Tests for Multiple Breaks in the Trend with Stationary or Integrated Shocks

Transcription

1 ests for Mutipe Breaks in the rend with Stationary or Integrated Shocks Nuno Sobreira and Luis C. Nunes Juy 24, 213 Abstract In this paper, we propose new tests of the presence of mutipe breaks in the trend of a univariate time-series where the number and dates of the breaks are unknown and that are vaid in the presence of stationary or unit root shocks. hese tests can aso be used to sequentiay estimate the number of breaks. he behavior of the proposed tests is studied through Monte Caro experiments. We iustrate the appicabiity of the proposed tests to ong historica time series of various U.S. macroeconomic time series. Keywords: Structura Change, rend Breaks, Stationarity, Unit Root. JEL: C12, C22. We are gratefu to Iiyan Georgiev, Pauo M. M. Rodrigues, Luis F. Martins, Vasco Gabrie and participants in seminars at Universiteit Van Amsterdam and Nova Schoo of Business and Economics, in the QED Conference Amsterdam, May 29, in the Econometric Society European meeting joint congress with the European Economic Association; Barceona, August 29 and in the XXXV Simposio de a Asociación Españoa de Economía Madrid, December 21 for hepfu comments and suggestions on earier versions of this paper. We aso thank David Harvey and Mohitosh Kejriwa for providing us with their Gauss programs. Financia support from Fundação para a Ciência e ecnoogia is aso acknowedged. Insper, Rua Quatá, 3, São Pauo, Brazi. Emai: nsobreira@insper.edu.br. Nova Schoo of Business and Economics, Campus de Campoide, P Lisboa, Portuga. Emai: cnunes@novasbe.pt 1

2 1. Introduction Many macroeconomic time series are characterized by a cear tendency to grow over time, that is, as having a deterministic time trend component. here has been a arge debate in the iterature regarding the appropriate methods to infer about the inearity and stabiity of the trend function and the nature of the shocks affecting a time series. his is a particuary important issue when it comes to make accurate economic forecasts or test economic hypotheses. In fact, there are many interesting economic appications that invove statistica inference on the parameters of the trend function, namey, in the continuous time macroeconomic modeing see Bergstrom et a., 1992, Nowman, 1998, in internationa trade, for exampe, with the Prebish-Singer hypothesis testing see Bunze and Vogesang, 25, in the empirica debate regarding regiona convergence in per capita income see Sayginsoy and Vogesang, 24, or in environmenta economics on the future consequences of goba warming see Vogesang and Franses, 25. he stationarity properties of the shocks have important impications on the appropriate methods to make inferences about the trend function. In particuar, the correct approach to make inferences about the stabiity or the existence of breaks in the trend depends on whether the shocks are I or I1. In the first case one shoud use regressions on the eves, whie for the atter the correct approach is to mode the first-differences of the series. However, it is often not known a priori whether the shocks are stationary or contain a unit-root. Moreover, stationarity or unit-root tests aso suffer from simiar probems since their properties are in turn affected by the stabiity of the trend function. Ony recenty some soutions to this diemma have been proposed in the iterature. hese resort to statistica tests of the nu hypothesis of a constant inear trend against the aternative of one break at some unknown date that do not require a priori knowedge of whether the noise is I or I1. Sayginsoy and Vogesang 24 proposed a Mean Wad and a Sup Wad statistic scaed by a factor based on unit root tests to smooth the discontinuities in the asymptotic distributions of the test statistics as the errors go from I to I1. he scaing factor approach is based on Vogesang 1998 who proposed test statistics for genera inear hypothesis regarding the parameters of the trend function which do not require knowedge as to whether the innovations are I or I1. Perron and Yabu 29 proposed a Feasibe Quasi Generaized Least Squares approach to estimate the sope of the trend function. By truncating the estimate of the sum of the autoregressive coefficients of the disturbance term to take the vaue of one whenever the estimate is in a neighborhood of one, they have shown that the imiting distribution of the t-statistic becomes Norma regardess of the persistence of the error term. Kejriwa and Perron 21 proposed a sequentia testing procedure based on Perron and Yabu 29. Harvey et a. 29 hereafter HL empoyed a weighted average of the appropriate regression t-statistics used to test the existence of a broken 2

3 trend when the errors are I and I1. However, as Lumsdaine and Pape 1997 point out with an exampe of Jones 1995, aowing for ony one break is not aways the best characterization of a macroeconomic variabe, speciay when anayzing ong historica time series. his paper extends the resuts from HL by providing tests of the nu hypothesis of no trend breaks against the aternative of one or more breaks in the trend sope which do not require knowedge of the form of seria correation in the data and are robust as to whether the underying shocks are stationary or have a unit-root. We buid on the framework proposed by HL for the case of a singe break, and construct test statistics that are weighted averages of the appropriate F-statistics to test the existence of mutipe trend breaks when the disturbance term is I and I 1.We adopt the weight function used in HL and prove that it has the same arge sampe properties regardess of the number of trend breaks being tested. We start by considering the case where the true break fractions are known and prove that the proposed statistics converge in distribution to a chi-square distribution under the nu. Next, we consider the case where the trend break fractions are unknown and need to be estimated. We transform our statistic in the same spirit as Andrews 1993 and Bai and Perron 1998 and take the supremum of the F statistic over a possibe break fractions except those that are activey restricted by the trimming parameter. Here, the weight function is evauated at the estimated break fractions and we prove that its arge sampe behavior is simiar regardess of the number of break fractions estimated and the number of structura breaks in the trend function. However, the asymptotic nu distributions of the appropriate F-statistics for I and I 1 environments are different and so, foowing Vogesang 1998, we provide a scaing factor that makes the asymptotic critica vaues invariant to the degree of persistence of the shocks. Finay, we propose doube maximum tests and a sequentia test procedure that can be used to estimate the number of trend breaks and that are aso robust to the order of integration of the error term. In both the known and unknown break dates settings, our proposed tests are made robust to short memory seria correation in the shocks via the use of standard non-parametric estimators of the ong run variance of the errors. he outine of this artice is as foows. Section 2 describes the broken trend mode, presents the test statistics for both known and unknown break fractions and estabishes the asymptotic behavior of these statistics. he sequentia testing procedure to estimate the number of breaks is aso described. In Section 3 we discuss practica issues reated to the appication of the test statistics proposed, namey, the critica vaues and the choice of the scaing constants. Size and power properties in finite sampes from appying these procedures are aso discussed in this section. Section?? provides an empirica appication to various U.S. macroeconomic time series data. Section 4 concudes the paper with a discussion of some issues raised by our anaysis and suggests possibe paths for future 3

