Likelihood ratio tests for multiple structural changes

Transcription

1 Journa of Econometrics 91 (1999) 299}323 Likeihood ratio tests for mutipe structura changes Jushan Bai* Department of Economics, E52-274B, Massachusetts Institute of Technoogy, Cambridge, MA 02139, USA Received 1 Apri 1997; received in revised form 1 June 1998; accepted 12 August 1998 Abstract This paper proposes a ikeihood-ratio-type test for mutipe structura changes in regression modes. The mode aows for agged-dependent variabes and trending regressors. The imiting distribution of the test is derived. We show that asymptotic critica vaues can be obtained anayticay. In addition, the number and the ocations of change points can be consistenty determined via the test procedure. The method is straightforward to impement Esevier Science S.A. A rights reserved. JEL cassixcation: C12; C22; C13; C52 Keywords: Structura change; Mutipe change points; Hypothesis testing; Dynamic modes; Trending regressors; Limiting distribution 1. Introduction Testing for structura change has aways been an important issue in econometrics because a myriad of poitica and economic factors can cause the reationships among economic variabes to change over time. Since the eary work of Chow (1960) and Quandt (1960), numerous studies have been undertaken on the issue of structura changes. However, most of the existing work focuses on testing for a singe change. In this paper, we propose a ikeihood-ratio-type test for mutipe changes. The proposed test is an exact * Te. (617) ; fax: (617) ; e-mai: jbai@mit.edu This is evidenced by the two specia voumes of this Journa edited, respectivey, by Broemeing (1982), and Dufour and Ghyses (1996), as we as by a number of monographs on this subject, e.g., Poirier (1979), Kramer (1989), and Hack and Westund (1991) /99/$ - see front matter 1999 Esevier Science S.A. A rights reserved. PII: S ( 9 8 )

2 300 J. Bai / Journa of Econometrics 91 (1999) 299}323 ikeihood-ratio test under the assumption of normaity. Since normaity is not assumed, the test may be considered a pseudo-gaussian ikeihood-ratio test. The imiting distribution of the test is derived in this paper. Critica vaues can be cacuated anayticay. When testing for a singe break, the imiting distribution of the test statistic is derived in the absence of breaks (typicay by direct appication of the functiona centra imit theorem under the nu of no change). In this paper, both nu and aternative hypotheses aow for the existence of break points and thus invove estimating these points. As a resut, this anaysis di!ers consideraby from the existing iterature. In particuar, functiona centra imit theorems cannot be directy appied to derive the imiting distribution. The behavior of break-point estimators under both the nu and the aternative hypotheses must be taken into account. In addition to having the capabiity of detecting the presence of a break, the proposed test can be used to determine the number and ocations of breaks in the data. This is of both theoretica and practica interest. Under the current practice, when the nu hypothesis is rejected, a singe break point is estimated. In many cases, there is no reason to assume that ony one break point is present, especiay for data covering an extended period of time. The possibiity of mutipe changes is consistent with the notion of mutipe equiibria proposed by economic theory, with each change (regime) representing a new equiibrium condition. Rudebusch and Diebod (1994) expain the inks between the structura-change mode and mutipe equiibria of economic theory. Hegwood and Pape (1996) suggest the existence of mutipe regimes in the rea exchange-rate data. In an interesting appication of the structura change method, Wiard et a. (1996) study how peope during the Civi War period responded to various events that were happening around them and compare the reative importance of those events (as impied by the data) to the accounts of traditiona historians. The nu hypothesis of our test assumes break points, whereas the aternative hypothesis assumes #1 break points. (When "0, the test reduces to the usua test of no change against a singe change.) Due to the format of these hypotheses, when the test is performed repeatedy whie augmenting the vaue of, the number of break-points can be consistenty estimated. A reated test is set out in Bai and Perron (1994). They aso propose to test versus #1 breaks. Their test does not have a ikeihood-ratio interpretation. Whenever an additiona break point is to be estimated, it is obtained conditiona on the previousy obtained break points. For exampe, suppose that break points have been computed, the (#1)th break point is estimated from one of the subsampes separated by the given break points. In contrast, in the ikeihood-ratio setting, the mode is estimated optimay under both nu and aternative hypotheses. This entais estimating breaks simutaneousy under the nu and estimating #1 breaks simutaneousy under the aternative. The imiting distribution of the LR test is shown to be of a simiar } but not identica

3 J. Bai / Journa of Econometrics 91 (1999) 299} } form to that of the Bai}Perron test. Athough its theoretica justi"cation requires considerabe e!ort, the LR procedure itsef is straightforward to impement. This paper makes two additiona contributions to the existing iterature. First, the procedure can dea with dynamic regressors (agged dependent variabes). The existing iterature contains many resuts admitting dynamic regressors, for exampe, Kramer et a. (1988). In those studies, the mode, and particuary the regressors, are stationary under the nu. In a dynamic setting, once a shift occurs, the regressors cease to be stationary. Since our nu hypothesis aows for shifts, both the dependent and the expanatory variabes are non-stationary. Second, we consider testing mutipe breaks in poynomia trends. We not ony characterize the imiting distribution, we aso derive anaytica expressions for asymptotic critica vaues. The remainder of this paper is organized as foows. In Section 2, we state the mode and assumptions. Section 3 de"nes the test statistic and gives the main resut. The assumption of regime-wise asymptotic stationarity is assumed in this section. Section 4 extends the test to incude poynomia trends. Limited Monte Caro resuts are reported in Section 5, and some remarks are given in Section 6. Technica proofs are provided in the Appendix. 2. Mode and assumption Consider the foowing m-break mode: y "zδ #u, t"1, 2,2, k, y "zδ #u, t"k#1,2, k, y "zδ #u, t"k#1,2, ¹, (1) where y is the observed dependent variabe at time t; z (q1) is a vector of covariates; δ ( j"1,2, m#1) is a vector of coe$cients with δ Oδ (i"1,2, m); and u is the disturbance at time t. The break points k,2, k as we as the number of breaks m are aso unknown. When m is known, a parameters incuding the break points can be easiy estimated. Their statistica properties are investigated by Bai and Perron (1994). In practice, the number of Some exact tests may not require stationarity, but require more speci"c distributiona assumptions such as normaity, e.g., Dufour and Kiviet (1996). This paper, however, does not consider integrated regressors. For this, readers are referred to Vogesang (1994), Banerjee et a. (1992), Hansen (1992), and Zivot and Andrews (1992).

