Joint hypothesis speci cation for unit root tests with. a structural break

Transcription

1 Joint hypothesis speci cation for unit root tests with a structural break Josep Lluís Carrion-i-Silvestre Grup de Recerca AQR Departament d Econometria, Estadística i Economia Espanyola Universitat de Barcelona Andreu Sansó Rosselló Departament d Economia Aplicada Universitat de les Illes Balears July 23, 2005 Abstract Several tests based on a t-ratio have been proposed in the literature to decide the order of integration of a time series allowing for a structural break. However, another approach based on testing a joint hypothesis of unit root and the irrelevance of some nuisance parameters is also feasible. This paper proposes new unit root tests consistent with the presence of a structural break applying this second perspective. Our approach deals both with the case where the break is not allowed under the null hypothesis, and where it is allowed. Simulations investigate the performance of this proposal compared to the existing tests. Keywords: Unit root tests, structural change, joint hypotheses, pseudo F-tests JEL classi cation: C12, C22 We would like to thank Pierre Perron and two anonymous referees for helpful comments. Corresponding author: Av. Diagonal, Barcelona, Spain. Tel: ; Fax: ; carrion@ub.edu. J. Ll. Carrion-i-Silvestre gratefully acknowledge the nancial port of the Ministerio de Ciencia y Tecnología under grant SEJ /ECON. A. Sansó gratefully acknowledge the nancial port of the Ministerio de Ciencia y Tecnología under grant SEC and Conselleria d Economia, Hisenda i Innovació del Govern Balear under grant PRIB

2 1 Introduction The unit root test most widely used in empirical applications is the Dickey-Fuller (DF) test, which is based on the individual signi cance in an auxiliary regression of one parameter. Notwithstanding, the irrelevance under the null hypothesis of unit root of certain deterministic regressors drove Dickey and Fuller (1981) to specify procedures to test the joint null hypothesis of unit root and the non-signi cance of these deterministic regressors. These authors proposed three F-type tests to test di erent joint hypotheses. An important feature of their approach is that the tests take into account more information when testing for the unit root hypothesis and, hence, more power is expected when deciding if a stochastic process is I(1) or I(0). However, the main objection to their proposal arises from the fact that the limiting distribution of these tests is derived assuming uncorrelated disturbances. On the other hand, if tests are to be valid when the disturbance term presents an autocorrelation scheme certain transformations are needed. To overcome this drawback Dickey and Fuller (1981) proposed the application of a parametric correction. A few years later, Perron (1990b) derived the same tests as Dickey and Fuller (1981) but from a less restricted framework assuming the strong-mixing conditions for the disturbance term de ned in Phillips (1987) and Phillips and Perron (1988). Due to the fact that under the null hypothesis the limiting distributions of these test statistics are function of nuisance parameters, Perron speci es non-parametric corrections to eliminate this dependence. The introduction of structural breaks in the integration analysis of time series gives rise to a new set of test statistics. Perron (1989) s paper is the reference point in this eld. As in the case of the standard DF test statistic, the consideration of structural breaks in the integration analysis involves some deterministic regressors that are irrelevant under the null hypothesis of unit root. This situation was studied by Banerjee, Dolado, and Galbraith (1990), and Banerjee, Lumsdaine, and Stock (1992), among others, and, more recently, in Hatanaka (1996), Montañés and Reyes (1997) and Montañés (1998), where pseudo F tests that take into account structural breaks were designed. Although all these papers rely on the fundamentals given in Perron (1989, 1990a) they do not exploit 2

3 all the possibilities. Hence, the goal of this paper is to design a procedure to test the joint hypothesis of unit root and the irrelevance of some regressors that complements and improves the existing approaches. The statistics presented in this paper test the unit root hypothesis allowing for one structural break in the same way as Perron (1989, 1990a) using the Innovational Outlier speci cation. Thus, we consider the four models of Perron (1989, 1990a) specifying the break point as unknown. Our analysis also distinguishes two di erent null hypotheses, depending on whether or not the structural break is allowed under the null hypothesis. Furthermore, we propose a procedure to test the unit root hypothesis allowing for a slope shift under the null hypothesis that o ers good properties in terms of empirical size and power. This feature is important since the limit distribution of some of the tests in the literature diverges in this case. The article proceeds as follows. In section 2 the various auxiliary regressions are presented with reference to the nature of the null and the alternative hypotheses. Section 3 o ers the limiting distribution for the tests assuming both known and unknown date of the break. Finite-sample performance of the tests is analysed through Monte Carlo experiments in section 4. In section 5 we illustrate the proposal with an empirical application to the Nelson-Plosser data set. Finally, section 6 concludes. All the mathematical derivations are outlined in the Appendix. 2 The statistical models The tests that we deal with are grouped according to the speci cation of the null hypothesis. First of all, we are concerned with the null hypothesis that only consider an integrated process that does not su er from a structural break, against the alternative hypothesis of a breaking-trend stationary process. Hence, the rst stage of the analysis follows the approximations de ned in Banerjee, Lumsdaine, and Stock (1992), Zivot and Andrews (1992), Perron (1997) and Vogelsang and Perron (1998) where the joint hypothesis of unit root and absence of structural breaks are speci ed under the null. Hatanaka 3

