fλ, k/k with

5 Kejriwal and Perron: Testing for Multiple Structural Changes 57 Downloaded by [Purdue University] at 9:23 24 August 212 Fλ, k defined as follows for the various cases: For case 1, Fλ, k = k [ h W b1,i,,λ i h W b1,i1,,λ i1 h bi1,i1 ],λ i,λ i1 g W1,λ i,λ i1. For case 2, Fλ, k = f f i,i h f 1,k1,, 1 k gw 1,λ i,λ i1, where W f r = ff 1/2 B f r. For case 3, Fλ, k = f P b i h b1,k1,, 1 where P b i r = 1 λ i [,λ i ]. For case 4, Fλ, k = f W Mi,i W b W hw b,,λ i, λ i W b Wb 1 W b h W fb1,k1,, 1 h W bi,i with W fb r = W f r, W b r, and where W Mi,i r = f i,i r λi r, forr k,,λ i gw 1,λ i,λ i1, f i,i λi W bi,i bi,i bi,i 1 bi,i r. For case 5, Fλ, k = f P i h fb1,k1,, 1 W k1 hwb,,λ i, where P i r = P b i r, P fb i r with P fb i r = Wf r λ i W f W b λ i W b Wb 1 W b r. Theorem 1 shows that it is possible to make inference in models involving I1 variables using the sup-wald test. Moreover, the limiting distributions differ depending on whether the intercept and/or the I1 coefficients are allowed to change. Note that for cases 2, 4, and 5, the limit distributions depend on the number of I1 coefficients that are not allowed to change. This is different from a stationary framework, in which the limit distribution is independent of the number of regressors whose coefficients are not allowed to change. We now consider the limit distributions of the test for the various cases in category b in which both I1 and I regressors are present. Theorem 2. Assume that Assumptions A1 A6 hold and that f 1 = b 1 = and let W xb1 = W xb, W 1. For the cases in category b, the limiting distributions of sup λ k ɛ F T λ, k under the null hypothesis are given by sup λ k ɛ Fλ, k/k with Fλ, k defined as follows: For case 1, Fλ, k = k gw xb, λ i,λ i1. For case 2, the limit distribution is the same as for case 3 in category a. For case 3, Fλ, k = f P b i h b1,k1,, 1 W hw b,,λ i k gw xb,λ i,λ i1. For cases 4 and 8, Fλ, k = k [ h W b1,i,,λ i h W b1,i1,,λ i1 h bi1,i1 ],λ i,λ i1 g W xb1,λ i,λ i1. For cases 5 and 6, the limit distributions are the same as for cases 1 and 2, respectively, in category a. For cases 7 and 9, Fλ, k = f f i,i h W f 1,k1,, 1 For case 1, Fλ, k = f W Mi,i k g W xb1,λ i,λ i1. h W fb1,k1,, 1 h W bi,i k,,λ i g W xb1,λ i,λ i1. For case 11, Fλ, k = f P i h fb1,k1,, 1 W hw b,,λ i k gw xb,λ i,λ i1. The practical implications of Theorem 2 are as follows. As shown in case 1, if the intercept and the I1 variables are held fixed and only the coefficients on the I variables are allowed to change, then the same limit distribution as given by Bai and Perron 1998 applies. But this equivalence with the case of stationary regressors holds only if the constant is not allowed to change. As shown in case 7, the limit distribution differs when the intercept is allowed to change and depends on the number of I1 variables present. The effect of allowing or not allowing the intercept to change can also be seen by comparing cases 3 and 4. The limit distributions are different, and, as expected, both depend on the number of I1 and I variables whose coefficients are allowed to change. A similar feature also applies when the regression involves I1 and I variables whose coefficients are not allowed to change, as shown in cases 1 and 11. Comparing these with cases 3 and 4 again shows that having I1 variables whose coefficients are not allowed to change alters the limit distributions. Finally, comparing cases a1 and b6, a2 and b5, a3 and b2, b4 and b8, and b7 and b9 shows that including I regressors whose coefficients are not allowed to change does not alter the limit distribution. Remark 1. For case 4 in category b, the limit distribution of sup λ k ɛ F T λ, k is: { k sup S λ i,λ i1 Vλ i,λ i1 1 S λ i,λ i1 λ 1,...,λ k k ɛ k λi W xb λ i1 λ i1 W xb λ i λ i W xb λ i1 λ i1 W xb λ i /λ i1 λ i λ i1 λ i }

