Distinguishing between trend-break models: method and empirical evidence

Transcription

1 Econometrics Journal (21), volume 4, pp Distinguishing between trend-break models: method and empirical evidence CHIH-CHIANG HSU 1 AND CHUNG-MING KUAN 2 1 Department of Economics, National Central University, aiwan cchsu@mgt.ncu.edu.tw 2 Institute of Economics, Academia Sinica, aiwan ckuan@ieas.econ.sinica.edu.tw Received: March 2 Summary We demonstrate that in time trend models, the likelihood-based tests of partial parameter stability have size distortions and cannot be applied to detect the changing parameter. A two-step procedure is then proposed to distinguish between different trend-break models. his procedure involves consistent estimation of break dates and properly-sized tests for changing coefficient. In the empirical study of the Nelson Plosser data set, we find that the estimated change points and trend-break specifications resulting from the proposed procedure are quite different from those of Perron (1989, 1997), Chu and White (1992), and Zivot and Andrews (1992). In another application, our procedure provides formal support for the conclusion of Ben-David and Papell (1995) that real per capita GDPs of most OECD countries exhibit a slope change in trend. Keywords: Change point, Partial parameter stability, ime trend model, rend-break model. 1. INRODUCION It has been widely accepted that many macroeconomic time series have a unit root since the seminal work of Nelson and Plosser (1982). Perron (1989) challenged the unit-root hypothesis and suggested that those series are better characterized by a deterministic trend with a break (structural change). In the simple time trend model, there are three types of break: an intercept change, a slope change, and changes in both parameters. It is of theoretical and practical interest to distinguish between trend-break models because they bear different economic interpretations. For example, Ben-David and Papell (1995) argued that a time trend model with a slope change is compatible with the Romer-type endogenous growth model, but not otherwise. Although Perron (1989) determined trend-break models by visual inspection of data rather than statistical tests, his choices of break types were never questioned in subsequent studies such as Banerjee et al. (1992), Zivot and Andrews (1992), Lumsdaine and Papell (1997), Nunes et al. (1997) and Perron (1997). As far as hypothesis testing is concerned, identifying break types amounts to testing which parameter in the model has changed. his is, however, not a straightforward job in the time trend model. For example, the tests of Chu and White (1992) check the constancy of each coefficient, but their tests reject the null hypothesis of partial parameter stability far too often, as observed by Kuan (1998, 1999). herefore, the Chu White tests cannot correctly identify the changing parameter.. Published by Blackwell Publishers Ltd, 18 Cowley Road, Oxford OX4 1JF, UK and 35 Main Street, Malden, MA, 2148, USA.

2 172 Chih-Chiang Hsu and Chung-Ming Kuan In this paper, we first show analytically and by simulations that the Wald-type tests of partial parameter stability in trend-break models depend on all the coefficients. As the critical values of such tests are determined by assuming all the coefficients remain constant, these tests suffer from serious size distortions when the coefficient being tested is stable but the other coefficient changes. We then propose a two-step procedure to identify the changing coefficient so as to distinguish between different trend-break models. In the first step, we jointly test the constancy of all the coefficients in the model and proceed to estimate the change point by the least-squares method when the null is rejected. In the second step, the standard likelihood ratio test of parameter stability is applied to each coefficient, taking the estimated point as the true change point. he resulting tests have a limiting χ 2 distribution. It is shown that such tests are sensitive only to the changes of the coefficient being tested and hence have proper size asymptotically, regardless of the behaviour of the other coefficient. he proposed procedure is also readily extended to models with a polynomial trend. In the empirical analysis of the Nelson Plosser data, we find that our estimated change points, which are consistent estimates, are quite different from those of Zivot and Andrews (1992) and Perron (1997). Moreover, the trend-break models identified by our method are very different from the choices of Perron (1989) and Chu and White (1992). In the second application to real per capita GDP of 16 countries, our testing results suggest that these data are consistent with the Romer-type growth model; this conclusion agrees with the finding of Ben-David and Papell (1995). his paper is organized as follows. We introduce the Wald-type tests of partial parameter stability in Section 2 and analyse their performance in Section 3. In Section 4, we propose a procedure that can properly distinguish between trend-break models. Section 5 presents empirical applications to the data sets of Nelson and Plosser (1982) and Ben-David and Papell (1995). Section 6 concludes the paper. All proofs are deferred to the Appendix. 2. WALD-YPE ESS OF PARIAL PARAMEER SABILIY Suppose that y t is generated according to y t = a t + b t ( t ) + ε t = x t β t + ε t, t = 1, 2... (1) where x t = [1 t/ ], β t = [a t b t ], and ε t is a random disturbance. We write the trend function as t/ to facilitate the subsequent analysis; our results will not be altered if we write (1) as y t = a t + b t t + ε t. o allow for serial correlations in the data, we assume the following condition holds. [A] ε t = j=1 φ j u t j, where j=1 j φ j <, and {u t } is a white noise with finite 4 + ζ moment, ζ >. Under [A], {ε t } obeys the functional central limit theorem: ( ) [ 1 τ] ε t, τ 1 (σ ε B(τ), τ 1), (2)

