Dealing with structural breaks

Dealing with Structural Breaks∗ Pierre Perron Boston University This version: April 20, 2005 Abstract This chapter is concerned with methodological issues related to estimation, testing and computation in the context of structural changes in the linear models. A central theme of the review is the interplay between structural change and unit root and on methods to distinguish between the two. The topics covered are: methods related to estimation and inference about break dates for single equations with or without restrictions, with extensions to multi-equations systems where allowance is also made for changes in the variability of the shocks; tests for structural changes including tests for a single or multiple changes and tests valid with unit root or trending regressors, and tests for changes in the trend function of a series that can be integrated or trendstationary; testing for a unit root versus trend-stationarity in the presence of structural changes in the trend function; testing for cointegration in the presence of structural changes; and issues related to long memory and level shifts. Our focus is on the conceptual issues about the frameworks adopted and the assumptions imposed as they relate to potential applicability. We also highlight the potential problems that can occur with methods that are commonly used and recent work that has been done to overcome them. ∗ This paper was prepared for the Palgrave Handbook of Econometrics, Vol. 1: Econometric Theory. For useful comments on an earlier draft, I wish to thank Jushan Bai, Songjun Chun, Ai Deng, Mohitosh Kejriwal, Dukpa Kim, Eiji Kurozumi, Zhongjun Qu, Jonathan Treussard, Tim Vogelsang, Tatsuma Wada, Tomoyoshi Yabu, Yunpeng Zhang, Jing Zhou. 1 Introduction This chapter is concerned with methodological issues related to estimation, testing and computation for models involving structural changes. The amount of work on this subject over the last 50 years is truly voluminous in both the statistics and econometrics literature. Accordingly, any survey article is bound by the need to focus on specific aspects. Our aim is to review developments in the last fifteen years as they relate to econometric applications based on linear models, with appropriate mention of prior work to better understand the historical context and important antecedents. During this recent period, substantial advances have been made to cover models at a level of generality that allows a host of interesting practical applications. These include models with general stationary regressors and errors that can exhibit temporal dependence and heteroskedasticity, models with trending variables and possible unit roots, cointegrated models and long memory processes, among others. Advances in these contexts have been made pertaining to the following topics: computational aspects of constructing estimates, their limit distributions, tests for structural changes, and methods to determine the number of changes present. These recent developments related to structural changes have paralleled developments in the analysis of unit root models. One reason is that many of the tools used are similar. In particular, heavy use is made in both literatures of functional central limit theorems or invariance principles, which have fruitfully been used in many areas of econometrics. At the same time, a large literature has addressed the interplay between structural changes and unit roots, in particular the fact that both classes of processes contain similar qualitative features. For example, most tests that attempt to distinguish between a unit root and a (trend) stationary process will favor the unit root model when the true process is subject to structural changes but is otherwise (trend) stationary within regimes specified by the break dates. Also, most tests trying to assess whether structural change is present will reject the null hypothesis of no structural change when the process has a unit root component but with constant model parameters. As we can see, there is an intricate interplay between unit root and structural changes. This creates particular difficulties in applied work, since both are of definite practical importance in economic applications. A central theme of this review relates to this interplay and to methods to distinguish between the two. The topics addressed in this review are the following. Section 2 provides interesting historical notes on structural change, unit root and long memory tests which illustrate the intricate interplay involved when trying to distinguish between these three features. Section 1 3 reviews methods related to estimation and inference about break dates. We start with a general linear regression model that allows multiple structural changes in a subset of the coefficients (a partial change model) with the estimates obtained by minimizing the sum of squared residuals. Special attention is given to the set of assumptions used to obtain the relevant results and their relevance for practical applications (Section 3.1). We also include a discussion of results applicable when linear restrictions are imposed (3.2), methods to obtain estimates of the break dates that correspond to global minimizers of the objective function (3.3), the limit distributions of such estimates, including a discussion of benefits and potential drawbacks that arise from the adoption of a special asymptotic framework that considers shifts of shrinking magnitudes (3.4). Section 3.5 briefly discusses an alternative estimation strategy based on estimating the break dates sequentially, and Section 3.6 discusses extensions of most of these issues to a general multi-equations system, which also allows changes in the covariance matrix of the errors. Section 4 considers tests for structural changes. We start in Section 4.1 with methods based on scaled functions of partial sums of appropriate residuals. The CUSUM test is probably the best known example but the class includes basically all methods available for general models prior to the early nineties. Despite their wide appeal, these tests suffer from an important drawback, namely that power is non-monotonic, in the sense that the power can decrease and even go to zero as the magnitude of the change increases (4.2). Section 4.3 discusses tests that directly allow for a single break in the regression underlying their construction, including a class of optimal tests that have found wide appeal in practice (4.3.1), but which are also subject to non-monotonic power when two changes affect the system (4.3.2), a result which points to the usefulness of tests for multiple structural changes discussed in Section 4.4. Tests for structural changes in the linear model subject to restrictions on the parameters are discussed in Section 4.5 and extensions of the methods to multivariate systems are presented in Section 4.6. Tests valid when the regressors are unit root processes and the errors are stationary, i.e., cointegrated systems, are reviewed in Section 4.7, while Section 4.8 considers recent developments with respect to tests for changes in a trend function when the noise component of the series is either a stationary or a unit root process. Section 5 addresses the topic of testing for a unit root versus trend-stationarity in the presence of structural changes in the trend function. The motivation, issues and frameworks are presented in Section 5.1, while Section 5.2 discusses results related to the effect of changes in the trend on standard unit root tests. Methods to test for a unit root allowing for a change 2 at a known date are reviewed in Section 5.3, while Section 5.4 considers the case of breaks occurring at unknown dates including problems with commonly used methods and recent proposals to overcome them (Section 5.4.2). Section 6 tackles the problem of testing for cointegration in the presence of structural changes in the constant and/or the cointegrating vector. We review first single equation methods (Section 6.1) and then, in Section 6.2, methods based on multi-equations systems where the object of interest is to determine the number of cointegrating vectors. Finally, Section 7 presents concluding remarks outlining a few important topics for future research and briefly reviews similar issues that arise in the context of long memory processes, an area where issues of structural changes (in particular level shifts) have played an important role recently, especially in light of the characterization of the time series properties of stock return volatility. Our focus is on conceptual issues about the frameworks adopted and the assumptions imposed as they relate to potential applicability. We also highlight problems that can occur with methods that are commonly used and recent work that has been done to overcome them. Space constraints are such that a detailed elicitation of all procedures discussed is not possible and the reader should consult the original work for details needed to implement them in practice. Even with a rich agenda, this review inevitably has to leave out a wide range of important work. The choice of topic is clearly closely related to the author’s own past and current work, and it is, accordingly, not an unbiased review, though we hope that a balanced treatment has been achieved to provide a comprehensive picture of how to deal with breaks in linear models. Important parts of the literature on structural change that are not covered include, among others, the following: methods related to the so-called on-line approach where the issue is to detect whether a change sas occurred in real time; results pertaining to non-linear models, in particular to tests for structural changes in a Generalized Method of Moment framework; smooth transition changes and threshold models; non parametric methods to estimate and detect changes; Bayesian methods; issues related to forecasting in the presence of structural changes; theoretical results and methods related to specialized cases that are not of general interest in economics; structural change in seasonal models; and bootstrap methods. The reader interested in further historical developments and methods not covered in this survey can consult the books by Clements and Hendry (1999), Csörgő and Horváth (1997), Krämer and Sonnberger (1986), Hackl and Westlund (1991), Hall (2005), Hatanaka 3 and Yamada (2003), Maddala nd Kim (1998), Tong (1990) and the following review articles: Bhattacharya (1994), Deshayes and Picard (1986), Hackl and Westlund (1989), Krishnaiah and Miao (1988), Perron (1994), Pesaran et al. (1985), Shaban (1980), Stock (1994), van Dijk et al. (2002) and Zacks (1983). 2 Introductory Historical Notes It will be instructive to start with some interesting historical notes concerning early tests for structural change. Consider a univariate time series, {yt ; t = 1, ..., T }, which under the null hypothesis is independently and identically distributed with mean µ and finite variance. Under the alternative hypothesis, yt is subject to a one time change in mean at some unknown date Tb , i.e., (1) yt = µ1 + µ2 1(t > Tb ) + et where et ∼ i.i.d. (0, σ 2e ) and 1(·) denotes the indicator function. Quandt (1958, 1960) had introduced what is now known as the Sup F test (assuming normally distributed errors), i.e., the likelihood ratio test for a change in parameters evaluated at the break date that maximizes the likelihood function. However, the limit distribution was then unknown. Quandt (1960) had shown that it was far from being a chi-square distribution and resorted to tabulate finite sample critical values for selected cases. Following earlier work by Chernoff and Zacks (1964) and Kander and Zacks (1966), an alternative approach was advocated by Gardner (1969) steemming from a suggestion by Page (1955, 1957) to use partial sums of demeaned data to analyze structural changes (see more on this below). The test considered is Bayesian in nature and, under the alternative, assigns weights pt as the prior probability that a change occurs at date t (t = 1, ..., T ). Assuming Normal errors and an unknown value of σ 2e , this strategy leads to the test " T #2 T X X −1 (yj − ȳ) pt Q = σ̂−2 e T t=1 j=t+1 PT 2 −1 2 where ȳ = T t=1 yt , is the sample average, and σ̂ e = T t=1 (yt − ȳ) is the sample variance of the data. With a prior that assigns equal weight to all observations, i.e. pt = 1/T , the test reduces to " T #2 T X X −2 (yj − ȳ) Q = σ̂ −2 e T PT −1 t=1 j=t+1 Under the null hypothesis, the test can be expressed as a ratio of quadratic forms in Normal variates and standard numerical method can be used to evaluate its distribution (e.g., Imhof, 4 1961, though Gardner originally analyzed the case with σ 2e known). The limit distribution of the statistic Q was analyzed by MacNeill (1974). He showed that Z 1 B0 (r)2 dr Q⇒ 0 where B0 (r) = W (r) − rW (1) is a Brownian bridge, and noted that percentage point had already been derived by Anderson and Darling (1952) in the context of goodness of fit tests. MacNeill (1978) extended the procedure to test for a change in a polynomial trend function of the form p X β i,t ti + et yt = i=0 where β i,t = β i + δ i 1(t > Tb ) The test of no change (δ i = 0 for all i) is then −2 Qp = σ̂ −2 e T T X t=1 " T X êj j=t+1 #2 P with σ̂2e = T −1 Tt=1 ê2t and êt the residuals from a regression of yt on {1, t, ..., tp }. The limit distribution is given by Z 1 Q⇒ Bp (r)2 dr 0 where Bp (r) is a generalized Brownian bridge. MacNeill (1978) computed the critical values by exact numerical methods up to six decimals accuracy (showing, for p = 0, the critical values of Anderson and Darling (1952) to be very accurate). The test was extended to allow dependence in the errors et by Perron (1991) and Tang and MacNeill (1993) (see also Kulperger, 1987a,b, Jandhyala and MacNeill, 1989, Jandhyala and Minogue, 1993, and Antoch et al., 1997). In particular, Perron (1991) shows that, under general conditions, the same limit distribution obtains using the statistic Q∗p = ĥe (0)−1 T −2 T X t=1 " T X j=t+1 êj #2 where ĥe (0) is a consistent estimate of (2π times) the spectral density function at frequency zero of et . 5 Even though, little of this filtered through the econometrics literature, the statistic Q∗p is well known to applied economists. It is the so-called KPSS test for testing the null hypothesis of stationarity versus the alternative of a unit root, see Kwiatkowski et al. (1992). More precisely, Qp is the Lagrange Multiplier (LM) and locally best invariant (LBI) test for testing the null hypothesis that σ 2u = 0 in the model yt = p X β i,t ti + rt + et i=0 rt = rt−1 + ut with ut ∼ i.i.d. N (0, σ 2u ) and et ∼ i.i.d. N (0, σ 2e ). Q∗p is then the corresponding large sample counterpart that allows correlation. Kwiatkowski et al. (1992) provided critical values for p = 0 and 1 using simulations (which are less precise than the critical values of Anderson and Darling, 1952, and MacNeill, 1978). In the econometrics literature, several extensions of this test have been proposed; in particular for testing the null hypothesis of cointegration versus the alternative of no cointegration (Nyblom and Harvey, 2000) and testing whether any part of a sample shows a vector of series to be cointegrated (Qu, 2004). Note also that the same test can be given the interpretation of a LBI for parameter constancy versus the alternative that the parameters follow a random walk (e.g., Nyblom and Mäkeläinen, 1983, Nyblom, 1989, Nabeya and Tanaka, 1988, Jandhyala and MacNeill, 1992, Hansen, 1992b). The same statistic is also the basis for a test of the null hypothesis of no-cointegration when considering functional of its reciprocal (Breitung, 2002). So what are we to make of all of this? The important message to learn from the fact that the same statistic can be applied to tests for stationarity versus either unit root or structural change is that the two issues are linked in important ways. Evidence in favor of unit roots can be a manifestation of structural changes and vice versa. This was indeed an important message of Perron (1989, 1990); see also Rappoport and Reichlin (1989). In this survey, we shall return to this problem and see how it introduces severe complications when dealing with structural changes and unit roots. It is also of interest to go back to the work by Page (1955, 1957) who had proposed to P use partial sums of demeaned data to test for structural change. Let Sr = rj=1 (yj − ȳ), his procedure for a two-sided test for change in the mean is based on the following quantities ¸ ∙ ¸ ∙ max Sr − min Si and max min Si − Sr 0≤r≤T 0≤i<r 0≤r≤T 0≤i<r and looks whether either exceeds a threshold (which, in the symmetric case, is the same). So we reject the null hypothesis if the partial sum rises enough from its previous minimum 6 or falls enough from its previous maximum. Nadler and Robbins (1971) showed that this procedure is equivalent to looking at the statistic ¸ ∙ RS = max Sr − min Sr 0≤r≤T 0≤r≤T i.e., to assess whether the range of the sequence of partial sums is large enough. But this is also exactly the basis of the popular rescaled range procedure used to test the null hypothesis of short-memory versus the alternative of long memory (see, in particular, Hurst, 1951, Mandelbrot and Taqqu, 1979, Bhattacharya et al., 1983, and Lo, 1991). This is symptomatic of the same problem discussed above from a slightly different angle; structural change and long memory imply similar features in the data and, accordingly, are hard to distinguish. In particular, evidence for long memory can be caused by the presence of structural changes, and vice versa. The intuition is basically the same as the message in Perron (1990), i.e., level shifts induce persistent features in the data. This problem has recently received a lot of attention, especially in the finance literature concerning the characteristics of stock returns volatility (see, in particular, Diebold and Inoue, 2001, Gourieroux and Jasiak, 2001, Granger and Hyung, 2004, Lobato and Savin, 1998, and Perron and Qu, 2004). 3 Estimation and Inference about Break Dates In this section we discuss issues related to estimation and inference about the break dates in a linear regression framework. The emphasis is on describing methods that are most useful in applied econometrics, explaining the relevance of the conditions imposed and sketching some important theoretical steps that help to understand particular assumptions made. Following Bai (1997a) and Bai and Perron (1998), the main framework of analysis can be described by the following multiple linear regression with m breaks (or m + 1 regimes): yt = x0t β + zt0 δ j + ut , t = Tj−1 + 1, ..., Tj , (2) for j = 1, ..., m + 1. In this model, yt is the observed dependent variable at time t; both xt (p × 1) and zt (q × 1) are vectors of covariates and β and δj (j = 1, ..., m + 1) are the corresponding vectors of coefficients; ut is the disturbance at time t. The indices (T1 , ..., Tm ), or the break points, are explicitly treated as unknown (the convention that T0 = 0 and Tm+1 = T is used). The purpose is to estimate the unknown regression coefficients together with the break points when T observations on (yt , xt , zt ) are available. This is a partial 7 structural change model since the parameter vector β is not subject to shifts and is estimated using the entire sample. When p = 0, we obtain a pure structural change model where all the model’s coefficients are subject to change. Note that using a partial structural change models where only some coefficients are allowed to change can be beneficial both in terms of obtaining more precise estimates and also in having can be more powerful tests. The multiple linear regression system (2) may be expressed in matrix form as Y = Xβ + Z̄δ + U, where Y = (y1 , ..., yT )0 , X = (x1 , ..., xT )0 , U = (u1 , ..., uT )0 , δ = (δ 01 , δ 02 , ..., δ 0m+1 )0 , and Z̄ is the matrix which diagonally partitions Z at (T1 , ..., Tm ), i.e. Z̄ = diag(Z1 , ..., Zm+1 ) with Zi = (zTi−1 +1 , ..., zTi )0 . We denote the true value of a parameter with a 0 superscript. In 0 0 particular, δ 0 = (δ 01 , ..., δ 0m+1 )0 and (T10 , ..., Tm0 ) are used to denote, respectively, the true values of the parameters δ and the true break points. The matrix Z̄ 0 is the one which diagonally partitions Z at (T10 , ..., Tm0 ). Hence, the data-generating process is assumed to be Y = Xβ 0 + Z̄ 0 δ 0 + U. (3) The method of estimation considered is based on the least-squares principle. For each mpartition (T1 , ..., Tm ), the associated least-squares estimates of β and δ j are obtained by minimizing the sum of squared residuals 0 (Y − Xβ − Z̄δ) (Y − Xβ − Z̄δ) = m+1 X Ti X [yt − x0t β − zt0 δ i ]2 . i=1 t=Ti−1 +1 Let β̂({Tj }) and δ̂({Tj }) denote the estimates based on the given m-partition (T1 , ..., Tm ) denoted {Tj }. Substituting these in the objective function and denoting the resulting sum of squared residuals as ST (T1 , ..., Tm ), the estimated break points (T̂1 , ..., T̂m ) are such that (T̂1 , ..., T̂m ) = argmin(T1 ,...,Tm ) ST (T1 , ..., Tm ), (4) where the minimization is taken over some set of admissible partitions (see below). Thus the break-point estimators are global minimizers of the objective function. The regression parameter estimates are the estimates associated with the m-partition {T̂j }, i.e. β̂ = β̂({T̂j }), δ̂ = δ̂({T̂j }). This framework includes many contributions made in the literature as special cases depending on the assumptions imposed; e.g., single change, changes in the mean of a stationary 8 process, etc. However, the fact that the method of estimation is based on the least-squares principle implies that, even if changes in the variance of ut are allowed, provided they occur at the same dates as the breaks in the parameters of the regression, such changes are not exploited to increase the precision of the break date estimators. This is due to the fact that the least-squares method imposes equal weights on all residuals. Allowing different weights, as needed when accounting for changes in variance, requires adopting a quasi-likelihood framework, see below. 