4 302 J. Bai / Journa of Econometrics 91 (1999) 299}323 breaks, m, is unknown. It is of importance to make an inference about the vaue of m. We propose a test statistic for testing the nu hypothesis of versus the aternative of #1 breaks. We show how this test can ead to a consistent estimate for m. Before discussing the test statistics, we make additiona assumptions as foows. Assumption 1. The regressors are regime-wise asymptoticay stationary. More speci"cay, et k "k!k, for i"0,2, m, pim 1 z z"vq k uniformy in v3[0, 1], where Q is a positive de"nite matrix. Throughout, we adopt the convention that k"0 and k "¹. Assumption 2. u,f forms a sequence of martingae di!erences where F "σ- "ed z, u ; s)t with Eu "σ for a t, and sup Eu (R. Assumption 3. The functiona centra imit theorem hods for z u in each regime. That is, for i"0,2,m, 1 ( k) z u N= (v), where = (v) is a q-dimensiona Wiener process on [0, 1] with variance} covariance matrix σvq, and =,2, = are independent of each other. Assumption 4. δ Oδ, k "[¹τ ], τ 3(0,1) with τ (τ (i"1,2, m). Assumption 1 is satis"ed by within-regime i.i.d. regressors having a positivede"nite variance}covariance matrix. Assumption 1 is aso satis"ed by any within-regime second-order stationary process such that the strong aw of arge numbers hods for z z. In these cases, Q "Ez z. Assumption 1 is ceary weaker than the goba asymptotic stationarity of the foowing: Pim 1 z z"vq, (2) ¹ uniformy in v3[0, 1]. Autoregressive modes with structura changes wi not satisfy the goba asymptotic stationarity (goba-stationary) (2) but do satisfy Assumption 1. In this case, Q is the second-moment matrix of a stationary autoregressive process with autoregressive parameter δ. Assumptions 2 and 3

5 are standard. But Assumption 2 rues out seriay correated disturbances. The resut of this paper can be extended to dependent disturbances such that u forms a sequence of mixingaes with respect to an increasing sequence of sigma-"eds but uncorreated with the regressors. For simpicity, we sha focus on uncorreated errors, making it possibe to incude agged dependent variabes. Assumption 4 is a standard technica device for asymptotic purposes. We next introduce some terminoogy and notation. A subsampe [i, j] means a subsampe consists of observations from i to j (incusive). An m-partition is a vector of m integers, (k,2, k ), such that 1(k (k (2k (¹. Each partition divides the whoe sampe into m#1 subsampes (segments). For a sma positive number π'0, we de"ne a set of partition, Λ,by Λ "(k,2, k ): k!k *π¹; i"1,2, m#1, (3) where k "0 and k "¹. Each of the m#1 subsampes resuting from a partition in Λ consists of at east a positive fraction of the observations. We assume π is sma so that (k,2, k )3Λ. Let S (k,2, k ) denote the sum of squared residuas corresponding to the partition (k,2, k ). That is, S (k,2, k )" J. Bai / Journa of Econometrics 91 (1999) 299} inf (y!z ), where k "0 and k "¹. The optima SSR is de"ned as S (kk, 2, kk )" min S (k,2, k ), 2 where the minimization is taken over Λ. In practice, π is set to 0.05, or Assuming there are m breaks, then (kk, 2, kk ) forms an estimator for (k, 2, k). If we de"ne τl "kk /¹, the break fraction, then τl is ¹ consistent for τ. That is, ¹(τL!τ )"kk!k "O (1), see Bai and Perron (1994). For simpicity, we sha say kk is ¹ consistent for k, with the understanding that we are referring to the break fractions. In the next section, we discuss the test statistics based on the optima SSRs. 3. The test statistic This section considers testing the hypothesis H : m" against H : m"#1. The test statistic is based on the di!erence between the optima SSR associated with breaks and the optima SSR associated with #1 breaks. Let

6 304 J. Bai / Journa of Econometrics 91 (1999) 299}323 (kk, 2, kk ) be the estimator of the break point (k,2, k ) under the nu. Let (kk *,2, kk * ) be the point at which the sum of squared residuas is minimized, when #1 break points are aowed. The test statistic is de"ned as sup LR (#1)" S (kk, 2, kk )!S (kk *,2, kk * ), (4) σl (#1) where σ(#1)"s (kk *,2, kk * /¹). When the errors are i.i.d. norma random variabes, the above test is a ikeihood ratio (LR) test. Because normaity is not assumed, Eq. (4) may be considered a pseudo-ikeihood ratio test. Using dynamic programming agorithm, the computation of Eq. (4) is straightforward and fast even for arge ¹ and arge. In the Monte Caro section beow, we use Bai and Perron's (1994) program to compute (4). ¹heorem 1. et Assumptions 1}4 hod. ¹hen under H : m", sup LR (#1) P maxξ,2,ξ, where ξ,2, ξ are independent and have the representation ξ " sup B (s) s(1!s) (5) here η "π/(τ!τ ), and the B (μ) are independent and standard Brownian bridges on [0, 1]. Athough the ξ 's are independent, they do not necessariy have the same distribution because the η 's may not be the same. Regardess of identica distribution, critica vaues are easy to obtain, as the imiting distribution has a known anaytica density function. Coroary 1. ;nder the assumptions of ¹heorem 1, im P(sup LR (#1)'c)"1! [1!G (c)], The computer program in GAUSS is avaiabe upon request.