3 Distinguishing trend break 173 where [ τ] is the integer part of τ, stands for weak convergence, B is the standard Brownian motion, and ( ) 2 σε 2 = lim 1 E ε t. In what follows we also write d = for equality in distribution, D P for convergence in probability, for convergence in distribution, and B for the two-dimensional Brownian bridge such that its elements, B1 and B 2, are two independent Brownian bridges. We are interested in testing the null hypothesis: H : Rβ t = Rβ, for all t, (3) where R is a q 2 (q = 1, 2) selection matrix. his is the joint hypothesis of parameter constancy in (1) when R = I 2, the 2 2 identity matrix. his is a hypothesis of partial parameter stability when R is a Cartesian unit vector; in particular, (3) is intercept (slope) constancy when R = e 1 = [1 ] (R = e 2 = [ 1]). Existing tests focus on the joint hypothesis of parameter constancy, e.g. MacNeil (1978), Vogelsang (1997), and Kuan (1998). Although Chu and White (1992) proposed separate tests for the intercept and slope coefficients, Kuan (1999) showed that their tests are in effect tests of joint parameter constancy. Less attention was given to the hypothesis of partial parameter stability, however. A leading alternative is a one-time change in the trend function: H 1 : Rβ t = { Rβ, t = 1, 2,..., k, Rβ + Rλ, t = k + 1,...,, (4) where k is the true, unknown change point and λ denotes the vector of parameter changes. As in Perron (1989), (4) with R = e 1, e 2 and I 2, are referred to as the crash, changing growth, and mixed hypotheses, respectively. Distinguishing between these trend breaks amounts to identifying which parameter in (1) has changed. When k is unknown, one can compute the Wald test against (4) at each hypothetical change point k = [ τ], [ τ] + 1,..., [ τ], where [τ, τ] is a set with closure in (, 1). Let M k := k x t x t, M k := For each k, the resulting Wald test is W (k; R) = [( ˆβ 1 (k) ˆβ 2 (k)] R [s 2 (k)r(m 1 k t=k+1 x t x t. + M 1 k )R ] 1 R[ ˆβ 1 (k) ˆβ 2 (k)], (5) where ˆβ 1 (k) = Mk 1 k x t y t and ˆβ 2 (k) = M 1 t=k+1 k x t y t are, respectively, the pre- and post-change ordinary least-squares estimators of β t, and s 2 (k) is a consistent estimator of σ ε 2. he maximal, mean and exponential Wald tests considered by Vogelsang (1997) are just functionals of W (k; I 2 ), analogous to those of Andrews (1993) and Andrews and Ploberger (1994) for stationary regressions. In view of the tests of Vogelsang (1997), it seems reasonable to construct the Wald-type tests of partial parameter stability based on W (k; e i ), i = 1, 2. Unfortunately, such tests are unable to identify the changing parameter, as shown in the next section.

4 174 Chih-Chiang Hsu and Chung-Ming Kuan 3. ES PERFORMANCE o evaluate the performance of the Wald-type tests of partial parameter stability, consider the following local sequence: β t = β + 1/2 λ1(t/ > τ ), (6) where λ is the vector of parameter changes, 1 is the indicator function, and τ = k / is the relative position of k in the sample. Note that λ may contain a zero element so that only one parameter is subject to change. It is easy to verify that the convergence results below hold uniformly in τ: [ 1 τ] τ x t x t z(h)z(h) dh Q(τ), (7) where z(h) = [1 h], h [, 1], so that the (i, j)th element of Q(τ) is τ (i+ j 1) /(i + j 1). Moreover, [ 1 τ] τ x t x t λ1(t/ > τ ) Q 1 (h)λ1(h > τ ) dh, (8) where Q 1 (h) = dq(h)/dh. By the functional central limit theorem (2), where G(τ) = τ ( ) [ 1 τ] x t ε t, τ 1 (σ ε G(τ), τ 1), (9) z(h) db(h). Let C(τ) be the orthogonal matrix such that C(τ) Q(1) 1/2 Q(τ) Q(1) 1/2 C(τ) = diag[ω 1 (τ), ω 2 (τ)], where ω j (τ), j = 1, 2, are the eigenvalues of Q(1) 1/2 Q(τ) Q(1) 1/2 ; see the proof of heorem 3.1 in the appendix for the exact form of ω j (τ). It is then clear that C(τ) {Q(1) 1/2 [Q(τ) Q(τ) Q(1) 1 Q(τ)]Q(1) 1/2 } C(τ) = diag [ω 1 (τ)(1 ω 1 (τ)), ω 2 (τ)(1 ω 2 (τ))] (τ). he following theorem gives the weak limit of (5) under the local sequence (6). heorem 3.1. Given (1) and (6), we have W ([ τ]; R) [U(τ) + (τ)] R (τ) [R (τ) R (τ) ] 1 R (τ)[u(τ) + (τ)] (1) H(τ), where U is a 2 1 standardized and time-rescaled Brownian bridge with U(τ) = (τ) 1/2 B (ω(τ)),

5 Distinguishing trend break 175 and R (τ) and (τ) are two non-stochastic functions such that R (τ) = R Q(1) 1/2 C(τ) (τ) 1/2, (τ) = 1 σ ε (τ) 1/2 C(τ) Q(1) 1/2 [ τ 1 ] Q(τ)Q(1) 1 Q 1 (h) λ1(h > τ ) dh. Q 1 (h) λ1(h > τ ) dh When both parameters are constant, λ is identically zero, and so is. Hence, (1) becomes W ([ τ]; R) U(τ) R (τ) [R (τ) R (τ) ] 1 R (τ) U(τ) H(τ). (11) For a given τ, R (τ) [R (τ) R (τ) ] 1 R (τ) is an idempotent matrix, and U(τ) has the bivariate standard normal distribution. Hence, H(τ) has a χ 2 (q) distribution so that the process H may be referred to as a chi-squared process. Similarly, H of heorem 3.1 may be referred to as a noncentral chi-squared process. Moreover, when R = I 2, R (τ) [R (τ) R (τ) ] 1 R (τ) = I 2 so that H(τ) of (11) simplifies to 2 j=1 B j (ω j(τ)) 2 ω j (τ)(1 ω j (τ)), a time-rescaled tied-down Bessel process. his extends Proposition 6 of Bai and Perron (1998) to the time trend model with one-time change. Note that the likelihood ratio test considered by Bai (1999) has the same limiting distribution. Remark. In heorem 3.1, the limit of W ([ τ]; R) depends on the behaviour of both parameters because R is determined by the entire vector λ. Consider the special case where (6) holds with λ i = and λ j. hen for R = e i, Rλ = so that Rβ t = Rβ, which is just the null of partial parameter constancy (3). In this case, heorem 3.1 still applies, and the null limit of W ([ τ]; e i ) depends also on λ j, the change of the jth coefficient. hus, the null limit of W ([ τ]; e i ) cannot be determined unless the behaviour of λ j is known a priori. If we take (11) with R = e i as the null limit, then as long as there are changes in the other coefficient, tests based on W ([ τ]; e i ) would reject the null hypothesis too often and therefore have size distortions. Note that the corresponding Lagrange Multiplier test and likelihood ratio test suffer from the same problem. he result of heorem 3.1 is, however, in sharp contrast with stationary regressions with breaks. Suppose that x t in (1) are stochastic regressors such that [ 1 τ] x t x t P τ Q. his condition is quite typical in the literature; see e.g. Ploberger et al. (1989) and Andrews (1993). It is shown in the appendix that R (τ) (τ) simplifies to [ R 1 τ 1 ] (τ) (τ) = τ(1 τ) R λ1(h > τ ) dh τ λ1(h > τ ) dh. (12)