3.1 The assumptions and their relevance To obtain theoretical results about the consistency and limit distribution of the break dates, some conditions need to be imposed on the regressors, the errors, the set of admissible partitions and the break dates. To our knowledge, the most general set of assumptions, as far as applications are concerned, are those in Perron and Qu (2005). Some are simply technical (e.g., invertibility requirements), while others restrict the potential applicability of the results. Hence, it is useful to discuss the latter. • Assumption on the regressors: Let wt = (x0t , zt0 )0 , for i = 0, ..., m, (1/li ) Qi (v) a non-random positive definite matrix uniformly in v ∈ [0, 1]. PTi0 +[li v] t=Ti0 +1 wt wt0 →p This assumption allows the distribution of the regressors to vary across regimes. It, however, requires the data to be weakly stationary stochastic processes. It can, however, be relaxed substantially, though the technical proofs then depend on the nature of the relaxation. For instance the scaling used forbids trending regressors, unless they are of the form {1, (t/T ), ..., (t/T )p }, say, for a polynomial trend of order p. Casting trend functions in this form can deliver useful results in many cases. However, there are instances where specifying trends in unscaled form, i.e., {1, t, ..., tp }, can deliver much better results, especially if level and trend slope changes occur jointly. Results using unscaled trends with p = 1 are presented in Perron and Zhu (2005). A comparison of their results with other trend specifications is presented in Deng and Perron (2005). Another important restriction is implied by the requirement that the limit be a fixed matrix, as opposed to permitting it to be stochastic. This, along with the scaling, precludes integrated processes as regressors (i.e., unit roots). In the single break case, this has been relaxed by Bai, Lumsdaine and Stock (1998) who considered, among other things, structural changes in cointegrated relationships. Consistency still applies but the rate of convergence and limit distributions of the estimates are different. Another context in which integrated 9 regressors play a role is the case of changes in persistence. Chong (2001) considered an AR(1) model where the autoregressive coefficient takes a value less than one before some break date and value one after, or vice versa. He showed consistency of the estimate of the break date and derived the limit distribution. When the move is from stationarity to unit root, the rate of convergence is the same as in the stationary case (though the limit distribution is different), but interestingly, the rate of convergence is faster when the change is from a unit root to a stationary process. No results are yet available for multiple structural changes in regressions involving integrated regressors, though work is in progress on this issue. The problem here is more challenging because the presence of regressors with a unit root, whose coeffients are subject to change, implies break date estimates with limit distributions that are not independent, hence all break dates need to be evaluated jointly. The sequence {wt ut } satisfies the following set of conditions. • Assumptions on the errors: Let the Lr -norm of a random matrix X be defined by P P kXkr = ( i j E |Xij |r )1/r for r ≥ 1. (Note that kXk is the usual matrix norm or the Euclidean norm of a vector.) With {Fi : i = 1, 2, ..} a sequence of increasing σ-fields, it is assumed that {wi ui , Fi } forms a Lr -mixingale sequence with r = 2 + δ for some δ > 0. That is, there exist nonnegative constants {ci : i ≥ 1} and {ψj : j ≥ 0} such that ψj ↓ 0 as j → ∞ and for all i ≥ 1 and j ≥ 0, we have: (a) kE(wi ui |Fi−j )kr ≤ ci ψj , (b) kwi ui − E(wi ui |Fi+j )kr ≤ ci ψj+1 . Also assume (c) maxi ci ≤ K < ∞, (d) P∞ 1+k ψj < ∞, (e) kzi k2r < M < ∞ and kui k2r < N < ∞ for some K, M, N > 0. j=0 j This imposes mild restrictions on the vector wt ut and permits a wide class of potential correlation and heterogeneity (including conditional heteroskedasticity) and also allows lagged dependent variables. It rules out errors that have unit roots. In this latter case, if the regressors are stationary (or satisfy the Assumption on the regressors stated above), the estimates of the break dates are inconsistent (see Nunes et al., 1995). However, unit root errors can be of interest; for example when testing for a change in the deterministic component of the trend function for an integrated series, in which case the estimates are consistent (see Perron and Zhu, 2005). The set of conditions listed above are not the weakest possible. For example, Lavielle and Moulines (2000) allow the errors to be strongly dependent, i.e., long memory processes such as fractionally integrated ones are permitted. They, however, consider only the case of multiple changes in the mean. Technically, what is important is to be able to establish a generalized Hájek-Rényi (1955) type inequality for the zero mean variables zt ut , as well as a Functional Central Limit Theorem and a Law of Large Numbers. 10 • Assumption on the minimization problem: The minimization problem defined by (4) is taken over all possible partitions such that Ti − Ti−1 ≥ T for some > 0. This requirement was introduced in Bai and Perron (1998) only for the case where lagged dependent variables were allowed. When serial correlation in the errors was allowed they introduced the requirement that the errors be independent of the regressors at all leads and lags. This is obviously a strong assumption which is often violated in practice. The assumption on the errors listed above are much weaker, in particular concerning the relation between the errors and regressors. This weakening comes at the cost of a mild strengthening on the assumption about the regressors and the introduction of the restriction on the minimization problem. Note that the latter is also imposed in Lavielle and Moulines (2000), though they note that it can be relaxed with stronger conditions on zt ut or by constraining the estimates to lie in a compact set. • Assumption on the break dates: Ti0 = [T λ0i ] , where 0 < λ01 < ... < λ0m < 1. This assumption specifies that the break dates are asymptotically distinct. While it is standard, it is surprisingly the most controversial for some. The reason is that it dictates the asymptotic framework adopted. With this condition, when the sample size T increases, all segments increase in length in the same proportions to each other. Oftentimes, an asymptotic analysis is viewed as a thought experiment about what would happen if we were able to collect more and more data in the future. If one adheres to this view, then the last regime should increase in length (assuming no other break will occur in the future) and all other segments then become a negligible proportion of the total sample. Hence, as T increases, we would find ourselves with a single segment, in which case the framework becomes useless. The fact is that any asymptotic analysis is simply a device to enable us to get useful information about the structure, which can help us understand the finite sample distributions, and hopefully to deliver good approximations. The adoption of any asymptotic framework should only be evaluated on this basis, no matter how ad hoc it may seem at first sight. Here, with say a sample of size 100 and 3 breaks occurring at dates 25, 50 and 75, all segments are a fourth of the total sample. It therefore makes sense to use an asymptotic framework whereby this feature is preserved. The same comments apply to contexts in which some parameters are made local to some boundary as the sample size increases. No claim whatsoever is made that the parameter would actually change if more data were collected, yet such a device has been found to be of great use and to provide very useful approximations. This applies to local 11 asymptotic power function, roots local to unity or shrinking size of shifts as we will discuss later. Having said that, it does not mean that the asymptotic framework that is adopted in this literature is the only one useful or even the best. For example, it is conceivable that an asymptotic theory whereby more and more data are added keeping a fixed span of data would be useful as well. However, such a continuous time limit distribution has not yet appeared in the structural change context. Under these conditions, the main theoretical results are that the break fractions λ0i are consistently estimated, i.e., λ̂i ≡ (T̂i /T ) →p λ0i and that the rate of convergence is T . More precisely, for every ε > 0, there exists a C < ∞, such that for large T , P (|T (λ̂i − λ0i )| > C∆−2 i ) < ε (5) for every i = 1, ..., m, where ∆i = δ i+1 − δi . Note that the estimates of the break dates are not consistent themselves, but the differences between the estimates and the true values are bounded by some constant, in probability. Also, this implies that the estimates of the other parameters have the same distribution as would prevail if the break dates were known. Kurozumi and Arai (2004) obtain a similar result with I(1) regressors for a cointegrated model subject to a change in some parameters of the cointegrating vector. They show the estimate of the break fraction obtained by minimizing the sum of squared residuals from the static regression to converge at a fast enough rate for the estimates of the parameters of the model to asymptotically unaffected by the estimation of the break date. 3.2 Allowing for restrictions on the parameters Perron and Qu (2005) approach the issues of multiple structural changes in a broader framework whereby arbitrary linear restrictions on the parameters of the conditional mean can be imposed in the estimation. The class of models considered is y = Z̄δ + u where Rδ = r with R a k by (m + 1)q matrix with rank k and r, a k dimensional vector of constants. The assumptions are the same as discussed above. Note first that there is no need for a distinction between variables whose coefficients are allowed to change and those whose coefficients are not allowed to change. A partial structural change model can be obtained as a special case 12 by specifying restrictions that impose some coefficients to be identical across all regimes. This is a useful generalization since it permits a wider class of models of practical interests; for example, a model which specifies a number of states less than the number of regimes (with two states, the coefficients would be the same in odd and even regimes). Or it could be the case that the value of the parameters in a specific segment is known. Also, a subset of coefficients may be allowed to change over only a limited number of regimes. Perron and Qu (2005) show that the same consistency and rate of convergence results hold. Moreover, an interesting result is that the limit distribution (to be discussed below) of the estimates of the break dates are unaffected by the imposition of valid restrictions. They document, however, that improvements can be obtained in finite samples. But the main advantage of imposing restrictions is that much more powerful tests are possible. 3.3 Method to Compute Global Minimizers We now briefly discuss issues related to the estimation of such models, in particular when multiple breaks are allowed. What are needed are global minimizers of the objective function (4). A standard grid search procedure would require least squares operations of order O(T m ) and becomes prohibitive when the number of breaks is greater than 2, even for relatively small samples. Bai and Perron (2003a) discuss a method based on a dynamic programming algorithm that is very efficient. Indeed, the additional computing time needed to estimate more than two break dates is marginal compared to the time needed to estimate a two break model. The basis of the method, for specialized cases, is not new and was considered by Guthery (1974), Bellman and Roth (1969) and Fisher (1958). A comprehensive treatment was also presented in Hawkins (1976). Consider the case of a pure structural change model. The basic idea of the approach becomes fairly intuitive once it is realized that, with a sample of size T , the total number of possible segments is at most T (T + 1)/2 and is therefore of order O(T 2 ). One then needs a method to select which combination of segments (i.e., which partition of the sample) yields a minimal value of the objective function. This is achieved efficiently using a dynamic programming algorithm. For models with restrictions (including the partial structural change model), an iterative procedure is available, which in most cases requires very few iterations (see Bai and Perron, 2003, and Perron and Qu, 2005, who make available Gauss codes to perform these and other tasks). Hence, even with large samples, the computing cost to estimate models with multiple structural changes should be considered minimal. 13 3.4 The limit distribution of the estimates of the break dates With the assumptions on the regressors, the errors and given the asymptotic framework adopted, the limit distributions of the estimates of the break dates are independent of each other. Hence, for each break date, the analysis becomes exactly the same as if a single break has occurred. The intuition behind this feature is first that the distance between each break increases at rate T as the sample size increases. Also, the mixing conditions on the regressors and errors impose a short memory property so that events that occur a long enough time apart are independent. This independence property is unlikely to hold if the data are integrated but such an analysis is yet to be completed. We shall not reproduce the results in details but simply describe the main qualitative feature and the practical relevance of the required assumptions. The reader is referred to Bai (1997a) and Bai and Perron (1998, 2003a), in particular. Also, confidence intervals for the break dates need not be based on the limit distributions of the estimates. Other approaches are possible, for example by inverting a suitable test (e.g., Elliott and Müller, 2004, for an application in the linear model using a locally best invariant test). For a review of alternative methods, see Siegmund (1988). The limit distribution of the estimates of the break dates depends on: a) the magnitude of the change in coefficients (with larger changes leading to higher precision, as expected), b) the (limit) sample moment matrices of the regressors for the segments prior to and after the true break date (which are allowed to be different); c) the so-called ‘long-run’ variance of {wt ut }, which involves potential serial correlation in the errors (and which again is allowed to be different prior to and after the break); d) whether the regressors are trending or not. In all cases, all relevant nuisance parameters can be consistently estimated and the appropriate confidence intervals constructed. A feature of interest is that the confidence intervals need not be symmetric given that the data and errors can have different properties before and after the break. To get an idea of the importance of particular assumptions needed to derive the limit distribution, it is instructive to look at a simple case with i.i.d. errors ut and a single break (for details, see Bai, 1997a). Then the estimate of the break satisfies, £ ¤ T̂1 = arg min SSR(T1 ) = arg max SSR(T10 ) − SSR(T1 ) Using the fact that, given the rate of convergence result (5), the inequality |T̂1 − T10 | < C∆−2 is satisfied with probability one in large samples (here, ∆ = δ 2 − δ 1 ). Hence, we can restrict 14 the search over the compact set C(∆) = {T1 : |T1 − T10 | < C∆−2 }. Then for T1 < T10 , 0 SSR(T10 ) − SSR(T1 ) = −∆0 T1 X 0 T1 X zt zt0 ∆ + 2∆0 t=T1 +1 zt ut + op (1) (6) zt ut + op (1) (7) t=T1 +1 and, for T1 > T10 , SSR(T10 ) 0 − SSR(T1 ) = −∆ T1 X zt zt0 ∆ t=T10 +1 T1 X 0 − 2∆ t=T10 +1 The problem is that, with |T1 − T10 | bounded, we cannot apply a Law of Large Numbers or a Central Limit Theorem to approximate the sums above with something that does not depend on the exact distributions of zt and ut . Furthermore, the distributions of these sums depend on the exact location of the break. Now let 0 W1 (m) = −∆ 0 X zt zt0 ∆ 0 + 2∆ t=m+1 for m < 0 and W2 (m) = −∆0 m X 0 X zt ut t=m+1 zt zt0 ∆ + 2∆0 m X zt ut t=1 t=1 for m > 0. Finally, let W (m) = W1 (m) if m < 0, and W (m) = W2 (m) if m > 0 (with W (0) = 0). Now, assuming a strictly stationary distribution for the pair {zt , ut }, we have that SSR(T10 ) − SSR(T1 ) = W (T1 − T10 ) + op (1) i.e., the assumption of strict stationarity allows us to get rid of the dependence of the distribution on the exact location of the break. Assuming further that (∆0 zt )2 ± (∆0 zt )ut has a continuous distribution ensures that W (m) has a unique maximum. So that T̂1 − T10 →d arg max W (m). m An important early treatment of this result for a sequence of i.i.d. random variables is Hinkley (1970). See also Feder (1975) for segmented regressions that are continuous at the time of break, Bhattacharya (1987) for maximum likelihood estimates in a multi-parameter case and Bai (1994) for linear processes. Now the issue is that of getting rid of the dependence of this limit distribution on the exact distribution of the pair (zt , ut ). Looking at (6) and (7), what we need is for the 15 difference T1 − T10 to increase as the sample size increases, then a Law of Large Numbers and a Functional Central Limit Theorem can be applied. The trick is to realize that from the convergence rate result (5), the rate of convergence of the estimate will be slower if the change in the parameters ∆i gets smaller as the sample size increases, but does so slowly enough for the estimated break fraction to remain consistent. Early applications of this framework are Yao (1987) in the context of a change in distribution for a sequence of i.i.d. random variables, and Picard (1985) for a change in an autoregressive process. Letting ∆ = ∆T to highlight the fact the change in the parameters depends on the sample size, this leads to the specification ∆T = ∆0 vT where vT is such that vT → 0 and T (1/2)−α vT → ∞ for some α ∈ (0, 1/2). Under these specifications, we have from (5) that T̂1 − T10 = Op (T 1−2α ). Hence, we can restrict the search to those values T1 such that T1 = T10 + [svT−2 ] for some fixed s. We can write (6) as 0 0 SSR(T10 ) − SSR(T1 ) = −∆00 vT2 T1 X zt zt0 ∆ + 2∆00 vT T1 X zt ut + op (1) t=T1 +1 t=T1 +1 The next steps depend on whether the zt includes trending regressors. Without trending regressors, the following assumptions are imposed (in the case with ut is i.i.d.) 0 • Assumptions for limit distribution: Let ∆Ti0 = Ti0 − Ti−1 , then as ∆Ti0 → ∞: a) 0 0 0 0 PTi−1 +[s∆Ti ] 0 PTi−1 +[s∆Ti ] 2 (∆Ti0 )−1 t=T zt zt →p sQi , b) (∆Ti0 )−1 t=T ut →p sσ 2i 0 +1 0 +1 i−1 i−1 These imply that 0 +[s∆T 0 ] Ti−1 i (∆Ti0 )−1/2 X 0 +1 t=Ti−1 zt ut ⇒ Bi (s) where Bi (s) is a multivariate Gaussian process on [0, 1] with mean zero and covariance E[Bi (s)Bi (u)] = min{s, u}σ 2i Qi . Hence, for s < 0 SSR(T10 ) − SSR(T10 + [svT−2 ]) = −|s|∆00 Q1 ∆0 + 2(∆00 Q1 ∆0 )1/2 W1 (−s) + op (1) where W1 (−s) is a Weiner process defined on (0, ∞). A similar analysis holds for the case s > 0 and for more general assumptions on ut . But this suffices to make clear that under these assumptions, the limit distribution of the estimate of the break date no longer depends on the exact distribution of zt and ut but only on quantities that can be consistently estimated. For details, see Bai (1997) and Bai and Perron (1998, 2003a). With trending regressors, the assumption stated above is violated but a similar result is still possible (assuming trends of 16 the form (t/T )) and the reader is referred to Bai (1997a) for the case where zt is a polynomial time trend. So, what do we learn from these asymptotic results? First, for large shifts, the distributions of the estimates of the break dates depend on the exact distributions of the regressors and errors even if the sample is large. When shifts are small, we can expect the distributions of the estimates of the break dates to be insensitive to the exact nature of the distributions of the regressors and errors. The question is then, how small do the changes have to be? There is no clear cut solution to this problem and the answer is case specific. The simulations in Bai and Perron (2005) show that the shrinking shifts asymptotic framework provides useful approximations to the finite sample distribution of the estimated break dates, but their simulation design uses normally distributed errors and regressors. The coverage rates are adequate, in general, unless the shifts are quite small in which case the confidence interval is too narrow. The method of Elliott and Müller (2004), based on inverting a test, works better in that case. However, with such small breaks, tests for structural change will most likely fail to detect a change, in which case most practitioners would not pursue the analysis further and consider the construction of confidence intervals. On the other hand, Deng and Perron (2005) show that the shrinking shift asymptotic framework leads to a poor approximation in the context of changes in a linear trend function and that the limit distribution based on a fixed magnitude of shift is highly preferable. 3.5 Estimating Breaks one at a time Bai (1997b) and Bai and Perron (1998) showed that it is possible to consistently estimate all break fractions sequentially, i.e., one at a time. This is due to the following result. When estimating a single break model in the presence of multiple breaks, the estimate of the break fraction will converge to one of the true break fractions, the one that is dominant in the sense that taking it into account allows the greatest reduction in the sum of squared residuals. Then, allowing for a break at the estimated value, a second one break model can be applied which will consistently estimate the second dominating break, and so on (in the case of two breaks that are equally dominant, the estimate will converge with probability 1/2 to either break). Fu and Cornow (1990) presented an early account of this property for a sequence of Bernoulli random variables when the probability of obtaining a 0 or a 1 is subject to multiple structural changes (see also, Chong, 1995). Bai (1997b) considered the limit distribution of the estimates and shows that they are not the same as those obtained when estimating all break dates simultaneously. In particular, 17 except for the last estimated break date, the limit distributions of the estimates of the break dates depend on the parameters in all segments of the sample (when the break dates are estimated simultaneously, the limit distribution of a particular break date depends on the parameters of the adjacent regimes only). To remedy this problem, Bai (1997b) suggested a procedure called ‘repartition’. This amounts to re-estimating each break date conditional on the adjacent break dates. For example, let the initial estimates of the break dates be denoted by (T̂1a , ..., T̂ma ). The second round estimate for the ith break date is obtained by fitting a a a (with + 1 and ending at date T̂i+1 one break model to the segment starting at date T̂i−1 a = T ). The estimates obtained from this repartition the convention that T̂0a = 0 and T̂m+1 procedure have the same limit distributions as those obtained simultaneously, as discussed above. 