7 J. Bai / Journa of Econometrics 91 (1999) 299} where c exp(!c/2) G (c)" 2Γ(q/2) 1!q c og(1!η ) # 2 η c #o(c). Given a size α, the corresponding critica vaue c can be easiy computed from the above formua. For arge c and sma η the "rst term inside the brackets of G (c) dominates, and thus the term 2/c may be ignored. In practice, the η "π/(τ!τ ) is unknown, even though π is under the contro of a researcher. It can be repaced, however, by its estimated vaue such that ηl "π/(τl!τl ) for i"1, 2,, where τl "0 and τl "1. Under the nu hypothesis, τl converges to τ at rate ¹, yieding good estimates for η. The anaytica expression is convenient for automation (computer programming). The expression G (c) is derived in DeLong (1981). Bai and Perron (1994) suggest an aternative (conditioning) procedure to test versus #1 breaks. Their test is aso based on the di!erence between the SSR associated with breaks and that associated with #1 breaks. However, the #1 break points are obtained not simutaneousy but sequentiay. More speci"cay, the (#1)th break point is obtained conditiona on the break points obtained previousy, and the (#1)th break point is picked from the #1 subsampes separated by the break points kk, 2, kk. This additiona break point is chosen in the subsampe where the sum of squared residuas achieves the greatest reduction. The imiting distribution of their test is the maximum of #1 i.i.d. random variabes of the form (5). The test proposed in this paper has a di!erent imiting distribution from that of Bai and Perron's sequentia test. The present test is conceptuay simper, and has a ikeihoodratio interpretaion. Athough the technica proof is more demanding than the sequentia test of Bai and Perron (1994), the test is easy to compute and impement Consistency of the test The test procedure proposed in this paper is consistent. That is, under the aternative hypothesis that m*#1, then sup LR (#1)PR with probabiity tending to 1. Therefore, when more than breaks exist, the nu hypothesis of breaks wi be rejected with probabiity tending to 1. We can The study of this conditioning procedure rather than LR was party due to our more primitive understanding of the LR procedure back then. Particuary, it was ess we understood in terms of the form of LR's imiting distribution, as we as the technica apparatus needed in justifying its imiting distribution. The LR's deeper understanding came ony recenty as re#ected in this paper. However, Bai and Perron's procedure does have an advantage that no simutaneous estimation of the break points is necessary when we use it repeatedy as increases.

8 306 J. Bai / Journa of Econometrics 91 (1999) 299}323 actuay show that sup LR (#1)"O (¹) under the aternative hypothesis. To see this, suppose there are #1 breaks. When ony breaks are aowed in estimation, at east one break point cannot be consistenty estimated. Lemma 2 of Bai and Perron (1994) then impies S (kk, 2, kk )! u"o (¹)PR. On the other hand, when #1 breaks are aowed in estimation, u!s (kk *,2, kk * )"O (1). Thus the numerator of the test statistic is O (¹) and the denominator is O (1). So the test statistic is O (¹) under the aternative hypothesis. This concusion remains true when more than #1 breaks exist under the aternative. The detais are omitted Estimating the number of breaks One advantage of our test is its capabiity of identifying the number of breaks in the data. The procedure is described in Bai and Perron (1994), athough they use a di!erent test. Here we give a brief summary. One begins with a test of no-break versus a singe break. If the hypothesis is rejected, one proceeds to test the nu of a singe break versus two breaks, and so forth. This process is repeated unti the test sup LR (#1) fais to reject the nu hypothesis of no additiona breaks. The estimated number of breaks is equa to the number of rejections. Let ml denote this number, and et m denote the true number of breaks. We have the foowing resut. ¹heorem 2. et Assumptions 1}4 hod. If the size α converges to zero sowy enough ( for the test based on sup LR (#1) to remain consistent), then P(mL "m )P1. One may start with H : " '0 if at east breaks are known to exist. Aternativey, this procedure may be used in a reverse order. That is, instead of increasing the vaue of, we decrease its vaue (starting with a reasonabe initia vaue) unti the nu of no-change is rejected. The number of break points is equa to the number speci"ed by the aternative hypothesis of the "rst rejection. The theoretica property of this reversed procedure, however, remains to be studied.