6 176 Chih-Chiang Hsu and Chung-Ming Kuan he difference between trending and stationary regressions is now clear. For the latter, the component of R is related only to the corresponding element of λ, and the limit of W ([ τ]; e i ) depends solely on the changes of the ith coefficient. he resulting Wald-type tests of partial parameter stability thus should have proper size asymptotically, regardless of the behaviour of the other coefficient. 1 Our simulations confirm the asymptotic analysis above. For the time trend model, we generate y t according to { 3 + t + εt, t = 1,..., [ τ y t = ], (3 + λ 1 ) + (1 + λ 2 )t + ε t, t = [ τ ] + 1,...,, where ε t are i.i.d. N(, 1) random variables. For stationary regressions, { 3 + xt + ε y t = t, t = 1,..., [ τ ], (3 + λ 1 ) + (1 + λ 2 )x t + ε t, t = [ τ ] + 1,...,, where x t are i.i.d. N(2, 1) and ε t are i.i.d. N(, 1) random variables. In all simulations, = 2, τ =.1,.2,...,.9, and the number of replications is 5. For each model, the maximal, mean, and exponential Wald tests with R = e 1 and e 2 are computed. he testing results at 5% level are summarized in ables 1 and 2. he critical values for the tests in able 1 are obtained from simulating (11) with R = e 1, e 2 and different functionals, where distributions are approximated using samples of = 1 observations and the number of replications is 1; the critical values for able 2 are taken from Andrews (1993) and Andrews and Ploberger (1994). When there is only an intercept change, a sensible test of intercept constancy (i.e. R = e 1 ) should have high empirical power, but a test of slope constancy (i.e. R = e 2 ) should have empirical size close to 5%. Similarly, when the slope changes but the intercept remains a constant, a test of intercept constancy should have empirical size close to 5%, but a test of slope constancy should have high power. Bearing these in mind, we immediately see the difference between ables 1 and 2. For the time trend model, those intercept (slope) tests reject far too often when the intercept (slope) is in fact a constant but the other coefficient changes. In stationary regressions, on the other hand, the Wald-type tests maintain proper sizes. Note that the intercept and slope tests of Chu and White (1992) suffer from the same problem; see Kuan (1998, 1999). Remark. he above result extends straightforwardly to models with a polynomial trend, i.e. (1) with x t = [1 t/... (t/ ) p 1 ]. For those models, equations (7) (9) still hold with z(h) = [1 h... h p 1 ], and heorem 3.1 remains valid. hus, any test based on W ([ τ]; R), where R is now a q p selection matrix, is sensitive to the changes of all parameters in the model and therefore cannot identify the changing parameter(s). 4. HE PROPOSED PROCEDURE When the change point τ is known, it is easily verified that R (τ ) (τ ) in heorem 3.1 is R λ, the change of the parameter(s) being tested. Since U(τ ) has a bivariate standard normal distribution, the asymptotic distribution of (1) becomes D W ([ τ ]; R) χ 2 (q; σ 2 λ R 1 Rλ), (13) 1 Some comments we received earlier argued that this result holds only when xt has mean zero. his is not true. he result (12) does not depend on the mean of x t ; see also the simulation results. ε R

7 Distinguishing trend break 177 able 1. he Wald-type tests of partial parameter stability: ime trend model. Intercept change: λ 1 = 2 and λ 2 = τ Max W (e 1 ) Max W (e 2 ) Mean W (e 1 ) Mean W (e 2 ) Exp W (e 1 ) Exp W (e 2 ) Slope change: λ 1 = and λ 2 =.2 Max W (e 1 ) Max W (e 2 ) Mean W (e 1 ) Mean W (e 2 ) Exp W (e 1 ) Exp W (e 2 ) Note: All entries are rejection probabilities in percentages with the nominal size set to 5%. he critical values of the maximal Wald tests for intercept and slope constancy are, respectively, and 1.155; those of the mean Wald tests are and 2.499; those of the exponential Wald tests are 2.17 and hese values are obtained from simulating (11). able 2. he Wald-type tests of partial parameter stability: Stationary regression. Intercept change: λ 1 = 2 and λ 2 = τ Max W (e 1 ) Max W (e 2 ) Mean W (e 1 ) Mean W (e 2 ) Exp W (e 1 ) Exp W (e 2 ) Slope change: λ 1 = and λ 2 =.5 Max W (e 1 ) Max W (e 2 ) Mean W (e 1 ) Mean W (e 2 ) Exp W (e 1 ) Exp W (e 2 ) Note: All entries are rejection probabilities in percentages with the nominal size set to 5%. he critical values are taken from Andrews (1993) and Andrews and Ploberger (1994). a noncentral chi-squared distribution with q degrees of freedom and the noncentrality parameter σε 2 λ R 1 R Rλ, where R = R [Q(τ ) 1 + [Q(1) Q(τ )] 1 ]R.