3.6 Estimation in a system of regressions The problem of estimating structural changes in a system of regressions is relatively recent. Bai et al. (1998) considered asymptotically valid inference for the estimate of a single break date in multivariate time series allowing stationary or integrated regressors as well as trends. They show that the width of the confidence interval decreases in an important way when series having a common break are treated as a group and estimation is carried using a quasi maximum likelihood (QML) procedure. Also, Bai (2000) considers the consistency, rate of convergence and limiting distribution of estimated break dates in a segmented stationary VAR model estimated again by QML when the breaks can occur in the parameters of the conditional mean, the covariance matrix of the error term or both. Hansen (2003) considers multiple structural changes in a cointegrated system, though his analysis is restricted to the case of known break dates. To our knowledge, the most general framework is that of Qu and Perron (2005) who consider models of the form yt = (I ⊗ zt0 )Sβ j + ut for Tj−1 + 1 ≤ t ≤ Tj (j = 1, ..., m + 1), where yt is an n-vector of dependent variables and zt is a q-vector that includes the regressors from all equations. The vector of errors ut has mean 0 and covariance matrix Σj . The matrix S is of dimension nq by p with full column rank. Though, in principle it is allowed to have entries that are arbitrary constants, it is usually a selection matrix involving elements that are 0 or 1 and, hence, specifies which regressors appear in each equation. The set of basic parameters in regime j consists of the p vector β j and of Σj . They also allow for the imposition of a set of r restrictions of 18 the form g(β, vec(Σ)) = 0, where β = (β 01 , ..., β 0m+1 )0 , Σ = (Σ1 , ..., Σm+1 ) and g(·) is an r dimensional vector. Both within- and cross-equation restrictions are allowed, and in each case within or across regimes. The assumptions on the regressors zt and the errors ut are similar to those discussed in Section 3.1 (properly extended for the multivariate nature of the problem). Hence, the framework permits a wide class of models including VAR, SUR, linear panel data, change in means of a vector of stationary processes, etc. Models with integrated regressors (i.e, models with cointegration) are not permitted. Allowing for general restrictions on the parameters β j and Σj permits a very wide range of special cases that are of practical interest: a) partial structural change models where only a subset of the parameters are subject to change, b) block partial structural change models where only a subset of the equations are subject to change; c) changes in only some element of the covariance matrix Σj (e.g., only variances in a subset of equations); d) changes in only the covariance matrix Σj , while β j is the same for all segments; e) ordered break models where one can impose the breaks to occur in a particular order across subsets of equations; etc. The method of estimation is again QML (based on Normal errors) subject to the restrictions. They derive the consistency, rate of convergence and limit distribution of the estimated break dates. They obtain a general result stating that, in large samples, the restricted likelihood function can be separated in two parts: one that involves only the break dates and the true values of the coefficients, so that the estimates of the break dates are not affected by the restrictions imposed on the coefficients; the other involving the parameters of the model, the true values of the break dates and the restrictions, showing that the limiting distributions of these estimates are influenced by the restrictions but not by the estimation of the break dates. The limit distribution results for the estimates of the break dates are qualitatively similar to those discussed above, in particular they depend on the true parameters of the model. Though only root-T consistent estimates of (β, Σ) are needed to construct asymptotically valid confidence intervals, it is likely that more precise estimates of these parameters will lead to better finite sample coverage rates. Hence, it is recommended to use the estimates obtained imposing the restrictions even though imposing restrictions does not have a first-order effect on the limiting distributions of the estimates of the break dates. To make estimation possible in practice, for any number of breaks, they present an algorithm which extends the one discussed in Bai and Perron (2003a) using, in particular, an iterative GLS procedure to construct the likelihood function for all possible segments. The theoretical analysis shows how substantial efficiency gains can be obtained by casting 19 the analysis in a system of regressions. In addition, the result of Bai et al. (1998), that when a break is common across equations the precision increases in proportion to the number of equations, is extended to the multiple break case. More importantly, the precision of the estimate of a particular break date in one equation can increase when the system includes other equations even if the parameters of the latter are invariant across regimes. All that is needed is that the correlation between the errors be non-zero. While surprising, this result is ex-post fairly intuitive since a poorly estimated break in one regression affects the likelihood function through both the residual variance of that equation and the correlation with the rest of the regressions. Hence, by including ancillary equations without breaks, additional forces are in play to better pinpoint the break dates. Qu and Perron (2005) also consider a novel (to our knowledge) aspect to the problem of multiple structural changes labelled “locally ordered breaks”. Suppose one equation is a policy-reaction function and the other is some market-clearing equation whose parameters are related to the policy function. According to the Lucas critique, if a change in policy occurs, it is expected to induce a change in the market equation but the change may not be simultaneous and may occur with a lag, say because of some adjustments due to frictions or incomplete information. However, it is expected to take place soon after the break in the policy function. Here, the breaks across the two equations are “ordered” in the sense that we have the prior knowledge that the break in one equation occurs after the break in the other. The breaks are also “local” in the sense that the time span between their occurrence is expected to be short. Hence, the breaks cannot be viewed as occurring simultaneously nor can the break fractions be viewed as asymptotically distinct. An algorithm to estimate such models is presented. Also, a framework to analyze the limit distribution of the estimates is introduced. Unlike the case with asymptotically distinct breaks, here the distributions of the estimates of the break dates need to be considered jointly. 4 Testing for structural change In this section, we review testing procedures related to structural changes. The following issues are covered: tests obtained without modelling any break, tests for a single structural change obtained by explicitly modelling a break, the problem of non monotonic power functions, and tests for multiple structural changes, tests valid with I(1) regressors, and tests for a change in slope valid allowing the noise component to be I(0) or I(1). 20 4.1 Tests for a single change without modelling the break Historically, tests for structural change were first devised based on procedures that did not estimate a break point explicitly. The main reason is that the distribution theory for the estimates of the break dates (obtained using a least-squares or likelihood principle) was not available and the problem was solved only for few special cases (see, e.g., Hawkins, 1977, Kim and Siegmund, 1989). Most tests proposed were of the form of partial sums of residuals. We have already discussed in Section 2, the Q test based on the average of partial sums of residuals (e.g., demeaned data for a change in mean) and the rescaled range test based on the range of partial sums of similarly demeaned data. Another statistic which has played an important role in theory and applications is the CUSUM test proposed by Brown, Durbin and Evans (1975). This test is based on the maximum of partial sums of recursive residuals. More precisely, for a linear regression with k regressors yt = x0t β + ut it is defined by ¯ ¯ Pr ¯ t=k+1 e v t¯ ¯ /(1 + 2 r − k ) CUSU M = max ¯¯ √ k+1<r≤T σ̂ T − k ¯ T −k where σ̂ 2 is a consistent estimate of the variance of ut (usually the sum of squared OLS residuals although, to increase power, one can use the sum of squared demeaned recursive residuals, as suggested by Harvey, 1975) and e vt are the recursive residuals defined by vet = (yt − x0t β̂ t−1 )/ft 0 ft = (1 + x0t (Xt−1 Xt−1 )xt )1/2 where Xt−1 contains the observations on the regressors up to time t − 1 and β̂ t−1 is the OLS estimate of β using data up to time t − 1. For an extensive review of the use of recursive methods in the analysis of structural change, see Dufour (1982) (see also Dufour and Kiviet, 1996, for finite sample inference in a regression model with a lagged dependent variable). The limit distribution of the CUSUM test can be expressed in terms of the maximum of a weighted Wiener process, i.e., ¯ ¯ ¯ W (r) ¯ ¯ CUSUM ⇒ sup ¯¯ ¯ 0≤r≤1 1 + 2r where W (r) is a unit Wiener process defined on (0, 1), see Sen (1982). Also, it was shown by Kramer, Ploberger and Alt (1988) that the limit distribution remains valid even if lagged 21 dependent variables are present as regressors. Furthermore, Ploberger and Kramer (1992) showed that using OLS residuals instead of recursive residuals yields a valid test, though the limit distribution under the null hypothesis is different (expressed in terms of a Brownian bridge, W (r) − rW (1), instead of a Wiener process). Their simulations showed the OLS based CUSUM test to have higher power except for shifts that occur early in the sample (the standard CUSUM tests having small power for late shifts). An alternative, also suggested by Brown, Durbin and Evans (1975), is the CUSUM of squares test. It takes the form: ¯ ¯ ¯ (r) r − k ¯ ¯ CUSSQ = max ¯¯ST − k+1<r≤T T − k¯ where (r) ST = à r X t=k+1 ! à vet2 / T X t=k+1 vet2 ! Ploberger and Kramer (1990) considered the local power functions of the CUSUM and CUSUM of squares. The former has non-trivial local asymptotic power unless the mean regressor is orthogonal to all structural changes. On the other hand, the latter has only trivial local power (i.e., power equal to size) for local changes that specify a one-time change in the coefficients (see also Deshayes and Picard, 1986). This suggests that the CUSUM test should be preferred, a conclusion we shall revisit below. Another variant using partial sums is the fluctuations test of Ploberger, Kramer and Kontrus (1989) which looks at the maximum difference between the OLS estimate of β using the full sample and the OLS estimates using subsets of the sample from the first observation to some date t, ranging from t = k to T . A similar test for a change in the slope of a linear trend function is analyzed in Chu and White (1992). Also, Chu, Hornik and Kuan (1995) looked at the maximum of moving sums of recursive and least-squares residuals. 4.2 Non monotonic power functions in finite samples All tests discussed above are consistent for given fixed values in the relevant set of alternative hypotheses. All (except the CUSUM of squares) are, however, subject to the following problem. For a given sample size, the power function can be non monotonic in the sense that it can decrease and even reach a zero value as the alternative considered becomes further away from the null value. This was shown by Perron (1991) for the Q statistic and extended to a wide range of tests in a comprehensive analysis by Vogelsang (1999). 22 This was illustrated using a basic shift in mean process or a shift in the slope of a linear trend (for some statistics designed for that alternative). In the change in mean case, with a single shift occurring, it was shown that the power of the tests discussed above eventually decreases as the magnitude of the shift increases and can reach zero. This decrease in power can be especially pronounced and effective with smaller mean shifts when a lagged dependent variable is included as a regressor to account for potential serial correlation in the errors. The basic reason for this feature is the need to estimate the variance of the errors (or the spectral density function at frequency zero when correlation in the errors is allowed) to properly scale the statistics. Since no break is directly modelled, one needs to estimate this variance using least-squares or recursive residuals that are ‘contaminated’ by the shift under the alternative. As the shift gets larger, the estimate of the scale gets inflated with a resulting loss in power. With a lagged dependent variable, the problem is exacerbated because the shift induces a bias of the autoregressive coefficient towards one (Perron, 1989, 1990). See Vogelsang (1999) for a detailed treatment that explains how each test is differently affected, that also provides empirical illustrations of this problem showing its practical relevance. Crainiceanu and Vogelsang (2001) also show how the problem is exacerbated when using estimates of the scale factor that allow for correlation, e.g., weighted sums of the autocovariance function. The usual methods to select the bandwidth (e.g., Andrews, 1991) will choose a value that is severely biased upward and lead to a decrease in power. With change in slope, the bandwidth increases at rate T and the tests become inconsistent. This is a troubling feature since tests that are consistent and have good local asymptotic properties can perform rather badly globally. In simulations reported in Perron (2005), this feature does not occur for the CUSUM of squares test. This leads us to the curious conclusion that the test with the worst local asymptotic property (see above) has the better global behavior. Methods to overcome this problem have been suggested by Altissimo and Corradi (2003) and Juhl and Xiao (2005). They suggest using non-parametric or local averaging methods where the mean is estimated using data in a neighborhood of a particular data point. The resulting estimates and tests are, however, very sensitive to the bandwidth used. A large one leads to properly sized tests in finite samples but with low power, and a small bandwidth leads to better power but large size distortions. There is currently no reliable method to appropriately chose this parameter in the context of structural changes. 23 4.3 Tests that allow for a single break The discussion above suggests that to have better tests for the null hypothesis of no structural change versus the alternative hypothesis that changes are present, one should consider statistics that are based on a regression that allows for a break. As discussed in the introduction, the suggestion by Quandt (1958, 1960) was to use the likelihood ratio test evaluated at the break date that maximizes this likelihood function. This is a non-standard problem since one parameter is only identified under the alternative hypothesis, namely the break date (see Davies, 1977, 1987, King and Shively, 1993, Andrews and Ploberger, 1994, and Hansen, 1996). The problem raised by Quandt was treated under various degrees of specificity by Deshayes and Picard (1984b), Worsley (1986), James, James and Siegmund (1987), Hawkins (1987), Kim and Siegmund (1989), Horvath (1995) and generalized by Andrews (1993a). The basic method advocated by Davies (1977), for the case in which a nuisance parameter is present only under the alternative, is to use the maximum of the likelihood ratio test over all possible values of the parameter in some pre-specified set as a test statististic. In the case of a single structural change occurring at some unknown date, this translates into the following statistic sup LRT (λ1 ) λ1 ∈Λ where LR(λ1 ) denotes the value of the likelihood ratio evaluated at some break point T1 = [T λ1 ] and the maximization is restricted over break fractions that are in Λ = [ 1 , 1 − 2 ], some subset of the unit interval [0, 1] with 1 being the lower bound and 1 − 2 the upper bound. The limit distribution of the statistic is given by sup LRT (λ1 ) ⇒ sup Gq (λ1 ) λ1 ∈Λ λ1 ∈Λ where Gq (λ1 ) = [λ1 Wq (1) − Wq (λ1 )]0 [λ1 Wq (1) − Wq (λ1 )] λ1 (1 − λ1 ) (8) with Wq (λ) a vector of independent Wiener processes of dimension q, the number of coefficients that are allowed to change (this result holds with non-trending data). Not surprisingly, the limit distribution depends on q but it also depends on Λ . This is important since the restriction that the search for a maximum value be restricted is not simply a technical requirement. It influences the properties of the test in an important way. In particular, Andrews (1993a) shows that if 1 = 2 = 0 so that no restrictions are imposed, the test diverges to 24 infinity under the null hypothesis (an earlier statement of this result in a more specialized context can be found in Deshayes and Picard, 1984a). This means that critical values grow and the power of the test decreases as 1 and 2 get smaller. Hence, the range over which we search for a maximum must be small enough for the critical values not to be too large and for the test to retain descent power, yet large enough to include break dates that are potential candidates. In the single break case, a popular choice is 1 = 2 = .15. Andrews (1993a) tabulates critical values for a range of dimensions q and for intervals of the form [ , 1 − ]. This does not imply, however, that one is restricted to imposing equal trimming at both ends of the sample. This is because the limit distribution depends on 1 and 2 only through the parameter γ = 2 (1 − 1 )/( 1 (1 − 2 )). Hence, the critical values for a symmetric trimming are also valid for some asymmetric trimmings. To better understand these results, it is useful to look at the simple one-time shift in mean of some variable yt specified by (1). For a given break date T1 = [T λ1 ], the Wald test is asymptotically equivalent to the LR test and is given by WT (λ1 ) = SSR(1, T ) − SSR(1, T1 ) − SSR(T1 + 1, T ) [SSR(1, T1 ) + SSR(T1 + 1, T )]/T where SSR(i, j) is the sum of squared residuals from regressing yt on a constant using data from date i to date j, i.e. à ! ! à j j j j X X 1 X 1 X et − yt − yt = et SSR(i, j) = j − i j − i t=i t=i t=i t=i Note that the denominator converges to σ 2 and the numerator is given by T X t=1 = ∙ à T 1X et − et T t=1 !2 − T1 X t=1 à T1 1 X et − et T1 t=1 !2 − T X t=T1 +1 à T X 1 et − et T − T1 t=T 1 !2 µ ¶¸−1 à T1 T T1 T1 −1/2 X T1 T − T1 −1/2 X T T 1− et et − T T T T t=1 t=T +1 1 !2 P1 P after some algebra. If T1 /T → λ1 ∈ (0, 1), we have T −1/2 Tt=1 et ⇒ σW (λ1 ), T −1/2 Tt=T1 +1 et = P P 1 T −1/2 Tt=1 et − T −1/2 Tt=1 et ⇒ σ[W (1) − W (λ1 )] and the limit of the Wald test is WT (λ1 ) ⇒ = 1 [λ1 W (1) − λ1 W (λ1 ) − (1 − λ1 )W (λ1 )]2 λ1 (1 − λ1 ) 1 [λ1 W (1) − W (λ1 )]2 λ1 (1 − λ1 ) 25 which is equivalent to (8) for q = 1. Andrews (1993a) also considered tests based on the maximal value of the Wald and LM tests and shows that they are asymptotically equivalent, i.e., they have the same limit distribution under the null hypothesis and under a sequence of local alternatives. All tests are also consistent and have non trivial local asymptotic power against a wide range of alternatives, namely for which the parameters of interest are not constant over the interval specified by Λ . This does not mean, however, that they all have the same behavior in finite samples. Indeed, the simulations of Vogelsang (1999) for the special case of a change in mean, showed the sup LMT test to be seriously affected by the problem of non monotonic power, in the sense that, for a fixed sample size, the power of the test can rapidly decrease to zero as the change in mean increases 1 . This is again because the variance of the errors is estimated under the null hypothesis of no change. Hence, we shall not discuss it any further. In the context of Model (2) with i.i.d. errors, the LR and Wald tests have similar properties, so we shall discuss the Wald test. For a single change, it is defined by (up to a scaling by q): µ ¶ 0 T − 2q − p δ̂ H 0 (H(Z̄ 0 MX Z̄)−1 H 0 )−1 Rδ̂ (9) sup WT (λ1 ; q) = sup k SSRk λ1 ∈Λ λ1 ∈Λ where H is the conventional matrix such that (Hδ)0 = (δ 01 −δ 02 ) and MX = I −X(X 0 X)−1 X 0 . Here SSRk is the sum of squared residuals under the alternative hypothesis, which depends on the break date T1 . One thing that is very useful with the sup WT test is that the break point that maximizes the Wald test is the same as the estimate of the break point, T̂1 ≡ [T λ̂1 ], obtained by minimizing the sum of squared residuals provided the minimization problem (4) is restricted to the set Λ , i.e., sup WT (λ1 ; q) = WT (λ̂1 ; q) λ1 ∈Λ When serial correlation and/or heteroskedasticity in the errors is permitted, things are different since the Wald test must be adjusted to account for this. In this case, it is defined by µ ¶ 1 T − 2q − p 0 0 ∗ δ̂ H (H V̂ (δ̂)H 0 )−1 H δ̂, (10) WT (λ1 ; q) = T k where V̂ (δ̂) is an estimate of the variance covariance matrix of δ̂ that is robust to serial correlation and heteroskedasticity; i.e., a consistent estimate of 1 V (δ̂) = plimT →∞ T (Z̄ 0 MX Z̄)−1 Z̄ 0 MX ΩMX Z̄(Z̄ 0 MX Z̄)−1 (11) Note that what Vogelsang (1998b) actually refers to as the sup Wald test for the static case is actually the sup LM test. For the dynamic case, it does correspond to the Wald test. 26 For example, one could use the method of Andrews (1991) based on weighted sums of autocovariances. Note that it can be constructed allowing identical or different distributions for the regressors and the errors across segments. This is important because if a variance shift occurs at the same time and is not taken into account, inference can be distorted (see, e.g., Pitarakis, 2004). In some instances, the form of the statistic reduces in an interesting way. For example, consider a pure structural change model where the explanatory variables are such that plimT −1 Z̄ 0 ΩZ̄ = hu (0)plimT −1 Z̄ 0 Z̄ with hu (0) the spectral density function of the errors ut evaluated at the zero frequency. In that case, we have the asymptotically equivalent P test (σ̂ 2 /ĥu (0))WT (λ1 ; q), with σ̂ 2 = T −1 Tt=1 û2t and ĥu (0) a consistent estimate of hu (0). Hence, the robust version of the test is simply a scaled version of the original statistic. This is the case, for instance, when testing for a change in mean as in Garcia and Perron (1996). The computation of the robust version of the Wald test (10) can be involved especially if a data dependent method is used to construct the robust asymptotic covariance matrix of δ̂. Since the break fractions are T -consistent even with correlated errors, an asymptotically equivalent version is to first take the supremum of the original Wald test, as in (9), to obtain the break points, i.e. imposing Ω = σ2 I. The robust version of the test is obtained by evaluating (10) and (11) at these estimated break dates, i.e., using WT∗ (λ̂1 ; q) instead of supλ1 ∈Λ WT∗ (λ1 ; q), where λ̂1 is obtained by minimizing the sum of squared residuals over the set Λ . This will be especially helpful in the context of testing for multiple structural changes. 4.3.1 Optimal tests The sup-LR or sup-Wald tests are not optimal, except in a very restrictive sense. Andrews and Ploberger (1994) consider a class of tests that are optimal, in the sense that they maximize a weighted average power. Two types of weights are involved. The first applies to the parameter that is only identified under the alternative. It assigns a weight function J(λ1 ) that can be given the interpretation of a prior distribution over the possible break dates or break fractions. The other is related to how far the alternative value is from the null hypothesis within an asymptotic framework that treats alternative values as being local to the null hypothesis. The dependence of a given statistic on this weight function occurs only through a single scalar parameter c. The higher the value of c, the more distant is the alternative value from the null value, and vice versa. The optimal test is then a weighted function of the standard Wald, LM or LR statistics for all permissible fixed break dates. 