Testing jointly for structural changes in the error variance and coefficients of a linear regression model

Transcription

1 Testing jointly for structural changes in the error ariance and coefficients of a linear regression model Jing Zhou Boston Uniersity Pierre Perron Boston Uniersity September 25, 27 Abstract We proide a comprehensie treatment of the problem of testing jointly for structural changes in both the regression coefficients and the ariance of the errors in a single equation system inoling stationary regressors. Our framework is quite general in that we allow for general mixing-type regressors and the assumptions on the errors are quite mild. Their distribution can be non-normal and conditional heteroskedasticity is permitted. Extensions to the case with serially correlated errors are also treated. We proide the required tools to address the following testing problems, among others: a) testing for gien numbers of changes in regression coefficients and ariance of the errors; b) testing for some unknown number of changes within some pre-specified maximum; c) testing for changes in ariance (regression coefficients) allowing for a gien number of changes in the regression coefficients (ariance); d) sequential procedures to estimate the number of changes present. These testing problems are important for practical applications as witnessed by recent interests in macroeconomics and finance where documenting structural changes in the ariability of shocks to simple autoregressions or Vector Autoregressie Models has been a concern. Applications to such macroeconomic time series reinforces the prealence of changes in both their mean and ariance and the fact that for most series an important reduction in ariance occurred in the 8s. In many cases, howeer, the so-called great moderation can instead be iewed as a great reersion. JEL Classification: C22 Keywords: Change-point; Variance shift; Conditional heteroskedasticity; Likelihood ratio tests; the Great moderation. Perron acknowledges financial support from the National Science Foundation under Grant SES We are grateful to Zhongjun Qu for comments and for pointing out an error in a preious draft. Department of Economics, Boston Uniersity, 27 Bay State Rd., Boston, MA, 2215 (jzhou112@bu.edu). Department of Economics, Boston Uniersity, 27 Bay State Rd., Boston, MA, 2215 (perron@bu.edu).

2 1 Introduction Both the statistics and econometrics literature contain a ast amount of work on issues related to structural changes with unknown break dates, most of it specifically designed for the case of a single change (for an extensie reiew, see Perron, 26). The problem of multiple structural changes has receied more attention recently mostly in the context of a single regression. Bai and Perron (1998, 23a) proide a comprehensie treatment of arious issues: consistency of estimates of the break dates, tests for structural changes, confidence interals for the break dates, methods to select the number of breaks and efficient algorithms to compute the estimates. Perron and Qu (24) extend this analysis to the case where arbitrary linear restrictions are imposed on the coefficients of the model. Related contributions include Hawkins (1976) who presents a comprehensie treatment of estimation based on a dynamic programming algorithm. Also, Liu, Wu and Zidek (1997) consider multiple structural changes in the context of a more general threshold model and propose an information criterion for the selection of the number of changes. Bai, Lumsdaine and Stock (1998) consider asymptotically alid inference for the estimate of a single break date in multiariate time series allowing stationary or integrated regressors as well as trends with estimation carried using a quasi maximum likelihood (QML) procedure. Also, Bai (2) considers the consistency, rate of conergence and limiting distribution of estimated break dates in a segmented stationary VAR model estimated again by QML when the break can occur in the parameters of the conditional mean, the ariance of the error term or both. Kejriwal and Perron (26a,b) proide a comprehensie treatment of issue related to testing and inference with multiple structural changes in a single equation cointegrated model. With respect to testing for structural change in the ariance of the regression error, the results are quite sparse. Qu and Perron (27a) consider a multiariate system estimated by quasi maximum likelihood which proides methods to estimate models with structural changes in both the regression coefficients and the coariance matrix of the errors. They proide a limit distribution theory for inference about the break dates and also consider testing for multiple structural changes, though, in this case, their analysis is restricted to models with Normally distributed errors and a prior that the breaks in coefficients and in thearianceoccuratdifferent dates. Horáth (1993) considers a change in the mean and ariance (occurring at the same time) of a sequence of i.i.d. random ariables with moments corresponding to those of a Normal distribution. Dais, Huang, and Yao (1995) extend the analysis to an autoregressie process under similar conditions. 1

3 We build on the work of Qu and Perron (27a) to proide a comprehensie treatment of the problem of testing jointly for structural changes in both the regression coefficients and the ariance of the errors in a single equation system inoling stationary regressors, allowing thebreakdatestobedifferent or oerlap. Our framework is quite general in that we allow for general mixing-type regressors and the assumptions on the errors are quite mild. Their distribution can be non-normal and conditional heteroskedasticity is permitted. Extensions to the case with serially correlated errors are also treated. We proide the required tools to address the following testing problems, among others: a) testing for gien numbers of changes in regression coefficients and ariance of the errors; b) testing for some unknown number of changes within some pre-specified maximum; c) testing for changes in ariance (regression coefficients) allowing for a gien number of changes in the regression coefficients (ariance); d) sequential procedures to estimate the number of changes present. These testing problems are important for practical applications as witnessed by recent interests in macroeconomics and finance where documenting structural changes in the ariability of shocks to simple autoregressions or Vector Autoregressie Models has been a concern; see, among others, Blanchard and Simon (21), Herrera and Pesaento (25), Kim and Nelson (1999), McConnell and Perez-Quiros (2), Sensier and an Dijk (24) and Stock and Watson (22). Gien the lack of proper testing procedures, a common approach is to apply standard sup-wald type tests (e.g., Andrews, 1993, Bai and Perron, 1998) for changes in the mean of the absolute alue of the estimated residuals; see, e.g., Herrera and Pesaento (25) and Stock and Watson (22). This is a rather ad hoc procedure. For the problem of testing for a change in ariance only (imposing no change in the regression coefficients), Deng and Perron (26) hae recently extended the CUSUM of squares test of Brown, Durbin and Eans (1975) allowing ery general conditions on the regressors and the errors (as suggested by Inclán and Tiao (1994) for Normally distributed time series). This test is, howeer, adequate only if no change in coefficient is present. As documented by, e.g., Stock and Watson (22), it is often the case that changes in both coefficients and ariance occur and the break dates need not be the same. A common method is to first test for changes in the regression coefficients and conditioning on the break dates found, then test for changes in ariance. This is clearly inappropriate as in the first step the tests suffers for seere size distortions (see Section 2). Also, neglecting changes in regression coefficient when testing for changes in ariance induces both size distortions and a loss of power. Hence, what is needed is a joint approach. To do so, our testing procedures are based on quasi likelihood ratio tests constructed using a likelihood function appropriate for identically 2