4 research. A our key resuts are proved in a Mathematica Appendix. 2. he Broken rend Mode We consider a time-series process {y t } with a first-order inear trend and m possibe time changes simutaneousy in the eve and in the sope: m m y t = α + βt + δ j DU t τ j + γ j D t τ j + ut, t = 1,..., 1 j=1 j=1 and u t = ρu t 1 + ε t, t = 2,...,, u 1 = ε 1, 2 where DU t τ j := 1 t > j and Dt τ j := 1 t > j t j capture, respectivey, the eventua j th break in the eve and in the sope occurring at date j := τj with associated break fraction τj, 1 and < τ1 <... < τm < 1. he sope coefficient changes from β to β + γ 1 and the eve shifts from α to α + δ 1 at time 1. At break point 2 the sope coefficient changes from β + γ 1 to β + γ 1 + γ 2 and the eve goes from α + δ 1 j 1 to α + δ 1 + δ 2. Generay, in period j the sope coefficient changes from β + γ i to j 1 j β + γ i whie the eve shifts from α + δ i to α + i=1 function is discontinuous at a break date j if δ j. i=1 i=1 j δ i for j = 1,..., m. he trend We assume that ε t in 2 satisfies Assumption 1 of Sayginsoy and Vogesang 24, pp. 2-3: Assumption 1. he stochastic process ε t is such that: i=1 ε t = CLη t, C L = c i L i i= with C1 2 > and i c i <, and where η t is a martingae difference sequence with i= unit conditiona variance and sup t E η 4 t <. he error term u t can have one unit root or none. If ρ < 1, u t is an I process. But if ρ = 1 then u t turns out to be an I1 process. We are interested in testing if there are trend breaks in y t and in estimating the number of breaks in the time series process, independenty of whether u t is I or I1. herefore, we woud ike to test the nu hypothesis H : γ 1 = γ 2 =... = γ m = against the two sided aternative: H 1 : γ 1 γ 2... γ m. 4

5 2.1. Known Break Fractions We start by considering the case where the number of breaks m and the vector of true break fractions τ = τ 1, τ 2,..., τ m are known. Simiary to HL, we partition H 1 into two oca aternatives H 1, : γ = κ 3/2 when u t is I and H 1,1 : γ = κ 1/2 when u t is I 1 where γ = γ 1, γ 2,..., γ m and κ is a k-dimensiona vector of finite constants, κ = κ 1, κ 2,..., κ m. Suppose one knows that u t is I, with ρ < 1. hen, to test the nu hypothesis H, we shoud use the F-statistic from the regression in eves which we denote by Ϝ τ 1 : [ ] 1 Ϝ τ = γ τ ω2 τ x D,t τ x D,t τ t=1 m+3:2m+2,m+3:2m+2 1 γ τ /m where x D,t τ := {1, t, DU t τ 1,..., DU t τ m, D t τ 1,..., D t τ m}, γ τ are the estimators for γ based on OLS regression of 1 and ω 2 τ is the Newey West estimator of the ong-run variance based on the corresponding OLS residuas û t Newey and West, 1987: ω 2 τ := γ τ j/ + 1 γ j τ, γ j τ = 1 û t τ û t j τ, j=1 with ag truncation = O 1/4. t=j+1 Now suppose that u t is known to be I 1, with ρ = 1 such that u t is I. test if the sope of the trend function is constant against the aternative of m breaks we shoud use the F-statistic after differentiating the data. So by appying first-differences to equation 1 we have: y t = β + m m γ j D t τ j + γ j DU t τ j + vt, t = 2,..., 5 j=1 j=1 where D t τ j := 1 t = j + 1 and v t = u t. he Ϝ 1 τ statistic is then given by: [ ] 1 Ϝ 1 τ = γ τ ω2 τ x DU,t τ x DU,t τ t=2 m+2:2m+1,m+2:2m+1 1 γ τ /m where x DU,t τ := {1, D t τ 1,..., D t τ m, DU t τ 1,..., DU t τ m}, γ τ are the estimators for γ from the OLS regression of equation 5 and ω 2 τ is the Newey West 1 he notation [.] i:j,i:j [.] j is used to denote a submatrix scaar formed by rows and coumns i unti j the j th eement from the matrix vector within the squared brackets 3 4 o 6 5

6 estimator for the ong run variance based on the corresponding OLS residuas ṽ t : ω 2 τ := γ τ +2 1 j/ + 1 γ j τ, γ j τ = 1 1 ṽ t τ ṽ t j τ, j=1 t=j+1 We now estabish the asymptotic distribution of the Ϝ τ and Ϝ 1 τ statistics under H 1, and H 1,1 with δ unrestricted: heorem 1. Let the time series process be generated by 1 and 2, and et Assumption 1 hod. i If u t is I ρ < 1 then, under H 1, : a Ϝ τ, where O p 7 d 1 m J τ, κ, and b Ϝ 1 τ = J τ, κ:= { κ + ω u Q τ 1 V τ } [ ] Q τ {κ + ωu Q τ 1 V τ }, Q τ = 1 ω 2 u R r, τ R r, τ dr, V τ = ii If u t is I1 ρ = 1 then, under H 1,1 : a Ϝ τ = O p 1 m J 1 τ, κ, where 1 R r, τ dw r, and b Ϝ 1 τ J 1 τ, κ := { κ + ω ε Q 1 τ 1 V 1 τ } [ ] Q1 τ {κ + ωε Q 1 τ 1 V 1 τ }, Q 1 τ = 1 ω 2 ε RU r, τ RU r, τ dr, V 1 τ = 1 RU r, τ dw r with R r, τ = R r, τ 1, R r, τ 2,..., R r, τ m where R r, τ i is the continuous time residua from the projection of r τ i 1 r > τ i onto the space spanned by {1, r} and RU r, τ = RU r, τ 1,..., RU r, τ m where RU r, τ i is the continuous time residua from the projection of 1 r > τ i onto {1}. Remark 1. It is straightforward to show that J τ, κ and J 1 τ, κ correspond to a noncentra chi-square distribution with m degrees of freedom and non centraity parameter d κ [ Q τ /ωu] [ 2 κ and κ Q 1 τ /ωε] 2 κ, respectivey. Furthermore, from heorem 1 we concude that, under H : γ = m 1 or κ = m 1, we have m Ϝ τ d χ 2 m if u t is I and aso m Ϝ 1 τ d χ 2 m if u t is I1. If we knew a the true potentia break dates and aso the order of integration of the error term u t, we coud use the appropriate F-statistic to test if the potentia m changes in sope are statisticay significant or not using critica vaues from the chi-square distribution with m degrees of freedom. Aso note that in the 6