27 Using either of the three basic statistics leads to tests that are asymptotically equivalent. Here, we shall proceed with the version based on the Wald test (and comment briefly on the version based on the LM test). The class of optimal statistics is of the following exponential form: ¾ ½ Z 1 c −q/2 WT (λ1 ) dJ(λ1 ) exp Exp-WT (c) = (1 + c) 21+c where we recall that q is the number of parameters that are subject to change, and WT (λ1 ) is the standard Wald test defined in our context as in (9). To implement this test in practice, one needs to specify J(λ1 ) and c. A natural choice for J(λ1 ) is to specify it so that equal weights are given to all break fractions in some trimmed interval [ 1 , 1− 2 ]. For the parameter c, one version sets c = 0 and puts greatest weight on alternatives close to the null value, i.e., on small shifts; the other version specifies c = ∞, in which case greatest weight is put on large changes. This leads to two statistics that have found wide appeal. When c = ∞, the test is of an exponential form, viz. ⎛ ⎞ µ µ ¶¶ T −[T 2 ] X T1 1 ⎠ WT exp Exp-WT (∞) = log ⎝T −1 2 T T1 =[T 1 ]+1 When c = 0, the test takes the form of an average of the Wald tests and is often referred to as the Mean-WT test. It is given by Mean-WT = Exp-WT (0) = T −1 T −[T X2 ] T1 =[T 1 ]+1 WT µ T1 T ¶ The limit distributions of the tests are µZ 1− 2 ¶ ¶ 1 exp Gq (λ1 ) dλ1 2 µ Exp-WT (∞) ⇒ log Z 1− 2 1 Gq (λ1 )dλ1 Mean-WT ⇒ 1 Andrews and Ploberger (1994) presented critical values for both tests for a range of values for symmetric trimmings 1 = 2 , though as stated above they can be used for some non symmetric trimmings as well. Simulations reported in Andrews, Lee and Ploberger (1996) show that the tests perform well in practice. Relative to other tests discussed above, the Mean-WT has highest power for small shifts, though the test Exp-WT (∞) performs better for moderate to large shifts. None of them uniformly dominates the Sup-WT test and they 28 recommend the use of the Exp-WT (∞) form of the test, referred to as the Exp-Wald test below. As mentioned above both tests can equally be implemented (with the same asymptotic critical values) with the LM or LR tests replacing the Wald test. As noted by Andrews and Ploberger (1994), the Mean-LM test is closely related to Gardner’s test (discussed in Section 2). This is because, in the change in mean case, the LM test takes the form of a scaled partial sums. Given the poor properties of this test, especially with respect to large shifts when the power can reach zero, we do not recommend the asymptotically optimal tests based on the LM version. In our context, tests based on the Wald or LR statistics have similar properties. Elliott and Müller (2003) consider optimal tests for a class of models involving nonconstant coefficients which, however, rule out one-time abrupt changes. The optimality criterion relates to changes that are in a local neighborhood of the null values, i.e., for small changes. Their procedure is accordingly akin to locally best invariant tests for random variations in the parameters. The suggested procedure does not explicitly model breaks and the test is then of the ‘function of partial sums type’. It has not been documented if the test suffers from non-monotonic power. They show via simulations, with small breaks, that their test also has power against a one-time change. The simulations can also be interpreted as providing support for the conclusion that the Sup, Mean and Exp tests tailored to a one-time change also have power nearly as good as the optimal test for random variation in the parameter. For optimal tests in a Generalized Method of Moments framework, see Sowell (1996). 4.3.2 Non monotonicity in power The Sup-Wald and Exp-Wald tests have monotonic power when only one break occurs under the alternative. As shown in Vogelsang (1999), the Mean-Wald test can exhibit a nonmonotonic power function, though the problem has not been shown to be severe. All of these, however, suffer from some important power problems when the alternative is one that involves two breaks. Simulations to that effect are presented in Vogelsang (1997) in the context of testing for a shift in trend. This suggests a general principle, which remains, however, just a conjecture at this point. The principle is that any (or most) tests will exhibit non monotonic power functions if the number of breaks present under the alternative hypothesis is greater than the number of breaks explicitly accounted for in the construction of the tests. This suggests that, even though a single break test is consistent against multiple 29 breaks, substantial power gains can result from using tests for multiple structural changes. These are discussed below. 4.4 Tests for multiple structural changes The literature on tests for multiple structural changes is relatively scarce. Andrews, Lee and Ploberger (1996) studied a class of optimal tests. The Avg-W and Exp-W tests remain asymptotically optimal in the sense defined above. The test Exp-WT (c) is optimal in finite samples with fixed regressors and known variance of the residuals. Their simulations, which pertain to a single change, show the test constructed with an estimate of the variance of the residuals to have power close to the known variance case. The problem, however, with these tests in the case of multiple structural changes is practical implementation. The Avg-W and Exp-W tests require the computation of the W -test over all permissible partitions of the sample, hence the number of tests that need to be evaluated is of the order O(T m ), which is already very large with m = 2 and prohibitively large when m > 2. Consider instead the Sup-W test. With i.i.d. errors, maximizing the Wald statistic with respect to admissible break points is equivalent to minimizing the sum of squared residuals when the search is restricted to the same possible partitions of the sample. As discussed in Section 3.3, this maximization problem can be solved with a very efficient algorithm. This is the approach taken by Bai and Perron (1998) (an earlier analysis with two breaks was given in Garcia and Perron, 1996). To this date, no one knows the extent of the power loss, if any, in using the sup-W type test compared with the Avg-W and Exp-W tests. To the author’s knowledge, no simulations have been presented, presumably because of the prohibitive cost of constructing the Avg-W and Exp-W tests. In the context of model (2) with i.i.d. errors, the Wald test for testing the null hypothesis of no change versus the alternative hypothesis of k changes is given by WT (λ1 , ..., λk ; q) = µ T − (k + 1)q − p k ¶ 0 δ̂ H 0 (H(Z̄ 0 MX Z̄)−1 H 0 )−1 H δ̂ SSRk where H now is the matrix such that (Hδ)0 = (δ 01 − δ 02 , ..., δ 0k − δ 0k+1 ) and MX = I − X(X 0 X)−1 X 0 . Here, SSRk is the sum of squared residuals under the alternative hypothesis, which depends on (T1 , ..., Tk ). Note that one can allow different variance across segments when construction SSRk , see Bai and Perron (2003a) for details. The sup-W test is defined by WT (λ1 , ..., λk ; q) = WT (λ̂1 , ..., λ̂k ; q) sup (λ1 ,...,λk )∈Λk, 30 where Λ = {(λ1 , ..., λk ); |λi+1 − λi | ≥ , λ1 ≥ , λk ≤ 1 − } and (λ̂1 , ..., λ̂k ) = (T̂1 /T, ..., T̂k /T ), with (T̂1 , ..., T̂k ) the estimates of the break dates obtained by minimizing the sum of squared residuals by searching over partitions defined by the set Λ . This set dictates the minimal length of a segment. In principle, this minimal length could be different across the sample but then critical values would need to be computed on a case by case basis. When serial correlation and/or heteroskedasticity in the residuals is allowed, the test is µ ¶ 1 T − (k + 1)q − p 0 0 ∗ δ̂ H (H V̂ (δ̂)H 0 )−1 H δ̂, WT (λ1 , ..., λk ; q) = T k with V̂ (δ̂) as defined by (11). Again, the asymptotically equivalent version with the Wald test evaluated at the estimates (λ̂1 , ..., λ̂k ) is used to make the problem tractable. The limit distribution of the tests under the null hypothesis is the same in both cases, namely, def W (λ1 , ..., λk ; q) sup supWT (k; q) ⇒ sup Wk,q = (λ1 ,...,λk )∈Λ with def W (λ1 , ..., λk ; q) = k 1 X [λi Wq (λi+1 ) − λi+1 Wq (λi )]0 [λi Wq (λi+1 ) − λi+1 Wq (λi )] . k i=1 λi λi+1 (λi+1 − λi ) again, assuming non-trending data. Critical values for = 0.05, k ranging from 1 to 9 and for q ranging from 1 to 10, are presented in Bai and Perron (1998). Bai and Perron (2003b) present response surfaces to get critical values, based on simulations for this and the following additional cases (all with q ranging from 1 to 10): = .10 (k = 1, ..., 8), = .15 (k = 1, ..., 5), = .20 (k = 1, 2, 3) and = .25 (k = 1, 2). The full set of tabulated critical values is available on the author’s web page (the same sources also contain critical values for other tests discussed below). The importance of the choice of for the size and power of the test is discussed in Bai and Perron (2003a, 2005). Also discussed in Bai and Perron (2003a) are variations in the exact construction of the test that allow one to impose various restrictions on the nature of the errors and regressors, which can help improve power. 4.4.1 Double maximum tests Often, one may not wish to pre-specify a particular number of breaks to make inference. For such instances, a test of the null hypothesis of no structural break against an unknown 31 number of breaks given some upper bound M can be used. These are called the ‘double maximum tests’. The first is an equal-weight version defined by UD max WT (M, q) = max1≤m≤M WT (λ̂1 , ..., λ̂m ; q), where λ̂j = T̂j /T (j = 1, .., m) are the estimates of the break points obtained using the global minimization of the sum of squared residuals. This UD max test can be given a Bayesian interpretation in which the prior assigns equal weights to the possible number of changes (see, e.g., Andrews, Lee and Ploberger, 1996). The second test applies weights to the individual tests such that the marginal p-values are equal across values of m and is denoted W D max FT (M, q) (see Bai and Perron, 1998, for details). The choice M = 5 should be sufficient for most applications. In any event, the critical values vary little as M is increased beyond 5. Double Maximum tests can play a significant role in testing for structural changes and it are arguably the most useful tests to apply when trying to determine if structural changes are present. While the test for one break is consistent against alternatives involving multiple changes, its power in finite samples can be rather poor. First, there are types of multiple structural changes that are difficult to detect with a test for a single change (for example, two breaks with the first and third regimes the same). Second, as discussed above, tests for a particular number of changes may have non monotonic power when the number of changes is greater than specified. Third, the simulations of Bai and Perron (2005) show that the power of the double maximum tests is almost as high as the best power that can be achieved using the test that accounts for the correct number of breaks. All these elements strongly point to their usefulness. 4.4.2 Sequential tests Bai and Perron (1998) also discuss a test of versus + 1 breaks, which can be used as the basis of a sequential testing procedure. For the model with breaks, the estimated break points denoted by (T̂1 , ..., T̂ ) are obtained by a global minimization of the sum of squared residuals. The strategy proceeds by testing for the presence of an additional break in each of the ( + 1) segments (obtained using the estimated partition T̂1 , ..., T̂ ). The test amounts to the application of ( + 1) tests of the null hypothesis of no structural change versus the alternative hypothesis of a single change. It is applied to each segment containing the observations T̂i−1 + 1 to T̂i (i = 1, ..., + 1). We conclude for a rejection in favor of a model with ( + 1) breaks if the overall minimal value of the sum of squared residuals (over all segments where an additional break is included) is sufficiently smaller than the sum of squared residuals from the breaks model. The break date thus selected is the one associated 32 with this overall minimum. More precisely, the test is defined by: WT ( + 1| ) = {ST (T̂1 , ..., T̂ ) − min inf ST (T̂1 , ..., T̂i−1 , τ , T̂i , ..., T̂ )}/σ̂ 2 , 1≤i≤ +1 τ ∈Λi,η (12) where ST (·) denotes the sum of squared residuals, and Λi, = {τ ; T̂i−1 + (T̂i − T̂i−1 ) ≤ τ ≤ T̂i − (T̂i − T̂i−1 ) }, (13) and σ̂ 2 is a consistent estimate of σ 2 under the null hypothesis and also, preferably, under the alternative. Note that for i = 1, ST (T̂1 , ..., T̂i−1 , τ , T̂i , ..., T̂ ) is understood as ST (τ , T̂1 , ..., T̂ ) and for i = + 1 as ST (T̂1 , ..., T̂ , τ ). It is important to note that one can allow different distributions across segments for the regressors and the errors. The limit distribution of the test is related to the limit distribution of a test for a single change. Bai (1999) considers the same problem of testing for versus + 1 breaks while allowing the breaks to be global minimizers of the sum of squared residuals under both the null and alternative hypotheses. This leads to the likelihood ratio test defined by: sup LRT ( + 1| ) = ST (T̂1 , ..., T̂ ) − ST (T̂1∗ , ..., T̂ ∗+1 ) ST (T̂1∗ , ..., T̂ ∗+1 )/T where {T̂1 , ..., T̂ } and {T̂1∗ , ..., T̂ ∗+1 } are the sets of and +1 breaks obtained by minimizing the sum of squared residuals using and + 1 breaks models, respectively. The limit distribution of the test is different and is given by: sup LRT ( + 1| ) ⇒ max{ξ 1 , ..., ξ where ξ 1 , ..., ξ +1 +1 } are independent random variables with the following distribution q X Bi,j (s) ξ i = sup s(1 − s) ηi ≤s≤1−η i j=1 with Bi,j (s) independent standard Brownian bridges on [0, 1] and η i = /(λ0i − λ0i−1 ). Bai (1999) discusses a method to compute the asymptotic critical values and also extends the results to the case of trending regressors. These tests can form the basis of a sequential testing procedure. One simply needs to apply the tests successively starting from = 0, until a non-rejection occurs. The estimate of the number of breaks thus selected will be consistent provided the significance level used decreases at an appropriate rate. The simulation results of Bai and Perron (2005) show 33 that such an estimate of the number of breaks is much better than those obtained using information criteria as suggested by, among others, Liu et al. (1997) and Yao (1998) (see also, Perron, 1997b). But for the reasons discussed above (concerning the problems with tests that allow a number of breaks smaller than the true value), such a sequential procedure should not be applied mechanically. It is easy to have cases where the procedure stops too early. The recommendation is to first use a double maximum test to ascertain if any break is at all present. The sequential tests can then be used starting at some value greater than 0 to determine the number of breaks. An alternative sequential method is provided by Altissimo and Corradi (2003) for the case of multiple changes in mean. It consists in testing for a single break using the maximum of the absolute value of the partial sums of demeaned data. One then estimate the break date by minimizing the sum of squared residuals and continue the procedure conditional on the break date previously found, until a non-rejection occurs. They derive an appropriate bound to use a critical values for the procedure to yield a strongly consistent estimate of the number of breaks. It is unclear, however, how the procedure can be extended to the more general case with general regressors. 4.5 Tests for restricted structural changes As discussed in Section 3.2, Perron and Qu (2005) consider estimation of structural change models subject to restrictions. Consider testing the null hypothesis of 0 break versus an alternative with k breaks. Recall that the restrictions are Rδ = r. Define 0 δ H 0 (H Ve (e δ)H 0 )− He δ, WT (λ1 , ..., λk ; q) = e (14) δ) is where e δ is the restricted estimate of δ obtained using the partition {λ1 , ..., λk }, and Ve (e an estimate of the variance covariance matrix of e δ that may be constructed to be robust to heteroskedasticity and serial correlation in the errors. As usual, for a matrix A, A− denotes the generalized inverse of A. Such a generalized inverse is needed since, in general, the covariance matrix of e δ will be singular given that restrictions are imposed. Again, instead of using the sup WT (λ1 , ..., λk ; q) statistic where the supremum is taken over all possible partitions in the set Λ , we consider the asymptotically equivalent test that evaluates the e1 , ..., λ ek ; q). Wald test at the restricted estimate, i.e., WT (λ The restrictions can alternatively be parameterized by the relation δ = Sθ + s where S is a q(k + 1) by d matrix, with d the number of basic parameters in the column 34 vector θ, and s is a q(k + 1) vector of constants. Then WT (λ̂1 , ..., λ̂k ; q, S) ⇒ sup W (λ1 , ..., λk ; q, S) |λi −λi−1 |>ε with W (λ1 , ..., λk ; q, S) = W ∗0 S[S 0 (Λ ⊗ Iq )S]−1 S 0 H 0 [HS(S 0 (Λ ⊗ Iq )S 0 )−1 H 0 S 0 ]− HS[S 0 (Λ ⊗ Iq )S]−1 S 0 W ∗ where Λ = diag(λ1 , λ2 − λ1 , ..., 1 − λk ), Iq is the standard identity matrix of dimension q and the q(k + 1) vector W ∗ is defined by W ∗ = [Wq (λ1 ), Wq (λ2 ) − Wq (λ1 ), ..., Wq (1) − Wq (λk )] with Wq (r) a q vector of independent unit Wiener processes. The limit distribution depends on the exact nature of the restrictions so that it is not possible to tabulate critical values that are valid in general. Perron and Qu (2005) discuss a simulation algorithm to compute the relevant critical values given some restrictions. Imposing valid restrictions results in tests with much improved power. 4.6 Tests for structural changes in multivariate systems Bai et al. (1998) considered a sup Wald test for a single change in a multivariate system. Bai (2000) and Qu and Perron (2005) extend the analysis to the context of multiple structural changes. They consider the case where only a subset of the coefficients is allowed to change, whether it be the parameters of the conditional mean, the covariance matrix of the errors, or both. The tests are based on the maximized value of the likelihood ratio over permissible partitions assuming uncorrelated and homoskedastic errors. As above, the tests can be corrected to allow for serial correlation and heteroskedasticity when testing for changes in the parameters of the conditional mean assuming no change in the covariance matrix of the errors. The results are similar to those obtained in Bai and Perron (1998). The limit distributions are identical and depend only on the number of coefficients allowed to change, and the number of times that they are allowed to do so. However, when the tests involve potential changes in the covariance matrix of the errors, the limit distributions are only valid assuming a Normal distribution for these errors. This is because, in this case, the limit distributions of the tests depend on the higher-order moments of the errors’ distribution. Without the 35 assumption of Normality, additional parameters are present which take different forms for different distributions. Hence, testing becomes case specific even in large samples. It is not yet known how assuming Normality affects the size of the tests when it is not valid. An important advantage of the general framework analyzed by Qu and Perron (2005) is that it allows studying changes in the variance of the errors in the presence of simultaneous changes in the parameters of the conditional mean, thereby avoiding inference problem when changes in variance are studied in isolation. Also, it allows for the two types of changes to occur at different dates, thereby avoiding problems related to tests for changes in the paremeters when, for example, a change in variance occurs at some other date (see, e.g., Pitarakis, 2004). Tests using the quasi-likelihood based method of Qu and Perron (2005) are especially important in light of Hansen’s (2000) analysis. First note that, the limit distribution of the Sup, Mean and Exp type tests in a single equation system have the stated limit distribution under the assumption that the regressors and the variance of the errors have distributions that are stable across the sample. For example, the mean of the regressors or the variance of the errors cannot undergo a change at some date. Hansen (2000) shows that when this condition is not satisfied the limit distribution changes and the test can be distorted. His asymptotic results pertaining to the local asymptotic analysis show, however, the sup-Wald test to be little affected in terms of size and power. The finite sample simulations show that if the errors are homoskedastic, the size distortions are quite mild (over and above that applying with i.i.d. regressors, given that he uses a very small sample of T = 50). The distortions are, however, quite severe when a change in variance occurs. But both problems of changes in the distribution of the regressors and the variance of the errors can easily be handled using the framework of Qu and Perron (2005). If a change in the variance of the residuals in a concern, one can perform a test for no change in some parameters of the conditional model allowing for a change in variance since the tests are based on a likelihood ratio approach. If changes in the marginal distribution of some regressors is a concern, one can use a multi-equations system with equations for these regressors. Whether this is preferable to Hansen’s (2000) bootstrap method remains an open question. Note, however, that in the context of multiple changes it is not clear if that method is computationaly feasible, especially for the heteroskedastic case. 36 4.7 Tests valid with I(1) regressors With I(1) regressors, the case of interest is that of a system of cointegrated variables. The goal is then to test whether the cointegrating relationship has changed and to estimate the break dates and form confidence intervals for them. Consider, for simplicity, the following case with an intercept and m I(1) regressors y2t : y1t = a + βy2t + ut (15) where ut is I(0) so that y1t and y2t are cointegrated with cointegrating vector (1, −β). To our knowledge, the only contribution concerning the consistency and limit distribution of the estimates of the break dates is that of Bai et al. (1998). They consider a single break in a multi-equations system and show the estimates obtained by maximizing the likelihood function to be consistent. They also obtain a limit distribution under a shrinking shifts scenario with the shift in the constant a decreasing at rate T b1 for some b1 ∈ (0, 1/2) and the shift in β decreasing at rate T b2 for some b2 ∈ (1/2, 1). Under this scenario the rate of convergence is the same as in the stationary case (since the coefficients on the I(1) variables are assumed to shrink at a faster rate). For testing, an early contribution in this area is Hansen (1992a). He considers tests of the null hypothesis of no change in both coefficients (for an extension to partial changes, see Kuo, 1998, who considers tests for changes in intercept only and tests for changes in all coefficients of the cointegrating vector). The tests considered are the sup and Mean LM tests directed against an alternative of a one time change in the coefficients. He also considers a version of the LM test directed against the alternative that the coefficients are random walk processes denoted Lc . The latter is an extension of Gardner’s (1969) Q-test to the multivariate cointegration context, which is based on the average of the partial sums of the scores and the use of a full sample estimate of the conditional variance of these scores. For related results with respect to LM tests for parameter constancy in cointegrated regressions, see Quintos and Phillips (1993). Gregory et al. (1996) study the finite sample properties of Hansen’s (1992a) tests in the context of a linear quadratic model with costs of adjustments. They show that power can be low when the cost of adjustment is high and suggest a simple transformation of the dependent variable that can increase power. They also consider the behavior of standard residuals based tests of the null hypothesis of no cointegration and show that their power reduces considerably when structural breaks are present in the cointegrating relation. Again, this is simply a manifestation of the fact that unit root tests have little power when the process 37 is stationary around a trend function that changes. Moreover, since Hansen’s (1992a) tests can also be viewed as a test for the null hypothesis of stationarity, in this context it can also be viewed as a test for the null hypothesis of cointegration versus the alternative of no cointegration. Note, however, that the sup and Mean Wald test will also reject when no structural change is present and the system is not cointegrated. Hence, the application of such tests should be interpreted with caution. No test are available for the null hypothesis of no change in the coefficients a and β allowing the errors to be I(0) or I(1). This is because when the errors are I(1), we have a spurious regression and the parameters are not identified. To be able to properly interpret the tests, they should be used in conjunction with tests for the presence or absence of cointegration allowing shifts in the coefficients (see, Section 6). The same comments apply to other tests discussed below. Consider now a cointegrated VAR system written in the following error correction format 0 0 ) of dimension n = m + 1, with yt0 = (y1t , y2t 0 ∆yt = µ + αB yt−1 + p X Γi ∆yt−i + ut (16) i=1 where B (n×r) is the cointegrating matrix and α (n×r) the adjustment matrix (hence, there are r cointegrating vectors). Under the null hypothesis, both are assumed constant, while under the alternative either one or both are assumed to exhibit a one time change at some unknown date T1 . For the case of a triangular system with the restriction that B 0 − [Ir , B ∗ ], Seo (1998) considers the Sup, Mean and Exp versions of the LM test for the following three cases: 1) the constant vector µ is excluded (and the data are assumed non-trending), 2) the constant µ is included but the data are not trending, 3) the constant µ is included and the data are trending. The Sup and Mean LM tests in this cointegrated VAR setup are shown to have a similar asymptotic distribution as the Sup and Mean LM tests of Hansen (1992a) for the case of a change in all coefficients. See also Hao (1996) who also considers the Lc tests for no cointegration allowing for a one time change in intercept at some unknown date using the maximal value overall possible break dates. Hansen and Johansen (1999) also consider a VAR process 2 . Then, the MLE (based on Normal errors) of the cointegrating matrix B are the eigenvectors corresponding to the r largest eigenvalues of the system −1 S01 | = 0 |λS11 − S10 S00 2 A contribution related to multiple structural changes occurring at known dates in the context of cointegrated VAR processes is Hansen (2003), in which case all tests have the usual chi-square distribution. 