4 and Koto (1995) followed this approach, and proposed pseudo-f test statistics assuming that the date of the break was known. As mentioned above, in this paper we also deal with endogenous break point. At the second stage of the analysis we adopt the proposal in Perron (1989, 1990a) and allow for a structural break under both the null and alternative hypotheses, an approach that was also used in Montañés and Reyes (1997) and Montañés (1998). Montañés and Reyes (1997) derived a pseudo F-test for non-trending variables and one level shift, while the analysis is extended to two level shifts in Montañés (1998). Besides, Hatanaka and Koto (1995) considered the case of a structural break that a ects both the level and the slope, but they restricted themselves to the known break case. Thus, although there are some contributions in the literature that are based on pseudo-f tests, these proposals do not o er a complete coverage of all possible situations. In what follows, we highlight the main di erences between the existing proposals and our own. 2.1 Joint null hypothesis of unit root and no break Let y t be the stochastic process generated according to the following Data Generation Processes (DGP): kx y t = + y t 1 + c i y t i + u t ; (1) i=1 where u t is a martingale di erence sequence, with = 0 for non-trending variables and 6= 0 for trending variables. Under the alternative hypothesis y t is described by a breaking-trend stationary process, that can a ect the time series in all di erent ways depending on the speci cation of the deterministic component. Speci cally, those that are of interest in applied research are 4

5 the following: y t = DU t + y t 1 + kx c i y t i + u t ; (2) i=1 y t = + DU t + y t 1 + kx c i y t i + u t ; (3) i=1 y t = + t + DU t + y t 1 + y t = + t + DT t + y t 1 + kx c i y t i + u t ; (4) i=1 kx c i y t i + u t ; (5) i=1 y t = + t + DU t + DT t + y t 1 + kx c i y t i + u t ; (6) with DU t = 1 and DT t = (t T b ) for t > T b, and 0 otherwise, T b denotes the break point. Notice that these speci cations corresponds to Innovational Outlier model used in Perron (1989, 1990a, 1997), Zivot and Andrews (1992) and Vogelsang and Perron (1998), among others. In fact, (6) is model C of Zivot and Andrews (1992), although the other possible speci cations referred to as models A and B can be obtained as constrained versions of (6). Thus, the model A without a time trend is given by (3), the model A with a time trend is (4) and, nally, the model B is given by (5). Moreover, equations (2) to (6) di er from those in Perron (1989, 1990a) since they do not include the dummy variable that captures the e ect of a structural break under the null hypothesis. Panel A in Table 1 summarizes for each regression the set of parametric restrictions that can be imposed and the notation used to indicate the corresponding test statistic. For instance, the statistic denoted above as 1 () tests the unit root hypothesis in regression (2) when (1) with = 0 is the assumed DGP. As a consequence, 1 () tests the joint hypothesis that (; ) = (0; 1). The same applies for the other regressions and vectors of constraints. Note that the joint hypothesis that corresponds to the tests 3;2 (), 4;2 () and () are considered by Hatanaka and Koto (1995). Hence, our proposal completes the set of statistical tools that can be applied to test for the unit root hypothesis through the consideration of the di erent speci cations that can arise when conducting an empirical analysis. i=1 5

6 2.2 Null hypothesis of unit root with a structural break Let us now consider the stochastic process y t de ned by: y t = + D (T b ) t + DU t + y t 1 + kx c i y t i + u t ; (7) i=1 where D (T b ) t = 1 for t = T b + 1, 0 otherwise, with = = 0 for non-trending variables, and 6= 0 and/or 6= 0 for trending variables, that is to say, a random walk that might be a ected by a structural break shifting the drift. In this case, practitioners are interested in auxiliary regressions such as the following: y t = DU t + dd (T b ) t + y t 1 + kx c i y t i + u t ; (8) i=1 y t = + DU t + dd (T b ) t + y t 1 + kx c i y t i + u t ; (9) i=1 y t = + t + DU t + dd (T b ) t + y t 1 + y t = + t + DT t + y t 1 + kx c i y t i + u t ; (10) i=1 kx c i y t i + u t ; (11) i=1 y t = + t + DU t + dd (T b ) t + DT t + y t 1 + kx c i y t i + u t : (12) Regressions (8) to (12) are likely to introduce the parametric restrictions shown in panel B in Table 1. Regression (12) shows a clear correspondence with those for the Innovational Outlier model in Perron (1989, 1990a). In fact, (12) represents model C in Perron (1989), and models A and B are simple particularizations. Notice that the speci cation given by regression (11) is the same as the one de ned when the break is not present under the null. This is because the introduction of DU t in (11) has the property of making the models given by (11) and (12) asymptotically indistinguishable see Perron (1994). To avoid unnecessary duplications, we have decided to exclude this speci cation from our discussion. It should be stressed that the joint hypothesis speci ed for the 2 () test is closely related to the proposal in Montañés and Reyes (1997), but it is not exactly i=1 the same, since they consider H 0 : (; ) = (0; 1), i.e. two restrictions, and we also 6

7 take into account the constraint in the constant term, i.e. H 0 : (; ; ) = (0; 0; 1), i.e. three restrictions. As before, we stress that Hatanaka and Koto (1995) de ne the joint hypotheses that are associated with the () test assuming the break point as known. 3 Asymptotic distribution Now that we have presented the various hypotheses and models we will focus on the limiting distribution of the test statistics assuming, except when indicated, that the break date (T b ) that is used in the regressions is the correct break date (Tb c), i.e. = c, where = T b =T and c = Tb c=t, 0 < ; c < 1, denote the break fraction parameters. The pseudo F-tests for the hypotheses collected in Table 1 are computed as: i () = (SSR SSR)/ q ; (13) SSR/ (T m) i = f1; 2; (3; 1) ; (3; 2) ; (4; 1) ; (4; 2) ; (5; 1) ; (5; 2)g where SSR is the sum of squared residuals under the null hypothesis, SSR is the sum of squared residuals for the unconstrained model, q is the number of constraints and m is the number of regressors under the alternative hypothesis. The tests that allow for a structural break under the null hypothesis are computed in a similar way, although they are denoted using the Greek character. Although Hatanaka and Koto (1995) specify some of the joint hypotheses de ned in the previous sections, we have decided to present the derivations for all the test statistics for the sake of completeness. The limiting distributions of these statistics are summarized in the following theorem. Theorem 1 Let T b = T for all T and 0 < < 1, where as T b! 1 and T! 1, remains constant, and fu t g T 0 is a martingale di erence sequence. a) Let fy t g T 0 be a stochastic process with the DGP given by (1) with k = 0. Thus, as T! 1: a.1) if = 0, then i () ) A i () and i () ) A i () for i = f1; 2, (3; 1), (3; 2), (4; 1), (4; 2), (5; 1), (5; 2)g 7