7 particuar case of ony one break, m = 1, heorem 1 is basicay equivaent to heorem 1 in HL by the equivaence between the F-statistic and the squared t-statistic when testing ony one coefficient. Remark 2. From the resuts of part i of heorem 1 it is seen that when u t is I, Ϝ 1 τ converges in probabiity to zero, regardess of the vaue of κ. Simiary, from the resuts in part ii of heorem 1 it is seen that when u t is I1, Ϝ τ diverges irrespective of the vaue of κ. Since, in practice, the order of integration is not known we woud ike to find a procedure that, at east asymptoticay, converges to the asymptotic distribution of Ϝ τ when u t is I and to the asymptotic distribution of Ϝ 1 τ when u t is I1. More p specificay, we woud ike to find a weight function, ca it λ., such that λ. 1 if p u t is I and λ. if u t is I1 ensuring that the appropriate statistic with nondegenerate distribution is seected. We empoy the soution proposed by HL and et λ. be a function of the KPSS statistic of the origina data S τ and of the differenced data S 1 τ : t S τ t=1 i=1 := ûi τ 2 2 ω, S 2 τ 1 τ := Lemma 1. Let the conditions of heorem 1 hod: t=2 t i=1 ṽi τ ω 2 τ 8 i If u t is I, then: a S τ = O p 1, and b S 1 τ = O p /. ii If u t is I1, then: a S τ = O p /, and b S 1 τ = O p 1. Note that the product of the two KPSS statistics, S τ S 1 τ, is O p / when u t is I and is O p / when u t is I1. Since by Lemma 1, S τ and S 1 τ, have the same asymptotic rates of convergence independenty of the number of trend breaks up to m breaks we can use the same weight function from HL: λ S τ, S 1 τ := exp [ {gs τ S 1 τ } v ] 9 where g and v are positive constants. Now we are abe to form the Ϝ λ statistic and study its asymptotic distribution: Ϝ λ τ := {λ S τ, S 1 τ Ϝ τ } + {[1 λ S τ, S 1 τ ] Ϝ 1 τ } 1 Notice that a higher g gives more weight to Ϝ 1 keeping everything ese constant. Using the resuts from heorem 1 and Lemma 1, we get the foowing resut. Coroary 1. Let the conditions of heorem 1 hod. 7

8 i If u t is I, then: λ S τ, S 1 τ p 1 under both H and H 1,, and Ϝ λ τ = Ϝ τ + o p 1 d 1 m J τ, κ; ii If u t is I1, then: λ S τ, S 1 τ Ϝ 1 τ + o p 1 d 1 m J 1 τ, κ; with J, and J 1, as defined in heorem 1. p, under both H and H 1,1, and Ϝ λ τ = Remark 3. From Coroary 1 we observe that we have constructed a test statistic to test the presence of m candidate trend breaks at known break dates that is vaid regardess of the order of integration of the errors. If u t is I, Ϝ λ τ is asymptoticay equivaent to Ϝ τ, whie if u t is I1, Ϝ λ τ becomes asymptoticay equivaent to Ϝ 1 τ. Since, given these conditions, both m Ϝ τ and m Ϝ 1 τ converge in distribution to a chisquare distribution with m degrees of freedom under the nu we can use the critica vaues of the centra chi-square distribution for Ϝ λ τ irrespective of whether the disturbances, u t, are I or I Unknown Break Fractions In this section, we consider tests of mutipe structura changes in the trend function with unknown change points. Suppose that the true break fractions τ are unknown but the number of breaks, m, is known. Proceeding in the same way as Andrews 1993 and Bai and Perron 1998 we can form F type statistics to test the nu hypothesis of no trend breaks against the aternative hypothesis that there are m trend breaks. Let τ m := τ 1,..., τ m and Λ m = {τ 1,..., τ m : τ i+1 τ i η, τ 1 η, τ m 1 η} and assume throughout that τ Λ m. If we knew that u t was I the F-statistic woud be defined as: Ϝ m := and if we knew that u t was I1 the statistic woud be given by: Ϝ 1 m := sup Ϝ τ m 11 τ m Λ m sup τ m Λ m Ϝ 1 τ m, 12 where the associated vectors of estimated break fractions of τ are given by and τ m := arg sup τ m Λ m Ϝ τ m 13 τ m := arg sup τ m Λ m Ϝ 1 τ m, 14 respectivey, such that Ϝ m = Ϝ τ m and Ϝ 1 m = Ϝ 1 τ m. o sove the probem of an unknown order of integration of the error term we foow the same strategy as in 8

9 the known break fraction case and write the anaogue of the Ϝ λ τ statistic: Ϝ λ m := {λ τ m, τ m Ϝ m } + b m ξ {[1 λ τ m, τ m ] Ϝ 1 m } 15 where λ τ m, τ m := λ S τ m, S 1 τ m and b m ξ is a positive finite constant such that, as wi be expained beow, for any significance eve ξ, the critica vaue of Ϝ λ m is the same regardess of whether u t is I or I1. he foowing heorem states the asymptotic distribution of Ϝ m and Ϝ 1 m under the nu hypothesis γ = when the innovation sequence {u t } is either I or I1. heorem 2. Let the time series process be generated by 1 and 2 under H : γ = δ = m 1 and et Assumption 1 hod. i If u t is I, then: a Ϝ m ii If u t is I1, then: a Ϝ m = O p with J, and J 1, as defined in heorem 1. d 1 m sup J τ m,, and b Ϝ 1 m = O p τ m Λ m, and b Ϝ 1 m ; d 1 m sup τ m Λ m J 1 τ m, ; Remark 4. Athough our objective is ony to test for changes in sope, we have to set additionay δ = in order to obtain a pivota imiting nu distribution for our test statistic. Hence, as in HL the nu hypothesis must be restated as H : γ = δ =. Remark 5. HL estabished the divergence rates for the 1 break case under a fixed aternative H 1 : γ using sup t instead of sup Ϝ statistics. Since Ϝ i τ 1 = t i τ 1 2 and Ϝ i τ 1 2Ϝ i τ 1, τ 2 mϝ i τ 1,..., τ m, i =, 1, the consistency of Ϝ and Ϝ 1 foow immediatey from heorem 3 from HL. Next, we estabish the arge sampe behavior of the weight function λ S τ m, S 1 τ m. For this purpose, we need to know the asymptotic behavior of the KPSS statistics S τ m and S 1 τ m when the disturbances u t are either I or I1 and the vector of break points, τ, is estimated, i.e., for the cases τ m = τ m and τ m = τ m. Lemma 2. Let the conditions of heorem 1 hod. i If u t is I, then: a S τ m = O p 1, and b S 1 τ m = O p /. ii If u t is I1, then: a S τ m = O p /, and b S 1 τ m = O p 1. From Lemma 2 it is seen that the resuts from Lemma 1 are unchanged and so the arge sampe behavior of the KPSS statistics is the same regardess of whether the trend break dates are known or unknown. We conjecture that Lemma 2 hods independenty of assuming the nu hypothesis H : γ 1 = γ 2 =... = γ m = or the aternative H 1 : γ j, 9