8 178 Chih-Chiang Hsu and Chung-Ming Kuan his shows that, provided that τ is known, the standard Wald test W ([ τ ]; e i ) depends only on the behaviour of the parameter being tested. In particular, under the null hypothesis that Rλ =, the limit of (13) becomes χ 2 (q), where q is the number of restrictions. Using the critical values from this chi-squared distribution, W ([ τ ]; e i ) has non-trivial local power and should not suffer from size distortions, at least asymptotically. When τ is unknown, it is natural to replace τ with a consistent change-point estimator and base the test of partial parameter stability on this estimator. A leading change-point estimator is the least-squares estimator. For a given hypothetical change point k = [ s], let x t denote an augmented collection of regressors: x t ([ s]; R ) = [x t 1(t/ > s)x t R ], where for R = e 1 and e 2, R are e 2 and e 1, respectively, such that R x t is the variable whose associated parameter is not tested under the null hypothesis (3), and for R = I 2, R = so that x t ([ s]; R ) is simply x t. he regression of y t on x t ([ s]; R ) is the constrained regression corresponding to the null hypothesis (3); the constrained residual sum of squares is denoted as RSS([ s]; R ). When there is no constraint, R = I 2, and regressing y t on x t ([ s]; I 2 ) = [x t 1(t/ > s)x t ] yields the unconstrained regression. he least-squares estimator of the change point τ is obtained by minimizing the unconstrained residual sum of squares RSS([ s]; I 2 ) with respect to s: ˆτ = inf{τ : τ = argmin s [τ,τ] RSS([ s]; I 2 )}, where < τ < τ < 1. One may also compute ˆτ as ˆk/, where ˆk = argmin k [k,k] RSS(k; I 2 ), with k = [ τ] and k = [ τ]; see e.g. Nunes et al. (1995). Such estimation would not be meaningful if [τ, τ] is too small. In practice, it is typical to set τ close to (e.g. τ =.5 or.1) and τ close to 1 (e.g. τ =.95 or.9). In what follows, we consider the likelihood ratio test of partial parameter stability, taking the least-squares change-point estimator ˆτ (or ˆk) as the true change point. We do not consider the Wald test because, as pointed out by Dufour (1997) and Critchley et al. (1996), when there are unidentified nuisance parameters (e.g. the change point), the likelihood ratio tests are pivotal functions, but the Wald tests are not and hence may have poor performance. 2 he likelihood ratio test of partial parameter stability can be expressed as L R (k; R) = (RSS(k; R ) RSS(k; I 2 ))/s 2 (k; I 2), where s 2 (k; I 2) = RSS(k; I 2 )/ is the variance estimator of the unconstrained regression. Based on the estimated change point ˆτ, the likelihood ratio test of the null hypothesis (3) with R = e i is L R ([ ˆτ]; R) = (RSS([ ˆτ]; R ) RSS([ ˆτ]; I 2 ))/s 2 ([ ˆτ]; I 2). (14) his test cannot be directly used to test partial parameter stability, however. Note that when the null hypothesis (3) is true with R = e i, there are two possibilities: (i) the ith coefficient is a constant but the jth coefficient changes; (ii) both coefficients remain constant. We shall show 2 We thank Professor Karim Abadir, the editor, for drawing our attention to these related works.

9 Distinguishing trend break 179 below that the LR statistic (14) behaves differently in these two cases and hence must be applied with care. For case (i), Bai (1997) and Bai and Perron (1998) showed that ˆτ is a -consistent estimator of τ (, 1). In the light of heorem 3.1 and (13), it can be shown that under the null hypothesis that the ith coefficient is a constant (i.e. e i λ = ), L R ([ ˆτ]; e i ) D χ 2 (1). his distribution holds whether or not the jth coefficient changes. hus, the likelihood ratio test based on the estimated change point ˆτ should not suffer from the size distortion problem discussed in the preceding section. Moreover, under the alternative (4) that the ith coefficient changes by Rλ = λ, we can show that 1 L R ([ ˆτ]; e i ) = O p (σε 2 (λ ) 2 e 1 i (τ )), (15) where ei (τ ) is the ith diagonal element of Q(τ ) 1 +[Q(1) Q(τ )] 1. hat is, L R ([ ˆτ]; e i ) is consistent against (4), and its power performance depends only on λ, the change of the parameter being tested. We therefore conclude that, as long as there is a change in some parameter, L R ([ ˆτ]; e i ), i = 1, 2, are capable of detecting the changing parameter. In case (ii), both coefficients are constant so that the change point τ is either or 1. When the change point is estimated over the set [τ, τ] [, 1], it is not easy to derive the limiting distribution of L R ([ ˆτ]; R). 3 o see how the test behaves in this case, we simulate the asymptotic distributions of L R ([ ˆτ]; e i ), i = 1, 2. In our simulations, [τ, τ] = [.1,.9], and the distributions are approximated using samples of = 1 observations and 2 replications. he resulting empirical densities are smoothed using an Epanechnikov kernel with a bandwidth calculated according to Silverman (1986, pp ). hese densities, together with the χ 2 (1) distribution are plotted in Figure 1. One can easily see that the simulated distributions of L R ([ ˆτ]; e i ) have much fatter right tails than χ 2 (1). his suggests that when both parameters remain constant, L R ([ ˆτ]; e i ) would reject the null hypothesis more often than it should if χ 2 (1) is taken as the null distribution. In other simulations (not reported), we also find that the larger the set [τ, τ], the more severe is the size distortion. hese results taken together indicate that χ 2 (1) can be the asymptotic null distribution of L R ([ ˆτ]; e i ) only when the ith coefficient is a constant and the jth coefficient changes, but not otherwise. As such, we must rule out the possibility that both parameters are constant before we can apply L R ([ ˆτ]; e i ) to test partial parameter stability. his immediately motivates the following two-step procedure. Procedure: [P1] est the joint hypothesis of parameter constancy. If the null hypothesis is not rejected, stop the procedure; otherwise, compute the least-squares estimator ˆτ and then go to [P2]. [P2] est the null hypothesis (3) with L R ([ ˆτ]; e i ), i = 1, 2, using the critical values from χ 2 (1) distribution. 3 It is, however, relatively easy to analyse the limiting behaviour of L R ([ ˆτ]; R) when ˆτ is obtained from [, 1]. Nunes et al. (1995) showed that ˆτ converges in probability to the set {, 1} when both coefficients are stable. Andrews (1993) also showed that L R ([ τ]; R) diverges in probability when τ approaches or 1. herefore, we would expect that L R ([ ˆτ]; R) also diverges in probability.