8 a.2) if 6= 0, then i () ) A i () for i = f(3; 2); (4; 2); (5; 2)g i () ) A i () for i = f(3; 2); (5; 2)g; b) Let fy t g T 0 be a stochastic process with the DGP given by (7) with k = 0. Thus, as T! 1: b.1) if = = 0, then i () ) A i () for i = f1; 2; (3; 1); (3; 2); (5; 1); (5; 2)g b.2) if 6= 0 and = 0, then i () ) A i () for i = f(3; 2); (5; 2)g b.3) if = 0 and 6= 0, then 8 >< B i () for i = f(5; 1); (5; 2)g with = c i () ) ; >: O p (T ) for i = f(5; 1); (5; 2)g with 6= c b.4) if 6= 0 and 6= 0, then 8 >< B i () for i = f(5; 2)g with = c i () ) ; >: O p (T ) for i = f(5; 2)g with 6= c with c = Tb c =T denoting the correct break fraction, ) for weak convergence in distribution, and A i and B i are expressions given in the Appendix de ned as functions of Brownian motions. The outline of the proof of Theorem 1 is given in the Appendix. Asymptotic critical values and a small set of nite sample critical values for di erent values of T when is known were computed in Carrion-i-Silvestre (1999). Some remarks are in order. First of all, the limiting distributions for i () and i () ; i ={1, 2, (3,1), (3,2)}, are equivalent. As mentioned above, this is because of the asymptotic irrelevance of D (T b ) t under the null hypothesis, although some di erences might exist in nite samples see Perron (1994). This equivalence does not hold for i () and i () when i ={(5,1), (5,2)}. For = c 8

9 they converge to di erent distributions whereas for 6= c it is shown in the Appendix that the limit distribution of T 1 i () is O p (1) and depends on. Consequently the i () tests, i ={(5,1), (5,2)}, are consistent when there is no error in the location of the break point, i.e. when = c, but diverge when the break point is misspeci ed, i.e. 6= c, and, therefore, the size of the tests goes towards one as T grows with 6= 0. Second, we can state that the resulting limiting distribution remains the same when the statistics are constructed with one of the data-dependent methods to select k to deal with autocorrelation. This holds provided k 3 max=t! 0 as T! 1 and k max! 1, where k max denotes the maximum number of lags that are allowed for the parametric correction see Banerjee, Lumsdaine, and Stock (1992), Zivot and Andrews (1992), Perron (1997) and Vogelsang and Perron (1998). Finally, the limiting distributions depend on the break fraction (), which usually is unknown. To estimate the break point we follow the procedures developed in Banerjee, Lumsdaine, and Stock (1992), Perron and Vogelsang (1992), Zivot and Andrews (1992), and Perron (1994, 1997), among others. All these approaches select the breaking date by either maximizing or minimizing a sequence of tests, i.e. applying the remum or in mum functional. Here we follow this stream of the literature and use the functional to estimate the break point. Henceforth, the test statistics are denoted as i and i, i ={1, 2, (3,1), (3,2), (4,1), (4,2), (5,1), (5,2)}. The following theorem summarizes the asymptotic distribution for the tests. Theorem 2 Let T b = T for all T and 0 < < 1, where as T b! 1 and T! 1, remains constant, and fu t g T 0 is a martingale di erence sequence. a) Let fy t g T 0 be a stochastic process with the DGP given by (1) with k = 0. Thus, as T! 1: a.1) if = 0, then i (3; 2), (4; 1), (4; 2), (5; 1), (5; 2)g ) SupA i () and i 2 ) SupA i () for i = f1; 2, (3; 1), 2 9

10 a.2) if 6= 0, then i ) SupA i () for i = f(3; 2); (4; 2); (5; 2)g 2 i ) SupA i () for i = f(3; 2); (5; 2)g; 2 b) Let fy t g T 0 be a stochastic process with the DGP given by (7) with k = 0. Thus, as T! 1: b.1) if = = 0, then i b.2) if 6= 0 and = 0, then i b.3) if 6= 0, then i ) SupA i () for i = f1; 2; (3; 1); (3; 2); (5; 1); (5; 2)g 2 A i () for i = f(3; 2); (5; 2)g ) Sup 2 = O p (T ) for i = f(5; 1); (5; 2)g; A i being the function of Brownian motions and of the nuisance parameters reported in Theorem 1, and where = [a; b] is a closed subset of (0; 1). The proof follows from Theorem 1, showing the continuity of the functionals that we have considered in the derivations see Zivot and Andrews (1992). As mentioned above, when autocorrelation is present in the residuals the resulting limiting distribution remains valid provided the parametric correction bases on k 3 max=t! 0 as T! 1 and k max! 1. Some important features should be highlighted. First of all, the test converges to a limit distribution that does not depend on the break fraction. The same follows for the i tests, i = f1, 2, (3; 1), (3; 2)g, given the asymptotic equivalence. Note that this makes our proposal more powerful since Hatanaka and Koto (1995) assume the break point as known, but we have shown that it is important to analyse the situation of a misspeci cation error when taking into account a slope shift under the null. Second, i when i = f(5; 1); (5; 2)g does not provide a consistent estimate of the break point when 6= 0, since the test diverges under the null hypothesis when 6= c. Thus, given 6= 0 in (12) size distortions will appear if T is large enough. In order to overcome this drawback and following the suggestion in Vogelsang and Perron (1998), we propose to proceed in two stages: (i) obtain a consistent estimate of the break fraction, and (ii) test the unit root hypothesis conditional on the break point of the rst stage. 10

