Spurious rejections by Dickey-Fuller tests in the presence of an endogenously determined break under the null

Badillo Amador, Rosa; Belaire Franch, Jorge; Reverte Maya, Carmelo Article Spurious rejections by Dickey-Fuller tests in the presence of an endogenously determined break under the null Revista de Métodos Cuantitativos para la Economía y la Empresa Provided in Cooperation with: Universidad Pablo de Olavide, Sevilla Suggested Citation: Badillo Amador, Rosa; Belaire Franch, Jorge; Reverte Maya, Carmelo (2010) : Spurious rejections by Dickey-Fuller tests in the presence of an endogenously determined break under the null, Revista de Métodos Cuantitativos para la Economía y la Empresa, ISSN 1886-516X, Universidad Pablo de Olavide, Sevilla, Vol. 09, pp. 3-16 This Version is available at: https://hdl.handle.net/10419/59081 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-sa/3.0/es/ REVISTA DE M´ ETODOS CUANTITATIVOS PARA LA ECONOM´ IA Y LA EMPRESA (9). P´aginas 3–16. Junio de 2010. ISSN: 1886-516X. D.L: SE-2927-06. URL: http://www.upo.es/RevMetCuant/art34.pdf Spurious Rejections by Dickey-Fuller Tests in the Presence of an Endogenously Determined Break under the Null Badillo Amador, Rosa Departamento de Econom´ıa Universidad Polit´ecnica de Cartagena Correo electr´onico: [email protected] Belaire Franch, Jorge Departamento de Fundamentos del An´alisis Econ´omico Universidad de Valencia Correo electr´onico: [email protected] Reverte Maya, Carmelo Departamento de Econom´ıa Financiera y Contabilidad Universidad Polit´ecnica de Cartagena Correo electr´onico: [email protected] ABSTRACT Leybourne et al. (1998) have proved the possibility of a ‘converse Perron phenomenon’ when conventional Dickey-Fuller tests are applied to determine the order of integration of a time series. That is, if the true generating process is I(1) but with a break, frequent spurious rejections of the null hypothesis can occur. Although Leybourne et al. (1998) suggest it would be appropriate to use procedures in which the break date was treated as endogenous, they consider it as exogenous. Thus, this paper analyses whether their results change when the structural break is identified endogenously, that is, if the break point is gleaned from the data. In this sense, applying a recursive tDF test to a unit root process which has a break in its level, there is no virtually evidence of the ‘converse Perron phenomenon’. For the rest of the endogeneization procedures (i.e., rolling and sequential) and for the two types of breaks considered (in level or in drift), we find, in line with Leybourne et al. (1998), some distortion in the Dickey-Fuller tDF test size, which depends on the break size, the location of the break point in the sample and the sample size. Keywords: unit roots; structural breaks; Dickey-Fuller tests. JEL classification: C12; C15; C22. MSC2010: 62P20. Art´ıculo recibido el 16 de noviembre de 2009 y aceptado el 29 de enero de 2010. 3 Rechazos espurios de los test de Dickey-Fuller en presencia de una ruptura bajo la hip´otesis nula end´ogenamente determinada RESUMEN Leybourne et al. (1998) muestran el cumplimiento del denominado “fen´omeno inverso de Perron” cuando se aplican los test convencionales de Dickey- Fuller para determinar el orden de integraci´on de una serie temporal. Este fen´omeno consiste en que, si el verdadero proceso generador es I(1) pero con una ruptura, pueden producirse rechazos espurios frecuentes de la hip´otesis nula. Aunque Leybourne et al. (1998) sugieren que ser´ıa apropiado utilizar procedimientos en los que la ruptura sea tratada como end´ogena, ellos la consideran como ex´ogena. As´ı, este trabajo analiza si sus resultados cambian cuando la ruptura estructural se determina end´ogenamente, es decir, a partir de los datos. En este sentido, aplicando el procedimiento tDF recursivo a un proceso de ra´ız unitaria con una ruptura en el nivel, no encontramos pr´acticamente evidencia del “fen´omeno inverso de Perron”. Para el resto de procedimientos de endogeneizaci´on (rolling y secuencial) y para los dos tipos de rupturas considerados (en nivel o en deriva) encontramos, en l´ınea con Leybourne et al. (1998), alguna distorsi´on en el tama˜no del test tDF de Dickey-Fuller, la cual depende de la magnitud de la ruptura, de su ubicaci´on en la muestra y del tama˜no de la misma. Palabras clave: ra´ıces unitarias; cambios estructurales; test Dickey-Fuller. Clasificaci´on JEL: C12; C15; C22. MSC2010: 62P20. 