10 j = 1,..., m, as shown in HL for the 1 break case. his impies that we can continue to use the same λ. function as defined above for the case of known break dates since if u t is I then λ τ m, τ m p 1 whie if u t is I1 we have λ τ m, τ m p, under both H and H 1, and so the F statistic that we woud ike to be chosen depending on the order of integration of u t is actuay seected asymptoticay. herefore we can state the foowing coroary: Coroary 2. Let the conditions of heorem 2 hod. i If u t is I, then: Ϝ λ m = Ϝ m + o p 1 ii If u t is I1, then: Ϝ λ m = b m ξ F 1 m + o p 1 d 1 m sup τ m Λ m J τ m,. d b m ξ 1 m sup τ m Λ m J 1 τ m,. Notice that contrary to the known break fraction case, the asymptotic distribution of F m is different from F 1 m and both no onger converge to a chi-square distribution with m degrees of freedom. In this case using the same reasoning as HL, we can choose a constant b m ξ errors. such that the critica vaues become the same for both I and I Doube Maximum ests he tests discussed above require the specification of the number of trend breaks, m, under the aternative hypothesis. However, in most appications, one is not sure about the number of breaks. herefore, we consider tests of the nu of no trend break against the aternative hypothesis of an unknown number of breaks in the trend sope up to some maximum M. Foowing Bai and Perron 1998, we use the cass of doube maximum tests which are generay written as: and D max Ϝ := D max Ϝ 1 := max 1 m M a,mϝ m = max a 1,mϜ 1 m = 1 m M max 1 m M a,m max 1 m M a 1,m sup Ϝ τ m 16 τ m Λ m sup Ϝ 1 τ m 17 τ m Λ m with a,1,..., a,m and a 1,1,..., a 1,M fixed weights that may be chosen in a way that refects some prior knowedge regarding the ikeihood that the data has a certain number of trend breaks. We use the same weight function to obtain a doube maximum test that is vaid for both I and I1 errors: D max Ϝ λ := { λ τ M, τ M D max Ϝ } + b M ξ { [1 λ τ M, τ M ] D max Ϝ 1 } 18 he b M ξ denote a constant that can be chosen, as before, in a way that guarantees the same critica vaues for both I and I1 cases. he asymptotic distribution of the 1

11 D max Ϝ λ test statistic is easiy obtained as an appication of the Continuous Mapping heorem and by noting that the weak convergence resuts of heorem 2 hod jointy for m = 1,..., M. Coroary 3. Let the conditions of heorem 2 hod. i If u t is I, then: D max Ϝ λ = ii If u t is I1, then: D max Ϝ λ = b M ξ max a,m Ϝ d m + o p 1 1 m M max max a 1,mϜ d 1 m + o p 1 b M ξ 1 m M 1 m M a,m max 1 m M a 1,m 1 m sup J τ m,. τ m Λ m 1 m sup J 1 τ m,. τ m Λ m We consider as in Bai and Perron 1998 two cases: the UD max Ϝ type of test where the weights are chosen uniformy across a possibe number of breaks, a d,1 =... = a d,m = 1, d =, 1, and the W D max Ϝ where the weights are defined in such a way that the margina p-vaues are equa across vaues of m, i.e., a d,1 = 1 and for m > 1, a d,m = C d ξ, 1 C d ξ, m where C d ξ, m is the asymptotic critica vaue of the test Ϝ d for a significance eve ξ and m breaks Sequentia ests and Estimation of the Number of Breaks As in Bai and Perron 1998, we aso extend our methodoogy to a test of the nu hypothesis of breaks in the trend against the aternative of + 1 breaks. Let τ = τ 1,..., τ and τ = τ 1,..., τ denote the vectors of estimated break fractions assuming breaks in the I and I1 cases, respectivey, as defined in equations 13 and 14. Let Ϝ τ 1,..., τ i 1, ζ, τ i,..., τ be the standard F-statistic for testing H : γ +1 = versus the aternative H 1 : γ +1 in the Mode: y t = α + βt + δ j DU t τ j + j=1 γ j D t τ j + δ +1 DU t ζ + γ +1 D t ζ + u t j=1 Simiary, et Ϝ 1 τ 1,..., τ i 1, ζ, τ i,..., τ be the standard F-statistic for testing H : γ +1 = versus the aternative H 1 : γ +1 in the Mode : y t = β + δ j D t τ j + j=1 γ j DU t τ j + δ +1 D t ζ + γ +1 DU t ζ + v t j=1 When the break dates are not known, we use the Ϝ + 1 and Ϝ test statistics defined as Ϝ 1 := sup τ 1 Λ 1 Ϝ τ, Ϝ 1 1 := sup τ 1 Λ 1 Ϝ 1 τ for = ; and for > as Ϝ + 1 := max sup Ϝ τ 1,..., τ i 1, ζ, τ i,..., τ 1 i +1ζ Λ,i 11

12 Ϝ := max sup Ϝ 1 τ 1,..., τ i 1, ζ, τ i,..., τ 1 i +1ζ Λ 1,i where the possibe eigibe break fractions ζ are contained in the foowing sets in which η is the trimming parameter: Λ,i = {ζ : τ i 1 + τ i τ i 1 η ζ τ i τ i τ i 1 η} 19 and Λ 1,i = {ζ : τ i 1 + τ i τ i 1 η ζ τ i τ i τ i 1 η}. 2 with τ = and τ +1 = 1. he next heorem estabishes the asymptotic behaviour of Ϝ + 1 and Ϝ for different orders of integration of the error term u t. heorem 3. Let the time series process y t be generated according to 1 and 2 with m = breaks and et Assumption 1 hod. i If u t is I, then: a im P Ϝ + 1 x = G x +1, where G x is the distribution function of sup J τ m, for m = 1, and b Ϝ = O p /. τ m Λ m ii If u t is I1, then: a Ϝ + 1 = O p /, and b im P Ϝ x = G 1 x +1, where G 1 x is the distribution function of sup J 1 τ m, for m = 1. τ m Λ m Remark 6. he resuts in the previous heorem show that critica vaues for the sequentia tests can be computed from the quanties of the asymptotic distributions of the Ϝ and Ϝ 1 test statistics for the case of just one break m = 1. he Ϝ λ + 1 statistic is then given by: Ϝ λ + 1 := { λ τ +1, τ +1 Ϝ + 1 } + b +1 { ξ [1 λ τ +1, τ +1 ] Ϝ } 21 where τ +1 = τ 1,..., τ +1 and τ +1 = τ 1,..., τ +1 and b +1 ξ is a constant that ensures that for a given significance eve ξ and nu hypothesis of trend breaks the critica vaues of the asymptotic distribution of Ϝ λ + 1 is the same in both I and I1 cases. Using Lemma 2 and the fact that the order of probabiity of the KPSS statistics S τ +1 and S 1 τ +1 under I or I1 errors is unchanged both under the nu and the aternative hypothesis, it is readiy seen that the weight function has the same asymptotic behavior as in Coroary 1 and so we may state the foowing coroary: Coroary 4. Let the conditions of heorem 3 hod. i If u t is I, then λ τ +1, τ +1 im P Ϝ λ + 1 x = G x +1. p 1, Ϝ λ + 1 = Ϝ o p 1 and 12