11 Indeed, note that this is the methodology adopted in Saikkonen and Lütkepohl (2002), although they only covered the presence of a level shift and we are interested in the slope shift case. To take into account the situations where the shift a ects the slope of the time series we follow Perron and Zhu (2004). Those authors show that a consistent estimate of the break fraction can be obtained through the minimization of the SSR on the model given by y t = d t + u t ; where d t collects the deterministic speci cation that involves a slope shift with or without a level shift, and u t can be either an I(1) or an I(0) process. In fact, Perron and Zhu (2004) propose three di erent speci cations for d t : Model I denoted as the Joint broken trend model which de nes d t = + t + DT t ; (14) Model II denoted as the Local Disjoint broken trend model which uses d t = + t + DU t + DT t ; (15) and Model III denoted as the Global Disjoint broken trend model which speci es d t = + t + DU t + B t ; (16) B t = t for t > T b and 0 otherwise. Note that both Model II and Model III provide the same break point estimate since the regressors in the two models span the same space, although the convergence rates of the break fraction are quite di erent. Let us denote ^ as the estimated break fraction using either (15) or (16). Theorem 3 in Perron and Zhu (2004) shows that ^II c O p (T 1 ) for (15) whereas ^ III c op (T 3 ) for (16). Therefore, using Model III to estimate the break fraction implies that the limiting distribution of the i will be the same as that for i ( c ), i = f(5; 1); (5; 2)g. In this case the limit distribution of the i (), i = f(5; 1); (5; 2)g, tests will be the one in Theorem 1 when = c. Asymptotic critical values are the ones reported in Table 2. 11

12 Asymptotic critical values for and i, i = f1, 2, (3; 1), (3; 2)g, are provided in Table 2, while nite-sample critical values are presented in Table 3 for and i, i = f1, 2, (3; 1), (3; 2)g, tests in order to save space we only report nite sample critical values for T = 100 although tables for T = 50 are available upon request. Furthermore, we only provide one common set of critical values for the i and i, i = f1, 2, (3; 1), (3; 2)g, tests since simulations not reported here showed that the di erences in terms of empirical size and power derived of establishing this distinction are negligible. Note that the nite-sample critical values have been computed using three di erent methods for the selection of the order of the autoregressive correction. The rst method assumes the order of the autoregressive correction (k) to be exogenous and equal to k = f0; 2; 5g. The second method selects k following the individual signi cance of the lags, k (t sig) criteria, advocated by Ng and Perron (1995). The third method is based on the modi ed AIC (MAIC) and BIC (MBIC) information criteria in Ng and Perron (2001). To obtain the limit distribution in Theorem 2 we require the de nition of some trimming, which is given by the computation of the test statistic in all possible dates unfortunately, the results in Perron (1997), which do not require any trimming, are only applicable when T b is chosen to minimize the ADF test statistic. 4 Finite sample performance We analyse the nite sample properties by looking at two of the tests that have been proposed in this paper. Firstly, we compare the 3;2 test with the tests t (1), t ; (1) and t ;jj (1) in Perron (1997). In brief, t (1) is the minimum of the sequence of t-statistics to test that = 1 in (10) for all possible values of the time break; t ; (1) is the t-statistic to test that = 1 in (10) but selecting the time break as the argument that minimizes the sequence of t-ratios of, similarly to t ;jj (1) where the time break is selected according to the absolute value of the t-ratio of. Secondly, we address the nite sample performance of the test, which in turn has been compared with the tests t (2), t ; (2), t ;jj (2) and t ;; (2) in Perron (1997) and Vogelsang and Perron (1998). As before, the 12

13 t (2) test denotes the minimum of the sequence of t-statistics to test that = 1 in (12) for all possible values of the time break; t ; (2) is the t-statistic of = 1 in (12) but selecting the time break as the argument that minimizes the sequence of t-ratios of, while t ;jj (2) selects the time break according to maximization of the absolute value of the t-ratio of. Finally, t ;; (2) applies the joint signi cance F-test of and in (12) to estimate the break point. The DGP speci ed for this Monte Carlo is similar to the one used in Vogelsang and Perron (1998): y t = (L) + t + DUt c + DT t c + zt ( < 1) (17) y t = (L) + t + D (T b ) t + DU c t + zt ( = 1) ; where (L) = (1 ( + ) L + L 2 ) 1 (1 + #L), (L) = (1 L) 1 (1 + #L), z t = z t 1 +z t 1 + (1 + #L) " t, " t iid N (0; 1), with the initial conditions set at zero. The empirical size is analysed using = 1, while the power is assessed using = f0:8; 0:95g for the 3;2 test and = 0:8 for the test. A range of situations have been considered with the following sets of parameters: (E1) = # = 0, (E2) = 0:6, # = 0, (E3) = 0:6, # = 0, (E4) = 0, # = 0:5, and (E5) = 0, # = 0:5. Experiment E1 does not include any autocorrelation scheme. Experiment E2 considers positive autocorrelated errors, a common situation in applied research. E3 formulates a negative autocorrelation for errors. Finally, E4 and E5 analyse the performance of the test when there are MA(1) errors. In all the simulations the level and slope of the time series have been set equal to = 4:56 and = 0:033, which mimics the model estimated for the logarithm of the real GNP in Perron (1997). Although the limiting distribution of the tests are invariant to these parameters they play an important role for the power of the tests proposed in this paper. Regarding the magnitude of the structural break, for the 3;2 test we have speci ed = {0, 1, 2, 5}, while for the test we have followed Vogelsang and Perron (1998) in specifying the magnitudes of the break through the parameters = {0, 5, 10} and = {0, 1, 2}. Note that in fact we work with the parameters 13