6 58 Journal of Business & Economic Statistics, October 21 Downloaded by [Purdue University] at 9:23 24 August 212 with S λ i,λ i1 = Sλ i Mλ i Mλ i1 1 Sλ i1, Vλ i,λ i1 = Mλ i Mλ i Mλ i1 1 Mλ i, Sλ i = λi Z dw 1, Mλ i = λ i Z Z, and Z = 1, W b. The first summation corresponds to the distribution in case 1 of category a, whereas the second summation corresponds to the p b I regressors whose coefficients are allowed to change. With these theoretical results for the sup-f T λ, k, we can obtain the limit distribution of the UD max and SEQ T k 1 k tests. These are stated in the following corollary. Corollary 1. Under Assumptions A1 A6 and f 1 = b 1 =, for a particular testing problem denote the limit distribution of the test sup λ k ɛ F T λ, k by sup λ k ɛ Fλ, k/k, then: a UD max F T M = max 1 m M sup λ m ɛ F T λ, m max 1 m M sup λ m ɛ Fλ, m/m, b lim T PSEQ T k 1 k x = G ε x k1, with G ε x the distribution function of sup λ 1 ε Fλ, Trends in Regressors Suppose now that the I1 regressors have a trending nonstochastic component, that is, are generated by ft = ρ f t ft and bt = ρ bt bt, with q b > 1 and ρ b. The limiting distributions of the tests then differ from those in the nontrending case. The derivation of the required modifications follows the treatment of Hansen 1992a. Consider a q b q b 1 matrix ρ b that spans the null space of ρ b and let C 2 =[C 12, C 22 ]= ρ b ρ b ρ b 1, ρ b ρ b bb ρ b 1/2. Note that C 2 bt = C 12 bt t, C 22 bt. With W 2T = diagt, I qb 1T 1/2,wehave W 1 2T C 2 b[tr] = T 1 C 12 b[tr] T 1 [Tr] T 1/2 C 22 b[tr] r W b 1 r J b r, 4 where W b 1 r is a q b 1-dimensional vector of independent Wiener processes [a linear combination of W b r]. Note that when q b = 1, W b 1 r = r. It then follows that T 1 W 1 [Tr] 2T C 2 bt bt C 1 2 W t=1 t=1 2T r T 1/2 W 1 [Tr] 2T C 2 bt u t σ r J b Jb, 5 J b dw 1. 6 Note that 4 through 6 also hold for ft with Wb 1 r replaced by W f 1 r,aq f 1 dimensional vector of independent Wiener processes [a linear combination of W f r]. Here also, when q f = 1, W f 1 r = r. Therefore, with trending regressors, the limiting distributions of the tests are not the same as those without trends; however, we can obtain them by simply replacing W f and W b x by Jf and J b. 4.3 Asymptotic Critical Values Because the asymptotic distributions are nonstandard, we obtain critical values through simulations for models with and without trends in regressors. We approximate the Wiener processes by partial sums of iid normal random variables with N = 5 steps. The number of replications is 2,. For each replication, the supremum of Fλ, k with respect to λ 1,...,λ k over the set k ɛ is obtained via a dynamic programming algorithm see Bai and Perron 23. The I regressors are simulated as independent sequences of iid N, 1 variables, and the I1 regressors as independent random walks with iid N, 1 errors [also independent of the I regressors]. The trimming values used are ɛ =.5,.1,.15,.2, and.25. Critical values are presented for up to nine breaks and four regressors. The maximum number of breaks allowed is eight when ɛ =.1, five when ɛ =.15, three when ɛ =.2 and two when ɛ =.25. For the UD max test, M issetto5or the maximum number of breaks possible. For models involving both I1 and I variables, critical values are provided for all possible permutations up to two regressors of each type. For the limit distributions of the tests with trending regressors and for the sequential tests, we tabulated the critical values for ɛ =.15,.2, and.25. Because of the large number of results, we present critical values only for cases that allow the intercept to change and for ɛ =.15 in Tables 1 4. For other cases and trimming values, tables of critical values are available on our website. 5. EXTENSIONS We now extend the analysis of the previous section to the cases in which we can have either serially correlated errors in the cointegrating regression or endogenous regressors. We show that simple modifications yield tests with the same limit distributions described earlier. 5.1 Serially Correlated Errors: A Modified sup-wald Test With serially correlated errors, we use the following robust version of the scaled F test: FT λ, k = T k 1q b p b q f p f k ˆγ R RT ˆV ˆγ R 1 R ˆγ, 7 where ˆV ˆγ is an estimate of the covariance matrix of ˆγ that is robust to serial correlation and heteroscedasticity see Bai and Perron 1998 for details. Note that when testing for the stability of coefficients associated with I1 variables, whether or not I variables are included, we can simply apply the following transformation to the test in 3: FT λ; k = ˆσ u 2/ ˆσ 2 F T λ, k, where ˆσ u 2 = T 1 T t=1 û 2 t and ˆσ 2 is a consistent estimate of σ 2. Because the break fractions are consistent even with serially correlated errors, we can first take the supremum of the original F test to obtain the break points, then obtain the robust version of the test by evaluating FT λ; k at these estimated break dates. That is, the test considered is sup λ k ɛ FT λ, k = F T ˆλ, k, where ˆλ = ˆλ 1,...,ˆλ k are the estimates of the break fractions obtained by minimiing the global sum of squared residuals 2. A problem with the sup-wald test is that with persistent errors, the sie distortions can be substantial, due to the estimation of the long-run variance using residuals under the