9 J. Bai / Journa of Econometrics 91 (1999) 299} The proof of this theorem is identica to that of Proposition 8 in Bai and Perron (1998) and is thus omitted. Let α be the size of the test and c be the corresponding critica vaue cacuated from the imiting distribution in Theorem 1. Because 1! [1!G (c)]) G (c))kc exp(!c/2) exp(!c/3) for a arge c, we have α)exp(!c/3). This impies c)!3og (α) (for a sma α or equivaenty arge c). Suppose α is chosen such that α "1/¹, then c )3og ¹. Under the aternative hypothesis, we know sup LR (#1) is of order O (¹). Consequenty, we wi reject the nu hypothesis with probabiity tending to 1. This impies that one wi not underestimate the true number of break points for arge sampes. The probabiity of overestimating the number of breaks is no more than α (see Bai and Perron, 1998). So if α P0 sowy (say no more quicky than 1/¹), we can consistenty estimate the number of breaks. In practice, ¹ is "xed, and so is α. Simuation shows that α"0.05 works satisfactoriy. Recenty, Andrews et a. (1996) consider an optima test for regression modes under the assumption of normaity. The nu hypothesis is no-change, and the aternative hypothesis aows for mutipe changes. In contrast, the present test aows for structura changes under the nu. It is this feature that enabes us to determine the number of breaks via hypothesis testing. In addition, critica vaues of the Andrews}Lee}Poberger test are not avaiabe except for a singe change, and do not appear to be straightforward to obtain even for arge sampes. In comparison, the critica vaues of the LR test of this paper can be obtained anayticay. Finay, we point out that a ikeihood-ratio-type test for the nu of no-change (m"0) against aternative mutipe changes (m") is derived in Bai and Perron (1998) in the absence of trending regressors. However, rejection of the nu of no-change does not provide information as to the number of breaks in the data. 4. Trending regressors In the previous sections, regime-wise asymptotic stationarity is assumed, ruing out trending regressors. We now derive the corresponding resuts in the presence of such regressors. Consider the foowing mode with poynomia trends: y "w γ #z δ #u, k #1)t)k,(i"1, 2,2, m#1) (6) where w "(1, t,2, t), k "0, and k "¹. The regressor z (q1) satis"es the assumptions stated in the previous section. It is assumed that (γ, δ )O(γ, δ ). This paper does not address the issue of optimaity. For a genera treatment of this issue, see Sowe (1996).

10 308 J. Bai / Journa of Econometrics 91 (1999) 299}323 Test statistic for testing H : m" versus H : m"#1hasthesameformatas before: it is constructed using the di!erence between the optima SSR associated with breaks and that associated with #1 breaks.for s3[0, 1], de"ne the matrix 1 2 A(s)"s s 2 1 p#1 s 1 2 s 1 3 s 2 1 p#2 s 1 p#1 s 1 p#2 s 2 1 2p#1 s, which is the (appropriatey) normaized imit of w w. Let λ (s),2, λ (s) denote the eigenvaues of A(1)A(s)A(1). (They are aso the eigenvaues of A(s) A(1).) These eigenvaues are continuous and increasing functions of s with λ (0)"0, and λ (1)"1. The imiting distribution of the test statistic is characterized in the next theorem. ¹heorem 3. et Assumptions 1}4 hod. ;nder H of breaks, the test statistic satis,es sup LR (#1) P maxψ,2, ψ where ψ " sup B(λ (s)) λ (s)(1!λ (s)) # B(s) s(1!s), the B ( ) ) are standard Brownian bridges on [0, 1], and the η are de,ned in ¹heorem 1. Coroary 2. ;nder the assumptions of ¹heorem 1, P(sup LR (#1)'c)"1! [1!H (c)], (7) where c exp(!c/2) H (c)+ 2Γ(r/2) 1 r!1 c 1 2 og det A(1!η )det(a(1)!a(η )) det A(η )det(a(1)!a(1!η )) q og(1!η ) η, with r"p#q#1 and det B as the determinant of B.

11 Theorem 3 and its coroary hod for genera trending regressors beyond poynomia ones. A that is needed is that the imiting matrix A(s) be positive de"nite and increasing. That is, A(u)'A(v) for u'v. However, under the speci"c structure of poynomia trends, H (c) canbesimpi"ed. It can be shown that det(a(1)!a(1!s))"det A(s)"s det A(1). This impies that Therefore, J. Bai / Journa of Econometrics 91 (1999) 299} ogdet A(1!η )det(a(1)!a(η )) det A(η )det(a(1)!a(1!η )) "(p#1)og1!η η. c/2 exp(!c/2) H (c)+ 2Γ(r/2) 1 r!1 c [(p#1)#q]og(1!η ) η. (8) Because (1/r!1/c)[(p#1)#q]"(1!r/c)[(p#1)#q]/r, the vaue of H (c) is proportiona to G (c) of Coroary 1 corresponding to r"p#1#q stationary regressors. The proportionaity factor is [(p#1)#q]/r. This impies the foowing interesting fact. When testing for no-change versus a singe change ("0), the asymptotic critica vaue of a size α test for mode (6) is equa to that of a size α(p#1#q)/((p#1)#q) test with p#1#q stationary regressors. The atter is tabuated in many papers, see Andrews (1993). 5. Some numerica resuts To iustate the procedure and to indicate the "nite sampe performance of the LR test, we report resuts of a imited set of samping experiments. The samping experiment considers three di!erent sets of regressors, simpe inear regression, autoregression, and inear trend. For each set of regressors, a sampe of ¹"150 observations is generated from a mode with 2 breaks (3 regimes). The "rst break is set at k "50 and the second at k "100. We may write the data generating process as y "a #b z #u, k #1)t)k, i"1, 2, 3, where (a, b ) is regime i's regression parameter, and k "0, and k "¹. The three sets of regressors are (I): z i.i.d. norma N(1,1); (II) z "y, and (III): z "t. The disturbances are i.i.d. standard norma for a cases. Let δ"[(a,b ), (a,b ), (a,b )].