38 where Sij = T −1 T X Rit Rjt (17) j=1 with R0t (resp., R1t ) the residuals from a regression of ∆yt (resp., yt−1 ) on a constant and lags of ∆yt . Hansen and Johansen (1999) show that instability in α and/or B will manifest themselves in the form of instability eigenvalues’ estimates when evaluated using different samples. They therefore suggest the use of the recursive estimates of λ. Their test takes the form of the fluctuations test of Ploberger et al. (1989) and will have power when either α or B change (see also Quintos, 1997). They also suggest a test that allows the detection of changes in β, an extension of the Lc test of Hansen (1992a) that can be constructed using recursive estimates of λ. Interestingly, Quintos (1997) documents that such tests over-reject the null hypothesis of no structural change when the cointegrating rank is over specified, i.e., when the number of stochastic trends, or unit root components, is under specified. This is the multivariate equivalent of the problem discussed in Section 2, namely that structural change and unit roots can easily be confounded. She proposes a test for the stability of the cointegrating rank. However, when the alternative hypothesis is of a greater rank (less unit roots), the tests will not have power if structural change is present. This is the dilemma faced when trying to assess jointly the correct rank of a cointegrating system and whether structural change is present in the cointegrating vectors. Another contribution, again based on functions of partial sums, is Hao and Inder (1996) who consider the CUSUM test based on OLS residuals from a cointegrating regression. >From this brief review, most tests available are seen to be of the LM type. Given our earlier discussion, these can be expected to have non-monotonic power since they do not explicitly allow for any break. However, no simulation study is available to substantiate this claim and show its relevance in practice. More work is needed in that direction and in considering Wald or LR type tests in a multiple structural changes context. Also, these tests are valid if the cointegrating rank is well specified. As discussed above, a rejection can be due to an over specification of this rank. The problem of jointly determining whether the cointegrating rank is appropriate and whether the system is structurally stable is an important avenue of further research. A potential, yet speculative, approach to determining if the data suggest structural changes in a cointegrating relationship or a spurious regression is the following. Suppose that one is willing to put an upper bound M (say 5) on the possible number of breaks. One can then use a multiple structural change test as discussed in Section 4.4. The reason is that 39 if the system is cointegrated with less than M breaks, the tests can be used to consistently estimate the number of breaks. However, if the regression is spurious, the number of breaks selected will always (in large enough samples) be the maximum number of breaks allowed. The same occurs when an information criterion is used to select the number of breaks (see, Nunes et al., 1996, and Perron, 1997b). Hence, selecting the maximum permissible number of breaks can be symptomatic of the presence of I(1) errors. Of course, more work is needed to turn this argument into a rigorous procedure. 4.8 Tests valid whether the errors are I(1) or I(0) We now consider the issue of testing for structural change when the errors in (2) may have a unit root. In the general case with arbitrary regressors, this question is of little interest. If the regressors are I(0) and the errors I(1), the estimates of the break dates will be inconsistent and so will the tests. This is simply due to the fact that the variability in the errors masks any potential shifts. With I(1) regressors, we have a cointegrated system when the errors are I(0), and a spurious regression with I(1) errors. In general, only the former is of interest. The problem of testing for structural changes in a linear model with errors that are either I(0) or I(1) is, however, of substantial interest when the regression is on a polynomial time trend. The leading case is testing for changes in the mean or slope of a linear trend, a question of substantial interest with economic data. We shall use this example to illustrate the main issues involved. Consider the following structure for some variable yt (t = 1, ..., T ) yt = β 0 + β 1 t + ut (18) where the errors follow a (possibly nonstationary) AR(k) process A(L)ut = et with A(L) = (1 − αL)A∗ (L) and the roots of A∗ (L) all outside the unit circle. If α = 1, the series contains an autoregressive unit root, while if |α| < 1, it is trend-stationary. The Q statistic is defined by #2 " T T X X ûj Q∗1 = ĥe (0)−1 T −2 t=1 j=t+1 P PT −1 where ĥu (0) = m t=τ +1 ût ût−τ , w(m, τ ) is some weight τ −m w(m, τ )R̂u (τ ) with R̂u (τ ) = T function with m/T → 0 (e.g., w(m, τ ) = 1 − |τ |/m if |τ | < m and 0 otherwise) and ût are 40 the least-squares residuals from estimating (18) by OLS. Then if |α| < 1, Z 1 ∗ B1 (r)2 dr Q1 ⇒ (19) 0 where ∙ ¸ ∙ ¸ Z 1 Z 1 B1 (r) = W (r) + 2 W (1) − 3 W (s) ds r − 3 W (1) − 2 W (s) ds r2 0 0 On the other hand, if α = 1, (m/T ) Q∗1 ⇒ Z 1 ∙Z 0 r 0 W1∗ ¸2 Z 1 W1∗ (r)2 dr (s) ds drÁκ Z ¸ ∙Z 1 ¸ Z 1 sW (s) ds +6r W (s) ds − 2 sW (s) ds 0 with W1∗ (r) = W (r)−4 ∙Z 0 R1 1 W (s) ds − (3/2) 0 1 0 0 and κ = −1 K (s) ds where K (τ /m) = ω (m, τ ) (see, Perron, 1991). Hence, the limit distribution is not only different under both the I(1) and I(0) cases, but the scaling needed is different. If one does not have prior knowledge about whether the series is integrated or not, one would need to use the statistic (m/T ) Q∗1 and reject using the critical values in the I(1) case in order to have a test that has asymptotic size no greater than some prespecified level in all cases. But this would entail a test with zero asymptotic size whenever the series is stationary. As suggested by Perron (1991), a solution is to base the test on a regression that parametrically account for the serial correlation in ut , namely yt = β 0 + β 1 t + k X αj yt−j + et (20) j=1 Since the errors are uncorrelated, one uses the statistic #2 " T T X X −2 êj QD1 = σ̂ −2 e T t=k+1 j=t+1 P where σ̂ 2e = T −1 Tt=k+1 ê2t with êt the residuals from estimating (20) by OLS (since the dynamics is taken into account parametrically, there is no need to scale with an estimate of the “long-run” variance). When |α| < 1, result (19) still holds, while when α = 1, we have ¸2 Z 1∙ Z r ∗ B1 (r) + H (1) QD1 ⇒ W1 (s) ds dr 0 0 41 R1 R1 where H (1) = 0 W1∗ (s) dW (s) Á 0 W1∗ (s)2 ds. The limit distributions are different but now the scaling for the convergence is the same. Hence, a conservative procedure is to use the largest of the two sets of critical values, which correspond to those from the limit distribution that applies to the I(1) case. The test is then somewhat asymptotically conservative in the I(0) case but power is still non-trivial. Perron (1991) discusses the power function in details and shows that it is non-monotonic, in that the test has zero power for large shifts (of course, when testing for a shift in level, the test has little power, if any, when the errors are I(1)). A natural extension is to explicitly model breaks and consider a regression of the form (21) yt = β 0 + β 1 t + γ 1 DUt + γ 2 DTt + ut where DUt = 1(t > T1 ) and DTt = 1(t > T1 )(t − T1 ). One can then use any of the tests advocated by Andrews and Ploberger (1996), though they may not be optimal with I(1) errors. These tests, however, also have different rates of convergence under the null hypothesis for I(0) and I(1) errors when based on regression (21). To remedy this problem, one can use a dynamic regression of the form yt = β 0 + β 1 t + γ 1 DUt + γ 2 DTt + k X αj yt−j + et j=1 This is the approach taken by Vogelsang (1997). He considers the Sup, Mean and Exp Wald tests and shows that they have well defined limit distributions under both I(0) and I(1) errors, which are, however, different. Again, at any significance level, the critical values are larger in the I(1) case and these are to be used to ensure tests with an asymptotic size no greater than pre-specified in both cases. Interestingly, Vogelsang’s results show that the Sup and Exp Wald tests have monotonic power functions but that the Mean-Wald test does not, the decrease in power being especially severe in the case of a level shift (this is due to the fact that the Sup and Exp tests assign most weight to the correct date, unlike the Mean test). Banerjee et al. (1992) also consider a Sup Wald test for a change in any one or more coefficients in a regression of yt on {1, t, yt−1 , ∆yt−1 , ..., ∆yt−k } assuming yt to be I(1). Vogelsang (2001) takes a different approach to obtain a statistic that has the same rate of convergence under both the I(0) and I(1) cases (see also Vogelsang 1998a,b). Let WT (λ1 ) be the Wald statistic for testing that γ 1 = γ 2 = 0 in (21). The statistic considered is P SWT (λ1 ) = WT (λ1 )[s2u /(100T −1 s2z )] exp(−bJT (m)) where s2u = T −1 PT t=1 û2t with ût the OLS residuals from regression (21), s2z = T −1 42 PT t=1 v̂t2 where v̂t are the OLS residuals from the following partial sum regression version of (21) ytp = β 0 t + β 1 ((t2 + t)/2) + γ 1 DTt + γ 2 [DTt2 + DTt ]/2 + vt (22) P where ytp = tj=1 yj and JT (m) is a unit root test that has a non-degenerate limit distribution in the I(1) case and converges to 0 in the I(0) case. Consider first the case with I(0) errors. We have WT (λ1 ), s2u and T −1 s2z all Op (1), hence P SWT (λ1 ) = Op (1), which does not depend on b . If the errors are I(1), WT (λ1 ) = Op (T ), T −1 s2u = Op (T ) and T −1 s2z = Op (1), hence P SWT (λ1 ) = Op (1) again. The trick is then to set b at the value which makes the critical values the same in both cases for any prescribed significance level. One can then use the Sup, Mean or Exponential version of the Wald test, though neither of the three have any optimal property in this context (another version based directly on the partial sums regression (22) is also discussed). Perron and Yabu (2005) consider an alternative approach which leads to more powerful tests. Consider the following special case of (21) for illustration yt = β 0 + β 1 t + γ 2 DTt + ut (23) so that the goal is to test for a shift in the slope of the trend function with both segments joined at the time of break. Assume that the errors are generated by an AR(1) process of the form (24) ut = αut−1 + et (an extension to the more general case is also discussed). If α = 1, the errors are I(1) and if |α| < 1, the errors are I(0). Consider the infeasible GLS regression yt∗ = β ∗0 + β 1 t∗ + γ 2 DTt∗ + et where for any variable, a ∗ indicates the quasi-differenced data, e.g., yt∗ = (1 − αL)yt . For a fixed break point T1 , the Wald test would be the best test to use and the limit distribution would be chi-square in both the I(1) and I(0) cases. However, if one used a standard estimate of α to construct a test based on the feasible GLS regression (e.g., α̂ obtained by estimating (24) with ut replaced by ût , the OLS residuals from (23)), the limit distribution would be different in both cases. Perron and Yabu (2005) show, however, that the same chi-square distribution prevails if one replace α̂ by a truncated version given by ⎧ ⎨ α̂ if T δ |α̂ − 1| > d α̂S = ⎩ 1 if T δ |α̂ − 1| ≤ d 43 for some δ ∈ (0, 1) and some d > 0. Theoretical arguments presented in Perron and Yabu (2004) show that δ = 1/2 is the preferred choice. Also, finite sample improvements are possible if one replaces α̂ by a median unbiased estimate (e.g., Andrews, 1993b) or the estimate proposed by Roy and Fuller (2001; see also Roy et al., 2004). When the break date is unknown, the limit distributions of the Sup, Mean or Exp Wald tests are no longer the same for the I(0) and I(1) cases. However, for the Mean version, the asymptotic critical values are very close (for all common significance levels). Hence, with this version, there is no need for an adjustment. Simulations show that for this case, a value d = 2 leads to tests with good finite sample properties and a power function that is close to that which could be obtained using the infeasible GLS regression, unless the value of α is close to but not equal to one. The issue of testing for structural changes in the trend function of a time series without having to take a stand on whether the series is I(1) or I(0) is of substantial practical importance. As discussed above, some useful recent developments have been made. Much remains to be done, however. First, none of the procedures proposed have been shown to have some optimality property. Second, there is still a need to extend the analysis to the multiple structural changes case with unknown break dates. 4.9 Testing for change in persistence A problem involving structural change and the presence of I(0) and I(1) processes relates to the quite recent literature on change in persistence. What is meant, in most cases, by this is that a process can switch at some date from being I(0) to being I(1), or vice versa. This has been an issue of substantial empirical interest, especially concerning inflation rate series (e.g., Barsky, 1987, Burdekin and Siklos, 1999), short-term interest rates (e.g., Mankiw et., 1987), government budget deficits (e.g., Hakkio and Rush, 1991) and real output (e.g., Delong and Summers, 1988). As discussed in Section 3.1, Chong (2001) derived the limit distribution of the estimate of the break date obtained by minimizing the sum of squared residuals from a regression that allows the coefficient on the lagged dependent variable to change at some unknown date. However, he provided no procedure to test whether a change has occurred and in which direction. A discussed in this review, tests for structural change started with statistics based on partial sums of the data (or some appropriate residuals, in general) as in the Q test of Gardner (1969), and tests of the null hypothesis of stationarity versus a unit root process started with the same statistic. Interestingly, we are again back to Gardner (1969) when 44 devising procedures to test for a change in persistence. Kim (2000) and Busetti and Taylor (2001) consider testing the null hypothesis that the series is I(0) throughout the sample versus the alternative that it switches from I(0) to I(1) or vice versa. The statistic used is the ratio of the unscaled Gardner’s (1969) Q test over P the post and pre-break samples. With the partial sums Si,t = tj=i+1 ûj , where ût are the residuals from a regression of the data yt on a constant (non-trending series) or on a constant and time trend (for trending series), it is defined by P (T − T1 )−2 Tt=T1 +1 ST21 ,t (25) ΞT (T1 ) = P 1 2 T1−2 Tt=1 S1,t Under the null hypothesis of I(0) throughout the sample, both the numerator and denominator are Op (1). Consider an alternative with the process being I(0) in the first sample and I(1) is the second, the numerator is then Op (T 2 ) and one rejects for large values. If the alternative is reversed, the denominator is Op (T 2 ) and one rejects for small values. Hence, with a known break date, a two sided test provides a consistent test against both alternatives. For the case with an unknown break date, Kim (2001) considers the sup, Mean or Exp functionals of the sequence ΞT (T1 ) with, as usual, a set specifying a range for permissible values of T1 /T (he suggests [0.2, 0.8]). The test is then consistent for a change from I(0) to I(1) but inconsistent for a change from I(1) to I(0). Busetti and Taylor (2005) note that maximizing the reciprocal of the test ΞT (T1 ) provides a consistent test against the alternative of a change from I(1) to I(0). Hence, their suggestion is to use the maximum of the test based ΞT (T1 ) and its reciprocal (whether the sup, Mean of Exp functional is used). Interestingly, Leybourne and Taylor (2004) suggest scaling both the numerator and denominator of (25) by an estimate of the long-run variance constructed from the respective sub-samples, in which case the test is then exactly the ratio of the Q tests applied to each sub-samples. No version of the test will deliver a consistent estimate of the break date and they suggest using the ratio of the post-break to pre-break sample variances of the residuals ût . They show consistency of the estimate but no limit distribution is obtained, thereby preventing making inference about the break date. Another issue related to this class of tests is the fact that they reject the null hypothesis often when the process is actually I(1) throughout the sample. This is due to the fact that, though the statistic ΞT (T1 ) is Op (1) in this case, the limit distribution is quite different from that prevailing in the constant I(0) case, with quantiles that are greater. Harvey et al. (2004) use the same device suggested by Vogelsang (1998, 2001) to solve the problem by multiplying the test by exp(−bJT ) with JT a unit root test that has a non-degenerate limit 45 distribution in the constant I(1) case and that converges to zero in the constant I(0) case. For a given size of the test, one can then select b so that the critical values are the same in both cases (see Section 4.8). Busetti and Taylor (2005) also consider locally best invariant (LBI) tests. As discussed in this review, this class of tests has important problems (e.g., non monotonic power) and here is no exception. Consider the LBI test for a change from I(0) to I(1), the form of the statistic is then given by: T X −2 2 −2 St,T (26) σ̂ (T − T1 ) t=T1 +1 PT −1 2 where σ̂ 2 = T t=1 ût is the estimate of the variance of the residuals using the full sample. Under the alternative, σ̂ 2 is Op (T ) and, hence, the test is Op (T ) under the alternative. Busetti and Harvey (2005) also consider using the sup, Mean or Exp functionals of the original Q test applied to the post break data only. It is similar to the test (26) but with P a scaling based on σ̂ 21 (T1 ) = (T − T1 )−1 Tt=T1 +1 û2t , an estimate of the variance based on the post-break data. This test has similar properties. In fact using the Q test itself applied to the whole sample would be consistent against a change from I(0) to I(1), showing that this class of tests will reject the null hypothesis with probability one in large samples if the process is I(1) throughout the sample. Both the LBI and the post break Q tests have a scaling that is Op (T ) when the alternative is true whatever break date is used. Consider now P1 2 ût . The test would then have the same instead the statistic (26) scaled by σ̂ 20 (T1 ) = T1−1 Tt=1 limit distribution under the constant I(0) null hypothesis but would be Op (T 2 ) under the alternative and, hence, more powerful. This illustrates once again, a central problem with LBI or LM type tests in the context of structural changes. The scaling factor is evaluated under the null hypothesis, which implies an inflated estimate when the alternative is true and a consequent loss of power. Leybourne et al. (2003) consider instead the null hypothesis that the process is I(1) throughout the sample, with the same alternatives that it can switch from I(1) to I(0), or vice versa. Their test for a change from I(0) to I(1) is based on the minimal value of the unit root test ADF GLS (T1 ), the ADF test proposed by Elliott et al. (1996) constructed using observations up to time T1 (labelled recursive test). Since this test does not use all information in the data for any given particular break date, they also consider using a similar unit root test from a full sample regression in which the coefficient on the lagged level is constrained to be zero in the post-break sample (labelled sequential). To test against the alternative hypothesis of a change from I(1) to I(0), the same procedures are applied to the 46 data arranged in reverse order. When the direction of the change is unknown, they consider the minimal value of the pair of statistics for each case. These tests will, however, reject when the process has no change and is I(0) throughout the sample. To remedy this problem, Leybourne et al. (2003) consider an alternative procedure when the null hypothesis is I(1) throughout the sample. It is the ratio of the minimal value of the pre-break sample variance of the residuals constructed from the original series relative to the minimal value of the same statistic constructed using time reversed data. The test has a well defined limit distribution under the null hypothesis of constant I(1), rejects when there is a shift and has a limit value of 1 when the process is I(0) throughout, which implies a non-rejection asymptotically in the latter case. Kurozumi (2004) considers a test constructed upon the LM principle. He shows that the test is asymptotically equivalent to the sum of the t-statistics on α1 and α2 in the regression yt−1 + α2 1(t ≤ T1 )e yt−1 + ∆e yt = α1 1(t ≤ T1 )e k X ci ∆e yt−i + et i=1 where yet are OLS detrended data (he also considers a version with GLS detrended data but his simulations show no power improvement). This test performs rather poorly and he recommends using a regression with a fitted mean that is allowed to change at the break date, even though this results in a test with lower local asymptotic power. With an unknown break date, one takes the minimal value of the tests over the range of permissible break dates. Deng and Perron (2005b) take the null hypothesis to be I(1) throughout and they follow the approach suggested by Elliott et al. (1996) in specifying a the null hypothesis as involving an autoregressive parameter (in the I(0) subsample) that is local to unity. They derive the Gaussian local power envelop and a feasible test that achieves this power envelop. It is shown that the test has higher power than those of Leybourne et al. (2003) and Kurozumi (2004), according to both the local power function and to the finite sample power (via simulations). But they also find a curious feature. The test is consistent when only a constant is included but inconsistent when a constant and a time trend are included. This is really a theoretical artifact that has little impact on finite sample power but is interesting nevertheless. Under the null hypothesis, the support of the test is the positive real line and one reject for small values. When a fitted trend is included, the limit distribution of the test is exactly zero. Most of the mass it at zero but there is a very small tail to the right, so that the probability of rejecting does not go to one for all possible significance levels. 