4 and independently distributed Normal errors. We then apply corrections to these likelihood ratio tests such that their limit distributions are free of nuisance parameters in the presence of non-normal distribution and conditional heteroskedasticity. We also consider extensions that allow for serial correlation as well. We apply our testing procedures to arious macroeconomic time series studied by Stock and Watson (22). On the one hand, our results reinforce the prealence of changes in both mean, persistence and ariance of the shocks in simple autoregressions. Most series hae an important reduction in ariance that occurred in the 8s. For many series, howeer, the eidence points to two breaks in the ariance of the shocks with the feature that it increases at the first one and decreases at the second. Hence, the so-called great moderation may be qualifiedasaphenomenonwherethehigharianceleelofthe7stoearly8sareoer and we are back to the leel of (roughly) pre-7s; sometimes this reersion is exact (e.g., inflation), incomplete (e.g., interest rates) or magnified (real ariables). Hence, the so-called great-moderation may rather be qualified as a great-reersion. We also present a number of interesting results pertaining to changes in leel and persistence of the series. The paper is structured as follows. Section 2 proides some motiations which show that commonly used procedures that do not treat the problem of changes in regression coefficient and in ariance jointly suffer from important size distortions and power losses. Section 3 presents the class of models considered as well as the testing problems to be addressed. Section 4 presents the quasi-likelihood tests to be used as the basis of the arious testing procedures. Section 5 discusses the main assumptions needed on the regressors and errors, deries the releant limit distributions under the arious null hypotheses and proposes corrected ersions of the tests that hae a limit distribution free of nuisance parameters. Section 5.1 deals with the case of martingale difference errors, Section 5.2 extends the analysis to serially correlated errors, Section 5.3 coers the case with an unknown number of breaks under the alternatie hypothesis, Section 5.4 discusses tests for an additional break in either the regression coefficients or the ariance. Section 6 proides simulation results to assess the adequacy of the suggested procedures in terms of their finite sample size and power and proides some guidelines for particular options. Section 7 presents a specific to general method to estimate the number of breaks in each of the regression coefficient and the ariance. Section 8 proides empirical applications related to arious macroeconomic time series for which changes in both the mean and the ariance has been a concern. Section 9 proides brief concluding remarks and directions for future research, and a brief appendix contains some technical deriations. 3

5 2 Motiation To motiate the importance of considering jointly the problem of testing for changes in the regression coefficients and the ariance of the errors, we start with some simple simulation experiments. The data generating process (DGP) is a simple sequence of i.i.d. Normal random ariable with mean and ariance that can change at a single date. To analyze the effect of ignoring a ariance break when testing for a change in the regression coefficients, the null hypothesis is specified by y t = μ + e t (1) where e t i.i.d. N(, 1+δ 1 I(t >T )) with I( ), the indicator function. We consider 3 break dates, T = {[.25T ], [.5T ], [.75T ]} and ariance change δ 1 arying between and 1 in steps of.5. The sample size is set to T =1and 5 replications are used. The test considered is the standard Sup-LR test (see Andrews, 1993) for a one-time change in μ occurring at some unknown date. The size of the test is presented in Figure 1. The results show important size distortions unless the break occurs early at T =[.25T ], and these are increasing with δ 1. To assess the effect on power, the DGP is y t = μ + I(t >T c )+e t (2) with e t as specified aboe. We consider T = {[.5T ], [.75T ]}; T c = {[.3T ]}, T =1, δ 1 = {,.5, 1, 1.5, 2, 2.5, 3} and aries between and 2. The results are presented in Figure 2, which shows that power decreases as the magnitude of the unaccounted break in ariance increases. We now consider the effect of a change in mean on the size and power of tests for a change in ariance that do not take into account the former change. We consider two testing procedures. One is based on the CUSUM of squares tests as originally proposed by Brown, Durbin and Eans (1975) and adocated as a test for a change in ariance by Inclán and Tiao (1994), who showed that it is related to the likelihood ratio test for a change in ariance in a sequence of i.i.d. Normal random ariables (though the equialence is not exact in finite samples). It is defined by CUSQ = max T S (r) T r k k+1 r T T k where S (r) T =( P r t=k+1 e2 t )/( P T t=k+1 e2 t ), withe t the recursie residuals. Its limit distribution under the null hypothesis is the supremum (oer [, 1]) of a Brownian Bridge process, for the 4

6 DGP considered here. To analyze the size of the test, DGP (2) with δ 1 =is used and we set T c = {[.25T ], [.5T ], [.75T ]} with arying between and 1. The results are presented in Figure 3 which show that in all cases the size of the test increases to one rapidly as the magnitude of the change in mean increases. This is not surprising in iew of the fact that the CUSQ test has power against a change in the regression coefficients as originally argued by Brown, Durbin and Eans (1975). For power, the DGP used is again (2) with δ 1 arying between and 15 and = {, 1, 1.5, 2, 2.5, 3, 3.5}. The results are presented in Figure 4, which show that a change in mean that is unaccounted for can increase the power of the CUSQ test. This results is, howeer, of little help gien the large size distortions. Finally, we consider the two steps method used by Herrera and Pesaento (25) and Stock and Watson (22), among others, which applies a test for a change in the mean of the absolute alue of the estimated residuals. Again, DGP (2) is used to assess the size (δ 1 =)and power properties. For size, aries between and 1 and we set T c = {[.25T ], [.5T ], [.75T ]}, while for power aries between and 3.5 and we consider two sets of break dates, namely {T c =[.5T ],T =[.3T ]} and {T c =[.75T ],T =[.3T ]}. The results are presented in Figures 5 and 6. They show that unless the unaccounted for change in mean is at mid-sample, the test suffers from serious size distortions, which increase as the magnitude of the change in mean increases. For the case of a break in mean at mid-sample, which suffers from no size distortions, Figure 6 shows that power decreases as the magnitude of the coefficient break increases. While the setup considered is quite simple, it shows how inference can be misleading when changes in the coefficients of the conditional mean and changes in the ariance of the errors are not analyzed jointly. The rest of the paper proides the necessary tools to do so. 3 Model and testing problems We start with a description of the most general specification of the model considered where multiple breaks occur in both the coefficients of the conditional mean and the ariance of the errors, at possibly different times. This will also allow us to set up the notation used throughout the paper. The main framework of analysis can be described by the following multiple linear regression with m breaks (or m +1regimes) in the conditional mean equation: y t = x tβ + ztδ j + u t, t = Tj 1 c +1,..., Tj c, (3) for j =1,..., m +1. In this model, y t is the obsered dependent ariable at time t; both 5