10 18 Chih-Chiang Hsu and Chung-Ming Kuan (a) (b) Figure 1. he simulated distributions of L R ([ ˆτ]; e i ) and χ 2 (1). In the first step, the tests of Vogelsang (1997) or Kuan (1998) may be employed to test the joint hypothesis of parameter constancy. Yet a simpler approach is to combine the joint test and change-point estimation in [P1]. Since minimizing the sum of squared residuals is equivalent to maximizing the likelihood ratio statistics of joint parameter constancy, the least-squares estimator ˆk is also ˆk = argmax k [k,k] L R (k; I 2 ). hus, the maximal likelihood ratio statistic max k [k,k] L R (k; I 2 ) play dual roles: testing joint parameter constancy and estimating the change point. he first step [P1] is now readily simplified. [P1 ] est the joint hypothesis of parameter constancy by max k [k,k] L R (k; I 2 ). If the null hypothesis is not rejected, stop the procedure; otherwise, determine ˆk (and hence ˆτ = ˆk/ ) by max k [k,k] L R (k; I 2 ) and then go to [P2]. As the proposed procedure consists of two tests, its overall size must be properly controlled. A simple approach to control the size of multiple tests is based on the so-called Bonferroni inequality; see e.g. Savin (1984). Let α i, i = 1, 2, denote the size of the individual test at

11 Distinguishing trend break 181 able 3. he proposed tests of partial parameter stability: time trend model. Intercept change: λ 1 = 2 and λ 2 = τ L R ([ ˆτ]; e 1 ) L R ([ ˆτ]; e 2 ) Intercept change: λ 1 = 3 and λ 2 = L R ([ ˆτ]; e 1 ) L R ([ ˆτ]; e 2 ) Slope change: λ 1 = and λ 2 =.2 L R ([ ˆτ]; e 1 ) L R ([ ˆτ]; e 2 ) Slope change: λ 1 = and λ 2 =.3 L R ([ ˆτ]; e 1 ) L R ([ ˆτ]; e 2 ) Note: All entries are rejection probabilities in percentages. he nominal size of the test at each step is 2.5%; the simulated critical value of the maximal L R (k; I 2 ) test in the first step is at the 2.5% level. the ith step. o ensure the overall size of this procedure not exceeding α, a common choice of α i based on the Bonferroni inequality is that α i = α/2. Although there are other choices of α i (for example, α i = 1 (1 α) 1/n ), we set α i = α/2 in the subsequent simulations and empirical study. We now conduct simulations to evaluate the finite-sample performance of the proposed procedure. All simulation setups are the same as those for able 1. In addition, the nominal size of the test at each step is 2.5% so that the overall nominal size will not exceed 5%, and [τ, τ] = [.5,.95] for change-point estimation. he rejection frequencies (overall sizes) of this procedure are collected in able 3. From this table we observe that when the intercept changes by 2, the slope test L R ( ˆτ; e 2 ) maintains proper sizes except for τ =.1 and.9. he intercept test has very good power for τ.6, but its power performance deteriorates as τ moves further to the right. hese tests perform better when the magnitude of change increases. he slope test now has roughly proper sizes for all change points considered, and the intercept test has good power for τ.7. When the slope changes by.2, the slope test has pretty good power except for τ =.1,.2,.9, but the intercept test has proper sizes for all change points considered. When the magnitude of change increases, the slope test has better power except for τ =.1,.9, and the intercept test still maintains proper sizes for all change points. hese results are very different from those of able 1 and confirm that the proposed procedure has the desired properties. 4 4 he simulation results also verify the analytic result (15). In accordance with (15), we can show that when the intercept changes by λ 1, the intercept test is proportional to λ e 1 (τ ) = τ (τ 3 3τ 2 + 3τ 1) 4(4τ 2 2τ λ ), and that when the slope changes by λ 2, the slope test is proportional to λ e 2 (τ ) = τ 3(τ 3 3τ 2 + 3τ 1) 12(3τ 2 3τ λ ).