47 5 Unit Root Versus Trend Stationarity in the Presence of Structural Change in the Trend Function As discussed throughout this review, structural changes and unit root non-stationarity share similar features in the sense that most tests for structural changes will reject in the presence of a unit root in the errors, and vice versa, tests of stationarity versus unit root will reject in the presence of structural changes. We now discuss methods to test the null hypothesis of a unit root in the presence of structural changes in the trend function. 5.1 The motivation, issues and framework To motivate the problem addressed, it is useful to step back and look at some basic properties of unit root and trend-stationary processes. Consider a trending series generated by (27) yt = µ + βt + ut where ∆ut = C(L)et (28) P P ∞ j with et ∼ i.i.d. (0, σ 2e ) and C(L) = ∞ j=0 cj L such that j=1 j|cj | < ∞ and c0 = 1. A popular trend-cycle decomposition is that suggested by Beveridge and Nelson (1981). The trend is specified as the long run forecast of the series conditional on current information, which results in the following τ t = µ + βt + C(1) t X ej j=1 P∞ e e while the cycle is given by ct = C(L)e cj Lj where e cj = Σ∞ t with C(L) = i=j+1 ci . Here the j=0 e trend has two components, a deterministic one (a linear trend) and a stochastic one specified by a random walk weighted by C(1). Hence, the trend exhibits changes every period in the form of level shifts. Note that if one considered a process which is potentially integrated of order 2, the trend would exhibit changes in both level and slope every period. When the process has no unit root, C(1) = 0 and the trend is a linear deterministic function of time. Within this framework, one can view the unit root versus trend-stationary problem as addressing the following question: do the data support the view that the trend is changing every period or never? The empirical analysis of Nelson and Plosser (1982) provided strong evidence that, if the comparison is restricted to these polar cases, the data support the view that a trend which ‘always’ changes is a better description than a trend that ‘never’ 48 changes (using a variety of US macroeconomic variables, furthermore many other studies have reached similar conclusions for other series and other countries). The question is then why restrict the comparison to ‘never’ or ‘always’ ? Would it not be preferable to make a comparison between ‘always’ and ‘sometimes’ ? Ideally, then, the proper question to ask would be ‘what is the frequency of permanent shocks?’. This is a question for which no satisfactory framework has been provided and, as such, it still remains a very important item for further research. The basic motivation for the work initiated by Perron (1989, 1990) is to take a stand on what is ‘sometimes’ (see also Rappoport and Reichlin, 1989). The specific number chosen then becomes case-specific. His argument was that in many cases of interest, especially with historical macroeconomic time series (including those analyzed by Nelson and Plosser, 1982), the relevant number of changes is relatively small, in many cases only one. These changes are then associated with important historical events: the Great Crash (US and Canada, 1929, change in level), the oil price shock (G7 countries, 1973, change in slope); World War II (European countries, change in level and slope), World War I (United Kingdom, 1917, change in level), and so on. As far as statistical modelling is concerned, the main conceptual issue is to view such changes as possibly stochastic but of a different nature than shocks that occur every period, i.e., drawn from a different distribution. However, the argument that such large changes are infrequent makes it difficult to specify and estimate a probability distribution for them. The approach is then to model these infrequent large changes in the trend as structural changes. The question asked by unit root tests is then: ‘do the data favor a view that the trend is ‘always’ changing or is changing at most occasionally?’ or ‘if allowance is made for the possibility of some few large permanent changes in the trend function, is a unit root present in the structure of the stochastic component?’. Note that two important qualifications need to be made. First, the setup allows but does not impose such large changes. Second, by “permanent” what should be understood is not that it will last forever but that, given a sample of data, the change is still in effect. For instance, the decrease in the slope of the trend function after 1973 for US real GDP is still in effect (see, Perron and Wada, 2005). When allowance is made for a one-time change in the trend function, Perron (1989, 1990) specified two versions of four different structures: 1) a change in level for a non-trending series; and for trending series, 2) a change in level, 3) a change in slope, and 4) a change in both level and slope. For each of the four cases, two different versions allow for different transition effects. Following the terminology in Box and Tiao (1975), the first is labelled 49 the “additive outlier model” and specifies that the change to the new trend function occurs instantaneously. The second is labelled the “innovational outlier model” and specifies that the change to the new trend function is gradual. Of course, in principle, there is an infinity of ways to model gradual changes following the occurrence of a “big shock”. One way out of this difficulty is to suppose that the variables respond to the “big shocks” the same way as they respond to so-called “regular shocks” (shocks associated with the stationary noise component of the series). This is the approach taken in the modelization of the “innovational outlier model”, following the treatment of intervention analyses in Box and Tiao (1975). The distinction between the additive and innovational outlier models is important not only because the assumed transition paths are different but also because the statistical procedures to test for unit roots are different. The additive outlier models for each of the four specifications for the types of changes occurring at a break date T1 are specified as follows: Model (AO-0) Model (AO-A) Model (AO-B) Model (AO-C) yt = µ1 + (µ2 − µ1 ) DUt + ut yt = µ1 + βt + (µ2 − µ1 ) DUt + ut yt = µ1 + β 1 t + (β 2 − β 1 ) DTt∗ + ut yt = µ1 + β 1 t + (µ2 − µ1 ) DUt + (β 2 − β 1 ) DTt∗ + ut where DUt = 1, DTt∗ = t−T1 if t > T1 and 0 otherwise, and ut is specified by (28). Under the null hypothesis C(1) 6= 0, while under the alternative hypothesis, C(1) = 0. Alternatively, one can define the autoregressive polynomial A(L) = (1 − L)C(L)−1 . The null hypothesis then specifies that a root of the autoregressive polynomial is one, i.e., that we can write A(L) = (1 − L)A∗ (L) where all the roots of A∗ (L) are outside the unit circle. Under the alternative hypothesis of stationary fluctuations around the trend function, all the roots of A(L) are strictly outside the unit circle. Model (AO-B) was found to be useful for the analysis of postwar quarterly real GNP for the G7 countries and Model (AO-0) for some exchange rate series as well as the US real interest rate, among others. It is important to note that changes in the trend function are allowed to occur under both the null and alternative hypotheses. The innovational outlier models are easier to characterize by describing them separately under the null and alternative hypotheses. Note also that the innovational outlier versions have been considered only for Models (A) and (C) in the case of trending series. The basic reason is that the innovational outlier version of Model (B) does not lend itself easily to 50 empirical applications using linear estimation methods. Under the null hypothesis, we have: Model (IO-0-UR) yt = yt−1 + C (L) (et + δD (T1 )t ) Model (IO-A-UR) yt = yt−1 + b + C (L) (et + δD (T1 )t ) Model (IO-C-UR) yt = yt−1 + b + C (L) (et + δD (T1 )t + ηDUt ) where D (T1 )t = 1 if t = T1 + 1 and 0 otherwise. Under this specification, the immediate impact of the change in the intercept is δ while the long run impact is C (1) δ. Similarly, under Model (IO-C), the immediate impact of the change in slope is η while the long run impact is C (1) η. Under the alternative hypothesis of stationary fluctuations, the specifications are: Model (IO-0-TS) yt = µ + C (L)∗ (et + θDUt ) Model (IO-A-TS) yt = µ + βt + C (L)∗ (et + θDUt ) Model (IO-C-TS) yt = µ + βt + C (L)∗ (et + θDUt + γDTt∗ ) where C(L)∗ = (1 − L)−1 C(L). The immediate impact of the change in the intercept of the trend function is θ while the long run impact is C (1)∗ θ, and the immediate impact of the change in slope is γ while the long run impact is C (1)∗ γ. 5.2 The effect of structural change in trend on standard unit root tests A standard unit root test used in applied research is the so-called augmented Dickey-Fuller (1979) test, which is based on the t-statistic for testing that α = 1 in the following regression yt = µ + βt + αyt−1 + k X ci ∆yt−i + et i=1 with the trend regressor excluded when dealing with non-trending series. A central message of the work by Perron (1989, 1990) is that, when the true process involves structural changes in the trend function, the power of such unit root tests can dramatically be reduced. In particular, it was shown that if a level shift is present, the estimate of the autoregressive coefficient (α when k = 0) is asymptotically biased towards 1. If a change in slope is present, its limit value is 1. It was shown that this translates into substantial power losses. Simulations presented in Perron (1994) show the power reduction to increase as k is increased (see also the theoretical analysis of Montañés and Reyes, 2000, who also show that the power problem remains with the Phillips-Perron (1988) type unit root test). For a more precise and complete theoretical analysis, see Montañés and Reyes (1998, 1999). Under 51 the null hypothesis, the large sample distribution is unaffected by the presence of a level shift (Montañés and Reyes, 1999) and the test is asymptotically conservative in the presence of a change in slope. It can, however, have a liberal size if the break occurs very early in the sample (λ1 < .15) as documented by Leybourne et al. (1998) and Leybourne and Newbold (2000). Intuitively, the latter result can be understood by thinking about the early observations as outliers such that the series reverts back to the mean in effect for the rest of the sample. The latter problem is, however, specific to the Dickey-Fuller (1979) type unit root test, which is based on the conditional likelihood function, discarding the first observations (see, Lee, 2000) 3 . It has also been documented that the presence of stuctural breaks in trend affects tests of the null hypothesis of stationarity (e.g., the Q or KPSS test) by inducing size distortions towards rejecting the null hypothesis too often (e.g., Lee et al., 1997). This is consistent with the effect on unit root tests in the sense that when trying to distinguish the two hypotheses, the presence of structural changes induces a bias in favor of the unit root representation. It is important to discuss these results in relation to the proper way to specify alternative unit root tests. The main result is that large enough changes in level and/or slope will induce a reduction in the power of standard unit root tests. Small shifts, especially in level, are likely to reduce power only slightly. Hence, what is important is to account for the large shifts, not all of them if the others are small. Consider analyzing the US real GDP over, say, the period 1900-1980. Within this sample, one can identify two shifts related to the 1929 crash (change in level) and the post 1973 productivity slowdown (change in slope). However the post 73 sample would, here, consists of only a small proportion of the total sample and the shift in slope in this period is unlikely to induce a bias and need not be accounted for. Hence, the testing strategy discussed below need not make a statement about the precise number of changes. It should rather be viewed as a device to remove biases induced by shifts large enough to cause an important reduction in power. 5.3 Testing for a unit root allowing for changes at known dates. The IO models under the null and alternative hypotheses can be nested in the a way which specifies the regression from which the statistics will be constructed as follows: yt = µ + θDUt + βt + γDTt∗ + δD (T1 )t + αyt−1 + k X ci ∆yt−i + et (29) i=1 3 Kim et al, 2004, study what happens when the trend regressor is absent and the series has a broken trend with the coefficients on the trend and shift in slope shrinking to zero as the sample size increases. 52 for a value of the truncation lag parameter k chosen to be large enough as to provide a good approximation (for methods on how to choose k, see Ng and Perron, 1995, 2001). For Model (IO-0), the regressors (t, DTt∗ ) are not present, while for Model (IO-A), the regressor (DTt∗ ) is not present. The null hypothesis imposes the following restrictions on the coefficients. For Model (IO—0), these are α = 1, θ = µ = 0 and, in general, δ 6= 0 (if there is a change in the intercept). For Model (IO-A), the restrictions are α = 1, β = θ = 0 and again, in general, δ 6= 0, while for Model (IO-C), α = 1, β = γ = 0. Under the alternative hypothesis, we have the following specifications: |α| < 1 and, in general, δ = 0. These restrictions are, however, not imposed by most testing procedures. The test statistic used is the t-statistic for testing the null hypothesis that α = 1 versus the alternative hypothesis that |α| < 1, denoted tα (λ1 ) with λ1 = T1 /T . It is important to note that, provided the specified break date corresponds to the true break date, the statistic is invariant to the parameters of the trend-function, including those related to the changes in level and slope (for an analysis of the case when the break date is mis-specified, see Hecq and Urbain, 1993, Montañés, 1997, Montañés and Olloqui, 1999, and Montañés et al., 2005, who also consider the effect of choosing the wrong specification for the type of break). The limit distribution of the test under the null hypothesis is R1 ∗ W (r, λ1 )dW (r) (30) tα (λ1 ) ⇒ hR0 i1/2 1 ∗ (r, λ )2 dr W 1 0 where W ∗ (r, λ1 ) is the residual function from a projection of a Wiener process W (r) on the relevant continuous time versions of the deterministic components ({1, 1(r > λ1 )} for Model (IO-0), {1, 1(r > λ1 ), r} for Model (IO-A) and {1, 1(r > λ1 ), r, 1(r > λ1 )(r − λ1 )} for Model (IO-C)). Tabulated critical values can be found in Perron (1989, 1990). See also Carrion i Silvestre et al. (1999). For the additive outlier models, the procedures are different and consist of a two-step approach. In the first step, the trend function of the series is estimated and removed from the original series via the following regressions for Model (AO-0) to (AO-C), respectively: yt = µ + γDUt + yet yt = µ + βt + γDUt + yet yt = µ + βt + γDTt∗ + yet yt = µ + βt + θDUt + γDTt∗ + yet 53 where yet is accordingly defined as the detrended series. The next step differs according to whether or not the first step involves DUt , the dummy associated with a change in intercept. For Models (AO-0), (AO-A) and (AO-C), the test is based on the value of the t-statistic for testing that α = 1 in the following autoregression: yt−1 + yet = αe k X dj D (T1 )t−j + j=0 k X ai ∆e yt−i + et i=1 Details about the need to introduce the current value and lags of the dummies D (Tb )t can be found in Perron and Vogelsang (1992b). The limit distributions of the tests are then the same as for the IO case. There is no need to introduce the dummies in the second step regression for Model (AO-B) where no change in level is involved and the two segments of the trend are joined at the time of break. The limit distribution is, however, different; see Perron and Vogelsang (1993a, 1993b). Again, in all cases, the tests are invariant to the change in level or slope provided the break date is correctly specified. These unit root tests with known break dates have been extended in the following directions. Kunitomo and Sato (1995) derive the limit distribution of the likelihood ratio tests for multiple structural changes in the AO case. Amsler and Lee (1995) consider a LM type test in the context of a shift in level of the AO type. Saikkonen and Lütkepohl (2001) also consider cases with a level shift of the AO type, though they allow for general forms of shifts which can be indexed by some unknown parameter to be estimated. Following Elliott et al. (1996), they propose a GLS-type detrending procedure, which is however based on an AR(p) process for the noise. On the basis of simulation results, they recommend using GLS detrending under the null hypothesis instead of a local alternative as done in Elliott et al. (1996). Lanne et al. (2002) propose a finite sample modification which is akin to a pre-whitening device. Let the detrended series be e e−γ eDUt − βt ytGLS = yt − µ and the estimate of the autoregressive polynomial of the first difference ∆ut be eb(L) (all estimates being obtained from the GLS procedure). With the filtered series defined as w et = eb(L)ytGLS , the test is then the t-statistic for testing that α = 1 in the regression w et = µ + αw et−1 + πeb(L)D (T1 )t + k X GLS ai ∆yt−i + et . i=1 Note that the limit distribution does not depend on the break date. This is because the data are detrended using a GLS approach under the null hypothesis of a unit root (or more 54 generally under a sequence of alternatives that are local to a unit root) and the level shift regressor is, in the terminology of Elliott et al. (1996), a slowly evolving trend, in which case, the limit distribution is the same as it would be if it was excluded (loosely speaking, the level shift becomes a one-time dummy). Hence, the limit distribution of the test is the same as that of Elliott et al. (1996) for their unit root test when only a constant is included as deterministic regressor. Lanne and Lütkepohl (2002) show that this test has better size and power than the test proposed in Perron (1990) and the LM test of Amsler and Lee (1995). A similar procedure for level shifts of the IO type is presented in Lütkepohl, Müller and Saikkonen (2001). 5.4 Testing for a unit root allowing for changes at unknown dates The methodology adopted by Perron (1989, 1990) was criticized by, among others, Christiano (1992), on the ground that using a framework whereby the break is treated as fixed is inappropriate. The argument is that the choice of the break date is inevitably linked to the historical record and, hence, involves an element of data-mining. He showed that if one did a systematic search for a break when the series is actually a unit root process without break, using fixed break critical value would entail a test with substantial size distortions. While the argument is correct, it is difficult to quantify the extent of the ‘data-mining’ problem in Perron’s (1989) study. Indeed, no systematic search was done, the break dates were selected as obvious candidates (the Crash of 1929 and the productivity slowdown after 1973) and the same break date was used for all series. Given the intractability of correctly assessing the right p-values for the tests reported, the ensuing literature addressed the problem by adopting a completely agnostic approach where a complete and systematic search was done. While this leads to tests with the correct asymptotic size (under some conditions to be discussed), it obviously implies a reduction in power. We shall return to the practical importance of this point. An avenue taken by Banerjee et al. (1992) was to consider rolling and recursive tests. Both perform standard unit root tests without breaks, the former using a sample of fixed length (much smaller than the full sample) that moves sequentially from some starting date to the end of the sample. The latter considers a fixed starting date for all tests and increases the sample used (from some minimal value to the full sample). In each case, one then considers the minimal value of the unit root test and rejects the null hypothesis of a unit root if this minimal value is small enough. Asymptotically, such procedures will correctly reject the null hypothesis if the alternative is true but the fact that all tests are based on 55 sub-samples means that not all information in the data is used and consequently one can expect a loss of power. An alternative strategy, more closely related to the methodology of Perron (1989) was adopted by Zivot and Andrews (1992) as well as Banerjee et al. (1992). They consider the IO type specification and a slightly different regression that does not involve the one-time dummy when a shift in level is allowed under the alternative hypothesis. For example, for Model C, the regression is yt = µ + θDUt + βt + γDTt∗ + αyt−1 + k X ci ∆yt−i + et (31) i=1 and the test considered is the minimal value of the t-statistic for testing that α = 1 over all possible break dates in some pre-specified range for the break fraction [ , 1 − ] where a popular choice for is 0.15. Denote the resulting test by t∗α = inf λ1 ∈[ ,1− ] tα (λ1 ) where tα (λ1 ) is the t-statistic for testing α = 1 in (31) when the break date T1 = [T λ1 ] is used. The limit distribution of the test is R1 ∗ W (r, λ1 )dW (r) ∗ (32) tα ⇒ inf hR0 i1/2 λ1 ∈[ ,1− ] 1 ∗ 2 W (r, λ1 ) dr 0 with W ∗ (r, λ1 ) as defined in (30). Perron (1997a) extended their theoretical results by showing, using projection arguments, that trimming for the possible values of λ1 was unnecessary and that one could minimize over all possible break dates 4 . For the Nelson-Plosser (1982) data set, Zivot and Andrews (1992) reported fewer rejections compared to what was reported in Perron (1989) using a known break date assumption. These rejections should be viewed as providing stronger evidence against the unit root but a failure to reject does not imply a reversal of Perron’s (1989) conclusions. This is a commonly found mis-conception in the literature, which overlooks the fact that a failure to reject may simply be due to tests with low power. Zivot and Andrews’ (1992) extension involves, however, a substantial methodological difference. The null hypothesis considered is that of a unit root process with no break while the alternative hypothesis is a stationary process with a break. Hence, there is an asymmetric treatment of the specification of the trend under the null and alternative hypotheses. In particular, limit result (32) is not valid if a break is present under the null hypothesis. 