7 x t (p 1) andz t (q 1) are ectors of coariates and β and δ j (j =1,..., m +1) are the corresponding ectors of coefficients; u t is the disturbance at time t. The indices (T1, c..., Tm), c or the break points, are explicitly treated as unknown (the conention that T c =and Tm+1 c = T is used). This is a partial structural change model since the parameter ector β is not subject to shifts and is estimated using the entire sample. When p =, we obtain a pure structural change model where all the model s coefficients are subject to change. Note that using a partial structural change model where only some coefficients are allowed to change can be beneficial both in terms of obtaining more precise estimates and more powerful tests. We also allow for n breaks (or n +1 regimes) for the ariance of the errors occurring at unknown dates (T1,..., Tn). Accordingly, the error term u t has zero mean and ariance σ 2 i for Ti 1 +1 t Ti (i =1,..., n +1), where again we use the conention that T = and Tn+1 = T. We allow the breaks in the ariance and in the regression coefficients to happen at different times, hence the m-ector (T1, c..., Tm) c and the n-ector (T 1,..., Tn) can hae all distinct elements or they can oerlap partly or completely. We let K denote the total number of break dates and max[m, n] K m + n. When the the breaks oerlap completely, m = n = K. The multiple linear regression system (3) may be expressed in matrix form as Y = Xβ + Zδ + U, where Y =(y 1,..., y T ),X =(x 1,..., x T ), U =(u 1,...,u T ), δ =(δ 1,δ 2,..., δ m+1),and Z is the matrix which diagonally partitions Z at (T1 c,...,tm), c i.e., Z = diag(z 1,..., Z m+1 ) with Z i =(z T c i 1 +1,..., z T c i ). Wedenotethetruealueofaparameterwitha superscript. In particular, δ =(δ 1,..., δ m+1) and (T1 c,..., Tm c ) are used to denote, respectiely, the true alues of the parameters δ and the true break dates in the regression coefficients. The matrix Z c is the one which diagonally partitions Z at (T1 c,...,tm c ). Hence, in its most general form, the data-generating process is Y = Xβ + Z δ + U (4) with E(UU )=Ω,wherethediagonalelementsofΩ are σ 2 i for Ti 1 +1 t Ti (i =1,..., n +1). We also consider cases with serial correlation in the errors for which the off-diagonal elements of Ω need not be. ThismodelisaspecialcaseoftheclassofmodelsconsideredbyQuandPerron(27a). The method of estimation considered is quasi maximum likelihood (QML) assuming serially uncorrelated Gaussian errors. They proe consistency of the estimates of the break fractions (λ 1,..., λ K) (T1 /T,...,TK/T ), whereti (i =1,..., K) denotes the union of the elements 6

8 of (T1 c,..., Tm c ) and (T1,..., Tn ). This is done under general conditions on the regressors and the errors. Substantial heterogeneity in the distributions of the regressors is allowed across regimes, though unit root processes are not permitted. The series z t u t and u t are assumed to be short memory processes haing bounded forth moments. Otherwise, the conditions are mild in the sense that they allow for substantial conditional heteroskedasticity and autocorrelation. They also derie the limit distribution of the estimates of the break dates. The testing problems to be considered are the following: TP-1: H : {m = n =} ersus H 1 : {m =, n = n a }; TP-2: H : {m = m a,n=} ersus H 1 : {m = m a, n = n a }; TP-3: H : {m =,n= n a } ersus H 1 : {m = m a, n = n a }; TP-4: H : {m = n =} ersus H 1 : {m = m a, n = n a }, where m a and n a are some positie numbers selected a priori. We shall also consider testing problems where the alternaties specify some unknown numbers of breaks, up to some maximum. These are: TP-5: H : {m = n =} ersus H 1 : {m =, 1 n N}; TP-6: H : {m = m a,n=} ersus H 1 : {m = m a, 1 n N}; TP-7: H : {m =,n= n a } ersus H 1 : {1 m M, n = n a }; TP-8: H : {m = n =} ersus H 1 : {1 m M, 1 n N}. We shall also be concerned with the following testing problems: TP-9: {m = m a,n= n a } ersus H 1 : {m = m a +1, n = n a }; TP-1: {m = m a,n= n a } ersus H 1 : {m = m a, n = n a +1}, where m a and n a non-negatie integers. These are useful to assess the adequacy of a model with a particular number of breaks by looking at whether including one more break is warranted. In Section 7, we also consider sequential testing procedures that allow estimating the number of breaks in both the conditional mean regression and the ariance of the errors. 7

9 4 The quasi-likelihood ratio tests In this section we consider the likelihood ratio tests obtained assuming Normally distributed and serially uncorrelated errors, for the testing problems TP-1 to TP-4. We derie their limit distributions, which in general, are not free of nuisance parameters. We then propose, in Section 5, modifications whose asymptotic distributions are free of nuisance parameters. Results for the testing problems TP-5 to TP-8 follow as straightforward corollaries and are discussed in Section 5.3. Consider TP-1 where one specifies no change in the regression coefficients (m = q =) but tests for a gien number n a of changes in the ariance of the errors. Under the null hypothesis, the log-likelihood function is gien by where log e L T = T 2 (log 2π +1) T 2 log eσ2 (5) eσ 2 = 1 T t=1 TX (y t x e tβ) 2 t=1 TX TX eβ = ( x t x t) 1 ( x t y t ) Under the alternatie hypothesis, we estimate the model using the Quasi-Maximum likelihood estimation method (QMLE). For a gien partition {T1,..., Tn}, the log-likelihood alue is gien by log ˆL T (T1,..., Tn)= T n a +1 2 (log 2π +1) X Ti Ti 1 log ˆσ 2 i, 2 i=1 (6) where the QMLE jointly sole the system ˆβ = ˆσ 2 i = nx a +1 T i i=1 T ix x t x t ˆσ 2 t=ti 1 +1 i t=1 1 n a +1 X i=1 T 1 ix (y Ti 1 t x tˆβ) 2 t=ti 1 +1 TiX x t y t ˆσ 2 t=ti 1 +1 i for i =1,..., n a +1. Hence, the Sup-Likelihood ratio test considered is i sup LR 1,T (n a,ε m = n =) = sup 2 hlog ˆL T T 1,..., T na log el T (λ 1,...,λ na) Λ,ε i = 2 hlog ˆL T ( ˆT 1,..., ˆT na ) log el T 8