12 182 Chih-Chiang Hsu and Chung-Ming Kuan Remarks. 1. he two-step procedure is also readily extended to models with a polynomial trend. As [P1] or [P1 ], one tests joint parameter constancy and estimates the change point at the first step. By choosing a suitable selection matrix R (q p), one then tests the null hypothesis that a single coefficient (or a subset of coefficients) is constant using L R ([ ˆτ]; R) which has a limiting χ 2 (q) distribution. 2. An obvious drawback of the proposed procedure is that it allows for only one break. When multiple breaks are present, the proposed procedure would not be valid because ignored break points result in model misspecification. Even though we do not consider multiple breaks in this paper, our procedure is still interesting in its own right. In particular, it shows that to properly identify the changing parameter in the time trend model, we should first estimate the change point and then test for the changing parameter. 5. EMPIRICAL APPLICAIONS In this section we apply the proposed procedure to the Nelson Plosser and Ben-David Papell data sets. In an influential study of the Nelson Plosser data, Perron (1989) determined the break types by visual inspection of each series and set the break dates at the year of the Great Depression or the year of oil shock. Zivot and Andrews (1992) took Perron s choices of break types for granted but estimated the break dates via an inf-t statistic; a different inf-t statistic was considered by Perron (1997). Whether these break-date estimates are consistent remains unknown, however. On the other hand, Ben-David and Papell (1995) analysed real per capita GDP data of 16 countries and determined their trend-break patterns using a crude method. Our empirical studies therefore provide formal testing results for break types and yield consistent break-date estimates. We first consider 13 annual macroeconomic series in the Nelson Plosser data set, excluding the post-war quarterly real GNP. he sample period of each series is given in able 4. We take the natural logarithm of each series, except for the interest rate. We follow the steps [P1 ] and [P2] to test each series. In computing the likelihood ratio tests, we estimate σ 2 ε by s 2 (l) = 1 ˆε t l ω j (l) ˆε t ˆε t j, j=1 t= j+1 where {ˆε t } are the OLS residuals, ω j (l) = 1 j/(1 + l), and the bandwidth l is determined by Andrews data-dependent formula with an AR(1) specification for ˆε t ; see Andrews (1991). 5 In able 4 we summarize the break dates determined by Perron (1997) and Zivot and Andrews (1992), and the proposed least-squares method with [τ, τ] = [.5,.95]. Some important differences between our and previous results are: Real GNP and Real per capita GNP had a change It turns out that the latter function is roughly symmetric about τ =.5 but the former is not (the maximum of the former is at τ =.25). his explains the asymmetric power of the intercept test in able 3. Moreover, the power of both tests must increase when the magnitude of change λ increases. 5 A parametric approach that eliminates serial correlations in errors is to include lagged dependent variables as additional regressors in the model; see e.g. Vogelsang (1997). his approach is not adopted in the paper because its empirical model is different from the model analysed in preceding sections. he empirical results of this approach are somewhat similar to those reported in the paper and are available upon request.

13 Distinguishing trend break 183 able 4. Estimated break dates in the Nelson Plosser data. Series Sample Perron Zivot Least-Squares t α t λ Andrews Method Real GNP Nominal GNP Real p/c GNP Ind. Prod Employment GNP Deflator CPI Wages Money Stock Velocity Interest Rate Stock Prices Real Wages Note: he years in the columns under t α and t λ are the break dates estimated by Perron (1997) using two different inf-t statistics. during World War II instead of the Great Depression; Velocity changes during the Great Depression instead of 1946, 1949 or 188. he latter finding agrees with the monetarist view of the Great Depression. he break dates of CPI, Interest Rate, and Stock Prices are also quite different from those of Zivot and Andrews (1992) and Perron (1997). able 5 summarizes the trend-break models identified by Perron (1989) and Chu and White (1992), and the proposed method using the 1% level (5% at each step). he critical value of the likelihood ratio test used in the first step is with [τ, τ] = [.5,.95]. Our results agree with those of Perron (1989) for only five series: Nominal GNP (crash model), Industrial Production (crash model), Nominal Wages (crash model), Stock Prices (mixed model), and Real Wages (mixed model). Our method, however, fails to identify a trend break for Employment and Money Stock. For the remaining series, our procedure suggests the changing-growth model for Real GNP and Real per capita GNP and the mixed model for the others. Comparing to Chu and White (1992), our results agree with theirs for only five series: Nominal GNP, Employment, GNP Deflator, CPI, and Nominal Wages. Note that all three studies suggest the crash model for Nominal GNP and Nominal Wages. o summarize, the proposed procedure results in quite different conclusions for the break dates and types. Ben-David and Papell (1995) found that real per capita GDPs in most OECD countries exhibit a slope change in trend. his contradicts the prediction of the neoclassical growth model but is compatible with the Romer-type endogenous growth model (i.e. there is a changing growth rate in the steady state path). We summarize the models identified by our procedure in able 6. It can be seen that at the 1% level (5% at each step), all series but Canada exhibit a changing growth rate (the changing growth model for Switzerland and USA and the mixed model for all other countries). hus, our testing results agree with those of Ben-David and Papell (1995) except for Canada.

14 184 Chih-Chiang Hsu and Chung-Ming Kuan able 5. Empirical results of the Nelson Plosser data. Series Perron Chu Proposed test White ypes L R ([ ˆτ]; e 1 ) L R ([ ˆτ]; e 2 ) l Real GNP A A B (.1) (<.1) Nominal GNP A A A (<.1) (.18) Real p/c GNP A A B (.66) (<.1) Ind. Prod. A no break A (.1) (.29) Employment A no break no break (.85) (.1) GNP Deflator A C C (<.1) (.5) CPI A C C Wages A A A (<.1) (.13) Money Stock A A no break (.38) (.22) Velocity A no break C Interest Rate A no break C Stock Prices C B C Real Wages C A C (.5) (<.1) Note: A, B and C are the crash, changing growth, and mixed models, respectively. he models identified by Chu and White (1992) and our procedure are based on the tests at the 1% level (5% at each step); the numbers in parentheses are p values. As discussed in the preceding section, the proposed procedure is not applicable when multiple breaks are present. Given the long sample periods of the data studied here, it may be more reasonable to expect that there are multiple changes. hus, the purpose of these empirical studies is not to assert that there is only one break in these series. Instead, our empirical results serve to illustrate the differences between the proposed and existing methods. A complete analysis of these data requires extending the proposed procedure to accommodate multiple breaks. his may be done by combining the method of Chong (1995) and our procedure. Chong (1995) showed that the least-squares change-point estimator converges to one of the true change points even when the number of breaks is underspecified. In view of this result, we may estimate a change point from the full sample, split the sample according to the estimated change point, and estimate change points from the split subsamples. he number and locations of breaks are then determined by