4 5 I. INTRODUCTION Much conventional asymptotic theory for least-squares estimation assumes stationarity, I(0), of the explanatory variables. However, Nelson and Plosser (1982) argue that almost all macroeconomic time series are non-stationary, and typically do have a unit root (I(1) series). The presence or absence of unit roots helps to identify some features of the underlying data generating process of a series. If the series is stationary, it tends to return to its mean value and fluctuate around it within a more-or-less constant range (i.e., it has a finite variance which does not depend on time). On the other hand, non-stationary series have a mean and/or variance depending on time and thus have no tendency to return to long-run deterministic path. The method of estimation of the standard regression model, Ordinary Least Square (OLS) method, is based on the assumption that the means and variances of these variables being tested are constant over the time. One illustration of the difficulties that can arise when performing an OLS regression with clearly non-stationary series is the problem of nonsense regression, so named by Yule (1926), or spurious regression in the terminology of Granger and Newbold (1974). That is, given two completely unrelated but integrated series, regression of one on the other will tend to produce statistically significant relationships between the variables when the fact all that is obtained is evidence of contemporaneous correlations rather than meaningful causal relations. Instead, if variables are non-stationary, the estimation of long-run relationship between those variables should be based on the cointegration method. Since the testing of the unit roots of a series is a precondition to the existence of cointegration relationship, Dickey and Fuller (1979) devised a procedure to formally test for non-stationarity (DF test). The simplest form of the DF test amounts to estimating: 1ttt yy ρ ε − = +, (1) with the null being H0: ρ=1 (unit root) against the alternative H1:1 ρ < . The standard approach to testing such a hypothesis is to construct a t-test, however, under non-stationarity, the statistic computed does not follow a standard t-distribution but, rather, a Dickey-Fuller distribution. This fact justifies the use of Monte Carlo techniques1, which are developed in Sections II, III and IV of the paper. 1 These Monte Carlo techniques involve taking (1) as the underlying data generating process (DGP), imposing the null hypothesis by fixing ρ =1, and randomly drawing samples of the εt from the normal distribution; this then generates thousands of samples of yt, all of which are consistent with the DGP (1). Then for each of the yt a regression based on (1) is undertaken, with ρ now free to vary, in order to compute (on the basis of thousands of replications) the percentage of times the model will reject the null hypothesis of a unit root when the null is true. These are the critical values for rejecting the null of a unit root at various significance levels based on the DF distribution of ( ˆ ρ -1)/ ˆ ρ σ , been ˆ ρ σ the standard deviation of ˆ ρ . 6 Following the work of Perron (1989), it is well known, however, that the usual DF test of the unit root null hypothesis can have low power when the true generating process is stationary around a broken linear trend. Perron (1989)’s study was criticized on the grounds that he treated the date of the break as known. Subsequent works used a variety of tests endogenizing the break point (Christiano, 1992; Zivot and Andrews, 1992; Banerjee et al., 1992; Lumsdaine and Stock, 1992; Perron and Vogelsang, 1992; Perron, 1994, 1997 and Vogelsang and Perron, 1998, inter alia). The summary picture one gets from these studies is that endogenizing the break point reverses the conclusions arrived at by Perron (1989). Leybourne et al. (1998) have also proved the possibility of the so-called ‘converse Perron phenomenon’, that is, if the true generating process is I(1) but with a break, frequent spurious rejections of the null hypothesis can occur. They also proved that this phenomenon can lead to a very serious problem of spurious rejections of the unit root null hypothesis, especially if the break occurs early in the series. Leybourne et al. (1998) also point out that the practice of using data further back in time to enlarge the series, presumably in search of additional power and more precise estimates, could easily lead to erroneous conclusions if incorporating the additional data introduces a break. These authors consider, as in Perron (1989), the date of the break as known, that is, as an exogenous event. In this context, the main goal of this paper is to re-examine the Monte Carlo analysis of Leybourne et al. (1998) in order to analyse whether the ‘converse Perron phenomenon’ also holds when the break point is chosen endogenously. In other words, we focus our attention in analysing if endogenizing the break point reverses the conclusions arrived at by Leybourne et al. (1998). Therefore, this article considers the presumption that, if there is a break, its date is not known a priori but rather is gleaned from the data, as it would be appropriate if there was no strong exogenous reason to suspect a break at a particular time. Following Banerjee et al. (1992), we carry out in this paper a set of tDF tests that control endogenously for structural breaks. These are known as recursive, rolling and sequential tests. Not surprisingly, we obtain that the empirical critical values are well below the full-sample standard tDF test. In addition, we obtain, in some cases, proportions