10 where the estimates ( ˆT 1,..., ˆT n a ) are the QMLE obtained by imposing the restriction that there is no structural change in the coefficients and Λ,ε = λ 1,...,λ n a ; λ i+1 λ i ε (i =1,..., n a 1),λ 1 ε, λ n a 1 ε ª. The parameter ε acts as a truncation which imposes a minimal length for each segment and will affect the limiting distribution of the test. For the testing problem TP-2, there are m a breaks in the regression coefficients under both the null and alternatie hypotheses, so that the test pertains to assess whether there are or n a breaks in ariance. For a gien partition {T1, c..., Tm c a }, the likelihood function under the null hypothesis is: log el T (T c 1,..., T c m a )= T 2 (log 2π +1) T 2 log eσ2 where and eσ 2 = 1 T TX (y t x e tβ zt e δ j ) 2 t=1 eβ = (X M ZX) 1 X M ZY e δj = (Z jz j ) 1 Z j (Y j X j e β) where M Z = I Z 1 Z Z Z, Z = diag (Z1,..., Z ma+1), andz j =(z T c j 1 +1,..., z T c j ), Y j = (y T c j 1 +1,..., y T c j ), X j =(x T c j 1 +1,..., x T c j ) for Tj 1 c <t Tj c (j =1,..., m a +1). The loglikelihood alue under the alternatie hypothesis is, for gien partitions {T1, c..., Tm c a } and {T1,...,Tn a }, log ˆL na+1 T T c 1,..., Tm c a ; T1,..., Tn T a = 2 (log 2π +1) X where the QMLE soles the following equations i=1 T i Ti 1 log ˆσ 2 i, (7) 2 ˆσ 2 i = T i T 1 ix (y Ti 1 t x tˆβ ztˆδ t,j ) 2 t=ti 1 +1 for i =1,..., n a +1,and ˆβ =(X M Zσ X) 1 X M Zσ Y 9

11 where M Zσ = I Z σ Z σ Z 1 σ Z σ with Z σ = diag(z1 σ,...,zm σ a+1), Zj σ =(zt σ j 1 c +1,..., zσ Tj c), and zt σ =(z t /ˆσ i ),forti 1 <t Ti (i =1,..., n a +1). Using the same notation, ˆδt,j =(Zj σ Zj σ ) 1 Zj σ (Yj σ Xj σˆβ) for T c j 1 t T c j,wherey σ j =(y σ T c j 1 +1,..., yσ T c j ) and X σ j =(x σ T c j 1 +1,..., xσ T c j ) with x σ t = (x t /ˆσ i ) and y σ t =(y t /ˆσ i ). Hence, the quasi Sup-likelihood ratio test is sup LR 2,T (m a,n a,ε n =,m a ) = 2 sup log ˆL T (T1, c..., Tm c a ; T1,..., Tn a ) sup log el T (T1, c..., Tm c a ) (λ c 1,...,λc ma ;λ 1,...,λ na) Λ ε (λ c 1,...,λc ma) Λ c,ε h = 2 log ˆL T ( T e 1, c..., T e m c a ; T e 1,..., T e n a ) log L e T ( ˆT 1 c,..., ˆT i m c a ) where and Λ c,ε = (λ c 1,...,λ c m); λ c j+1 λ c j ε (j =1,..., m a 1),λ c 1 ε, λ c m a 1 ε ª Λ ε = {(λ c 1,..., λ c m,λ 1,..., λ n); for (λ 1,..., λ K )=(λ c 1,..., λ c m) (λ 1,..., λ n) (8) λ j+1 λ j ε (j =1,..., K 1),λ 1 ε, λ K 1 ε} Note that we denote the estimates of the break dates in coefficients and ariance by a when these are obatined jointly, as opposed to the estimates which are obtained separately and denoted by a ˆ. Remark 1 The set Λ ε which defines the possible alues of the break fractions in coefficients (λ c 1,..., λ c m) and in ariance (λ 1,..., λ m) allows them to hae some (or all) common elements or be completely different. What is important is that each break fraction be separated by a non-zero alue ε. This does complicate inference since many cases need to be considered. To illustrate, consider the case with m a = n a =1.WecanhaeK =1in which case it is a one break model with both the coefficients and the ariance of the errors changing at the same breakdate. Ontheotherhand,ifK =2, the break date for the change in coefficients is different from that for the change in ariance. This leads to two additional possible cases to consider: a) λ c 1 λ 1 ε (the break in the coefficients occurs before the break in the ariance), b) λ c 1 λ 1 + ε (the break in the coefficients occurs after the break in the ariance). The maximized likelihood function for these two cases can easily be ealuated using the algorithm 1

12 of Qu and Perron (27a) since it permits the imposition of restrictions. For example, if λ c 1 λ 1 ε, we hae a two break model and the restrictions needed are that the ariance of the errors in the first and second regimes is identical, and the regression coefficients are identical in the second and third regimes. Hence, for the case m a = n a =1, there are three maximized likelihood alues to construct and the test corresponds to the maximal alue oer these three cases. When m a or n a are greater than one, more cases need to be considered, but the principle remains the same. For the testing problem TP-3, the null hypothesis specifies n a breaks in ariance and zero break in the regression coefficients so that, for a gien partition {T1,..., Tn}, the likelihood function is gien by na+1 log el T T 1,...,Tn T a = 2 (log 2π +1) X i=1 T i Ti 1 log eσ 2 i, 2 where for i =1,..., n a +1,with eσ 2 i = T i T 1 ix (y Ti 1 t x tβ e z e tδ) 2 t=ti 1 +1 ( e β, e δ ) =(W σ W σ ) 1 W σ Y σ, where W σ =(w1 σ,..., wt σ ) with wt σ =(x σ t,zt σ ). Under the alternatie hypothesis, there are m a breaks in the regression coefficients and n a breaks in ariance so that the likelihood function is gien by (7). Hence, the Sup-Likelihood ratio test is sup LR 3,T (m a,n a,ε m =,n a ) = 2 sup log ˆL T (T1, c..., Tm c a ; T1,..., Tn a ) sup log el T (T1,..., Tn a ) (λ c 1,...,λc ma ;λ 1,...,λ na) Λ ε (λ 1,...,λ na) Λ,ε h i = 2 log ˆL T ( et 1, c..., et m c a ; et 1,..., et n a ) log el T ( ˆT 1,..., ˆT n a ) For the testing problem TP-4, the null hypothesis specifies no break under both the null and alternatie hypotheses and, hence, the log-likelihood alue under the null hypothesis is gien by log el T as specified by (5). The alternatie hypothesis specifies m a breaks in the regression coefficients and n a breaks in the ariance of the errors and the log likelihood alue 11