12 310 J. Bai / Journa of Econometrics 91 (1999) 299}323 The design coe$cients are (I) (regression): δ"[(1.0, 1.0), (1.5, 1.5), (2.0, 2.0)] (II) (autoregression): δ"[(10.0, 0.5), (10.0, 0.4), (10.0, 0.5)], (III) (inear trend): δ"[(1.0, 1.0), (1.1, 1.1), (1.2, 1.2)]. For design (II), the intercept is not subject to shifts. In addition, the autoregressive sope coe$cient has a sma shift in magnitude. But because of the arge vaue of the intercept, the expected vaue of z "y is arge. This impies that the break points can be accuratey estimated because the variance of the break-point estimator is inversey reated to (b!b )Ez (assume no intercept shift). For a designs, the mode is estimated by imposing the restriction that each regime has at east 5 observations. This de"nes π or Λ in Eq. (3) impicity. Tabe 1 reports the percentage rejections from 5000 repetitions for testing H : m" against H : m"#1 for "0, 1, 2. Two nomina sizes are considered: α"0.05 and α"0.10. The percentages in the ast two coumns can be viewed as actua sizes (after dividing by 100) because the nu hypothesis is true. The percentages in the "rst four coumns can be viewed as powers (again after dividing by 100). For the inear regression and autoregressive modes, the actua sizes correspond we with the nomina sizes. But for the inear trend mode, the actua sizes are somewhat beow the nomina ones. Tabe 2 gives the seected number of breaks via hypothesis testing with size α"0.05. The test sup LR(#1) is repeatedy performed by increasing the vaue of, unti the nu hypothesis is accepted. For each sampe, the estimated number of breaks, ml, is equa to the number of rejections. Tabe 2 shows the distribution of ml out of 5000 simuated sampes. For a three modes, over 90% of the time, the method eads to correct identi"cation of the number of breaks. We now consider the reative performance of the LR test with Bai and Perron's Conditiona Test sup F (#1). In the atter test, the (#1)th break point is estimated assuming the "rst break points are given (estimated in previous steps). The imiting distribution of the conditiona test is given in Tabe 1 Percentage rejections of sup LR(#1) (5000 repetitions) H m"0 m"1 m"2 H m"1 m"2 m"3 Size Mode (I) (II) (III)

13 J. Bai / Journa of Econometrics 91 (1999) 299} Tabe 2 Distribution of the estimated number of breaks based on sup LR(#1) (5000 repetitions) ml *4 Mode (I) (II) (III) Proposition 7 of Bai and Perron (1998). To assess the reative performance of the conditiona and LR procedures, Monte Caro experiments are conducted using the same data generating process described earier. Tabes 3 and 4 report the resuts associated with the conditiona test. These tabes show that the conditiona procedure aso does a satisfactory job. In comparion with Tabes 1 and 2, the sup LR test shows an improved preformance. Taken together, the sup LR test has reasonabe size and power properties. Its performance in determining the number of breaks is quite satisfactory. An added attraction ies in its feasibity and ease of use. 6. Discussion 6.1. Partia structura change Thus far, we aow a regression coe$cients to vary. That is, a coe$cients are re-estimated whenever a break occurs. Such a setup is caed fu structura change. If it is known that some of the coe$cients do not change, the constraint of no change shoud be imposed. This wi increase the e$ciency of the estimated regression parameters as we as the power of hypothesis testing. This setup is caed partia structura change. A genera partia-change mode is described by y "x β#z δ #u, k #1)t)k, for i"1,2, m#1. The coe$cient β is constant for the whoe sampe. Bai and Perron (1998) deveop a fast-computing agorithm for partia change modes. Under some genera conditions for x, z, as described in Bai and Perron (1998), a preceding resuts hod. In particuar, even though x contains trending regressors, Theorem 1 hods as ong as z is regime-wise asymptoticay stationary. That is, the presence of regressor x does not ater the theoretica resuts in Theorems 1}3. These caims can be proved rigorousy at a greater technica expense.

14 312 J. Bai / Journa of Econometrics 91 (1999) 299}323 Tabe 3 Percentage rejections of the conditiona test (5000 repetitions) H m"0 m"1 m"2 H m"1 m"2 m"3 Size Mode (I) (II) (III) Tabe 4 Distribution of the estimated number of breaks based on the conditiona test (5000 repetitions) ml *4 Mode (I) (II) (III) It is noted that the tests of this paper are constructed using the fu sampe, which can take advantage of partia changes. This is important particuary for mutipe changes because the savings in the number of degrees of freedom can be substantia. Sampe-spitting methods frequenty used in appied work are unabe to make use of partia changes Mutipe-threshod time series modes The procedure proposed here may be extended to threshod time series modes (Tong, 1990). One interesting topic is to identify the number of threshods. To date, the determination of the number of threshods in a mode is based on a ess forma approach. This is usuay accompished by potting and identifying the turning points in the cusum of recursive residuas. Hansen (1996) recenty suggests a sampe-spitting method to study mutipe-threshod modes. By arranging the data in an appropriate way, the threshod time series mode can be turned into a structura-change probem, at east computationay. Thus an LR-type test statistic for testing threshods versus #1 threshods can be easiy constructed. The underying theoretica distribution, however, is di!erent from the one given here. Chan (1991) considers the nu of inearity against a singe threshod. In view of the resut of this paper and that of Chan,

15 J. Bai / Journa of Econometrics 91 (1999) 299} conjectures can be made about the imiting distribution of the LR type test for mutipe threshods. However, the underying theoretica pinning needs a more coser examination. Acknowedgements This paper has bene"ted from seminar presentations at the MIT/Harvard Econometrics workshop as we as the 1996 ASSA meetings in New Oreans. Partia "nancia support is acknowedged from the Nationa Science Foundation under Grants SBR and SBR Appendix: Proofs Under the nu hypothesis of changes, (kk, 2, kk ) is ¹ consistent for (k,2, k ) (see Bai and Perron, 1998). When there are actuay breaks but #1 are aowed in the estimation, we sha show that of them wi be ¹ consistent. More speci"cay, et (kk *,2, kk * ) denote the estimated #1 break points. There exists a subset i, i,2, i of 1, 2,2, #1 such that kk *!k"op (1) (v"1,2, ). emma A.1. et Assumptions A1}A4 hod. ;nder the nu hypothesis of breaks, there are and exacty of kk *,2, kk * that wi be ¹ consistent for (k,2, k ). ¹hat is, kk *!k"op (1) for some i,2, i L1,2,2, #1. Proof. We"rst prove consistency. That is, there exists a subset i,2, i of 1, 2,2, #1 such that for every ε'0, we have P(kK *!k 'ε¹)( ε (v"1,2, ) for a arge ¹. Suppose such a subset does not exist, it then impies that there exists a break point which cannot be consistenty estimated. Then, from the resut of Bai and Perron (1998, Lemma 2) u!s (kk *,2, kk * )"!O (¹)P!R. This contradicts with the east squares principe that u!s (kk *,2, kk * )* u!s (kk, 2, kk )*0. Next suppose, without oss of generaity, that kk *,2, kk *, kk *,2, kk * are consistent. This impies that kk * is cose to k, and kk * is cose to k. By the

