A NOTE ON SPURIOUS BREAK

Transcription

1 Econometric heory, 14, 1998, Printed in the United States of America+ A NOE ON SPURIOUS BREAK JUSHAN BAI Massachusetts Institute of echnology When the disturbances of a regression model follow an I~1! process there is a tendency to estimate a brea point in the middle of the sample, even though a brea point does not actually exist+ In this note, we provide a mathematical proof for this phenomenon+ 1. INRODUCION Recently, Nunes, Kuan, and Newbold ~1995!~henceforth NKN! pointed out that when the disturbances of a regression model follow an I~1! process there is a tendency to estimate a brea point in the middle of the sample, even though a brea point does not actually exist+ his phenomenon is called a spurious brea by the authors and was discovered by a simulation experiment+ In this note, we provide a mathematical proof for this phenomenon+ It is of interest to as the following question, and it is, in fact, often ased+ Given a regression model with no brea point, and supposing that a brea point is entertained in estimation, how does the estimated brea point behave? Let Z denote the estimated brea point and the sample size+ Define Zl 0+ Z hus, Zl denotes the estimated brea fraction+ For I~0! disturbances, it can be shown that the estimated brea fraction converges to the boundary ~i+e+, either 0 or 1!+ Because boundary values imply no brea in the sample, the estimated brea point conforms to the true model of no brea+ However, a different phenomenon emerges for I~1! disturbances+ NKN found that the estimated brea point will stay in the middle of the sample, suggesting the existence of a brea point+ he authors call this phenomenon spurious brea in an analogy to spurious regression+ Although the problem of spurious regression is much better understood and is well documented ~see Granger and Newbold, 1974; Phillips, 1986; Durlauf and Phillips, 1988!, the problem of spurious brea is less well studied+ his note taes up the issue+ In particular, we shall deliver a mathematical proof that Zl does not converge to 0 or 1, corroborating and confirming the simulation findings of NKN+ It should be emphasized that our analysis assumes the absence of a brea in the data-generating process+ he result suggests that caution should be exercised in estimating a brea point when disturbances are I~1!+ Diagnostic testing should be performed prior to es- I than three anonymous referees and Bruce Hansen ~co-editor! for their comments and suggestions that led to this much improved version+ Partial financial support is acnowledged from the National Science Foundation under grant SBR Address correspondence to: Jushan Bai, Department of Economics, E52-274B, Massachusetts Institute of echnology, Cambridge, MA 02139, USA; jbai@mit+edu Cambridge University Press $

2 Z Z Z Z 664 JUSHAN BAI timating a brea point, and test statistics that are robust to I~1! errors should be used+ Vogelsang ~1994a, 1994b! proposed a number of test statistics useful for this purpose+ 2. NOAION AND ASSUMPION Consider the model y t x t b 1 «t t 1,2,+++, x t b 2 «t t1,+++,+ Let bz 1 ~! be the least squares estimator of b 1 based on the first observations and b 2 ~! be the least squares estimator of b 2 based on the last observations, i+e+, b 1 ~! ( 1( b 2 ~! ( t1 1 ( x t y t, t1 x t y t + Define the sum of squared residuals for the full sample as S ~! ( ~ y t x t bz 1 ~!! 2 ( t1 and define the brea point estimator as argmin 1 S ~! argmin 1 S ~! ( «t 2 ~ y t x t bz 2 ~!! 2 he second equality follows because ( «2 t does not depend on + Finally, let Zl min ~l : l argmin u@ tl, Nl# S ~@u#!( «t 2, where 0, tl Nl 1+ he behavior of l can be determined by exploring the limiting process of S ~! ( «2 t + In the absence of a brea, i+e+, b 1 b 2, it is easy to show that S ~! ( «2 t M ~!, where M ~! ( «t x t ( ( «t x t t1 1( ( t1 + x t «t 1 ( t1 x t «t + (1)

3 o obtain the limiting process for M ~!, we need the following assumptions, which are similar to those of NKN+ Assumption A1+ For an I~1! error process «t, we assume that 2 «2 t n c * l 0 W 2 ~u! du with W~u! being a standard Wiener process and c 0 a constant+ Assumption A2+ here exists a diagonal matrix D such D 102 n Q~l!+ D 102( he matrix Q~l! is assumed to be positive definite for all l 0, and Q~v! Q~u! is positive definite for all v u 0+ In addition, Q~0! 0+ Assumption A3+ For some a SPURIOUS BREAK 665 a D ( x t «t n G~l!, (2) where G~l!is a stochastic process having continuous sample path with G~0!0+ 1 he assumptions here are quite general, encompassing many models used in practice+ Various special cases of G~l! and Q~l! are given in NKN+ ypically, G~l! is a functional of a Gaussian process+ 3. MAIN RESUL In this section, we characterize the limiting behavior of M ~!+ When «t is I~0!, M ~! has a proper limit # with a,b ~0,1! and a b+ his will not be true when «t is I~1!+ A normalization is required+ Define M * ~! a M ~!+ We also note that Z argmax M ~! argmax M * ~! because a does not depend on + Under Assumptions A1 A3, it can be shown that ~see NKN! d Zl & argmax M * ~l!, l@ sl, Nl# where (3) M * ~l! G~l! Q~l! 1 Q~l!# G~l!#+ (4) When «t is I~0!, a similar limiting process will be obtained+ his implies that, in the absence of a brea point, the estimated brea point Zl is a random variable with support tl, Nl#+ his is true regardless of whether the error is I~0! or I~1!+ hus if the attention is focused on the compact tl, Nl#, not much difference between the two cases can be discerned+ It is important to explore the be-