13 is gien by (7). Hence, the Sup-Likelihood ratio test is then sup LR 4,T (m a,n a,ε n = m =) = 2 sup log ˆL T T c 1,..., Tm c a ; T1,..., Tn a log LT e (λ c 1,...,λc ma ;λ 1,...,λ na) Λ ε i = 2 hlog ˆL T ( et c1,..., et cma ; et 1,..., et na ) log el T (9) 5 The limiting distributions of the tests Wenowconsiderthelimitdistributionofthetests.Westartwiththecasewheretheerrors are martingale differences in Section 5.1 and consider extensions to serially correlated errors in Section The case with martingale difference errors. Since some testing problems imply a gien non-zero number of breaks under the null hypothesis, we need some conditions to ensure that the estimates of the break fractions are consistent at a fast enough rate and that the estimates of the parameters are also consistent. This problem was analyzed in Qu and Perron (27a) and we simply use the same set of assumptions. If breaks are allowed in the regression coefficients under both the null and alternatie hypotheses, we specify the following conditions: Assumption A1: The conditions stated in Assumptions A1-A4 and A6-A8 of Qu and Perron (27a) are assumed to hold. When the null hypothesis specifies no change in the regression coefficients, we shall assume, with w t =(x t,zt) : Assumption A2: T 1 P [Ts] t=1 w tw t p sq, uniformly in s [, 1], withq some positie definite matrix. Assumption A2 rules out trending regressors and imposes the requirement that the limit moment matrix of the regressors be homogeneous throughout the sample. Hence, we aoid thecasewherethemarginaldistribution of the regressors may change while the coefficients do not (see, e.g., Hansen, 2). This follows from our basic premise that regimes are defined by changes in some coefficients. When changes in the ariance of the errors are allowed under both the null and alternatie hypotheses, we specify: 12

14 Assumption A3: The conditions stated in Assumption A5 of Qu and Perron (27a) are assumed to hold with the addition that the errors {u t } form an array of martingale differences relatie to F t = σ-field {..., z t 1,z t,..., x t 1,x t,...,u t 2,u t 1 }. Whenthenullhypothesisimposesnochangesinariance,weshallneed: Assumption A4: The errors {u t } form an array of martingale differences relatie to F t = σ-field {..., z t 1,z t,...,x t 1,x t,..., u t 2,u t 1 }, and, additionally, E (u 2 t )=σ 2 for all t and T P 1/2 [Ts] t=1 z tu t σq 1/2 W q (s), where W q (s) is a q-ector of independent Wiener processes. Also, T P 1/2 [Ts] t=1 (u2 t /σ 2 1) ψw(s) where W (s) is a Wiener process independent of W q (s) and ψ =lim T ar(t P 1/2 T t=1 (u2 t /σ 2 ) 1). Assumption A4 rules out instability in the error process and states that a basic functional central limit theorem holds for the weighted partial sums of the errors and their squares. Note that A4 assumes no serial correlation in the errors u t. This will be relaxed later. The limiting distributions, under the releant null hypothesis, of the likelihood ratio tests for the testing problems TP-1 to TP-4 are stated in the following Theorem, where denotes weak conergence under the Skorohod topology and is the Eucledian norm. Theorem 1 Under the releant null hypothesis, we hae, as T, a) For TP-1, under A2 and A4: where sup LR 1,T (n a,ε m = n =) b) For TP-2, under A1 and A4: ψ sup (λ 1,...,λ na) Λ, 2 sup LR 2,T (m a,n a,ε n =,m a ) sup (λ 1,...,λ na) Λ c, Xn a i=1 ψ 2 ψ sup (λ 1,...,λ na) Λ, 2 λ i W λ i+1 λ Xn a i=1 Xn a i=1 λ i+1λ i i+1w (λ i ) 2 λ i+1 λ i λ i W λ i+1 λ λ i+1λ i i+1w (λ i ) 2 λ i+1 λ i λ i W λ i+1 λ λ i+1λ i i+1w (λ i ) 2 λ i+1 λ i Λ c,ε = (λ 1,..., λ n); for (λ 1,..., λ K )=(λ c 1,..., λ c m) (λ 1,..., λ n) λ j+1 λ j ε (j =1,..., K 1),λ 1 ε, λ K 1 ε} 13

15 and Λ,ε = λ 1,..., λ n a ; λ i+1 λ i ε (i =1,..., n a 1),λ 1 ε, λ n a 1 ε ª. where c) For TP-3, under A1-A3: sup LR 3,T (m a,n a,ε m =,n a ) sup Xm a (λ c 1,...,λc ma) Λ c, j=1 sup Xm a (λ c 1,...,λc ma) Λ c, j=1 λ c jw q (λ c j+1) λ c j+1w q (λ c j) 2 λ c j+1λ c j(λ c j+1 λ c j) λ c jw q (λ c j+1) λ c j+1w q (λ c j) 2 λ c j+1λ c j(λ c j+1 λ c (1) j) Λ c,ε = (λ c 1,..., λ c m); for (λ 1,...,λ K )=(λ c 1,..., λ c m) (λ 1,..., λ n ) λ j+1 λ j ε (j =1,..., K 1),λ 1 ε, λ K 1 ε} and Λ c,ε = (λ c 1,..., λ c m); λ c j+1 λ c j ε (j =1,...,m a 1),λ c 1 ε, λ c m a 1 ε ª d) For TP-4, under A2 and A4: sup LR 4,T (m a,n a,ε n = m =) where sup (λ c 1,...,λc ma ;λ 1,...,λ na) Λ ε P ma λ c j W q(λ c j+1 ) λc j+1 W q(λ c j ) 2 j=1 λ c j+1 λc j(λ c j+1 λc j) + ψ 2 P na (λ i W (λ i+1) λ i+1 W (λ i ))2 i=1 λ i+1 λ i (λ i+1 λ i ) Λ ε = {(λ c 1,..., λ c m,λ 1,..., λ n); for (λ 1,...,λ K )=(λ c 1,..., λ c m) (λ 1,..., λ n) λ j+1 λ j ε (j =1,...,K 1),λ 1 ε, λ K 1 ε} Remark 2 For the testing problems TP-2 and TP-3, the limit distributions depend on the true unknown alue of the releant break fractions corresponding to the break dates allowed under both the null and alternatie hypotheses. The results, howeer, indicate that these distributions can be bounded by limit random ariables which do not depend on such unknown alues. This follows since Λ c,ε Λ,ε and Λ c,ε Λ c,ε. Hence, a conseratie testing procedure is possible. As we shall see, the test is barely conseratie if the trimming parameter ε is small, though as ε gets large (e.g..2) the test will be somewhat undersized. 14