14 = (1 ) = (1 + #), = (1 ) = (1 + #), = (1 ) + + # = (1 + #) and = [ (1 ) + + #] = (1 + #) in (17). The true break fraction is set equal to c = 0:5 for a sample size of T = 100 observations. The Monte Carlo is based on n = 1; 000 replications. The GAUSS code that has been used in the simulations is available upon request. The order of the parametric correction (k) is chosen using the k (t sig) criteria of Ng and Perron (1995) for the tests in Perron (1997) and Vogelsang and Perron (1998). In addition, for the 3;2 and tests we also used the MAIC and MBIC information criteria in Ng and Perron (2001). For sake of brevity, we only report the results for the M AIC criterion since similar outcomes were obtained for the M BIC criterion. The maximum lag order is k max = 5. The notation that is used for the test indicates the method that is applied to account for non-iid disturbance terms: the erscript sig denotes the k (t sig) criteria of Ng and Perron (1995), while the erscript MAIC is for the information criterion. Throughout this section the corresponding nite-sample critical values are used Analysis for the 3;2 test Table 5 reports the results of the empirical size and power for the 3;2 test. When the disturbance term is iid and either there is no structural break ( = 0), the magnitude of the level shift is small ( = 1) or medium ( = 2), all the tests have exact size. However, some distortions on the size appear for large level shifts, i.e. for the = 5 case. This is a feature common to all the tests considered in this paper. It has also been reported in the literature and is due to the fact that as the magnitude of the break increases, the detection of the break point becomes easier. Consequently, the true break point is the one that is chosen the most and, hence, we should use the critical values that are computed assuming the break date to be known if the size distortion is to be overcome see Perron and Vogelsang (1992), Perron (1997) and Vogelsang and Perron (1998). Notwithstanding, this implies that we should get a conservative test since critical values from the known t ;; 1 Following Vogelsang and Perron (1998), the critical values for the t (2), t ; (2), t ;jj (2) and (2) tests are the ones that assume no break under the null hypothesis. 14

15 break date case are smaller than with an unknown break date. 2 Fortunately and as argued in Perron (1997), for most macroeconomic time series intercept shifts are less than 5 standard deviations, so that these size distortions due to large level shifts do not seem to be a problem in practice. The picture changes when we deal with non-iid disturbances. Let us rst focus on the method for selecting k for the 3;2 test. Thus and except for the negative MA case (experiment E5), the 3;2 test computed using the k (t sig) criterion, i.e. the one sig denoted as 3;2, o ers better performance in terms of size. Therefore, we suggest computing the test in this way. sig In what follows, we focus the discussion just on the sig 3;2 test. Besides, if we compare 3;2 with the tests in Perron (1997) we note that the empirical size of all the tests is close to the nominal one. However, for the MAIC experiment E5 the 3;2 test is the one that shows the least size distortion of all the tests considered in the simulations. sig In general, the power analysis reveals that 3;2 performs better than the existing sig tests. Thus, when the power is assessed with = 0:8 the 3;2 test outperforms the other tests for the DGPs given by (; #) = (0; 0), (; #) = ( 0:6; 0) and (; #) = (0; 0:5). For the positive AR error case (experiment E2) the t (1) is slightly more powerful than sig 3;2, although the di erences are not large and diminish as increases. For the experiment E5 t sig (1) and 3;2 o er similar power values, although these tests su er sig severe size distortions. The eriority of the 3;2 test is more evident for = 0:95, since it outperforms all the tests for whichever DGP. Moreover, it is noticeable that in some cases the tests based on t-ratios have no virtual power, but the test that we propose here shows reasonable power values even in the case where there is no level shift. For instance, for experiment E3 with = 1 the powers for the t sig (1) and 3;2 tests are and respectively. This evidences the gain in power that can be obtained when using the pseudo-f test. The fact that in some cases the power of the test increases with might seem counterintuitive, although we should bear in mind that the null hypothesis of the 3;2 test is the joint null hypothesis of unit root and statistical insigni cance of both 2 We thank a referee for raising this point. 15

16 the parameter that measures the magnitude of the level shift and the slope coe cient. Thus, while a decrease in power should be expected as approaches unity, the statistical signi cance of the other parameters compensates the loss of power note that the statistic has the correct empirical size. This explains why the 3;2 test shows good power while the others do not. Note that a similar e ect can be observed in Table VII in Dickey and Fuller (1981) for the joint test when the drift term is non-zero. sig In all, from the sample size analysis we suggest the application of the 3;2 test since it has low size distortion among the class of test statistics considered in the simulations sig and better power. Therefore, simulation evidence brings us to conclude that the 3;2 test outperforms the existing tests in all situations. 4.2 Analysis for the test Let us now focus on the situation in which the structural break a ects both the level and slope of the time series. Table 6 presents the empirical size and power for the test. First, notice that the t (2), t ; (2), t ;jj (2) and t ;; (2) tests have exact size when there is no break under the null hypothesis, while the test shows mild overrejection. This is not surprising if we take into account that we are not using the pseudo-f test and the corresponding critical values that suit the DGP in the best possible way. Notwithstanding, the size distortions are not problematic when compared to the results that are obtained with a conservative strategy. For instance, we might consider computing the test statistics using either the t sig or M AIC criteria as a way to obtain the correct size when there is no break under the null hypothesis. Simulation results reported in Table 6 show that in this case the test has correct size, although it diverges when 6= 0 when 6= 0 and = 0 the size distortions are similar to those evidenced by the t (2), t ; (2), t ;jj (2) and t ;; (2) tests. Given the poor performance of the conservative strategy, the size distortion presented by the test when there is no break under the null hypothesis is of little importance. The picture changes when there is a structural break under the null hypothesis since, in general, has the empirical size close to the nominal one, while the other 16