15 Distinguishing trend break 185 able 6. Empirical results of real per capita GDP data. Series Samples break Proposed test dates ypes L R ([ ˆτ]; e 1 ) L R ([ ˆτ]; e 2 ) l Australia C Austria C Belgium C Canada A (<.1) (.8) Denmark C Finland C France C Germany C Italy C Japan C Netherlands C Norway C Sweden C Switzerland B (.84) (.3) UK C USA B (.47) (.1) Note: A, B and C are the crash, changing growth, and mixed models, respectively. he models identified by our procedure are based on the tests at the 1% level (5% at each step); the numbers in parentheses are p values. repeating this (estimation and sample splitting) procedure. Once the break dates are estimated, we can, analogous to the second step of our procedure, construct the unconstrained regression model and test for various break types using the likelihood ratio test. A problem with this approach is that, when there are many change points, we must consider a large number of break types.

16 186 Chih-Chiang Hsu and Chung-Ming Kuan 6. CONCLUSIONS In this paper, we show that the likelihood-based tests of partial parameter stability in the time trend model depend on the changes of all coefficients, even though they are constructed using only the estimates of a particular coefficient. It is therefore inappropriate to rely on these tests to identify the changing parameter. We then suggest a two-step procedure that enables us to distinguish between trend-break models. his procedure involves consistent estimation of break points and properly-sized tests of partial parameter stability. hus, the proposed procedure should lead to more reliable trend-break specifications if there is a break in the series. he proposed procedure has some limitations. First, it is not applicable when there are multiple changes. his difficulty may be overcome by combining the sequential estimation method of Chong (1995) and the proposed procedure. Second, we do not consider the potential presence of a unit root, so that our procedure is only applicable to the series that are (known to be) trend stationary. Nevertheless, this procedure can still serve as a preliminary test to determine the alternatives of Perron s models for testing for unit roots in the presence of structural breaks. hese alternatives are important for deriving the limits of Perron s tests; see Montañés and Reyes (1998). Constructing a test procedure that is robust to the presence of a unit root is beyond the scope of this paper and will be left to future research. An interesting extension of this paper is to consider testing partial parameter stability in cointegrating regressions, as in Kuo (1998). Kuo (1998) notices from simulations that his tests reject too often when the coefficient being tested is a constant but the other coefficient changes. his is precisely the problem we discussed in Section 2. he extension of the proposed procedure to cointegrating regressions is currently being investigated. ACKNOWLEDGEMENS he authors are deeply indebted to Professor Karim Abadir (the editor) and two anonymous referees for their very constructive suggestions. We also thank seminar participants at the 1999 North American Summer Meeting and Far Eastern Meeting of the Econometric Society for helpful comments. C.-C. Hsu wishes to thank the National Science Council of the Republic of China for research support (NSC H 8 8). REFERENCES Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, Andrews, D. W. K. (1993). ests for parameter instability and structural change with unknown change point. Econometrica 61, Andrews, D. W. K. and W. Ploberger (1994). Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, Bai, J. (1997). Estimation of a change point in multiple regression models. he Review of Economic and Statistics 79, Bai, J. (1999). Likelihood ratio tests for multiple structural changes. Journal of Econometrics 91, Bai, J. and P. Perron (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66,

17 Distinguishing trend break 187 Banerjee, A., R. L. Lumsdaine and J. H. Stock (1992). Recursive and sequential tests of the unit root and trend break hypotheses: heory and international evidence. Journal of Business and Economic Statistics 1, Ben-David, D. and D. H. Papell (1995). he great wars, the great crash, and steady state growth: some new evidence about an old stylized fact. Journal of Monetary Economics 36, Chong,. (1995). Partial parameter consistency in a misspecified structural change model. Economics Letters 49, Chu, C.-S. J. and H. White (1992). A direct test for changing trend. Journal of Business and Economic Statistics 1, Critchley, F., P. Marriott and M. Salmon (1996). On the differential geometry of the Wald test with nonlinear restrictions. Econometrica 64, Dufour, J.-M. (1997). Some impossibility theorems in econometrics with applications to structural and dynamic models. Econometrica 65, Kuan, C.-M. (1998). ests for changes in models with a polynomial trend. Journal of Econometrics 84, Kuan, C.-M. (1999). A note on tests for partial parameter instability in the trend stationary model. Economics Letters 65, Kuo, B.-S. (1998). est for partial parameter instability in regressions with I (1) processes. Journal of Econometrics 86, Lumsdaine, R. L. and D. H. Papell (1997). Multiple trend breaks and the unit root hypothesis. he Review of Economic and Statistics 79, MacNeil, I. B. (1978). Properties of sequences of partial sums of polynomial regression residuals with applications to test for change of regression at unknown times. Annals of Statistics 6, Montañés, A. and M. Reyes (1998). Effect of a shift in the trend function on Dickey-Fuller unit root tests. Econometric heory 14, Nelson, C. R. and C. I. Plosser (1982). rends and random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics 1, Nunes, L. C., C.-M. Kuan and P. Newbold (1995). Spurious break. Econometric heory 11, Nunes, L. C., P. Newbold and C.-M. Kuan (1997). esting for unit roots with breaks Evidence on the great crash and the unit-root hypotheses reconsidered. Oxford Bulletin of Economics and Statistics 59, Perron, P. (1989). he great crash, the oil price shock and the unit root hypothesis. Econometrica 57, Perron, P. (1997). Further evidence on breaking trend functions in macroeconomic variables. Journal of Econometrics 8, Ploberger, W., W. Krämer and K. Kontrus (1989). A new test for structural stability in the linear regression model. Journal of Econometrics 4, Savin, N. E. (1984). Multiple hypothesis testing. In Z. Griliches and M. D. Intriligator (eds), he Handbook of Econometrics, vol. 2, pp North-Holland. Silverman, B. W. (1986). Density Estimation for Statistics and Analysis. New York: Chapman & Hall. Vogelsang,. J. (1997). Wald-type tests for detecting breaks in the trend function of a dynamic time series. Econometric heory 13, Zivot, E. and D. W. K. Andrews (1992). Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business and Economic Statistics 1,