16 314 J. Bai / Journa of Econometrics 91 (1999) 299}323 de"nition of Λ, kk * cannot be cose to either k or k. That is, there exists ε '0 such that with arge probabiity kk *!k *ε ¹, and k!kk **ε ¹. This impies that the subsampe [1, kk *] contains i!1 true break points. If we use this subsampe to estimate the true break points, the soution coincides with (kk *,2, kk * ). Since there are i!1 true break points, and we aow i!1 breaks in the estimation, by the resut of Bai and Perron (1998), the resuting estimator (kk *,2,kK * )is¹-consistent for (k,2, k ). A simiar argument appies to the subsampe [kk *,¹]. And so kk *,2, kk * are ¹-consistent for k,2, k. emma A.2. et Assumptions 1}4 and H hod. et j,2, j be integers (may be negative). For every M(R, uniformy over j )M (i"1,2, ) we have S (k#j,2, k #j )!S (k,2, k )! = ( j )"o (1), where for i"1,2,, = (0)"0 and = ( j)" (δ!δ ) z z (δ!δ )!2(δ!δ ) z u, j"!1,!2, 2 (δ!δ ) z z (δ!δ )#2(δ!δ ) z u, j"1, 2, 2 Proof. To be concrete, we sha anayze the case of j '0 for i"1,2,. Other cases can be anayzed simiary. Let (δk, 2, δk ) be the estimated regression parameters corresponding to the partition (k#j,2, k #j ). The resuting SSR for the tota sampe can be written as the summation of the SSRs of 2#1 segments, with each segment invoving observations from a singe true regime ony. More speci"cay, S (k#j,2, k #j )" [(y!z δk )# (y!z δk ), (A.1) with k"0 and j "0. Simiar to the above, we can express S (k,2, k ) as the summation of SSR's of the same (2#1) segments. Let (δk,2,δk ) be the Notice the fact that, min S (k,2, k, kk *, k,2, k ),S (kk *, 2, kk * ). 2 2

17 J. Bai / Journa of Econometrics 91 (1999) 299} estimated regression parameters based on the partition (k,2, k ). Then S (kk,2, kk ) is given by Eq. (A.1) with the "rst δk repaced by δk and the second repaced by δk (not δk because of no misspeci"cation in regime spes). Thus, S (k#j,2, k #j )!S (k,2, k )" [(y!z δk )!(y!z δk )]# [(y!zδk )!(y!z δk )]. (A.2) Because j is bounded for each i, a estimated regression parameters are root ¹ consistent for the true regression parameters. This, together with the boundedness of j, aows us to write the second term of the right hand side of Eq. (A.2) as [u #z(δ!δ )]!u#o (1)" = ( j )#o (1). This foows from y!z δk "u!z (δk!δ )"u!z (δ!δ )!z (δk!δ )" u!z (δ!δ )#z O (¹). Simiary, y!z δk "u!z O (¹). Next, we sha argue that the "rst term on the right-hand side of Eq. (A.2) converges to zero in probabiity. The said term is equa to!2 (δk!δk ) (z z)(δk!δk ). z u # [(δk!δ )#(δk!δ )] It is not di$cut to prove that δk!δk "O (1/¹) for a i (foows from the boundedness of j ). Using this fact we see that each of the two expressions above is O (¹). The next emma states that when an additiona break is aowed in estimation, Lemma 2 shoud be modi"ed by adding an extra term, which wi determine the imiting distribution of the test statistic. Since exacty estimated breaks wi be consistent by Lemma 1, one of them wi not be consistent. The ocation of this inconsistent break point can be in any of the (#1)-segments separated by the -consistent ones. The foowing emma gives the asymptotic behavior of the SSR when an extra break point (denoted by h) is in the ith segment

18 316 J. Bai / Journa of Econometrics 91 (1999) 299}323 (i"1, 2,2, #1). In the remainder of this paper, we use D(k, k ) to denote the sum of squared residuas for the subsampe [k #1, k ]. That is, D(k, k )"min (y!z δ). emma A.3. et Assumptions 1}4 hod. ;nder the assumption of breaks, we have S (h, k#j,2, k #j )!S (k,2, k )! = ( j )#Γ (h)"o (1), S (k#j, h, k#j,2, k #j )!S (k,2, k )! = ( j )#Γ (h)"o (1), S (k#j,2, k #j, h) where!s (k,2, k )! = ( j )#Γ (h)"o (1), Γ (h)"d(k, k )!D(k, h)!d(h, k ) (A.3) and the o (1) is uniform in j )M and in h3< for equation i, with < "h: h3[k #j #π¹, k #j!π¹]. (A.4) So for h3<, we have (k #j,2, k #j, h, k #j,2, k #j )3Λ. We note that Γ (h) is the di!erence between the restricted and unrestricted sums of squared residuas (aowing a break at h) for the subsampe [k #1, k ]. From the standard resut, if j "0 and j "0in<, then sup Γ (h)/σp ξ, (A.5) where ξ is de"ned in Eq. (5). When j and j are non-zero but bounded, the imiting distribution is the same. Lemma A.3 can be put in a more compact form, for i"1,2, #1 S (k #j,2, k #j, h, k #j,2, k #j )!S (k,2, k )" = ( j )!Γ (h)#o (1), (A.6)