17 tests show size distortions. The empirical size of the t (2), t ; (2) and t ;jj (2) tests grows (overrejection) as the magnitude of the break increases. On the other hand, the empirical size for the t ;; (2) test decreases with the magnitude of the break. The underrejection feature of t ;; (2) does not seem to be problematic since, as shown below, the power increases with the magnitude of the break. Note that these results are to be expected since the distribution of t (2), t ; (2), t ;jj (2) and t ;; (2) diverges as T! 1 with 6= 0 see Vogelsang and Perron (1998). The size of the pseudo-f test proposed in this paper remains at a reasonable level for all the cases considered of special relevance is the case where the variable is a ected by medium or large level shifts for which the empirical size of both statistics approaches the nominal value. In addition, the sig test performs well when either = 0 or = 0. These features are also found when facing the negative AR (experiment E3) and positive MA (experiment E4) models, for which sig and MAIC outperform the other tests. Finally, all the tests overreject the null hypothesis when the DGP introduces a negative MA, although in general the size distortion is less severe for the sig and MAIC tests. Regarding the power of the tests, neither proves to be er. For instance, for the experiments E1 and E2 the MAIC test o ers better global performance in terms of power, with the exception of the DGP which uses (; ) = (10; 1) for which the sig test is more powerful. This overall performance is also found for the DGP with the negative AR, i.e. for experiment E3. When the disturbance term includes a positive MA the MAIC test o ers better results, although we should take into account the size distortion (underrejection) mentioned above see for instance, the DGP with (; ) = (5; 1). The high power shown by all tests when there is a negative MA is due to their large size distortions. Moreover, when there is no break in the slope ( = 0) the break point is over estimated, which causes loss in power of the test. Nevertheless, the bias in the estimation of the break fraction disappears as T grows, which implies an increase in the empirical power. For instance, for = 0, = 10, = # = 0, = 0:8 and T = 100, the 17

18 rejection frequency for MAIC is see Table 6 and for T = 200, the rejection frequency is 0.36 whereas for T = 500 is As a consequence, the lack of power of the test in this scenario can be explained both by the adjustment of an over parameterised model and the small sample size. Note also that Vogelsang and Perron (1998) conduct simulation experiments where the DGP is given by an Innovational Outlier model but the break point and the statistical inference is carried out using an Additive Outlier model see Table 6B in Vogelsang and Perron (1998). Their results showed the same feature that has been found in our simulations. Finally, the simulations have shown that the empirical size of t ;; (2) tends to decrease with the magnitude of the structural break, although this does not a ect the power of the test. Nevertheless, t ;; (2) does not outperform in all scenarios. To sum up, evidence reported in this section indicates that the test statistics that account for the signi cance of some of the deterministic elements considerably improve the power of the statistical inference and have reasonable size, arguing in favor of their application in empirical research. Simulations indicate that the test presents a mild size distortion when = = 0, although it is our opinion that this is a minor problem given the bene ts that the test o ers for those situations in which 6= 0. Finally, for practical purposes, we suggest the use of the MAIC between the empirical size and power trade-o. test as a compromise 5 Empirical illustration The tests proposed in this paper are applied using the Nelson-Plosser data set, which has been analysed in many studies see Nelson and Plosser (1982), Perron (1989, 1997), Zivot and Andrews (1992) and Lumsdaine and Papell (1997) among others. Following these papers, the deterministic model considered for each time series is the same as the one speci ed in Perron (1989). Thus, we specify model A for Real GNP, Nominal GNP, Real per capita GNP, Industrial Production, Employment, GNP De actor, Consumer Prices, Nominal Wages, Money Stock, Velocity and Bond Yield. Model C is speci ed for Real 18

19 Wages and Common Stock Prices. All time series but the Bond Yield are transformed into logarithms. We compute the tests that do not introduce the restriction on the drift that is to say, we consider 3;2 and for the case where the structural break under the null is not allowed and apply 3;2 and for the case where the break under the null is permitted. The order of the autoregressive correction is chosen using the t for the 3;2 and 3;2 tests, while both the t sig and MAIC criteria are applied for the and tests we set k max = 10 for the maximum order of lags. Table 7 reports the test statistics for the data set. The null hypothesis of unit root is rejected at the 5% signi cance level for Real GNP, Nominal GNP and Industrial Production, where this conclusion is reached independently of whether or not the structural break is allowed under the null hypothesis for the Nominal GNP at the 10% level. In addition, the null hypothesis is rejected at the 10% signi cance level for Nominal Wages and Bond Yield, either using sig 3;2 or 3;2, and for the GNP de actor when using the test. For Stock Prices there is mild evidence against the unit root using the sig 3;2 test, although it is not rejected for the other combinations. Due to the evidence reported in the simulations, we stress the results using the MAIC criterion, so that Stock Prices is non-stationary in variance. Finally, for the other variables the null hypothesis cannot be rejected. These results can be compared with those obtained in Zivot and Andrews (1992) and Perron (1997). Notice that the exact comparison with Zivot and Andrews (1992) can only be established using the results for the 3;2 and tests, while the comparison with Perron (1997) is given for the 3;2 test. Consequently, the results for the speci cations that include a slope shift under the null hypothesis, i.e. those based on the test, are new in the literature Perron (1997) uses the limit distribution that assumes no break under the null hypothesis as an approximation. Zivot and Andrews (1992) concluded that the unit root hypothesis could be rejected for Real GNP, Nominal GNP, Industrial Production and Nominal Wages, which agree with the results presented in Table 7. However, they found that Real per capita GNP, Employment and Stock Prices were also trend-break stationary processes, a conclusion not ported by our tests. On 19