16 The proof of this Theorem is gien in the Appendix. For the testing problem TP-3, the bound is the same as the limit distribution in Bai and Perron (1998). Hence, the critical alues proided by Bai and Perron (1998, 23b) can be used. For the testing problems TP-1 and TP-2, the same limit distribution (for a one parameter change) applies except for the scaling factor (ψ/2). This quantity can neertheless still be consistently estimated. Consider the following class of estimates: ˆψ = 1 T XT 1 j= (T 1) ω (j, m) TX t= j +1 ˆη tˆη t j (11) where ˆη t =(û 2 t /ˆσ 2 ) 1 where ˆσ 2 = T P 1 T t=1 û2 t with û t the residuals under the null hypotheses. Here w(j, m) is a weight function and m some bandwidth which can be selected using one of the many alternatie ways that hae been proposed; see, e.g., Andrews (1991). The estimate ˆψ will be consistent under some conditions on the choice of w(j, m) and the rate of increase of m as a function of T. Following Kejriwal and Perron (26a), we use the residuals under the null hypothesis to construct ˆψ but the residuals under the alternatie hypothesis to select the bandwidth parameter m (see also Kejriwal, 27). Simulations showed that using the residuals under the alternatie hypothesis to select m and construct ˆψ leads to tests with important size distortions. Using the residuals under the null for both leads to conseratie and less powerful tests. Using the hybrid method permits, as we shall see, to control the exact size in small samples without significant loss of power. In our simulations and empirical applications, we use the Quadratic Spectral kernel and to select m we adopt the method suggested by Andrews (1991) with an AR(1) approximation. Remark 3 If the errors are i.i.d., ψ = μ 4 /σ 4 1, which can be consistently estimated using ˆψ =ˆμ 4 /ˆσ 4 1, whereˆσ 2 = T P 1 T t=1 û2 t and ˆμ 4 = T P 1 T t=1 û4 t with û t the residuals under the null or alternatie hypotheses. Also, if the errors are Normal, ψ =2so that no adjustment is necessary, a case that was coered by Qu and Perron (27a). Since these cases are of less releance in practical applications, we shall only consider a correction inoling ˆψ as defined by (11). But it is useful to note that a simpler correction is aailable if the i.i.d. assumption is reasonable. 15

17 We then hae the following corrected statistic with a nuisance parameter free limit distribution: Xn a λ i W λ i+1 λ i+1w (λ i ) 2 sup LR 1,T = (2/ˆψ)supLR 1,T sup (λ 1,...,λ na) Λ, i=1 sup LR 2,T = (2/ˆψ)supLR 2,T sup sup Xn a (λ 1,...,λ na) Λ, i=1 Xn a (λ 1,...,λ na) Λ c, i=1 λ i W λ i+1 λ i+1λ i λ i+1λ i λ i+1 λ i λ i W λ i+1 λ λ i+1λ i λ i+1w (λ i ) 2 λ i+1 λ i (12) i+1w (λ i ) 2 λ i+1 λ i For the testing problem TP-4, it is possible to obtain a transformation that has a limit distribution free of nuisance parameters but the procedure is more inoled. It is gien by sup LR 4,T =suplr 4,T ˆψ 2 ˆψ LR (13) where LR is the likelihood ratio test for no break in ariance ersus n a breaks ealuated using the estimates { et 1,..., et n a } obtained by maximizing the likelihood function jointly allowing for m a breaks in coefficients, i.e., i LR =2 hlog ˆL T ( et 1,..., et na ) log el T where log ˆL T ( ) and log el T are defined by (6) and (5), respectiely. Note that LR is not equialent to LR 1,T (n a,ε m = n =) which is based on the estimates of the break dates for the changes in ariance assuming no break in coefficients. Since { T e 1 /T,..., T e n a /T } are consistent estimates of the break fractions whether we hae m a breaks in coefficients or not, we deduce that Xn a (λ i W λ i+1 λ i+1w (λ i )) 2 and, hence, sup LR 4,T LR ψ 2 sup sup (λ 1,...,λ na) Λ ε i=1 (λ c 1,...,λc ma ;λ 1,...,λ na) Λ ε λ i+1λ i λ i+1 λ i P ma λ c j Wq(λc j+1 ) λc j+1 Wq(λc j ) 2 j=1 λ c j+1 λc j(λ c j+1 λc j) + P n a (λ i W(λ i+1) λ i+1 W (λ i ))2 i=1 λ i+1 λ i (λ i+1 λ i ) sup LR 4 (14) The limit distribution (14) is new. To obtain the releant critical alues we proceeded as follows. We first simulate a dependent ariable and q regressors as independent N(, 1) 16

18 random ariables. This is without loss of generality since the limit distribution does not depend on the distribution of the regressors and using Normally distributed series will ensure a closer correspondence with the asymptotic distribution for a gien sample size, which we set to T =5. The algorithm of Qu and Perron (27a) imposing appropriate restrictions is then used to obtain the estimates of the m a break dates in coefficients and the n a break dates in ariance using the trimming specified by Λ. We then simulate a q 1 ector of independent Wiener processes W q+1 ( ) as partial sums of independent N(, 1) random ariables, again with T =5, and ealuate the quantity Xm a λ c jw q λ c j+1 λ c j+1w q λ c j 2 n a X λ i W λ i+1 λ λ c j+1λ c j λ c j+1 λ c + i+1w (λ i ) 2 j λ i+1λ i λ i+1 λ i j=1 where W q ( ) contains the first q elements of the ector W q+1 ( ) and W ( ) is the q+1th element of the simulated ector W q+1 ( ). This is repeated 2, times to obtain the releant quantiles corresponding to the distribution of the sum of the two terms. The critical alues for tests of size 1%, 2.5%, 5% and 1% are presented in Table 1 for q between 1 and 5 and ε =.1,.15,.2 and.25. For ε =.1,.15,.2, m a =1, 2 and n a =1, 2. For ε =.25,m a =1, and n a =1gien that ε =.25 imposes a maximal number of 2 breaks. 5.2 Extensions to serially correlated errors i=1 We now consider the case where the errors u t can be serially correlated. Assumptions A3 and A4 are replaced by: To that effect Assumption A3 : The conditions stated in Assumption A5 of Qu and Perron (27a) are assumed to hold. and when the null hypothesis imposes no changes in ariance, we shall need: Assumption A4 : E (u 2 t )=σ 2 for all t and T 1/2 P [Ts] t=1 z tu t σq 1/2 W q (s), where W q (s) is a q-ector of independent Wiener processes. Also, T 1/2 P [Ts] t=1 (u2 t /σ 2 1) ψw(s) where W (s) is a Wiener process independent of W q (s) and 1/2 ψ = lim ar(t T TX (u 2 t /σ 2 ) 1). For the testing problems TP-1 and TP-2, the results are the same and the sup LR 1,T and sup LR 2,T t=1 are statistics that will be asymptotically inariant to non-normal errors, serial 17