19 J. Bai / Journa of Econometrics 91 (1999) 299} where for i"1, the "rst term on the eft-hand side shoud be understood as S (h, k #j,2, k #j ), and for i"#1, as S (k #j,2, k #j, h). Proof of emma. A.3. Again, consider the case of j *0 for a s. We sha prove the second equation of Lemma A.3. The rest are simiar (the "rst and ast are actuay simper). Note that S (k #j, h, k #j,2, k #j )"D(0, k #j )# D(k #j, h)#d(h, k #j )# D(k i #j, k ). By adding and subtracting D(k #j, k #j ), we have S (k #j, h, k #j,2, k #j )"D(k #j, h)#d(h, k #j )!D(k #j, k #j ) #S (k #j,2, k #j ). The ast term does not depend on h. By Lemma A.2, a we need is D(k #j, h)#d(h, k #j )!D(k #j, k #j ) "D(k, h)#d(h, k )!D(k, k )#o (1), (A.7) where o (1) is uniform in h3<. This resut wi impy that the test statistic based on the sampe [k #j, k #j ] wi be asymptoticay equivaent to the test based on the sampe [k,k ]. This shoud be obvious given the boundedness of j and j. Beow is the forma proof. Note that Eq. (A.7) is impied by the foowing two equations: D(k#j, h)!d(k#j, k#j )"D(k, h)!d(k, k#j )#o (1), (A.8) D(h, k #j )!D(k, k #j )"D(h, k )!D(k, k )#o (1) (A.9) because adding Eqs. (A.8) and (A.9) yieds Eq. (A.7). Consider Eq. (A.8). Let δk and δk be estimators of δ [k#1, h] and [k#1, k#j ], respectivey. Then using sampes D(k, h)" (y!zδk )# (y!zδk ) (A.10)

20 318 J. Bai / Journa of Econometrics 91 (1999) 299}323 and D(k, k #j )" (y!zδk )# (y!zδk ). (A.11) Since both δk and δk are consistent for δ, and because j is bounded, the di!erence between the two "rst terms of Eqs. (A.10) and (A.11) is o (1), and therefore, D(k, h)!d(k, k#j )" (y!zδk )! (y!zδk )#o (1). (A.12) Let δk be the estimator of δ based on the sampe [k#j #1, h]. Then D(k#j, h)" (y!z δk ). Because j is bounded, and because h!k*c¹ for some c'0, δk!δk "O (¹). This impies that the second term of Eq. (A.10) is D(k#j, h)#o. [See the proof of the "rst expression of Eq. (4).] Simiary, the second term of Eq. (A.11) is D(k#j, k#j )#o.in view of Eq. (A.12), we obtain Eq. (A.8). The proof of Eq. (A.9) is simiar. Proof of ¹heorem 1. If kk!k )M for a i, then S (kk, 2, kk )"min 2 S (k#j,2, k #j ), here and in what foows, the minimization with respect to j,2, j is taken over j )M, i. By Lemma A.2, min S (k#j,2, k #j )!S (k,2, k )" min = (j)#o (1). 2 (A.13) If kk *!k )M (v"1,2, ), then S (kk *,2, kk * )" min min S (k#j,2, k 2 _ #j, h, k#j,2, k #j ), where < is de"ned in Eq. (A.4). Note that for h3<, we have (k #j,2, k #j, h, k #j,2, k #j )3Λ. The above says that the

21 J. Bai / Journa of Econometrics 91 (1999) 299} ocation of the inconsistent break point wi be in the segment where the sum of squared residuas is reduced the most. Lemma A.3 is equivaent to min min S (k#j,2, k#j, h, k #j,2, k #j ) 2 _!S (k,2, k )" min min = ( j )!Γ (h) 2 _ #o (1). (A.14) Let <M "k: k3[k #π¹!m, k!π¹#m] and < "k: k3[k # π¹#m, k!π¹!m]. Then < L< L<M. Neither < nor <M depends on the js. We have min min = ( j )!Γ (h) 2 _ * min = ( j)#min and, repacing <M by <, min min = ( j )!Γ (h) 2 _ ) min = ( j)#min min!γ (h) (A.15) M min!γ (h) (A.16) Adding and subtracting S (k,2, k ) and using Eqs. (A.13), (A.14) and (A.15), we obtain S (kk, 2, kk )!S (kk *,2, kk * ))! min min!γ (h)#o (1), M (A.17) provided that kk!k )M and kk *!k )M. Simiary, repacing <M by <,we obtain S (kk, 2, kk )!S (kk *,2, kk * )* max max Γ (h)#o (1) (A.18)