7 Kejriwal and Perron: Testing for Multiple Structural Changes 59 Table 1. Asymptotic critical values [the entries are quantiles x such that Psup Fλ, k/k x = α] Downloaded by [Purdue University] at 9:23 24 August 212 Non trending case Number of breaks, k Trending case Number of breaks, k q b α UD max UD max Category a case 1, ɛ = Category a case 2, ɛ = Category a case 4, ɛ =.15 1, ,

8 51 Journal of Business & Economic Statistics, October 21 Non trending case Number of breaks, k Table 1. Continued Trending case Number of breaks, k q f, q b α UD max UD max 2, , Table 2. Asymptotic critical values [the entries are quantiles x such that Psup Fλ, k/k x = α] Downloaded by [Purdue University] at 9:23 24 August 212 Non trending case Number of breaks, k Trending case Number of breaks, k q b, p b α UD max UD max Category b cases 4 and 8, ɛ =.15 1, , , , Category b cases 7 and 9, ɛ =.15 1, , , ,

9 Kejriwal and Perron: Testing for Multiple Structural Changes 511 Table 2. Continued Downloaded by [Purdue University] at 9:23 24 August 212 Non trending case Number of breaks, k Trending case Number of breaks, k q f, q b, p b α UD max UD max Category b case 1, ɛ =.15 1, 1, , 1, , 2, , 2, , 1, , 1, , 2, , 2, Table 3. Asymptotic critical values of the sequential test SEQ T k 1 k Non trending case k Trending case q b α Category a case 1, ɛ = k

10 512 Journal of Business & Economic Statistics, October 21 Table 3. Continued Downloaded by [Purdue University] at 9:23 24 August 212 Non trending case k Trending case q b α q f Category a case 2, ɛ = Category a case 4, ɛ =.15 1, , , , k alternative hypothesis. On the other hand, Vogelsang 1999 showed through simulation experiments that estimation of the long-run variance under the null hypothesis leads to the problem of nonmonotonic power in finite samples. In related work, Crainiceanu and Vogelsang 27 showed that commonly used data-dependent bandwidths for estimation of the long-run variance based on the misspecified null model are too large under the alternative hypothesis. This in turn leads to a decrease in power as the magnitude of the change increases. As a solution to this sie power trade-off, we use a new estimator of the long-run variance constructed using a hybrid method that involves residuals computed under both the null and alternative hypotheses. In particular, the data-dependent bandwidth is selected based on the residuals obtained under the alternative hypothesis. With this particular value of the bandwidth, the estimate is computed using residuals obtained under the null hypothesis of no structural change. Specifically, the proposed estimator is T T 1 ˆσ 2 = T 1 ũ 2 t T 2T 1 wj/ĥ ũ t ũ t j, 8 t=1 j=1 t=j1 where ũ t are the residuals obtained imposing the null hypothesis. The kernel function w is the quadratic spectral, and,