19 correlation and conditional heteroskedasticity so that the limit distribution (12) still applies. For the testing problems TP-3 and TP-4, things are more complex. Consider first TP-3. When the errors u t are serially correlated, the likelihood ratio type tests for changes in the coefficients of the conditional mean depend on nuisance parameters and would be hard to implement in practice. In such a case, structural changes in the regression coefficients can still be tested using the following Wald type statistics taking into account the presence of serial correlation: sup c (λ1,...,λ c ma) Λ ε F 3,T (m a,n a,ε m =,n a ),where F 3,T (m a,n a,ε m =,n a )= (T (m a +1)q p) ³ ˆδ R R ˆV (ˆδ)R 1 Rˆδ (15) m a q with ˆδ =(δ 1,..., δ m a +1) is the quasi-maximum likelihood estimate of the coefficients that are subject to change, under a gien partition of the sample, R is the conentional matrix such that (Rδ) = δ 1 δ 2,..., δ m a δ m a +1 and ˆV (ˆδ) is an estimate of the ariance coariance matrix of ˆδ that is robust to serial correlation and heteroskedasticity, i.e, a consistent estimate of V (ˆδ) =plim T T Z σ Z σ 1 Ω Z σ Z σ Z σ 1,where Z σ = M Xσ Z σ, Ω = Z σ E( Z σ Ub U b Z σ), Ub = M X σ U σ, M Xσ = I T X σ (XσX σ ) 1 Xσ,with Z σ = diag Z1 σ,..., Zm a+1 σ, Zj σ =(zt σ j 1 c +1,...,zσ Tj c),u σ =(u σ 1,..., u σ T ),zt σ =(z t /σ i ) and u σ t =(u t /σ i ),forti 1 <t Ti (i =1,..., n a +1). Under A2, A3 and additional assumptions under which a consistent estimate of V (ˆδ) can be obtained using kernel based methods as in Andrews (1991), the limiting distribution of sup F 3,T (m a,n a,ε m =,n a ) is the same as in the case with martingale difference errors, i.e, as stated in (1). In practice, the computation of the aboe tests could be ery inoled, especially if a data dependent method is used to construct the robust asymptotic coariance, ˆV (ˆδ). Following Bai and Perron (1998), we suggest firsttousethedynamic programming algorithm to get the break points corresponding to the global maximization of the likelihood function defined by (7), then plug the estimates into (15) to construct the test. This will not affect the consistency of the test since the break fractions are consistently estimated. For the testing problem TP-4, things are more complex. We shall adopt a quasi Wald testing procedure. Note first that the information matrix is block diagonal with respect to δ and σ 2, hence the test will inole one component for changes in δ and one component for changes in σ 2.Thefirst is the same as discussed aboe, namely sup F 3,T as defined by (15), except that one can use z t instead of zt σ since the null hypothesis specifies no break in ariance. The difficulty is with the second component. The Wald test for the equality in ariance across regime is asymptotically different from the LR test een with martingale difference errors. In fact, its limit distribution is quite complex and would necessitate additional tables of critical 18

20 alues. A compromise that is simple and yet still leads to a consistent test is to sum the indiiduals Wald tests for each successie pairs of regimes. This leads to the component: sup F σ T = ˆψ 1 n Xa i=1 (ˆσ 2 i+1 ˆσ 2 i ) 2 Ã ˆσ 4 i+1 eλ i+1 e λ i ˆσ 4 i eλ i λ e i 1! 1 where ˆσ 2 i =(et i et i 1) P 1 T i T i 1 +1 û2 t and the estimates are constructed by maximizing the likelihood function (7) subject to the restrictions imposed by the set Λ ε The test statistic suggested is then sup F 4,T (m a,n a,ε m =,n a =)=supf 3,T +supft σ It is easy to show that, under A2 and A4, the limit distribution of sup F 4,T is the same as the modified LR test in the case of martingale difference errors, i.e., gien by the random ariable (14). 5.3 A double maximum test The tests discussed aboe need the prior information of the specification of the alternatie hypothesis, i.e., the number of breaks in regression parameters and in the ariance of the errors. Howeer, in practice, researchers may lack such information, hence the need for the testing problems TP-5 to TP-8. Bai and Perron (1998) proposed so-called double maximum tests to sole this problem in a model with only breaks in the parameters. They are tests of no structural break against an unknown number of breaks gien some upper bound. Bai and Perron (1998) suggested two ersions of such tests. The first is an equal-weight ersion labelled UDmax. It can be gien a Bayesian interpretation in which the prior assigns equal weights to the possible number of changes. The second test applies weights to the indiidual tests such that the marginal p-alues are equal across alues of m and n and is denoted WDmax. Bai and Perron (26) showed ia simulations that the two ersions hae similar finite sample properties. Hence, we shall only consider the UDmax test gien that it is simpler to construct. The Double Maximum test can play a significant role in testing for structural changes and it is arguably the most useful tests to apply when trying to determine if structural changes are present. While the test for one break is consistent against alternaties inoling multiple changes, its power in finitesamplescansometimesbepoor. First,therearetypesofmultiple structural changes that are difficult to detect with a test for a single change (for example, 19

21 two breaks with the first and third regimes the same). Second, tests for a particular number of changes may hae non monotonic power when the number of changes is greater than specified. Third, the simulations of Bai and Perron (26) show, in the context of testing for changes in the regression coefficients, that the power of the double maximum tests is almost as high as the best power that can be achieed using the test that accounts for the correct number of breaks. All these elements strongly point to their usefulness. For each testing problem, the tests and their limit distributions are presented in the following Theorem. Theorem 2 Under the releant null hypothesis, we hae, as T, a) For TP-5, under A2 and either A4 or A4 : UDmax LR 1,T = max sup 1 n a N LR 1,T (n a,ε m = n =) Xn a λ i W λ i+1 max 1 n a N sup (λ 1,...,λ na) Λ, i=1 b) For TP-6, under A1 and either A4 or A4 : (λ 1,...,λ na) Λ c, i=1 λ i+1λ i λ i+1w (λ i ) 2 λ i+1 λ i UDmax LR 2,T = max sup 1 n a N LR 2,T (m a,n a,ε n =,m a ) n a X λ i W λ i+1 λ max sup i+1w (λ i ) 2 1 n a N λ i+1λ i λ i+1 λ i c) For TP-7, under A1-A3: max 1 n a N sup Xn a (λ 1,...,λ na) Λ, i=1 λ i W λ i+1 λ λ i+1λ i UDmax LR 3,T = max 1 m a M sup LR 3,T (m a,n a,ε m =,n a ) max 1 m a M max 1 m a M sup Xm a (λ c 1,...,λc ma) Λ c, j=1 sup Xm a (λ c 1,...,λc ma) Λ c, j=1 i+1w (λ i ) 2 λ i+1 λ i λ c jw q (λ c j+1) λ c j+1w q (λ c j) 2 λ c j+1λ c j(λ c j+1 λ c j) λ c jw q (λ c j+1) λ c j+1w q (λ c j) 2 λ c j+1λ c j(λ c j+1 λ c j) 2