4 666 JUSHAN BAI havior of M * ~l! for l near the boundary+ For I~0! error process «t, NKN proved that M~l! r ` as l r 0or1,thus Zl r $0,1%, if tl r 0 and Nl r 1 ~also see Andrews, 1993!+ In their Remar 1 ~p+ 742!, NKN pointed out that they were unable to characterize the limiting behavior of M * ~l! for l near0or1+hrough simulation, they found that M * ~l! does not diverge to infinity as l decreases to zero or increases to 1+ In the following, we shall prove that M * ~l! is a well-defined process and is uniformly bounded in probability In Assumption A3, we shall assume that a 2+ his is true whenever x t contains a nonzero mean regressor ~e+g+, a constant or a trend!+ When all components of x t are I~0! and have zero means, it is possible that a 1 ~for an example, see NKN!+ ~Actually, an I~1! dependent variable with I~0! regressors is unliely to be a useful model+! In any case, our proof does not apply to the situation for which 0,a2 and thus is not considered in this paper+ HEOREM 1+ Assume that Assumptions A1 A3 hold+ For the a defined in ~2!, assume a 2+ We have sup l~0,1! M * ~l! O p ~1!+ (5) Proof of heorem 1+ For an arbitrary vector z and an arbitrary projection matrix P, we have z Pz z z+ Apply this inequality to M * ~! to obtain M * ~! a ( «2 t 2 2 ( «t for #, (6) from a 2+ Because 2 ( «2 t does not depend on, and it has a limit by Assumption A1, M * ~! is uniformly bounded in probability+ hus its limit, M * ~l!, is uniformly bounded in probability for l ~0,1!+ herefore, M * ~l! is stochastically bounded even when l r $0,1%+ his is in contrast to the case of I~0! errors for which the corresponding process grows without bound as l tends to the boundary+ o rule out the possibility that Zl r $0,1%, we need to further examine the behavior of M * ~l! for l near 0 and 1+ Strictly speaing, M * ~l! is not defined yet at l 0 and l 1+ As the limit of M * ~l! when l r 0, M * ~0! should be defined as M * ~0! G~1! Q~1! 1 G~1!, (7) which is obtained from ~4! by taing G~l! 0, Q~l! 0, and G~l! Q~l! 1 G~l! 0 for l 0+ Note that the term G~l! Q~l! 1 G~l! is the limit of the first term of ~1! on the right-hand side divided by a + Now a( x t «t ( 1( x t «t a ( «2 t 2 ( «2 t,

5 which converges to zero in probability for any given or with l r 0+ It follows that G~l! Q~l! 1 G~l! r 0 in probability as l r 0+ hus the definition of ~7! is the limit of M * ~l! as l r 0+ As a result, M * ~l! becomes continuous at l 0+ Similarly, we can define, as the limit of M * ~l! as l r 1, M * ~1! M * ~0!+ his extension of M~l! maes it continuous at l 1+ We next show that the maximum of M * ~l! is attained neither at 0 nor 1+ HEOREM 2+ Under the conditions of heorem 1, we have ~i! With probability 1, SPURIOUS BREAK 667 M * ~0! M * ~1! M * ~l!, for every 0 l 1+ (8) ~ii! If G~l! has a continuous distribution for each l, then with probability 1, M * ~0! M * ~1! M * ~l!, for every 0 l 1+ (9) heorem 2~i! implies that as long as M * ~l! is not a constant process, the maximum value of M * ~l!will not be attained at 0 or 1+ Let l * argmax l@ tl, Nl# M * ~l!+ Assume that there exists a l 1 such that M * ~l 1!M * ~0!~this is true if the process is not a constant!+ hen l * r0 0,1as tlr0or Nlr1+his assertion follows from the continuity of M * ~l! at 0 and 1 ~there exists a neighborhood N 0 of l 0 such that M * ~l! M * ~l 1! for all l N 0 ; the same is true for l 1!+ In summary, when M * ~{! is not a constant process, not only does M * ~{! not attain its maximum at0or1,but also the extreme point of M * ~{! on any subset is bounded away from 0 or 1+ From ~3!, Zl r l * + his implies that the estimated brea point will also be bounded away from 0 and 1, provided that M * ~l! is not a constant process+ Of course, if M * ~{! is a constant process, any point is an extreme point+ It is difficult to construct an example ~or model! such that M * ~{! does not depend on l+ For the various concrete examples given in NKN, M * ~{! is not a constant+ heorem 2~ii! gives a sufficient condition to guarantee the nonconstantness of M * ~{!+ he result of part ~ii! is much stronger than needed for the occurrence of spurious breas+ o prove heorem 2, we need the following lemma+ For a symmetric matrix A, we write A 0 if it is positive definite+ LEMMA1+ For arbitrary positive definite matrices A and B with A B ~ pp!, and arbitrary vectors x and y ~ p 1!, we have x A 1 x y B 1 y ~x y! ~A B! 1 ~x y! 0+ (10) Proof of Lemma 1+ Define the matrix H ~A B!1 A 1 ~A B! 1 ~A B! 1 ~A B! 1 B1 + (11)