20 the other hand, the unit root was not rejected for Consumer Prices, Money and Velocity, which is in agreement with our ndings. Notwithstanding, there are two discrepancies between their results and ours since they classi ed Bond Yield and GNP De actor as nonstationary, but our results provide some mild evidence against the unit root hypothesis for these series. Consider now the comparison with results obtained in Perron (1997). Our results agree, on the one hand, for Real GNP, Nominal GNP, Industrial Production and Nominal Wages, for which evidence in favour of the stationarity is found, and, on the other hand, for Consumer Prices, Velocity, Stock Prices and Real Wages for which the unit root cannot be rejected at 5% signi cance level. However, while in Perron (1997) an ambiguous situation was reached for Real per capita GNP and Money, the values for the tests presented in Table 7 indicate that the null cannot be rejected. Contradictory results are reached for Employment, GNP De actor and Bond Yield. Thus, the use of the statistic based on the minimization of the ADF test lead Perron (1997) to the non-rejection of the null hypothesis for Employment at the 5% level, while the test based on the t-statistic on the change in intercept rejected it. The test proposed in this paper suggests nonstationarity, so that, due to eriority shown in the simulations by t (1) and 3;2 tests over the t ; (1) test, we should conclude that Employment is non-stationarity in variance. For the GNP De actor series Perron (1997) concluded in favour of the unit root, and here we have shown that mild evidence can be found when allowing for a break under the null hypothesis. Regarding Bond Yield series, Perron (1997) characterized this variable as I(1) after concluding that the p-value associated with the di erent tests was about 0.99, but here the null is rejected at the 10% level with a break point estimation located in 1966 or 1967, depending on whether the structural break is excluded or included respectively under the null hypothesis. In order to decide on the stochastic properties of this variable, we should remember the outperformance in terms of power of the tests proposed here compared to the existing tests, especially for those situations with high autoregressive parameter. Thus, looking at results in Table 3 in Perron (1997), we realize that the 20

21 estimated autoregressive coe cient for Bond Yield is around 0.934, that is, a high value that suggests a problem of lack of power in the tests in Perron (1997). 6 Conclusions In the paper we have proposed several test statistics that complement and improve the existing procedures to test the integration order of a time series taking into account a structural break at an unknown time not only under the alternative but also under the null hypothesis. The goal of the paper is the speci cation of various sets of parametric restrictions along with the corresponding tests, which provides analysts with a procedure that considers more information when testing the unit root hypothesis with structural breaks. The performance of the di erent tests and approaches has been analysed through a set of Monte Carlo experiments. The main conclusion is that the test statistics based on the rema functional are more powerful than the existing tests for the null of unit root with a structural break, and have good size in almost all situations considered. However, it has been shown in the paper that the use of the functional is to be avoided when allowing for a slope shift under the null hypothesis since the tests diverge. In order to overcome this drawback, we suggest a testing procedure with good properties when a slope shift is present under the null hypothesis. The paper also suggests the best procedure to choose the lag length of the autoregressive correction. Therefore, our proposal introduces new tools to test the unit root hypothesis from di erent perspectives. 21

22 References Banerjee, A., J. J. Dolado, and J. W. Galbraith (1990): Recursive and Sequential Tests for Unit Roots and Structural Breaks in Long Annual GNP Series, Discussion Paper 9010, Banco de España. Servicio de Estudios. Banerjee, A., R. L. Lumsdaine, and J. H. Stock (1992): Recursive and Sequential Tests of the Unit-Root and Trend-Break Hypotheses: Theory and International Evidence, Journal of Business & Economic Statistics, (13, 3), Carrion-i-Silvestre, J. L. (1999): Integració I Estacionarietat de Sèries Temporals Amb Ruptures Estructurals, Ph.D. thesis, Departament d Econometria, Estadística i Economia Espanyola. Universitat de Barcelona. Dickey, D. A., and W. A. Fuller (1981): Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root, Econometrica, (49), Hatanaka, M. (1996): Time-Series-Based Econometrics. Oxford University Press. Hatanaka, M., and Y. Koto (1995): Are There Unit Root in Real Economic Variables? (An Encompassing Analysis of Di erence and Trend Stationarity), The Japanese Economic Review, 46(2), Lumsdaine, R. L., and D. H. Papell (1997): Multiple Trend Breaks and the Unit Root Hypothesis, Review of Economics and Statistics, (79), Montañés, A. (1998): Unit Roots, Level Shifts and Purchasing Power Parity, Discussion paper, Department of Economic Analysis. University of Zaragoza. Montañés, A., and M. Reyes (1997): Testing for Unit Roots in Variable with a Change in the Mean: a Pseudo F-ratio Test, Discussion paper, Department of Economic Analysis. University of Zaragoza. Nelson, C., and C. Plosser (1982): Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications, Journal of Monetary Economics, (10),

23 Ng, S., and P. Perron (1995): Unit Root Test in ARMA models with Data-Dependent Methods for the Selection of the Truncation Lag, Journal of the American Statistical Association, (90), (2001): Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power, Econometrica, 69(6), Perron, P. (1989): The Great Crash, the Oil Price Shock and the Unit Root Hypothesis, Econometrica, (57, 6), (1990a): Testing for a Unit Root in a Time Series with a Changing Mean, Journal of Business & Economic Statistics, (8, 2), (1990b): Tests of Joint Hypotheses for Time Series Regression with a Unit Root, in Advances in Econometrics. Co-Integration, Spurious Regression and Unit Roots, pp Jai Press, Inc. (1994): Trend, Unit Root and Structural Change in Macroeconmic Time Series, in Cointegration for the Applied Economist. Macmillan. (1997): Further Evidence on Breaking Trend Functions in Macroeconomic Variables, Journal of Econometrics, (80), Perron, P., and T. Vogelsang (1992): Nonstationarity and Level Shifts with an Application to Purchasing Power Parity, Journal of Business & Economic Statistics, (10, 3), Perron, P., and X. Zhu (2004): Structural Breaks with Deterministic and Strochastic Trends, Journal of Econometrics, p. Forthcoming. Phillips, P. C. B. (1987): Time Series Regression with a Unit Root, Econometrica, (55, 2), Phillips, P. C. B., and P. Perron (1988): Testing for a Unit Root in Time Series Regression, Biometrika, (75),