13 ii If u t is I1, then λ τ +1, τ +1 p, Ϝ λ + 1 = b +1 ξ Ϝ o p 1 and im P b +1 ξ Ϝ λ + 1 x = G 1 x +1. he Ϝ λ + 1 can be used to estimate the number of breaks in the trend sope without making any assumption about the errors being I or I1. he procedure starts with =, by using the Ϝ λ 1 to test for the presence of one break. If the nu hypothesis is rejected, we set = 1 and perform the Ϝ λ2 1 test. he procedure is repeated unti the Ϝ λ + 1 test cannot reject the nu hypothesis of breaks. Remark 7. In sma sampes, for some particuar combinations of the breaks in the trend sope, this sequentia procedure may not perform we. For instance, in the presence of two breaks of opposite signs, the Ϝ λ 1 may have ow power in identifying the two breaks, causing the sequentia estimation procedure to stop too soon. A simpe modification of this sequentia procedure that is abe to obviate to this probem consists in using the Ϝ λ with m = 2 or a doube maximum test D max Ϝ λ whenever the Ϝ λ 1 does not reject the nu hypothesis of no break. If the Ϝ λ with m = 2 or the doube maximum test does not reject H then we concude that there are no trend breaks. Otherwise we proceed to Ϝ λ 3 2. We ca these sequentia procedures SeqϜ λ 1, SeqϜ λ 2, SeqUD max Ϝ λ and SeqW D max Ϝ λ. Figure 4 summarizes the steps to impement in each type of sequentia test presented. Remark 8. he sequentia procedure to estimate the number of breaks can be made consistent by etting the significance eve of the Ϝ λ + 1 test converge to zero sowy enough as expained in Proposition 8 from Bai and Perron However, for a given sampe, this has no practica impications and the usua significance eves can be used. 3. Size and Power Simuations In this section we provide the resuts of severa Monte Caro simuations. he trimming parameter η was set equa to.15. Asymptotic critica vaues were obtained with discrete approximations =1 of the asymptotic distributions using 5 simuations and the rndn pseudo random number generator in Gauss. o appy these tests we need to choose constants g and v from the weight function and the bandwidth parameter from the ong run variance estimator. After considering severa combinations of the vaues of g and v constants in the weight function, and truncation ag in the ong run variance estimator we have chosen g = m 1, v = 6, = [4/1] 1/4 as these presented the best resuts in terms of size and power in the range of simuations considered. Hence these are the vaues which shoud be chosen in practica appications of these tests. hese resuts appy for both Modes A and B. abe 1 reports the obtained asymptotic critica vaues for the cass Ϝ λ m statistics for m = 1,..., 5 and for the UD max Ϝ λ and 13

14 W D max Ϝ λ statistics up to a maximum of 3 trend breaks. In abe 2 we present critica vaues for the Fλ + 1 statistic for different vaues of. Since the vaues provided are for the unknown break fraction case we aso provide the vaues of b m ξ.o anayse the power and size properties we used 5 simuations with 15 observations derived from the foowing DGP based on Mode B: m m y t = α + βt + δ j DU t τ j + γ j D t τ j + ut 22 with the error term given by: j=1 j=1 1 ρl u t = 1 θl ε t, t = 2,...,, u 1 = ε 1, ε t NIID, 1 23 We anaysed different eves of persistence on the error term u t measured by the autoregressive parameter ρ and moving average parameter θ. We use ρ = 1 c with c {, 1, 2, } and θ {.5,,.5}. For the power curves, we generated data from the DGP described by equations 22 and 23 for a grid of γ 1 = δ 1 /5 vaues covering the range [, 1] with steps of.1. Resuts for the size of the Ϝ λ m and D max Ϝ λ test statistics with the number of breaks under the aternative m = 1,..., 5 and upper bound M = 3 are presented in abe 3 for = 15. In the case of I1 c =, ρ = 1 shocks we see that the Ϝ λ m test is oversized speciay when θ =.5. Size distortions become speciay higher with m if θ {.5, } but in the case of θ =.5 the size remains fairy constant regardess of the number of trend breaks set under the aternative hypothesis. For ρ.93 c = 1 and ρ.87 c = 2 the Ϝ λ m test shows reasonabe size contro for θ {.5, } with a sight size depreciation towards the over-sizing region for ρ.93 and θ =.5. In the case of ρ = c = we observe that for m = 1 and m = 2 the Ϝ λ m is sighty oversized if θ {.5, } and undersized if θ =.5. Since in these cases the size decreases with m we have arge degree of under-size with a higher number of trend breaks under H 1. In genera the UD max Ϝ λ and W D max Ϝ λ statistics seem to have simiar finite sampe size performances for M = 3: the D max Ϝ λ cass is speciay under-sized in the case of pure MA shocks with θ =.5 and over-sized if the errors foow an I1 process with θ =.5, simiary to what was observed for the Ϝ λ m statistics. Unreported simuations show that these size distortions become worse with the increase of the number of trend breaks aowed under H 1. However, the W D max Ϝ λ is substantiay more sensitive than the UD max Ϝ λ to M. Consider now Figures 1, 2 and 3 that dispay the power of the tests for a DGP with 1 change point as a function of the magnitude of the break γ 1 occurring in the midde of the sampe, τ1 = 1/2, for different vaues of ρ and θ. he resuts show that the tests have 14