22 d) For TP-8, under A2 and A4: UDmax LR 4,T = max 1 n a N max sup 1 m a M LR 4,T (m a,n a,ε n = m =) max max 1 n a N 1 m a M sup (λ c 1,...,λc ma ;λ 1,...,λ na) Λ ε P ma λ c j W q(λ c j+1 ) λc j+1 W q(λ c j ) 2 j=1 λ c j+1 λc j(λ c j+1 λc j) + P n a (λ i W (λ i+1) λ i+1 W (λ i ))2 i=1 λ i+1 λ i (λ i+1 λ i ) For TP-5 to TP-7, the critical alues of the limit distributions are aailable from Bai and Perron (1998, 23b) for N or M equalto5. NotethatforthetestingproblemsTP-5 and TP-6, the results are alid whether the errors are martingale differences or whether serial correlation is allowed. This is not the case for TP-7 and TP-8 for the same reasons as discussed aboe that the likelihood ratio tests are not applicable when the errors are serially correlated. In this case, we consider the maximum of the Wald-type test and the results are presented in the following Theorem. Theorem 3 Under the releant null hypothesis, we hae, as T, a) For TP-7, under A2 and A3 : UDmax F 3,T = max 1 m a M sup F 3,T (m a,n a,ε m =,n a ) max 1 m a M max 1 m a M b) For TP-8, under A2 and A4 : UDmax F 4,T = max 1 n a N sup Xm a (λ c 1,...,λc ma) Λ c, j=1 sup Xm a (λ c 1,...,λc ma) Λ c, j=1 max sup F 4,T (m a,n a,ε n = m =) 1 m a M max max 1 n a N 1 m a M sup (λ c 1,...,λc ma ;λ 1,...,λ na) Λ ε λ c jw q (λ c j+1) λ c j+1w q (λ c j) 2 λ c j+1λ c j(λ c j+1 λ c j) λ c jw q (λ c j+1) λ c j+1w q (λ c j) 2 λ c j+1λ c j(λ c j+1 λ c j) P ma λ c j Wq(λc j+1 ) λc j+1 Wq(λc j ) 2 j=1 λ c j+1 λc j(λ c j+1 λc j) + P n a (λ i W(λ i+1) λ i+1 W (λ i ))2 i=1 λ i+1 λ i (λ i+1 λ i ) The limit distribution applicable for the testing problem TP-8 is new. We obtained critical alues using simulations as discussed aboe for the case of a fixed number of breaks under the alternatie hypothesis. These are presented in Table 1 for ε =.1,.15, and.2, and alues of M and N up to 2. 21

23 5.4 Testing for an additional break We now consider the testing problems TP-9 and TP-1, which looks at whether including an additional break is warranted. Let ( T e 1, c..., T e m; c T e 1,..., T e n) denote the estimates of the break dates in the regression coefficients and the ariance of the errors obtained jointly by maximizing the quasi-likelihood function assuming m breaks in the coefficients and n breaks in the ariance. For the testing problem TP-9, the issue is whether an additional break in the regression coefficients is present. Following Bai and Perron (1998) and Qu and Perron (27a), the test is sup Seq T (m +1,n m, n) = max LR T ( et 1, c..., et j 1,τ, c et j c,..., et m; c et 1,..., et n) 1 j m+1 τ Λ c j,ε LR( et 1, c..., et m; c et 1,..., et n) where Λ c j,ε = {τ; et c j 1 +(et c j et c j 1)ε τ et c j ( et c j et c j 1)ε} (16) This amounts to performing m +1tests for a single break in the regression coefficients for each of the m +1 regimes defined by the partition { T e 1 c,..., T e m}. c Notethattherearedifferent scenarios when allowing breaks in coefficients and in the ariance to happen at different dates, since ( et 1 c,..., et m) c and ( et 1,..., et n) can partly or completely oerlap or be altogether different. This implies two possible cases: 1) if the n break dates in ariance are a subset of the m break dates in coefficients, then there is no ariance break between et j 1 c and et j c ;2) otherwise, there is one or more ariance breaks between et j 1 c and et j c. In either cases, one can appeal to the results of part (c) of Theorem 1 with m a =1sinceanyalueofn a (the number of breaks in ariance) is allowed, including. Itistheneasytodeducethat, inthe case of martingale errors, the limit distribution of the test is, under Assumptions A2 and A3, lim P (sup Seq T (m +1,n m, n) x) =G q,ε (x) m+1 T where G q,ε (x) is the cumulatie distribution function of the random ariable (W q (λ) λw q (1)) 2 sup. (17) λ Λ 1,ε λ (1 λ) where Λ 1,ε = {λ; ε<λ<1 ε}. The critical alues of the distribution function G q,ε (x) m+1 can be found in Bai and Perron (1998, 23b). When serial correlation in the error, the 22

24 principle is the same except that the statistic is based on the robust Wald test sup F 3,T as defined by (15) applied for a one break test to each segment. For the testing problem TP-1, similar considerations apply. Here the issue is whether an additional break in the ariance is present. The test statistic is ³ sup Seq T (m, n +1 m, n) = 2/ˆψ max sup LR T ( et 1, c..., et m; c et 1,..., et j 1,τ, et j,..., et m) 1 j n+1 τ Λ j,ε LR( et 1, c..., et m; c et 1,..., et n) where Λ j,ε = {τ; et j 1 +(et j et j 1)ε τ et j ( et j et j 1)ε}. The correction factor (2/ˆψ) is needed to ensure that the limit distribution of the test is free of nuisance parameters when the errors are allowed to be non-normal, serially correlated and conditionally heteroskedastic. One can then use part (b) of Theorem 1 to deduce that, under A1 and A4, or A1 and A3 applied to each segments under the null hypothesis, 6 Monte Carlo experiments lim P (sup Seq T (m, n +1 m, n) x) =G 1,ε (x) n+1. T This section presents the results of simulation experiments to address the following issues: 1) which particular ersion of the correction factor ˆψ has better finite sample properties?; 2) whether applying a correction alid under more general conditions than needed is detrimental to the size and power of the test; 3) the finite sampe size and power of the arious tests proposed. Throughout, we use 1, replications. 6.1 The choice of ˆψ To address what specific ersion of the correction factor to use, we consider the size and power of the sup LR4,T test under the following simple Data Generating Process (DGP) with ARCH(1) errors: y t = μ 1 + μ 2 1(t >[.25T ]) + e t, e t = u t p ht, u t i.i.d. N(, 1), h t = δ 1 + 1(t>[.75T ]) + γe 2 t 1, 23