of rejections of the unit root null hypothesis, when it is true, lower than those obtained by Leybourne et al. (1998) when the break date is treated as exogeneous. One of these cases is when a break in level is occurred under the null and a recursive tDF test is applied. In this case, the spurious rejection of the null is so low that we can consider that there is virtually no evidence of the ‘converse Perron phenomenon’. For the rest of the endogeneization procedures (i.e., rolling and sequential) and for the two types of breaks considered (in level or in drift), we find, in line with Leybourne et 7 al. (1998), some distortion in the Dickey-Fuller DF t test size, which depends on the break size, the location of the break point in the sample and the sample size. The rest of the paper is organized as follows. Section II reviews a variety of tests, based on the standard tDF statistics, which treat the break date as unknown a priori. Section III reports finite critical values calculated by Monte Carlo experiments for these tests. In Section IV and V we analyse the possibility of spurious rejection of the unit root null hypothesis when an I(1) time series presents a structural change in either its level or its drift and when the tests analysed in Section II are applied. In Section VI the data of Leybourne et al. (1998) are re-examined to empirically illustrate the simulation results. Section VII concludes. II. THE MODELS AND STATISTICS We begin with a briefly review about the statistical procedures used to test for a unit root allowing for the presence of a structural change in the I(1) generated process. Three classes of standard DF statistics that control endogenously for structural breaks are considered. These are known as recursive, rolling and sequential tests2. A traditional DF regression, like this: 1 1,..., , ttt y t y tT μ βρ ε − = ++ + = (2) is estimated in this paper. We take subsamples t =1,…, κ , where κ = κ 0, κ 0+1,…,T, and using as criteria the minimum values3 of the t-ratio evaluating 1 ρ = . κ 0 is the starting value of the recursive estimation and T is the size of the full sample. This test is known as the recursive min DF t test ( min ˆDF t test). The rolling min DF t test ( min DF t) is based on subsamples of fixed size Ts, rolling through the sample. We choose the min DF t statistic between all subsamples. Finally, the sequential test statistic ( *min DF t test) is computed using the full sample and sequentially incrementing the date of the hypothetical break using a dummy variable and 2 For more details, see Banerjee et al. (1992). 3 We consider the minimal t-statistic criteria in all tDF tests that control endogenously for structural breaks due to the fact that we are interested in obtaining the highest spurious rejection frequency. It is clear that the use of other criteria used in Banerjee et al. (1992), such as the maximum Dickey Fuller t-statistic or a t-statistic based on the difference between its maximum and minimum values, would result in lower spurious rejection frequencies. 8 choosing the lowest value of the statistic. We consider a shift in mean, which is referred by Perron (1989, 1990) as the ‘crash’ model: 1 ( ) , 1,..., tt tt ydDty tT μ τβρ ε − = ++++= (3) where: 1, if t> T ( ) (0,1) 0, otherwise t D τ ττ ⎧ =∈ ⎨ ⎩ (4) and the break fraction is denoted as τ = κ /T. The t-stastistic testing d=0 provides information about whether there has been a break or jump in the mean. The DF t test evaluating 1 ρ = is used to test for the order of integration of the series. III. CRITICAL VALUES FOR THE RECURSIVE, ROLLING AND SEQUENTIAL TESTS This section reports finite critical values of recursive, rolling and sequential min DF t tests. All the calculations have been programmed in Ox 4.1 (http://www.doornik.com). The critical values are computed using data generated for the null model (0 1) = ttt y , iid N , Δ εε and are based on 10,000 Monte Carlo replications4 for the following finite sample sizes5: T=100, 75 and 50 (see Table 1). The recursive statistic, min ˆDF t, is computed by estimating (2), under both assumptions: β =0 or β ≠0 (see in Table 1 break in level or break in drift columns, respectively), over t=1,…, κ , for 0,...,T κκ =, with the following trimming parameter: τ 0=0.25. The rolling statistic, min DF t, is computed by estimating (2), also under both assumptions: β =0 or β ≠0, over t= κ -[T τ 0]+1,…, κ , κ =[T τ 0],…,T, being the trimming parameter: τ 0=1/3. The sequential statistic, *min DF t, is computed by estimating (3) sequentially, for 00 ,...,T κκ κ = −, under both assumptions: β =0 or β ≠0, with () t D τ given by (4). For the *min DF tstatistic the trimming parameter is τ 0=0.01. As pointed out by Banerjee et al. (1992), the choice of τ 0 for the previous statistics implies a trade-off between needing enough observations in the shortest regression and wanting to capture possible breaks early and late in the sample. As shown in Table 1, recursive, rolling and sequential critical values are, not surprisingly, well below the full-sample standard DF critical values6. 