6 668 JUSHAN BAI It suffices to prove H to be positive semidefinite because the left-hand side of ~10! is equal to z Hz for z ~x, y!+ Let D ~A B! 1 B 1 0+ Let C be a matrix with the first p rows ~I,~A B! 1 D 1! and second p rows ~0, I!+ Using the identity ~A B! 1 A 1 ~A B! 1 D 1 ~A B! 1 we obtain C HC diag~0, D! 0+ hus C HC is positive semidefinite+ his implies that H is positive semidefinite because C has full ran+ his proves the lemma+ Proof of heorem 2+ he inequality M * ~0! M * ~l! is equivalent to G~1! Q~1! 1 G~1! G~l! Q~l! 1 Q~l!# G~l!# 0+ Clearly, part ~i! of heorem 2 follows from Lemma 1 by letting AQ~1!, BQ~l!, xg~1!, and y G~l!+ Next, consider ~ii!+ Let A Q~1! and B Q~l! and let H be defined in ~11!+ hen M * ~0! M * ~l! is equivalent to j Hj 0, where j ~G~1!,G~l!! +Let G be an orthogonal matrix such that G HGdiag~l 1,+++,l 2p! with l 1 l 2 {{{ l 2p, where l i s are the eigenvalues of H+ Because H 0 and H 0, the maximum eigenvalue of H is positive+ It follows that j Hj ~Gj! diag~l 1,+++,l 2p!Gj h 2 l 1, where h is the first component of Gj+ When G~l! has a continuous distribution, so does j+ hus Gj is a vector of continuous random variables, implying h 2 l 1 0 with probability 1 because P~h 2 0! 0+ hat is, j Hj 0 with probability 1+ he preceding analysis applies to I~1! regressors with I~1! disturbances+ In this case, Q~l! in Assumption A2 is a random positive definite matrix; all the preceding argument applies+ he details can be found in Bai ~1996!+ his implies that for spurious regression models, if a brea is allowed in estimation, a spurious brea will occur+ All these results lend support to the observation that when it comes to I~1! processes, one should be careful about maing the hypothesis of a brea point+ It is well nown that a process with a brea point may be mistaen as I~1!~Perron 1989!; the converse is also true+ Our result is simply a rigorous proof of this fact+ However, when y t and x t are I~1! but are cointegrated, a spurious brea will not arise because the underlying disturbances are I~0!+ Furthermore, should there indeed exist a shift in the cointegrating relationship, the brea point can be estimated more precisely than I~0! models ~given the same magnitude of shift!+ Estimating a brea point in cointegrating relationship was studied by Bai, Lumsdaine, and Stoc ~1997! and by an earlier version of this paper+

7 SPURIOUS BREAK 669 NOE 1+ he corresponding assumption in NKN ~see their ~A3!! is stated in terms of y t rather than «t, which applies to y t being I~1!+ he current form allows y t to depend on deterministic regressors and on an additive I~1! error process+ REFERENCES Andrews, D+W+K+ ~1993! ests for parameter instability and structural change with unnown change point+ Econometrica 61, Bai, J+ ~1996! A Note on Spurious Brea and Regime Shift in Cointegrating Relationship+ Woring paper 96-13, Department of Economics, MI+ Bai, J+, R+L+ Lumsdaine, &J+H+Stoc ~1997! esting for and Dating Common Breas in Multiple ime Series+ Manuscript, Kennedy School of Government, Harvard University ~forthcoming in Review of Economic Studies!+ Durlauf, S+N+ &P+C+B+Phillips ~1988! rends versus random wals in time series analysis+ Econometrica 56, Granger, C+W+J+ &P+Newbold ~1974! Spurious regression in econometrics+ Journal of Econometrics 2, Nunes, L+C+, C+M+ Kuan, &P+Newbold ~1995! Spurious brea+ Econometric heory 11, Perron, P+ ~1989! he great crash, the oil price shoc and the unit root hypothesis+ Econometrica 57, Phillips, P+C+B+ ~1986! Understanding spurious regression in econometrics, Journal of Econometrics 33, Vogelsang, +J+ ~1994a! Wald-type tests for detecting shifts in the trend function of a dynamic time series+ CAE Woring paper 94-12, Cornell University+ Vogelsang, +J+ ~1994b! esting for a shift in mean without having to estimate serial correlation parameters+ Manuscript, Department of Economics, Cornell University+