15 simiar power for the case of I1 shocks with sma differences attributabe to unequa finite sampe size performances. However, in most cases with I shocks the Ϝ λ 1 has higher power than a the other tests which is not surprising since our DGP incudes ony 1 trend break. Aso, notice that the power Ϝ λ m definitey decreases as we increase the number of trend breaks set under H 1. his is expained by the fact that, as we increase m, we are aowing for more breaks than necessary to detect the singe break in the DGP. Finay, consider abe 4. Here we present the empirica reative frequency at 5% eve to seect,1,2 and more than 2 trend breaks of the proposed sequentia statistics together with the Kejriwa and Perron 21 sequentia procedure coumns headed SeqϜ F S 1. In our experiment, y t may have no breaks, 1 trend break and 2 trend breaks with the same magnitude and same sign,γ 1 = γ 2, or with opposite signs, γ 1 = γ 2. We considered a trend break of magnitudes γ 1 {.5, 1} occurring in the midde of the sampe if there is 1 break, τ1 =.5, and ocated at τ1 = 1/3 and τ2 = 2/3 if there are 2 trend breaks. he usua vaues of ρ were considered with no moving average effects, θ =. A sequentia tests have power to efficienty detect the presence one break in trend with a sma power advantage of the proposed sequentia procedures reative to SeqϜ F S 1. For a DGP with 2 trend breaks with same sign and magnitude the tests show simiar and reasonabe power to detect 2 breaks. his happens speciay as we decrease the persistence of the errors, ρ. However, the differences are quite considerabe when we ook for the 2 opposite breaks case. Here SeqϜ F S 1 and, speciay, SeqϜ λ 1 have ow power to detect breaks and are ceary outperformed by its competitors SeqϜ λ 2, SeqUD max Ϝ λ and SeqW D max Ϝ λ. For exampe, for the highest magnitude considered in the simuations γ 1 = 1, γ 2 = 1 and I 1 shocks the SeqϜ λ 1 and the SeqϜ F S 1 ony estimate 2 breaks with 41% and 54% power, respectivey, whie the other sequentia tests have probabiity of around 9% to detect 2 change points. Aso if u t is a highy persistent I process c = 1, 2 and for the same magnitudes the SeqϜ λ 1 and the SeqϜ F S 1 ony detect 2 breaks with, at most, 25% and 52% probabiity, respectivey, whereas its competitors dispay amost fu power. On the basis of the resuts on abe 4, we woud recommend the use of SeqϜ λ 2 when testing the nu of no trend break against an unknown number of trend breaks: this sequentia test has smaer and ony mid size distortions in comparison with the other sequentia tests and is abe to detect with high power changes in the trend function without suffering the opposite breaks probem. If the empirica researcher is sure about the number of trend breaks under the aternative then it shoud use the Ϝ λ m and specify the number of m trend breaks under H 1. However, it shoud be cautious if m is quite arge 4 and the number of observations is sma because simuation resuts show increasing size distortions with m. In that case, we recommend the use of the D max Ϝ λ statistics as a pre test to check if there are trend breaks and if the nu is rejected use Ϝ λ m to estimate the break dates. 15

16 4. Concusions In this paper we presented tests for the presence of mutipe structura change in the trend sope of a univariate time series which do not require knowedge of the form of seria correation in the data and are vaid regardess of the shocks being I or I1. We have considered two Modes: a Joint and a Disjoint Broken rends Mode. We have extended the test procedure proposed by Harvey et a. 29 and constructed a weighted average of two F-statistics, one standardy used when the data is I and the other usuay appied for data exhibiting a unit root. We start by considering the case in which the researcher is sure about the break dates if there is any structura change in the trend function. Next, we proposed tests for known number of trend breaks but unknown break dates under the aternative. Here, the break dates estimated are goba maximizers of the F statistics over a permissibe break fractions. Finay, we anaysed tests for the practitioner who is aso not sure about the number of break dates if trend changes have occurred. We anaysed doube maximum tests and aso 4 sequentia procedures that can be used to estimate the number of breaks. We have estabished the arge sampe properties of a these tests. Monte Caro evidence shows that our tests have reasonabe size and power properties and recommend the use of a modified sequentia approach where the doube maximum test or the F λ 2\ is used to detect breaks of opposite signs. An empirica exampe iustrated the usefuness of the proposed procedures. 16

17 References Andrews, D. W. K ests for parameter instabiity and structura change with unknown change point. Econometrica, 614: Bai, J. and Perron, P Estimating and testing inear modes with mutipe structura changes. Econometrica, 661: Bergstrom, A. R., Nowman, K. B., and Wymer, C. R Gaussian estimation of a second order continuous time macroeconometric mode of the uk. Economic Modeing, 94: Bunze, H. and Vogesang,. J. 25. Powerfu trend function tests that are robust to strong seria correation, with an appication to the prebisch-singer hypothesis. Journa of Business & Economic Statistics, 23: Gregory, A. and Hansen, B Residua-based tests for cointegration in modes with regime shifts. Journa of Econometrics, 7: Harvey, D. I., Leybourne, S. J., and ayor, A. R. 29. Simpe, robust, and powerfu tests of the breaking trend hypothesis. Econometric heory, 254: Jones, C. I ime series tests of endogenous growth modes. he Quartery Journa of Economics, 112: Kejriwa, M. and Perron, P. 21. A sequentia procedure to determine the number of breaks in trend with an integrated or stationary noise component. Journa of ime Series Anaysis, 315: Kwiatkowski, D., Phiips, P. C. B., Schmidt, P., and Shin, Y esting the nu hypothesis of stationarity against the aternative of a unit root : How sure are we that economic time series have a unit root? Journa of Econometrics, 541-3: Leybourne, S., ayor, R., and Kim,.-H. 27. Cusum of squares-based tests for a change in persistence. Journa of ime Series Anaysis, 283: Lumsdaine, R. L. and Pape, D. H Mutipe trend breaks and the unit-root hypothesis. he Review of Economics and Statistics, 792: Newey, W. K. and West, K. D A simpe, positive semi-definite, heteroskedasticity and autocorreation consistent covariance matrix. Econometrica, 553:73 8. Nowman, K. B Econometric estimation of a continuous time macroeconomic mode of the united kingdom with segmented trends. Computationa Economics, 123:

18 Perron, P Further evidence on breaking trend functions in microeconomic variabes. Journa of Econometrics, 8: Perron, P. and Yabu,. 29. esting for shifts in trend with an integrated or stationary noise component. Journa of Business & Economic Statistics, 273: Perron, P. and Zhu, X. 25. Structura breaks with deterministic and stochastic trends. Journa of Econometrics, : Sayginsoy, O. and Vogesang,. 24. Powerfu tests of structura change that are robust to strong seria correation. Discussion Papers 4-8, University at Abany, SUNY, Department of Economics. Stock, J. H. and Watson, M. 28. Forecasting in dynamic factor modes subject to structura instabiity. he Methodoogy and Practice of Econometrics, A Festschrift in Honour of Professor David F. Hendry. Vogesang,. J rend function hypothesis testing in the presence of seria correation. Econometrica, 661: Vogesang,. J. and Franses, P. H. 25. esting for common deterministic trend sopes. Journa of Econometrics, 1261:1 24. Zivot, E. and Andrews, D. W. K Further evidence on the great crash, the oiprice shock, and the unit-root hypothesis. Journa of Business & Economic Statistics, 13:

19 Appendix Proof of heorem 1. m Ϝ τ as: i a From the Frisch-Waugh-Love heorem we can write [ m Ϝ τ = {κ + 3/2 Rt τ R t τ ] 1 } Rt τ u t [ 3 R t τ R t τ ] ω 2 τ [ {κ + 3/2 Rt τ R t τ ] 1 } Rt τ u t with R t τ := R t τ 1,..., R t τ m where R t τ j is the vector of residuas from the regression of D t τ j on {1, t}. From standard weak convergence resuts, namey, the Continuous Mapping heorem CM and the Functiona Centra Limit r heorem FCL, 1 d 2 ω u W r, we can estabish that: t=1 u t [ 3/2 Rt τ R t τ ] 1 Rt τ d u t [ 1 ] 1 1 ω u R r, τ R r, τ dr R r, τ dw r := ω u Q τ 1 V τ where W r is the standard Brownian Motion, R r, τ is the continuous time residua vector whose j th eement is given by the projection of r τ j 1 r > τ j onto the space spanned by {1, r}, Q τ := R τ, r R τ, r dr is a nonsinguar matrix, and V τ := 1 1 R τ, r dw r. It is aso we known that the ong run variance estimator ω 2 τ is consistent, ω 2 τ p ω 2 u. With these resuts, the asymptotic distribution of m Ϝ τ can be written as : m Ϝ τ d { κ + ω u Q τ 1 V τ } [ ] Q τ {κ + ωu Q τ 1 V τ }. b Again appeaing to the Frisch-Waugh-Love heorem it is possibe to rewrite Ϝ 1 τ as: Ϝ 1 τ = 1 {κ 12 + [ 1 RU t τ RU t τ ] 1 } RUt τ u t m ω 2 u 19

20 [ 1 RU t τ RU t τ ] {κ 12 + [ 1 RU t τ RU t τ ] 1 } RUt τ u t ω 2 τ with RU t τ = RU t τ 1,..., RU t τ m where RU t τ j is the vector of residuas from the regression of DU t τ j on {1}. Now notice that RUt τ j can be simpified to: τ RU t τj j 1 = τ j, if t j, if t > j and so we get that RU t τ j ut = τ j u u j 1 τ j u1 = O p 1 since u t I. Aso, Leybourne et a. 27 proved that ω 2 τ has a finite and positive probabiity imit provided that = o 1/2 and Assumption 1 from their paper hods. Hence, we get ω 2 τ p 2 sγ s = O p 1, where γ s = E [ u t u t s ]. Finay, it is straightforward to see that 1 RU t τ RU t τ s= 1 RU τ, r RU τ, r dr = O 1 which is a non-singuar matrix given that τj, 1 for j = 1,..., m. So, since a terms from the right hand side of 24 are O p 1 we proved that Ϝ 1 τ = O p 1. ii a We have that Ϝ τ equas: Ϝ τ = 1 { κ + [ 3 R t τ R t τ ] 1 } 5/2 R t τ u t m [ 3 R t τ R t τ ] 1 ω 2 τ { κ + [ 3 R t τ R t τ ] 1 } 5/2 R t τ u t 24 Using standard weak convergence resuts we can prove that: 1 5/2 R t τ d u t ω u R r, τ dw r = O p 1 and 3 R t τ R t τ 1 R r, τ R r, τ dr = O 1 which is a non-singuar matrix given that τj, 1 for j = 1,..., m. Extending appropriatey formua 23 from Kwiatkowski et a to Mode A, it 2

21 is possibe to show that 1 ω 2 τ d ω 2 u 1 H r, τ 2 dr where H r, τ is the continuous time residua from the projection of W r onto the space spanned by {1, r, r τ1 1 r > τ1,..., r τm 1 r > τm}. Since a terms have nondegenerate distributions we can say that Ϝ τ = O p 1. b Foowing the same ines from the proofs of the previous resuts we can rewrite m Ϝ 1 τ as: { m Ϝ 1 τ = κ + [ 1 RU t τ RU t τ ] 1 } 1/2 RU t τ ε t [ 1 RU t τ RU t τ ] ω 2 τ { κ + [ 1 RU t τ RU t τ ] 1 } 1/2 RU t τ ε t From [ 1 RU t τ RU t τ ] 1 1/2 RU t τ d ε t [ 1 ] 1 1 RU τ, r RU t τ, r dr ω ε RU τ, r dw r := ω ε Q 1 τ 1 V 1 τ where Q 1 τ := 1 RU τ, r RU τ, r dr and V 1 τ := 1 RU τ, r dw r and using the fact that ω 2 τ p ω 2 ε, we can estabish the asymptotic distribution of m Ϝ 1 τ : m Ϝ 1 τ d { κ + ω ε Q 1 τ 1 V 1 τ } [ ] Q1 τ {κ + ωε Q 1 τ 1 V 1 τ } ω 2 ε Proof of Lemma 1. Resuts ia and ib foow from Kwiatkowski et a If the error term u t I then S τ = O p 1 and converges to a function of the Wiener process. Simiary, if u t I 1, then when we differentiate the mode the disturbances become stationary and so S 1 τ = O p 1.Resut iia foows from 1 ω 2 τ p ω 2 u 1 H r, τ 2 dr, an extension of expression 22 from Kwiatkowski et a to Mode A, and 4 t û i τ t=1 i=1 2 P ω 2 u 1 a 2 H r, τ dr da. Hence, we have that under u t I 1, S τ = O p /. Finay, from resuts from 21

22 Leybourne et a. 27 we get that if u t I, then ω 2 τ p 2 sγ s = O p 1 and 1 t 2 ṽ i τ = O p 1 which estabishes the resut t=2 i=1 s= S 1 τ = O p 1. Proof of heorem 2. he proof of ia and iib foows by appying the Continuous Mapping heorem as in Zivot and Andrews 1992 to show convergence in distribution from the space D[, 1] to the space C, 1 using the uniform metric as in Gregory and Hansen 1996 and Perron he sup-f test statistics Ϝ and Ϝ 1 can be written as continuous functionas of the processes 3 R t. R t., 3/2 R t. u t, ω 2. and 1 RU t. RU t.,, 1/2 RU t. ε t, ω. 2 respectivey. Using simiar arguments from Zivot and Andrews 1992 we can show the joint weak convergence of these processes: 3 R t. R t., 3/2 R t. u t, ω R., r R., r dr, R., r dw r, ωu. 2, 1 RU t. RU t., 1/2 RU t. ε t, ω RU., r RU., r dr, RU., r dw r, ωε 2. It foows that Ϝ m 1 m sup J τ m, and Ϝ 1m 1 τ m Λ m m sup J 1 τ m, by the τ m Λ m CM. From heorem 1ib and the fact that ω 2 τ m p 2 sγ s uniformy in τ m s= it foows that if u t I, then Ϝ 1 τ m = O p uniformy in τ m and so the resut in ib is proved. Finay, given the derivations in the proof of heorem 1iia it can be easiy seen that Ϝ m can be written as a continuous functiona of stochastic processes on, 1 so that by the arguments in Zivot and Andrews 1992, by appying the Continuous Mapping heorem, it foows that Ϝ m is O p and so the resut in iia is proved. Proof of heorem 3. hroughout the proof we empoy the foowing additiona notation: Let RSSR τ i 1, τi and USSR τ i 1, ζ, τi denote, respectivey, the restricted 22