4 The use of Monte Carlo method is justified in the Introduction of the paper. 5 We consider T= 50 and T=75 because a great number of annual macroeconomic time series have small sample sizes. T=100 is also chosen to compare our results with those obtained by Leybourne et al. (1998). 6 See Fuller (1976). 9 Table 1. Recursive, Rolling and Sequential min DF t Statistics: Critical Values. Break in level T Percentile Recursive min DF ˆ t Rolling min DF t Sequential *min DF t 0.010 -4.2865 -5.2763 -4.9464 0.025 -3.9356 -4.8736 -4.6571 0.050 -3.6393 -4.5247 -4.3866 100 0.100 -3.3446 -4.2341 -4.0984 0.010 -4.3915 -5.3762 -5.0532 0.025 -3.9982 -4.9242 -4.7051 0.050 -3.7039 -4.5907 -4.4135 75 0.100 -3.3678 -4.2551 -4.1251 0.010 -4.5631 -6.0037 -5.1457 0.025 -4.1142 -5.3701 -4.7674 0.050 -3.7624 -4.9114 -4.4688 50 0.100 -3.4080 -4.4835 -4.1421 IV. UNIT ROOT WITH A BREAK IN LEVEL We next analyse the possibility of spurious rejection of the unit root null hypothesis when recursive, rolling and sequential tests are applied, and when there is a break in an I(1) generating process. In line with Perron (1989) and Leybourne et al. (1998), we permit just a single break and we shall concentrate on additive outlier models, implying that the break in trend is abrupt. Particularly, we discuss in this section the simplest possible case, where monotonic trend or drift is assumed to be absent. In that case the alternative would be stationarity about a fixed mean, and the null would be I(1) with zero mean change. The experimental design of Leybourne et al. (1998) was employed. Thus, we consider a time series t y with the following data generation process (DGP): 1 ( ) , , 1,..., tt tttt y stT α τν νν ε − = +=+= , (6) where εt∼i.i.d.N(0,1). In Eq. (6): 0, t T ( ) (0,1) 1, t> T, t s τ ττ τ ≤ ⎧ =∈ ⎨ ⎩ (7) All simulations are based on 5,000 replications using sample sizes of 100 observations7. An additional initial 100 observations were discarded to remove the influence of the initial 7 Due to space restrictions, we report only the results for T=100. Results for T=75 and T=50 are available from the authors upon request. Break in drift T Percentile Recursive min DF ˆ t Rolling min DF t Sequential *min DF t 0.010 -4.9516 -5.8392 -5.4076 0.025 -4.6063 -5.4476 -5.1584 0.050 -4.3453 -5.1752 -4.9072 100 0.100 -4.0021 -4.8272 -4.6183 0.010 -5.1104 -6.1137 -5.4617 0.025 -4.7386 -5.6565 -5.1791 0.050 -4.4036 -5.2671 -4.9080 75 0.100 -4.0559 -4.8914 -4.6110 0.010 -5.4902 -6.9147 -5.6003 0.025 -4.8853 -6.2614 -5.2342 0.050 -4.5261 -5.7501 -4.9747 50 0.100 -4.1104 -5.2707 -4.6490 10 condition 00y=. In order to compare our results with those of Leybourne et al. (1998), the values α ∈ {2.5, 5, 10} were chosen for the break size. The break in level was therefore imposed after observation τ T= κ . For each replication, the min ˆDF t and min DF t tests are estimated using regression (2), under the assumption β =0, and the *min DF tstatistic is estimated using regression (3), under the same assumption and for D t ( τ ) defined in Eq. (4). The (false) rejections of the unit root hypothesis are noted at the 5% level of significance using the critical values calculated in Section III (see Table 1, break in level columns). The resulting empirical rejection frequencies are presented in Tables 2 to 4 for T=100 in level-break columns. It can be seen that, using min ˆDF ttest, the spurious rejection of the null hypothesis is below the nominal size, and is independent of the location of the break and its magnitude, but not of the sample size, as the higher is T the lower is the spurious rejection rate of the null hypothesis. However, in the case of the min DF t and *min DF ttests, ignoring the possibility of a break produces many rejections of the null, especially when α increases and when T decreases. For these two tests, the break location also influences on the spurious rejection rate. For example, in the case of the min DF t test, the spurious rejection rate is lower when 0 TT τ τ >− , since only the subsample from the 0 TT τ − observation to the last observation is the one capturing the break8. Regarding the *min DF ttest, there is a higher rejection rate when the break point is closer to the middle of the sample. Comparing our results with those obtained in Leybourne et al. (1998) where the break point is considered as exogenous, we obtain a lower proportion of rejections of the unit root null hypothesis only when using the min ˆDF ttest, and this lower rejection frecuency is occurred for all magnitudes of the break and for the different τ values considered in our study. In this case, the spurious rejection of the null is so low that we can consider that there is virtually no evidence of the ‘converse Perron phenomenon’. This finding suggests the use of the min ˆDF t test when there is a break in the level of the series t y and when its DGP could be given by expression (6). 8 This phenomenon is inherent to this procedure, which is based on subsamples of fixed size rolling through the sample.