4 Perron (1997a) also showed how the weak convergence result could be obtained using the usual sup metric instead of the hybrid metric adopted in Zivot and Andrews (1992). 56 Vogelsang and Perron (1998) show that, in this case, t∗α diverges to −∞ when a shift in slope is present. This implies that a rejection can be due to the presence of a unit root process with a breaking trend. The reason for this is the following. With a fixed break date, the statistic tα (λ1 ) from regression (29) is invariant to the values of the parameters of the trend function under both the null and alternative hypotheses. When searching over a range of values for the break date (only one of which corresponding to the true value), this invariance no longer holds. In the case of Model A with only a level shift and for non-trending series with a change in mean considered by Perron and Vogelsang (1992a), the statistic t∗α is asymptotically invariant to the value of the level shift but not in finite samples. Simulations reported by Perron and Vogelsang (1992b) show size distortions that increase with the magnitude of the level shift. They argue, however, that substantial size distortions are in effect only when implausibly large shifts occur and that the problem is not important in practice. Vogelsang and Perron (1998) make the same arguments for the case of a shift in slope. Even though in practice the distortions may be small, it nevertheless remains a problematic feature of this approach and we consider recent attempts below which do not have this problem. Perron and Vogelsang (1992a), for the non-trending case, and Perron (1997a), for the trending case, extend the analysis of Zivot and Andrews (1992). They consider tests for both the IO and AO cases based on the minimal value of the t-statistic for testing that α = 1, and also tests based on tα (λ1 ) with T1 selected by maximizing the absolute value of the t-statistic on the coefficient of the appropriate shift dummy, DUt if only a level shift is present and DTt∗ if a slope change is present (see also Christiano, 1992, and Banerjee et al., 1992). For the IO case, they also suggest using regression (29) instead of (31) which includes the one-time dummy D (Tb )t since that would be the right regression to use with a known break date. They derive the limit distribution under the null hypothesis of a unit root and no break (in which case it does not matter if the one-time dummy D (Tb )t is incorporated). Perron (1997a) also considers tests where the break date is selected by minimizing or maximizing the value of the t-statistic on the slope dummy, which allows one to impose a priori the restriction of a direction for the change in slope and provides a more powerful test. Carrion-i-Silvestre et al. (2004) consider statistics which jointly test the null hypothesis and the zero value of appropriate deterministic regressors, extending the likelihood ratio test of Dickey and Fuller (1981). 57 5.4.1 Extensions and other approaches We now briefly review some extensions and alternative approaches and return below to an assessment of the various methods discussed above. Unless stated otherwise, all work described below specifies the null hypothesis as a unit root process with no break in trend. Perron and Rodríguez (2003) consider tests for trending series with a shift in slope in the AO framework. Following Elliott et al. (1996), they derive the asymptotic local power envelop and show that using GLS detrended series (based on a local alternative) yields tests with power close to the envelop. For the non-trending case, Clemente, Montañés and Reyes (1998) extend the results of Perron and Vogelsang (1992a) to the case with two breaks. A similar extension is provided by Lumsdaine and Papell (1997) for the case of trending series. Generalizations to multiple breaks include the following. Ohara (1999) extends the Zivot and Andrews (1992) approach to the general case with m breaks, though only critical values for the two break case are presented. Ohara (1999) also proves an interesting generalization of a result in Perron (1989) to the effect that, if a unit root test allowing for m1 changes in slope is performed on a series having m0 changes with m0 > m1 , then the least-squares estimate of α converges to one. This provides theoretical support for Rule 6 stated in Campbell and Perron (1991), which states that ‘a non-rejection of the unit root hypothesis may be due to misspecification of the deterministic components included as regressors’. Kapetanios (2005) also deals with the multiple break case but considers the following strategy, based on the sequential method of Bai (1997b) and Bai and Perron (1998) (see section 3.5). First, denote the set of t-statistics for a unit root over all possible one break partitions by τ 1 . Choose the break date that minimizes the sum of squared residuals. Then impose that break and insert an additional break over all permissible values (given some imposed trimming) and store the associated unit root tests in the set τ 2 , then choose the additional break that minimizes the sum of squared residuals. Continue in this fashion until an m break model is fitted and m sets of unit root tests are obtained. The unit root test selected is then the one that is minimal over all m sets. The limit distribution is, however, not derived, and the critical values are obtained through simulation with T = 250. Saikonnen and Lütkepohl (2002) extend their tests for a level shift with a known break date (of a general form possibly indexed by some unknown parameter) to the case of a shift occurring at an unknown date. It can be performed in both the AO and IO frameworks and the resulting procedure is basically the same as discussed in Section 5.3 for the known break date case. This is because, with a GLS detrending procedure based on a specification that is local to a unit root, the limit distribution of the test is the same whatever the break point 58 is selected to be. Hence, one can substitute any estimate of the break date without affecting the limit null distribution of the test. They recommend using a unit root specification for the detrending (as opposed to using a local alternative as in Elliott et al., 1996) since it leads to tests with finite sample sizes that are robust to departures of the estimate of the break date from its true value. Of course, power is highly sensitive to an incorrectly estimated break date. Lanne et al. (2003) assess the properties of the tests when different estimates of the break date are used. A substantial drawback of their approach is that they found the test to have non-monotonic power, in the sense that the larger the shift in level the lower the power in rejecting the unit root. Also, the power is sensitive to departures from the exact specification for the type of change, and power can be reduced substantially if allowance is made for a general shift indexed by some parameter when the shift is actually an abrupt one. Consider now testing the null hypothesis of stationarity. Tests of the type proposed by Kwiatkowski et al. (1992) will reject the null hypothesis with probability one in large enough samples if the process is affected by structural changes in mean and/or slope but is otherwise stationary within regimes. This follows in an obvious way once one realizes that the KPSS test is also a consistent test for structural change (see, nevertheless, simulations in Lee et al., 1997). In order not to incorrectly reject the null hypothesis of stationarity, modifications are therefore necessary. Kurozumi (2002), Lee and Strazicich (2001b) and Busetti and Harvey (2001, 2003) consider testing the null hypothesis of stationarity versus the alternative of a unit root in the presence of a single break for the specifications described above (see also, Harvey and Mills, 2003). Their test is an extension of the Q-statistic of Gardner (1969), or equivalently the KPSS test as discussed in Section 2. The test is constructed using leastsquares residuals from a regression incorporating the appropriate dummy variables. They provide critical values for the known break date case. When the break is unknown, things are less satisfactory. To ensure the consistency of the test, Lee and Strazicich (2001b) and Busetti and Harvey (2001, 2003) consider the minimal value (as opposed to the maximal value) of the statistics over all permissible break dates. Since the test rejects for large values, this implies the need to resort to the value of the statistic at the break point that permits the least-favorable outcome against the alternative. Hence, it results in a procedure with low power. Kurozumi (2002) as well as Busetti and Harvey (2001, 2003) also consider using the estimate of the break date that minimizes the sum of squared residuals from the relevant regression under the null hypothesis. Since, the estimate of the break fraction is then consistent, one can use critical values corresponding to the known break date case. They 59 show, however, that the need to estimate the break date induces substantial power losses. Busetti (2002) extended this approach to a multivariate setting, where the null hypothesis is that a set of series all share a common trend subject to a change and a stationary noise function, the alternative being that one or more series have unit root noise components. Also of related interest is the study by Kim et al. (2002) who study unit root tests with a break in innovation variance following the work by Hamori and Tokihisa (1997). The issue of unit roots and trend breaks has also been addressed using a Bayesian framework with results that are generally in agreement with those of Perron (1989), see Zivot and Phillips (1994), Wang and Zivot (2000) and Marriott and Newbold (2000). 5.4.2 Problems and recent proposals Theoretical results by Vogelsang and Perron (1998) and simulation results reported in Perron and Vogelsang (1992a), Lee and Strazicich (2001a), Harvey et al. (2001) and Nunes et al. (1997) yield the following conclusions about the tests when a break is present under the null hypothesis. For the IO case when a slope shift is present, both versions using the break date by minimizing the unit root test or maximizing the absolute value of the t-statistic on the coefficient of the slope dummy, yield tests with similar features, namely an asymptotic size 100%. In the presence of a level shift, the asymptotic size is correct but liberal distortions occur when the level shift is large. When the one time dummy D (T1 )t is included in the regression, the source of the problem is that the break point selected with highest probability (which increases as the magnitude of the break increases) is T10 −1, i.e., one period before the true break; and it is for this choice of the break date that the tests have most size distortions. Lee and Strazicich (2001a) show that the problem is the same as if the one time dummy D (T1 )t was excluded when considering the known break date case. Their result also implies that, when unit root tests are performed using a regression of the form (31) without the one time dummy D (T1 )t , the correct break date is selected but the tests are still affected by size distortions (which was also documented by simulations). In cases with only a level shift, Harvey et al. (2001) suggest evaluating the unit root t-statistic at the break date selected by maximizing the absolute value of the t-statistic on the coefficient of the level shift plus one, and show that the tests then have correct size even for large breaks. For the AO type models, the following features apply. When the break date is selected by minimizing the unit root test, similar size distortions apply. However, when the break date is selected by maximizing the absolute value of the t-statistic on the relevant shift dummy, the tests have the correct size even for large breaks, and the correct break date is selected in 60 large samples. Vogelsang and Perron (1998) argue that the limit distribution of the unit root tests is then that corresponding to the known break date case. They suggest, nevertheless, to use the asymptotic critical values corresponding to the no break case since this leads to a test having asymptotic size no greater than that specified for all magnitudes of the break, even though this implies a conservative procedure when a break is present. An alternative testing procedure, which naturally follows from the structural change literature reviewed in Section 3, is to evaluate the unit root test at the break date selected by minimizing the sum of squared residuals from the appropriate regression. Interesting simulations pertaining to the IO case are presented in Lee and Strazicich (2001). They show that if one uses the usual asymptotic critical values that apply for the no break case under the null hypothesis, the tests are conservative when a break is present (provided the one time dummy D (T1 )t is included in the regression). They correctly note, however, that the limit null distribution when no break is present depends on the limit distribution of the estimated break date which may depend on nuisance parameters. Hatanaka and Yamada (1999) present useful theoretical results for the IO regression (though they specify the data generating process to be of the AO type). They show that, when a change in slope is present, the estimate of the break fraction λ1 , obtained by minimizing the sum of squared residuals, is consistent and that the rate of convergence is T in both the I(1) and I(0) cases. They also show that this rate of convergence is sufficient to ensure that the null limit distribution of the unit root test is the same as when the break date is known. Hence, one need only use the critical values for the known break date case that pertains to the estimated break date. The test has accordingly more power since the critical values are smaller in absolute value (they also consider a two break model and show the estimates of the break dates to be asymptotically independent). The problem, however, is that the results apply provided there is a break in the slope under the null hypothesis. Indeed, if no break is present, the known break date limit distribution no longer applies; and if the break is small, it is likely to provide a poor approximation to the finite sample distribution. Hatanaka and Yamada (1999) present simulations results calibrated to slope changes in Japanese real GDP that show the estimates of the break dates to have a distribution with fat tails and the unit root test accordingly shows size distortions. For the AO case, the work of Kim and Perron (2005) leads to the following results based on prior work by Perron and Zhu (2005). Under the null hypothesis of a unit root, if a slope change is present, the rate of convergence of the estimate of the break date obtained by minimizing the sum of squared residuals is not fast enough to lead to a limit distribution 61 for the unit root tests (evaluated at this estimate of the break date) that is the same as in the known break date case. They, however, show that a simple modification yields a similar result as in the IO case. It involves performing the unit root test by trimming or eliminating data points in a neighborhood of the estimated break date. This again leads to unit root tests with higher power. Let us summarize the above discussion. First, in the unknown break date case, the invariance properties with respect to the parameters of the trend no longer apply as in the known break date case. Popular methods based on evaluating the unit root test at the value of the break date that minimizes it or maximizes the absolute value of the t-statistic on the coefficient of the relevant dummy variable suffer from problems of liberal size distortions when a large break is present (except with the latter method to select the break date in the IO case) and little if any when the break is small. When the break is large, evaluating the unit root test at the break date that minimizes the sum of squared residuals leads to a procedure with correct size and better power. So this suggests a two step procedure that requires in the first step a test for a change in the trend function that is valid whether a unit root is present or not, i.e., under both the null and alternative hypotheses. In this context, the work of Perron and Yabu (2005) becomes especially relevant. This is the approach taken by Kim and Perron (2005). They use a pre-test for a change in trend valid whether the series is I(1) or I(0). Upon a rejection, the unit root test is evaluated at the estimate of the break date that minimizes the sum of squared residuals from the relevant regression. If the test does not reject, a standard Dickey-Fuller test is applied. This is shown to yield unit root tests with good size properties overall and better power. In cases where only level shifts are present, similar improvements are possible even though, with a fixed magnitude of shift, the estimate of the break date is not consistent under the null hypothesis of a unit root. 6 Testing for Cointegration Allowing for Structural Changes We now discuss issues related to testing for cointegration when allowing for structural changes. We first consider, in Section 6.1, single equation methods involving systems with one cointegrating vector. Here tests have been considered with the null hypothesis as nocointegration and the alternative as cointegration, and vice versa. In Section 6.2, we consider the multivariate case, where the issue is mainly determining the correct number of cointegrating vectors. Since many of the issues are similar to the case of testing for unit roots allowing structural breaks, our discussion will be brief and outline the main results and procedures suggested. 62 6.1 Single equation methods Consider an n dimensional vector of variables yt = (y1t , y2t ) with y1t a scalar, and y2t an n − 1 vector. We suppose that the sub-system y2t is not cointegrated. Then the issue is to determine whether or not there exists a cointegrating vector for the full system yt . Consider the following static regression (33) y1t = α + βy2t + ut The system is cointegrated if there exists a β such that the errors ut are I(0). Hence, a popular method is to estimate this static regression by OLS and perform a unit root test on the estimated residuals (see, Phillips and Ouliaris, 1990). Here the null hypothesis is no-cointegration and the alternative is cointegration. Another approach is to use the Error Correction Model (ECM) representation given by: ∆y1t = bzt−1 + k X di ∆y2t + et i=1 where zt = y1t − βy2t is the equilibrium error. In practice, one needs to replace β by an estimate that is consistent when there is cointegration. The test can then be carried using the t-statistic for testing that b = 0 (see, e.g., Banerjee et al., 1986). When adopting the reverse null and alternative hypotheses, a statistic that has been suggested is, again, Gardner’s (1969) Q test (see Shin, 1994). It can be constructed using the OLS residuals from the static regression when the regressors are strictly exogenous, or, more generally, the residuals from a regression augmented with leads and lags of the firstdifferences of the regressors, as suggested by Saikkonen (1991) and Stock and Watson (1993). Of course, many other procedures are possible. Here, structural changes can manifest themselves in several ways. First, there can be structural changes in the trend functions of the series without a change in the cointegrating relationship (i.e., a change in the marginal distributions of the series). Campos et al. (1996) have documented that shifts in levels do not affect the size of tests of the null hypothesis of no cointegration, for both the ECM based test and the test based on the residuals from the static regression. However, they affect the power of the latter, though not of the former. If all regressors have a common break in the slope of their trend function, the tests can be liberal and reject the null hypothesis of no-cointegration too often, though different tests are affected differently (Leybourne and Newbold, 2003). This is related to what has been labelled as co-breaking processes. Changes in the variance of the errors ut can also induce size distortions if it occurs early enough in the sample (e.g., Noh and Kim, 2003). 