18 188 Chih-Chiang Hsu and Chung-Ming Kuan APPENDIX Proof of heorem 3.1. For each k, the Wald statistic (5) can be expressed as W (k; R) = [ ˆβ 1 (k) ˆβ 2 (k)] R [ R(Mk 1 s We first note the following equality: + M 1 k )R ] 1 R[ ˆβ s 1 (k) ˆβ 2 (k)]. Q(τ) 1 + [Q(1) Q(τ)] 1 = [Q(τ) Q(τ)Q(1) 1 Q(τ)] 1. By (7), M k / Q(τ) and M k / Q(1) Q(τ). hus, R(M 1 k Given the definition of C(τ), + M 1 k )R R[Q(τ) Q(τ)Q(1) 1 Q(τ)] 1 R. [Q(τ) Q(τ)Q(1) 1 Q(τ)] 1 = Q(1) 1/2 C(τ) (τ) 1 C(τ) Q(1) 1/2, so that for R (τ) = RQ(1) 1/2 C(τ) (τ) 1/2, Under (6), ˆβ 1 (k) = β + M 1 k [ R(M 1 k ( ˆβ 2 (k) = β + M 1 k It follows from (8) and (9) that 1 ( R( ˆβ s 1 (k) ˆβ 2 (k)) ( ) ( 1 1 = R M k 1 1 s + M 1 k )R ] 1 [R (τ)r (τ) ] 1. ) k x t x t λ1(t/ > τ ) t=k+1 k x t x t λ1(t/ > τ ) + M 1 k ) x t x t λ1(t/ > τ ) + 1 s ( k x t ε t ), + M 1 k ( t=k+1 ) k x t ε t ( ) ( 1 1 R M 1 k x t x t s λ1(t/ > τ ) + 1 s t=k+1 ( 1 τ ) RQ(τ) 1 Q 1 (h) λ1(h > τ ) dh + G(τ) σ ε R[Q(1) Q(τ)] 1 ( 1 σ ε 1 τ t=k+1 x t ε t ). x t ε t ) ) Q 1 (h) λ1(h > τ ) dh + G(1) G(τ).

19 Distinguishing trend break 189 For this limit we observe that Q(τ) 1 G(τ) [Q(1) Q(τ)] 1 [G(1) G(τ)] = (Q(τ) 1 + [Q(1) Q(τ)] 1 ) [G(τ) (Q(τ) 1 + [Q(1) Q(τ)] 1 ) 1 [Q(1) Q(τ)] 1 G(1)] = (Q(τ) 1 + [Q(1) Q(τ)] 1 )(G(τ) Q(τ)Q(1) 1 G(1)); the other terms can be handled similarly. It follows that R( ˆβ 1 (k) ˆβ 2 (k)) s R[Q(τ) Q(τ)Q(1) 1 Q(τ)] 1 {[G(τ) Q(τ)Q(1) 1 G(1)] + 1 [ τ 1 ]} Q 1 (h) λ1(h > τ ) dh Q(τ)Q(1) 1 Q 1 (h) λ1(h > τ ) dh σ ε = R (τ)( (τ) 1/2 C(τ) Q(1) 1/2 [G(τ) Q(τ)Q(1) 1 G(1)] + (τ)). In the light of Bai (1999), (τ) 1/2 C(τ) Q(1) 1/2 [G(τ) Q(τ)Q(1) 1 G(1)] d = (τ) 1/2 B (ω(τ)). Specifically, for the trend stationary model, ω i (τ), i = 1, 2, are the roots of where ω 2 + ( 4τ + 6τ 2 4τ 3 )ω + τ 4 =, ω 1 (τ) = (2 3τ + 2τ 2 + 2(1 3τ + 4τ 2 3τ 3 + τ 4 ) 1/2 )τ, ω 2 (τ) = (2 3τ + 2τ 2 2(1 3τ + 4τ 2 3τ 3 + τ 4 ) 1/2 )τ, and B (ω()) = B () = = B (1) = B (ω(1)). Hence, B (ω(τ)) is a time-rescaled Brownian bridge. he assertion follows from these results and the continuous mapping theorem. Proof of Equation (12). For stationary regressions, the Wald statistic for testing partial parameter stability is ( ˆV W (k, R) = [ ˆβ 1 (k) ˆβ 2 (k)] R [R 1 (k) + ˆV ) ] 2 (k) 1 R R[ ˆβ k k 1 (k) ˆβ 2 (k)], where ˆV i (k), i = 1, 2, are the consistent estimators of Q 1 S Q 1 and ( 1 S = lim var x t ε t ). Under the local alternatives (6), (8) becomes [ 1 τ] x t x t λ1(t/ > τ ) τ P Q λ1(h > τ ) dh.

20 19 Chih-Chiang Hsu and Chung-Ming Kuan Since in stationary regressions C(τ) (τ) 1 C(τ) = {Q(1) 1/2 [Q(τ) Q(τ) Q(1) 1 Q(τ)]Q(1) 1/2 } 1 = [τ(1 τ)] 1, we obtain the following result: R (τ) (τ) [ τ = R Q 1/2 C(τ) (τ) 1 C(τ) Q 1/2 Q λ1(h > τ ) dh τ [ τ = R [τ(1 τ)] 1 λ1(h > τ ) dh τ 1 ] λ1(h > τ ) dh. Note that this proof will not alter even when x t has a nonzero mean. 1 ] λ1(h > τ ) dh