11 Kejriwal and Perron: Testing for Multiple Structural Changes 513 Table 4. Asymptotic critical values of the sequential test SEQ T k 1 k Downloaded by [Purdue University] at 9:23 24 August 212 Non trending case k Trending case q b, p b α Category b cases 4 and 8, ɛ =.15 1, , , , Category b cases 7 and 9, ɛ =.15 1, , , , k Category b case 1, ɛ =.15 1, 1, , 1, , 2,

12 514 Journal of Business & Economic Statistics, October 21 Table 4. Continued Downloaded by [Purdue University] at 9:23 24 August 212 Non trending case k Trending case q f, q b, p b α , 2, , 1, , 1, , 2, , 2, following Andrews 1991, the estimate of the bandwidth is given by ĥ = â2T 1/5, where â2 =[4 ˆρ 2 /1 ˆρ 4 ] and ˆρ = T t=2 û t û t 1 / T t=2 û 2 t 1, with û t the residuals from the model estimated under the alternative hypotheses. As we show later, the sup-wald test based on this estimator is able to bypass the problem of nonmonotonic power while maintaining an exact sie close to the nominal sie. For more details on the merits of this approach, see Kejriwal Endogenous I1 Regressors In general, the assumption of strict exogeneity is too restrictive, and the test statistics described in the previous section are not robust to the problem of endogenous regressors. In this section we use the linear leads and lags estimator dynamic OLS as proposed by Saikkonen 1991 and Stock and Watson 1993 and prove that the limiting distributions of the tests based on this estimator are the same as those obtained with the static regression under strict exogeneity. The modified regression is given by y t =ĉ i ft ˆδ f x ft ˆβ f bt ˆδ bi x bt ˆβ bi t j ˆ j ˆv t, 9 j= l T where t = ft, bt. Note that the numbers of leads and lags of t need not be the same, but here we specify the same values for simplicity. Alternatively, l T could be interpreted as the maximum of the number of leads and lags. To prove our results, we need a few additional assumptions, which are the same as those required to show the consistency of the estimate of the cointegrating vector in the case of a model with no structural change. l T Assumption A7. Let ζ t = u t, u f t, u b t and ζ t = u f t, u b t. The spectral density matrix f ζζ w is bounded away from so that f ζζ w αi n n = q f q b 1 for w [,π] where α>. In addition, the covariance function of ζ t is absolutely summable; that is, denoting Eζ t ζ tk = Ɣk, werequire that k= Ɣk <, where is the standard Euclidean norm. Denoting the fourth-order cumulants of ζ t by κ ijkl m 1, m 2, m 3, we assume that m 1 m 2 m 3 κ ijkl m 1, m 2, m 3 <, where the summations run from to. Assumption A7 states the same conditions used by Saikkonen 1991 and allows us to represent the error u t as u t = j= ζ t j j v t, with k= j < and where v t is a stationary process such that Eζ t v tk =, for all k, and f vv w = f uu w f uζ wf ζ ζ w 1 f ζ uw. The DGP under the null hypothesis is then y t = c ft δ f x ft β f l T k j= l T t j j v t, where v t = v t j >l T ζ,t j j v t e t. The last requirements pertain to the possible rate of increase of l T as T increases. Following Kejriwal and Perron 28a, these are given by the following: Assumption A8. As T, l T, l 2 T /T, and l T j >l T j. Note that Assumption A8 allows the use of information criteria, such as the Akaike information criterion and the Bayes information criterion. Because there can be serial correlation in the errors v t, we need to apply a correction for its presence. Thus we consider the statistic sup λ k ɛ FT Dλ, k = FD T ˆλ, k,