63 Second, structural changes can manifest themselves through changes in the long-run relationship (33), either in the form of a change in the intercept, or a change in the cointegrating vector. Here, the power of standard tests for the null hypothesis of no-cointegration can have substantially reduced power as documented by Gregory et al. (1996) and Gregory and Hansen (1996a). An early contribution that proposed tests for the null hypothesis of no-cointegration allowing for the possibility of a change in the long-run relation is that of Gregory and Hansen (1996a). They extend the residual-based tests by incorporating appropriate dummies in regression (33) and taking as the resulting test-statistic the minimal value over all possible break dates. Cases covered are: 1) allowing a change in the level α; 2) allowing for a similar change in level when regression (33) includes a time trend; 3) allowing for changes in both the level α and the cointegrating vector β (with no trend); 4) the case allowing for a change in the level and slope of an included trend and of the cointegrating vector is analyzed in Gregory and Hansen (1986b). The limit distributions of the various tests are derived under the null hypothesis that the series are not cointegrated and are individually I(1) processes with a stable deterministic trend component. As in the case of tests for unit roots, the value of the break date associated with the minimal value of a given statistic is not, in general, a consistent estimate of the break date if a change is present. Cook (2004) shows the size of the tests to be affected (towards excessive rejections) when the series are not cointegrated and are individually I(1) processes with a change in trend. The issue of allowing the possible change in trend under both the null and alternative hypotheses does arise in the context of testing the null hypothesis of no-cointegration. Indeed, under the null hypothesis, the model is a spurious one and the parameters of the cointegrating vector are not identified. It might be possible to identify a change in the slope of a trend under the null hypothesis, but this case is seldom of empirical interest. This means that no further gains in power are possible by trying to exploit the fact that a change in specification occurs under both the null and alternative hypotheses, as was done for unit root tests. Such gains are, however, possible, when adopting cointegration as the null hypothesis. Concerning tests that takes the null hypothesis to be cointegration, the contributions include Bartley et al. (2001), Carrion-i-Silvestre and Sanso (2004) and Arai and Kurozumi (2005). All are based on various modifications of Gardner’s (1969) Q statistic as used by Shin (1994) without structural breaks. The general framework used is to specify the cointegrating relationship by y1t = α1 + α2 (1 > T1 ) + γ 1 t + γ 2 (t − T1 )1(t > T1 ) + β 1 y2t + β 1 y2t 1(t > T1 ) + ut 64 (34) The required residuals to construct the Q test are based on transformed regressions that allow the construction of asymptotically optimal estimates of the cointegrating vector. Bartley et al. (2001) consider only a change in the level and slope of the trend and use the canonical cointegrating regression approach suggested by Park (1992) to estimate the cointegrating vector β. The break date is selected by minimizing the sum of squared residuals from the canonical cointegrating regression. They argue that the resulting estimate of the break fraction is consistent and that the limit distribution of the test corresponds to that applying in the known break date case. The simulations supports this assertion. Carrion-i-Silvestre and Sanso (2004) and Arai and Kurozumi (2005) extend the analysis to cover more cases, in particular allowing for a change in the cointegrating vector. In the case of strictly exogenous regressors, they construct the Q test using residuals from the static regression (34) (scaled appropriately with an estimate of the long-run variance of the errors, which allows for serial correlation). In the general case without strictly exogenous regressors, both recommend using the residuals from regression (34) augmented with leads and lags of the first-differences of y2t (Carrion-i-Silvestre and Sanso (2004) show that the use of the Fully Modified estimates of Phillips and Hansen (1990) leads to tests with very poor finite sample properties). Both select the break date by minimizing the sum of squared residuals from the appropriate regression, following the work of Kurozumi and Arai (2004) who show that the estimate of the break fraction in this model converges at least at rate T 1/2 . This permits limit critical values corresponding to the known break date case. They also consider selecting the break date as the value which minimizes the Q statistic but do not recommend its use given that the resulting tests then suffers from large size distortions in finite samples. A caveat about the approach discussed above is the fact that for the suggested methods to be valid, there must be a change in the cointegrating relationship, if cointegration actually holds. This is because the search for the potential break date is restricted to break fractions that are bounded, in large samples, from the boundaries 0 and 1. Hence, when there is no change the limit value cannot be 0 or 1, the estimate is inconsistent and has a non-degenerate limit distribution, which in turn affects the limit distribution of the test (i.e., it does not correspond to the one that would prevail if no break was present). But to ascertain whether a break is present, one needs to know if there is cointegration, which is actually the object of the test. Tests of whether a change in structure has occurred (as reviewed in Section 4.7) will reject the null hypothesis of no change when a change actually occurs in a cointegrating relationship, and will also reject if the system is simply not cointegrated. Hence, we are led to a circular argument. The test procedure needs to allow for the possibility of a change and 65 not impose it. It may be possible to relax the restriction on the search for the break date by allowing all possible values. In the context of cointegrated I(1) regressors, it is, however, unknown at this point, if the estimate of the break fraction would converge to 0 or 1 when no change is present. 6.2 Methods based on a mutivariate framework We now consider tests that have been proposed when the variables are analyzed jointly as a system. Here, the results available in the literature are quite fragmentary and much of it pertains to a single break at a known date. Also, different treatments are possible by allowing for a change in the trend function of the original series (i.e., the marginal processes), or in allowing for a change in the cointegrating relation. One of the early contribution is that of Inoue (1999). It allows for a one time shift in the trend function of the series at some unknown date, either in level for non-trending series and for both level and slope in trending series. He considers an AO type framework and also an IO type regression when only a shift in intercept is allowed in the VAR. The specification of the null and alternative hypotheses follow Zivot and Andrews (1992) and Gregory and Hansen (1996), in that the shifts are allowed only under the alternative hypothesis. Hence, the null hypothesis is that the system contains no break and no more than r cointegrating vectors, and the alternative hypothesis is that the data can exhibit a change in trend and that the cointegrating rank is r + 1, or greater than r. The breaks are assumed to occur at the same date for all series. Under the alternative hypothesis, the series are not assumed to be co-breaking, in the sense that the cointegrating vector that reduces the non-stationarity in the stochastic component also eliminates the non-stationarity in the deterministic trend. He considers the trace and maximal eigenvalue tests of Johansen (1988, 1991) with data appropriately detrended allowing for a shift in trend, and the resulting statistic is based on the maximal values over all permissible break dates. It is unclear what are the properties of the tests when the null hypothesis is true with data that have broken trends. Also, although the parameter r can be selected arbitrarily, the procedures cannot be applied sequentially to determine the cointegrating rank of the system. This is because the breaks are not allowed under the null hypothesis, only under the alternative. So if one starts with, say, r = 0, breaks are allowed for alternatives such that the cointegrating rank is greater than 0. But, upon a rejection, if one then wants to test the null of rank 1 versus an alternative with rank greater than 1, one needs to impose no break under the null hypothesis of rank 1, a contradiction from what was specified in the earlier step. 66 Saikkonen and Lütkepohl (2000a) also considers a test of the null hypothesis of r cointegrating vectors versus the alternative that this number is greater than r, allowing for a break in the trend function of the individual series under both the null and alternative hypotheses. They, however, only consider a level shift (in trending or non trending series) occurring at some known date. To estimate the coefficients of the trend component of the series, they use a similar GLS procedure, as discussed in Section 5.3, appropriately extended for the multivariate nature of the problem. This detrending method imposes the null hypothesis. Hence, the effect of level shifts is negligible in large samples and the limit distribution of the test is the same as the standard (no-break) cointegration test of Lütkepohl and Saikkonen (2000) and Saikkonen and Lütkepohl (2000b). Once the detrended data is obtained the test is based on the eigenvalues of a reduced rank problem where restrictions implied by the process and the breaks are not imposed. Johansen et al. (2000) consider a more general problem but still with known break dates. They consider multiple structural changes in the following VAR of order k, ⎞ ⎛ k−1 X yt−1 ⎠ + µj + Γi ∆yt−i + et ∆yt = (Π, Πj ) ⎝ t i=1 for Tj−1 + k < t ≤ Tj for j = 1, ..., m. Hence, there are m breaks which can affect the constant and the coefficients of the trend. Various tests for the rank of the cointegrating matrix are proposed (imposing or not various restrictions on the deterministic components). Since the estimates of the coefficients of the trend are estimated from a maximum-likelihood type approach (following Johansen, 1988, 1991), the limit distribution depends on the exact specification of the deterministic components and on the true break dates. Asymptotic critical values are presented via a response surface analysis. For the special case of a single shift in level, Lütkepohl et al. (2003) compare the two approaches of Saikkonen and Lütkepohl (2000a) and Johansen et al. (2000). They show that the former has higher local asymptotic power. However, the finite sample size-adjusted power is very similar. They recommend using the method of Saikkonen and Lütkepohl (2000a) on the basis of better size properties in finite samples and also on the fact that they view having a limit distribution free of the break dates to be advantageous. A problem with this argument is that the non-dependence of the limit distribution on the break date with the procedure of Saikkonen and Lütkepohl (2000a) no longer holds in more general models, especially when slope shifts are involved. Indeed, no result is yet available for this approach with a GLS type detrending procedure when slope shifts are present. 67 Lütkepohl et al. (2004) extend the analysis of Saikkonen and Lütkepohl (2002), which pertained to testing for a unit root allowing for a change in the level of a series occurring at an unknown date (see Section 5.4.1). The GLS type procedure discussed above is used to estimate the coefficients of the deterministic components. Once the series are detrended, the cointegration tests of Johansen (1988) can be used. In the unit root case, with a GLS detrending procedure that imposes the null hypothesis, the change in mean reduces to an outlier in the first-differenced series. Here, thing are more complex and a consistent estimate of the break date is preferable. Estimating the break date has, however, no effect on the limit null distribution of the test statistic since, here again, it does not depend on the true value of the break date. In it useful to consider in more detail the issue of estimating the break date. The n vector of data yt is assumed to be generated by yt = µ + θDUt + δt + xt where DUt = 1(t > T1 ) and xt is a noise component generated by a VAR, with the following ECM representation, k X ∆xt = Πxt−1 + Γi ∆xt−i + et i=1 Here, the presence of cointegration implies the decomposition Π = αβ 0 with β the n × r matrix of cointegrating vectors. Hence, we also have the following ECM representation for yt ∆yt = ν + αβ 0 (yt−1 − δ(t − 1) − θDUt−1 ) + k X i=1 Γi ∆yt−i + k X γ i ∆DUt−i + et (35) i=1 This ECM representation will be affected by a level shift if β 0 θ 6= 0, otherwise only the impulse dummies ∆DUt−i are present. In most cases of interest, we have β 0 δ = 0, which specifies that the same linear combinations that eliminate the stochastic non-stationarity also eliminate the non-stationarity induced by the trend. The condition β 0 θ = 0 can be interpreted in the same way, i.e., if some variables are affected by changes in trend, the linear combination of the data specified by the cointegrating vectors will be free of structural breaks. This is often referred to as ‘co-breaking’. Hence, the condition β 0 θ 6= 0 requires that the series be non co-breaking, which may be unappealing in many cases. Lütkepohl et al. (2004) estimate the break date by minimizing the determinant of the sample covariance matrix of the estimates of the errors et . They show the estimate of the break fraction to 68 converge at rate T , though no limit distribution is given since this rate is enough to guarantee that the limit distribution of the test be independent of the break date. Note that the search for the break date is restricted to an interval that excludes a break fraction occurring near the beginning or the end of the sample. This is important, since it makes the procedure valid conditional on shifts in level occurring. Without shifts, the true break fraction is 0 or 1, which are excluded from the search. Hence, in this case the estimated break fractions will converge to some random variable. But given that a GLS type detrending is done, this has no impact of the limit distribution of the rank test. A similar result holds when co-breaking shifts are present, though Lütkepohl et al. (2004) argue that if the shifts are large enough, they can be captured by the impulse dummies ∆DUt−i (for more details on estimation of break dates in this framework, see Saikkonen et al., 2004). All contributions discussed above do not address the problem of a potential shift in the cointegrating vector. A recent analysis by Andrade et al. (2005) deals with this in the context of a one-time change. The object is to test the null hypothesis of r cointegrating vectors versus the alternative that this value is greater than r. They allow the change in the cointegrating relationship to occur under both the null and alternative hypotheses and the number of cointegrating vectors is the same in both regimes. This allows a sequential procedure to determine the rank. The issues are addressed using the following generalized ECM ∆yt = 1(t ≤ T1 )[α0 β 00 yt−1 − δ 0 dt ] + 1(t > T1 )[α1 β 01 (yt−1 − yT1 ) − δ 1 dt ] + k X Γi ∆yt−i + et i=1 where dt is a vector of deterministic components (usually the null set or a constant). Note that the data is re-normalized after the break to start again at 0. This is done since otherwise the variance of β 01 yt−1 would increase after the break given that it depends on the value of β 01 yT1 . They note that the estimation of this model by maximum likelihood is quite involved and suggest a simpler principle components analysis. Let β i⊥ be a matrix such that β 0i β i⊥ = 0, and suppose that the loading factors (or adjustment matrices) are constant, i.e., α0 = α1 , the test for the null hypothesis that the cointegrating rank is r is based on testing that γ 0 = γ 1 = 0 in the following system h i h i 0 0 0 0 ∆yt = 1(t ≤ T1 ) γ 0 β̂ 0⊥ yt−1 + αβ̂ 0 yt−1 + 1(t > T1 ) γ 1 β̂ 1⊥ yt−1 + αβ̂ 1 (yt−1 − yT1 ) + k X Γi ∆yt−i + et i=1 69 0 0 where β̂ 1 and β̂ i⊥ are estimates obtained from the principle components analysis. The statistic is based on a multivariate Fisher-type statistic modified to eliminate the effect of nuisance parameters on the limit distribution under the null hypothesis. They also consider a version that is valid when the break date is unknown, based on the maximal values over a specified range for the break date, and present a test to evaluate how many cointegrating vectors are subject to change across regimes. When both the cointegrating matrix β and the loading factors α are allowed to change, a more involved testing procedure is offered, which applies, however, only to the known break date case. An interesting recent contribution is that of Qu (2004). It proposes a procedure to detect whether cointegration (or stationarity in the scalar case) is present in any part of the sample, more precisely whether there is evidence in any part of the sample that a system is cointegrated with a higher cointegrating rank than the rest of the sample. The test procedure is based on a multivariate generalization of Gardner’s (1969) Q test as used in Breitung (2002). The main device used is that if one or more sub-samples have a different cointegrating rank, one can find them by searching, in an iterative fashion, over all possible partitions of the sample with three segments or two breaks. The relevant limit distributions are derived allowing the possibility of imposing some structure if desired (e.g., that the change occurs at the beginning or end of the sample). He also discusses how to consistently estimate the break dates or the boundaries of the regimes when a change has been detected. A modification is also suggested to improve the finite sample performance of the test. This approach also permit testing for changes in persistence with the null hypothesis specified as an I(1) process throughout the sample. It also permits detecting whether cointegration is present when the cointegrating vector changes at some unknown possibly multiple dates. 7 Conclusions This review has discussed a large amount of research that has been done in the last fifteen years or so pertaining to issues related to structural changes and to try to distinguish between structural changes and unit roots. But still, some important questions remain to be addressed: limit distributions of estimates of break dates in a cointegrated system with multiple structural changes, issues of non-monotonic power functions for tests of structural change and how to alleviate the problems, evaluating the frequency of permanent shocks; just to name a few. Research currently under progress is trying to address these and other issues. One recent area of research where similar tools have been applied is related to distinguish70 ing between long-memory processes and short-memory processes with structural changes, in particular level shifts. This is especially important in financial economics, where it is widely documented that various measures of stock return volatility exhibit properties similar to those of a long-memory process (e.g., Ding et al., 1993, Granger and Ding, 1995 and Lobato and Savin, 1998). For reviews of the literature on purely long-memory processes, see Robinson (1994a), Beran (1994) and Baillie (1996). A mentioned in Section 2, a popular test for long-memory is the rescaled-range test. Yet, interestingly, Gardner’s (1969) Q test makes yet another appearance. Indeed, it was, along with a slight modification, also proposed to test this problem by Giraitis et al. (2003). So we have the same test acting with the null hypothesis of a stable short-run memory process versus an alternative that is either structural change, a unit root or long-memory. This goes a long way showing how the three problems are inter-related. One of the most convincing evidence that stock market volatility may be better characterized by a short-memory process affected by occasional level shifts is that of Perron and Qu (2004). They show that the behavior of the log-periodogram estimate of the long-memory parameter (the fractional differencing coefficient), as a function of the number of frequencies used in the regression, is very different for the two types of processes. The pattern found with data on daily SP500 return series (absolute or square root returns) is very close to what is expected with a short-memory process with level shifts. They also present a test which rejects the null hypothesis of long memory. Given that unit root and long memory processes share similar features, it is not surprising that many of the same problems are being addressed with similar findings. Along the lines of Perron (1989) for unit roots, it has been documented that short-memory processes with level shifts will exhibit properties that make standard tools conclude that long memory is present (e.g., Diebold and Inoue, 2001, Engle and Smith, 1999, Gourieroux and Jasiak, 2001, Granger and Ding, 1996, Granger and Hyung, 2004, Lobato and Savin, 1998, and Teverosovky and Taqqu, 1997). Some papers have also documented the fact that long-memory processes will induce, similar to unit root processes, a rejection of the null hypothesis of no-structural change when using standard structural change tests; for the CUSUM and the Sup-Wald test applied to a change in a polynomial trend, see Wright (1998) and Krämer and Sibbertsen (2002). Results about the rate of convergence of the estimated break fraction in a single mean shift model can be found in Kuan and Hsu (1998). When there is structural change, the estimate is consistent but the rate of convergence depends on d. When d ∈ (0, 1/2) and 71 there is no change, the limit value is not 0 or 1 but rather the estimate of the break fraction converges to a random variable, suggesting a spurious change, exactly as in the unit root case (see Nunes et al., 1995, and Bai, 1998). For results related to multiple structural changes in mean, see Lavielle and Moulines (2000). A test for a single structural change occurring at some known date in the linear regression model is discussed in Hidalgo and Robinson (1996). It is essentially a Wald test for testing that the coefficients are the same in both regimes, which accounts for the long-memory correlation pattern in the residuals. Lazarová (2005) presents a test for the case of a single change in the parameters of a linear regression model occurring at an unknown date. The test follows the “fluctuations tests” approach of Ploberger et al. (1989) with different metrics used to weight the differences in the estimates for each permissible break dates (giving special attention to the Sup and Mean functionals). The limit distribution depends on nuisance parameters and a bootstrap procedure is suggested to obtain the relevant critical values. Related to the problem of change in persistence (see Section 4.9), Beran and Terrin (1996) present a test for a change in the long-memory parameter, based on the maximal difference, across potential break dates, of appropriately weighted sums of autocovariances. Related to unit root tests allowing for a change in the trend function, Gil-Alana (2004) extends Robinson’s (1994b) test to allow for a one-time change occurring at a known date. For a review of some related results, see Sibbertsen (2004). The literature on structural changes in the context of long memory processes is quite new and few results are available. Still, there is a large demand for empirical applications. Given the nature of the problems and series analyzed, it is important to have procedures that are valid for multiple structural changes. For example, with many financial time series, it is the case that allowing for structural breaks reduces considerably the estimates of the long-memory parameters within regimes (e.g., Granger and Hyung, 2004, for stock return volatility). Are the reductions statistically significant? Are the reductions big enough that one can consider the process as being of a short-memory nature within regimes? Is there significant evidence of structural changes? Is the long-memory parameter stable across regimes? 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