22 320 J. Bai / Journa of Econometrics 91 (1999) 299}323 provided that kk!k )M and kk *!k )M. By the resut of Bai and Perron (1998), P(i s.t. kk!k 'M)(ε and by Lemma A.1, P(ν s.t. kk *!k 'M) (ε for arge M. These resuts and Eq. (A.17) impy that P(S (kk, 2, kk )!S (kk *,2, kk * )/σ*x) )2ε#P max max M Γ (h)/σ#o (1)*x (A.19) From Eq. (A.5), max max M Γ (h)/σ#o P max ξ. It foows from Eq. (A.19) that (because ε is arbitrary and σl (#1) P σ) im P(sup LR (#1)*x))P max ξ *x. (A.20) Simiary, using Eq. (A.18) and noting that Eq. (A.5) hods when < is repaced by <, we obtain im P(sup LR (#1))x))P max Inequaities Eqs. (A.20) and (A.21) impy ξ )x. (A.21) im P(sup LR (#1)*x)"P max ξ *x. This proves Theorem 1. Proof of ¹heorem 3. The more demanding part of the proof is the same as that of Theorem 1. In particuar, the proof of Eqs. (A.17) and (A.18) needs no change. It remains to show that sup Γ (h)/σp ψ, where ψ is speci"ed in the theorem, and Γ (h)"d(k, k)!d(k, h)!d(h, k), and the supremum with respect to h is taken over the set <M or <. We note that Γ (h) is simpy the di!erence between the restricted sum of squared residuas and the unrestricted sum of squared residuas (aowing for one break at h), computed for the segment [k #1, k ]. We consider the case of i"1. Other cases are the same. Write N for k. Let x "(w, z ), and X "(x, x,2, x ) (h"1,2, N). Let ; "(u,2, u ) (h"1,2,n). Then D(0, N)!D(0, h)!d(h, N)"< (X X!(X X )(X X )(X X ))<,

23 J. Bai / Journa of Econometrics 91 (1999) 299} where < "X;!(X X )(X X )X ;. Let B "diag(n, N,2, N, NI ), where I is the qq identity matrix. Because a constant is incuded in the regressor, we may assume Ez "0. Then B (X X )B P diag(a, sr ),Q, where R "pim Nz z and A "A(s)isde"ned in the text. Furthermore, B X ; Nσ=( ) ), where =(s) is a Gaussian process with E=(s)=(r)"Q. Let <(s)"=(s)!q Q=(1). We have D(0, N)!D(0, [Ns])!D([Ns], N)Nσ<(s)(Q!Q Q Q )<(s) "σ<*(s)(qq Q!(QQ Q))<*(s), (A.22) where <*(s)"q<(s). The variance}covariance matrix of <*(s) is QQ Q!(QQ Q). Let λ (s),2, λ (s) be the eigenvaues of AA A (they are chosen to be continuous and increasing in s), and et λ (s)"s for i"1,2, q. Then there exists an orthogona matrix C,suchthat C QQ QC"diag(λ (s), i"1,2, p#1#q). Thus C <*(s) isavectorof Gaussian processes with E[C <*(s)<*(r)c ]"diag[λ (sr)!λ (s)λ (r); i"1,2, p#1#q]. The ast q components of C <*(s) are standard Brownian bridges, and the "rst p#1 components are time-scaed Brownian bridges. Thus the imiting distribution in Eq. (A.22) has the foowing representation: B (λ (s)) λ (s)(1!λ (s)) # B (s) s(1!s), (A.23) where the B ( ) ) are standard Brownian bridges on [0, 1]. Finay note that, the smaest h in < (or in <M ) satis"es h/npη, and the argest h satis"es h/np1!η. This, together with Eq. (A.23), yieds the representation of ψ given in the theorem. The independence of ψ,2, ψ foows from the fact that they are derived from non-overapping segments. Proof of Coroary 2. We ony need to "nd the expression for H (c)"p(ψ 'c). Let t " og(λ (s)/(1!λ (s)). Using the argument of Chan (1991), we have P(ψ 'c)+1!exp!2χ (c) c p#1#q!1 dt ds ds, where χ (c) is the density function of a Chi-square random variabe with p#1#q degrees of freedom. From exp(!x)+1!x for sma x,

24 322 J. Bai / Journa of Econometrics 91 (1999) 299}323 and we have dt " 1 2 og λ (1!η )(1!λ (η )) λ (η )(1!λ (1!η )), P(ψ 'c)+2χ (c) c p#1#q!1 1 2 og λ (1!η )(1!λ (η )) λ (η )(1!λ (1!η )) #qog1!η η, since the ast q eigenvaues are identity functions. Using the fact that λ (s)2λ (s)"det(aa A)"det(A )deta(s) and (1!λ (s))2 (1!λ (s))"det(i!a A A )"det(a )det[a!a(s)], and substituting the density function for χ (c), we obtain the expression of H (c) given in the coroary. References Andrews, D.W.K., Testing for structura instabiity and structura change with unknown change point. Econometrica 61, 821}856. Andrews, D.W.K., Lee, I., Poberger, W., Optima changepoint tests for norma inear regression. Journa of Econometrics 70, 9}38. Bai, J., Perron, P., Estimating and testing for mutipe structura changes in inear modes. Econometrica 66, 47}78. Banerjee, A., Lumsdaine, R.L., Stock, J.H., Recursive and sequentia tests of the unit root and trend break hypothesis: theory and internationa evidence. Journa of Business and Economic Statistics 10, 271}287. Broemeing, L. (Ed.), Structura change in econometrics. Journa of Econometrics 19, 1}182. Chan, K.S., Percentage points of ikeihood ratio tests for threshod autoregression. Journa of Roya Statistica Society, Series B 53, 691}696. Chow, G.C., Tests of equaity between sets of coe$cients in two inear regressions. Econometrica 28, 591}605. DeLong, D.M., Crossing probabiities for a square root boundary by a Besse process. Communications in Statistics-Theory and Methods A 10 (21), 2197}2213. Diebod, F.X., Rudebusch, G.D., Measuring business cyces: a modern perspective. NBER Working paper, No Dufour, J.M., Ghyses, E. (Eds.), Recent deveopments in the econometrics of structura change. Journa of Econometrics, 70, 1}316. Dufour, J.M., Kiviet, J.F., Exact tests for structura change in "rst-order dynamic modes. Journa of Econometrics 70, 39}68. Hack, P.D., Westund, A.H., (Eds.), Economics Structura Change: Anaysis and Forecasting. Springer-Verag, New York.

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