23 and unrestricted sum of squared residuas for testing H : γ = in the mode: y t = α + β t i 1 + γdt ζ + u t, t = i 1 + 1,..., i with i := τi. Simiary, denote by RSSR 1 τ i 1, τi and USSR1 τ i 1, ζ, τi, respectivey, the restricted and unrestricted sum of squared residuas for testing H : γ = in the mode: i a Notice that: y t = β + γdu t ζ + v t, t = i 1 + 1,..., i Ϝ τ 1,..., τi 1, ζ, τi+1,..., τ RSSR τ = i 1, τi USSR τ i 1, ζ, τi ω 2 τ1,..., τi 1, ζ, τ i+1,..., τ + o p 1 Since,under H there are trend breaks occurring at dates 1,...,, it is we known that ω p 2 τ1,..., τi 1, ζ, τi+1,..., τ ω 2 u. Moreover, simiar arguments from the proof of heorem 2 can be used to prove that, under H, for each i = 1,..., + 1: RSSR τ sup i 1, τi USSR τ i 1, ζ, τi ζɛλ ω 2 τ,i 1,..., τi 1, ζ, τ i+1,..., τ d ζ τ sup J i 1, ζɛλ τ,i i τi 1 where Λ,i is as defined in 19with τ i 1 and τ i repaced by τ i 1 and τ i, respectivey. Since τ 1,..., τ have been obtained by the Ϝ statistic we get that, under H, τ i τi = O p 3 2 from heorem 3 of Perron and Zhu 25. his impies that i = i + O p 1 2 and so it is possibe to show that 25 hods with τ1,..., τi 1, τi,..., τ repaced by τ1,..., τ i 1, τ i,..., τ. Now notice that RSSR.,. and USSR.,. are computed on different and non overapping regimes which impies independence of the weak imits in 25. Since we are taking the maximum over + 1 independent random variabes we get that: im P Ϝ + 1 x = G x +1 where G x is the distribution function of sup τ 1 Λ 1 J τ 1, where we empoyed the change in variabe τ 1 = ζ τ i 1 / τ i τ i 1. b Notice that: sup ζɛλ 1,i Ϝ 1 τ 1,..., τi 1, ζ, τi+1,..., τ = sup ζɛλ 1,i 25 RSSR 1 τ i 1, τi USSR1 τ i 1, ζ, τi ω 2 τ1,..., τi 1, ζ, τ i+1,..., τ 26 23

24 where Λ 1,i is as defined in 2 with τ i 1 and τ i repaced by τ i 1 and τ i, respectivey. Simiar arguments from the proof of heorem 2 aow us to estabish that, under the nu, ω 2 τ1,..., τi 1, ζ, τi+1,..., τ = Op 1 and, furthermore, RSSR 1 τ i 1, τi USSR1 τ i 1, ζ, τi = Op 1 uniformy in ζ, given that u t I and the fact that there are no breaks between observations i and i. Hence, the F-statistic in 26 is O p uniformy. Since τ i τi is O p 1 2 by the proof of heorem 3 from HL, this is enough to estabish that, for each i = 1,..., + 1: RSSR 1 τ i 1, τ i USSR 1 τ i 1, ζ, τ i sup ζɛλ 1,i ω 2 τ 1,..., τ i 1, ζ, τ i+1,..., τ = O p uniformy in ζ. Since we are taking the maximum over + 1 i.i.d random variabes that are O p we obtain the desired asymptotic resut. ii a Simiar arguments from the proof of heorem 2 can be empoyed to show that, under the nu of no trend breaks, if u t I 1, then the eft hand side of 25 is O p uniformy over a ζ. Since τ i τi = O p 1 2 from heorem 3 of Perron and Zhu 25 the rate of convergence remains the same when we repace τ 1,..., τi 1, τi,..., τ by τ1,..., τ i 1, τ i,..., τ. Hence, we have that, for each i = 1,..., + 1: RSSR τ i 1, τ i USSR τ i 1, ζ, τ i sup ζɛλ,i ω 2 τ 1,..., τ i 1, τ i,..., τ = O p Since we are taking the maximum over +1 i.i.d random variabes that are O p we obtain the desired asymptotic resut. b Using the fact, under H, i = i + O p 1 from Bai and Perron 1998, the same arguments from ia can be used to show that: im P Ϝ x = G 1 x +1 where G 1 x +1 is the distribution function of sup τ 1 Λ 1 J 1 τ 1,. 24

25 abe 1: Asymptotic critica vaues and b m ξ vaues for the mutipe trend breaks Ϝ λ and Doube Maximum UD max Ϝ λ and W D max Ϝ λ tests. Mode A Ϝ m m = 1 m = 2 λ m = 3 m = 4 m = 5 UD max Ϝ λm = 3 W D max Ϝ λm = 3 c.v. b 1 c.v b 2 c.v b 3 ξ c.v b 4 c.v b 5 c.v b 3 ξ c.v b 3 ξ ξ ξ ξ ξ ξ Mode B Ϝ m m = 1 m = 2 λ m = 3 m = 4 m = 5 UD max Ϝ λm = 3 W D max Ϝ λm = 3 c.v. b 1 c.v b 2 c.v b 3 ξ c.v b 4 c.v b 5 c.v b 3 ξ c.v b 3 ξ ξ ξ ξ ξ ξ

26 abe 2: Asymptotic critica vaues and b ξ vaues of the sequentia trend breaks test Ϝ λ + 1 Mode A Fλ + 1 = = 1 = 2 = 3 = 4 ξ Critica vaue b 1 ξ Critica vaue b 2 1 ξ Critica vaue b 3 2 ξ Critica vaue b 4 3 ξ Critica vaue b 5 4 ξ Mode B Ϝλ + 1 = = 1 = 2 = 3 = 4 ξ Critica vaue b 1 ξ Critica vaue b 2 1 ξ Critica vaue b 3 2 ξ Critica vaue b 4 3 ξ Critica vaue b 5 4 ξ

27 abe 3: Empirica size of Ϝ λm and D max Ϝ λ tests, 5% nomina eve, = 15. ρ θ Ϝ λ m UD max Ϝ λ W D max Ϝ λ m = 1 m = 2 m = 3 m = 4 m =

28 abe 4: Size and Power of Sequentia ests, Mode B, =15 SeqϜF S 1 SeqϜ λ 1 SeqϜ λ 2 SeqUD max Ϝ λ SeqW D max Ϝ λ γ1 γ2 ρ 1 2 >2 1 2 >2 1 2 >2 1 2 >