24 Saikkonen, P., and H. Lütkepohl (2002): Testing for a Unit Root in a Time Series with a Level Shift at Unknown Time, Econometric Theory, 18, Vogelsang, T., and P. Perron (1998): Additional Tests for a Unit Root Allowing for a Break in the Trend Function at an Unknown Time, International Economic Review, 39(4), 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 & Economic Statistics, (10, 3),

25 A Mathematical appendix A.1 Proof of Theorem 1 The proof of Theorem 1 is carried out in two steps. In the rst we obtain the asymptotic distributions of the tests that do not consider a structural break under the null hypothesis. The proof is developed fully for the simplest case, i.e. the 1 () test, but the derivations can be applied for the other tests. In the second step we deal with the asymptotic distribution of the tests where the structural break is also considered under the null hypothesis. We develop the proof for the simplest case and describe step by step the way by which the asymptotic distribution of 1 () is obtained. However, in this case we have also decided to present the results for the 5;1 () and () tests. The derivations for the other tests are available upon request. To facilitate cross-comparisons, we have made extensive use of the projection notation in Perron (1997). Some words regarding notation. Let S t = P t j=1 u j, S 0 = 0, and X T (r) = 1 T 1=2 S [T r], (j 1) =T r < j=t, j = 1; : : : ; T, where [] denotes the integer part of the argument, and 2 = lim T!1 T 1 E (ST 2 ). In order to simplify notation, throughout this Appendix and unless otherwise indicated, we assume that the estimated break fraction equals the true (correct) break fraction, i.e. = c. A.1.1 The proof for 1 () Consider the following regression: y t = z i t;t () i () + u t ; (18) for models i = f1; 2; (3; 1) ; (3; 2) ; (4; 1) ; (4; 2) ; (5; 1) ; (5; 2)g. The vector zt;t i () contains the deterministic and stochastic regressors of the model and its de nition depends on and T. For example, for the simplest model, i = 1, z 1 t;t () = [DU t; y t for the more general one, i = (5; 2), z () t;t () = [1, t, DU t, DTt, y t z i t;t 1 ], whereas 1 ]. Note that the () vector for the other models follows from z() t;t (). Let us also de ne ZT i (; r) = 25

26 z[t i r];t () R, with R being the rescaling matrix. For instance, for the model i = (5; 2), R is given by R = diag(t 1=2, T 3=2, T 1=2, T 3=2, 1 T 1 ). We can de ne the limiting functions Z i (; r) in such a way that, for the model i = (5; 2), Z () (; r) = (1, du (; r), r, dt (; r), W (r)) 0, where du (; r) = 1 (r > ) and dt (; r) = 1 (r > ) (r ). In what follows we drop the erscript that indicates the model. Note that under the null hypothesis y t = u t and the sum of squared residuals is equal to: SSR = TX yt 2 = t=1 TX u 2 t : t=1 Hence, SSR, after rescaling by T 1, converges to the variance of the disturbance term. On the other hand, the sum of squared residuals for the unconstrained model can be obtained from SSR = u 0 Mz T () u, where Mz T () = I T z T () z T () 0 z T () 1 zt () 0 is the usual idempotent matrix of projections. Using these elements we can see that the numerator of (13) is given by: num = u 0 Z T () Z T () 0 Z T () 1 ZT () 0 u: (19) It is easy to see that 2 Rz 0 6 zr ) 4 (1 ) R 1 W (r) dr R 1 W (r) dr R 1 W 0 (r)2 dr = G () ; Rz 0 u ) (W (1) W ()) =2 W (1) = H (; ) ; elements that de ne the limiting distribution of the numerator as: num ) H (; ) 0 G () 1 H (; ) : Notice that the denominator of (13) corresponds to the estimator of the variance of the disturbance term for the unconstrained model which in the limit converges to 2 u. Now we can derive the limiting distribution of 1 () as: () ) (1=2) H (; ) G () 1 H (; ) = A 1 () : (20) 26

27 Similar developments can be followed to assess the asymptotic distribution of the other test statistics, which are available upon request form the authors. A.1.2 The proof for 1 () Notice that for 1 (), the stochastic process under the null hypothesis is given by: y t = dd (T b ) t + u t = z 1;t 1 + u t ; (21) but under the alternative it behaves according to: y t = DU t + dd (T b ) t + y t 1 + u t : The sum of the squared residuals under the null is computed through: SSR = u 0 Mu = u 0 u u 0 z 1 (z1z 0 1 ) 1 z1u 0 (22) TX = u 2 t u 2 T b +1; t=1 where z 1;t = [D (T b ) t ], whereas under the alternative the sum of the squared residuals is equal to: SQR = TX u 2 t u 0 z 2 R (Rz2z 0 2 R) 1 Rz2u; 0 (23) t=1 where z 2;t = [DU t ; D (T b ) t ; y t 1 ]. The application of the rescaling matrix R = diag(t 1=2, 1, 1 T 1 ) provides the following results 2 Rz2u 0 ) 6 4 (W (1) W ()) u Tb +1 / 2 W (1) = H (; ) ; Rz2z 0 2 R ) 6 4 (1 ) 0 R 1 W (r) dr 1 0 R 1 0 W (r)2 dr = G () : 27