Single-Equation Tests for Cointegration with GLS Detrended Data

Single-Equation Tests for Cointegration with GLS Detrended Data Abstract We provide GLS-based versions of two widely used approaches for testing whether or not non-stationary economic time series are cointegrated: single-equation static regression or residual-based tests and single-equation conditional error correction model (ECM) based tests. Our approach is to consider nearly optimal tests for unit roots and apply them in the cointegration context. Our GLS versions of the tests do indeed provide substantial improvements over their OLS counterparts. We derive the local asymptotic power functions of all tests considered for a DGP with weakly exogenous regressors. This allows obtaining the relevant non-centrality parameter to quasi-di¤erence the data. We investigate the e¤ect of non-weakly exogenous regressors via simulations. With weakly exogenous regressors strongly correlated with the dependent variable, the ECM tests are clearly superior. When the regressors are potentially non-weakly exogenous, the residuals-based tests are clearly preferred. 1 Introduction Even though they are applicable only under some speci…c conditions, single-equation static regression or residual-based tests for cointegration, proposed by Engle and Granger (1987) and developed by Phillips and Ouliaris (1990), have been quite popular in applied work mostly because of their computational simplicity. The statistics are designed to test the null hypothesis of no cointegration in a single equation setting assuming that the variables introduced as regressors are not cointegrated. These tests also have some appeal because they follow quite intuitively from the basic de…nition of cointegration as laid out in Engle and Granger (1987). If the system of variables is cointegrated, then there exists a linear combination (given by the cointegrating vector) that is stationary. In this case, the residuals from a simple static regression are stationary and, as shown by Stock (1987), this regression estimated by Ordinary Least Squares (OLS) will provide a consistent estimate of the cointegrating vector. In the absence of cointegration, the residuals from the static regression are nonstationary for any choice of the parameter vector and we have what has been labelled, following Granger and Newbold (1974) and later Phillips (1986), a spurious regression. Hence, an obvious testing strategy is to test the null hypothesis of no cointegration using some unit root test on the estimated residuals from the simple static regression. Another type of single-equation cointegration tests is based on estimates of a conditional error correction model (ECM); see Kremers, Ericsson and Dolado (1992), Banerjee et al. (1993), Boswijk (1994), Banerjee, Dolado and Mestre (1998), and Zivot (2000) among others1 ; and for critical values, see Ericsson and MacKinnon (2002). An analysis of the local asymptotic power function of various single-equation cointegration tests is presented in Pesavento (2004). Of course, there are many alternative approaches available, some applicable under less restrictive conditions; for example, the system based tests of Johansen (1991) and Stock and Watson (1988). The reader is referred to one of the many available surveys; for example, Perron and Campbell (1992), Banerjee, Dolado, Galbraith and Hendry (1993), Watson (1994), Lütkepohl (1999) and Johansen (2006). Elliott, Rothenberg and Stock (1996, henceforth ERS), following the work of Dufour and King (1991), showed that several unit root tests constructed using Generalized Least Squares (GLS) or quasi-di¤erenced data2 have asymptotic power functions close to the 1 See also Harbo et al. (1998), Kiviet and Phillips (1992), Mackinnon, et al. (1999) and Pesaran et al. (2000). 2 There is some argument to the fact that the use of the terminology “GLS detrending”is not appropriate 1 Gaussian local asymptotic power envelope. Hence, they enjoy some optimal properties over tests constructed using OLS detrended data and the simulations in ERS showed substantial power gains in …nite samples. If such a detrending device is bene…cial for unit root tests, it is natural to think that it would also be for cointegration tests. Our aim, accordingly, is to analyze residual-based and ECM tests for cointegration when they are constructed using GLS detrended or quasi-di¤erenced data. We consider the standard ADF and the Z^ and Zt tests analyzed in Phillips and Ouliaris (1990) as well as the class of modi…ed unit root tests analyzed in Stock (1999), Perron and Ng (1996) and Ng and Perron (2001). We also consider two ECM-based tests. Unfortunately, in the context of testing for cointegration, the problem of …nding the Gaussian local asymptotic power envelope, as done by ERS for unit root tests, is intractable (at least to us) given the fact that the cointegrating vector is not identi…ed under the null hypothesis. It nevertheless seems sensible to argue that tests that perform best in the unit root case would also allow power gains in the cointegration context. This is the approach we follow in this paper. Hence, we consider nearly optimal tests for unit roots and apply them in the cointegration context. We cannot claim that any of the tests achieves the best power possible. Given the approach taken, we also cannot claim that the non-centrality parameters c suggested correspond exactly to the values that would yield a 50% local asymptotic power for the tests. While we agree that it would be of theoretical interest to be able to remedy these caveats, we believe that, for all practical purposes, it would make little di¤erences. Our GLS versions of the tests do indeed provide substantial improvements over their OLS counterparts and further gains in power for the classes of tests considered are likely to be minor. The end-products do provide improvements over currently available methods and are therefore useful for empirical applications. We derive the asymptotic distributions of all tests considered for a Data Generating Process (DGP) with weakly exogenous regressors for the relevant parameter in the ECM. Ideally, one would adopt a DGP that is the most general possible, not imposing weakly exogenous regressors. The problem is that in this general case, the local asymptotic power functions of the tests are not only di¢ cult to obtain but very complex functions of many nuisance parameters. On the other hand, with weakly exogenous regressors, the local asgiven that the procedure does not consider a full GLS transformation (since it only considers the leading root modelled as local to unity). An alternative terminology is that of “quasi-di¤erenced” data. We still believe, for reasons that will become clear later in the text, that the use of “GLS detrending”is meaningful since it is this feature of the procedure that is of importance, even if constructed in a partial fashion. We shall use both terminology interchangeably. 2 ymptotic power functions will depend on a single nuisance parameter, namely R2 , which represents the long-run contribution of the regressors in the cointegrating relation, as well as the number of regressors m. Also, as will be shown in the simulations the power functions of the residual-based tests are not much in‡uenced by nuisance parameters other than R2 and m, even though they are not invariant to them. However, the power functions of ECM-based tests are very sensitive to the degree of endogeneity of the regressors, with power that can often be zero. Using this DGP will allow obtaining the relevant local asymptotic power functions in this base case, from which we can select the relevant non-centrality parameter to quasi-di¤erence the data. These may not correspond exactly to the value that yields 50% local asymptotic power functions but given the relative robustness of the residual-based tests to departures from the weakly exogenous regressors case, the di¤erences should have little practical impact on the power of the tests. As we shall show, even with this less than optimal approach to select the non-centrality parameter, the residual-based tests will have much improved power over their OLS-based versions in all DGPs considered including those with non-weakly exogenous regressors. The GLS versions of the ECM based tests will have higher power in the case of weakly exogenous regressors but will be very de…cient when the DGP involves non-weakly exogenous regressors. Our work is related to that of Lütkepohl and Saikkonen (2000), Saikkonen and Lütkepohl (2000a,b) and Xiao and Phillips (1999) who considered the use of GLS or quasi-di¤erenced data when testing for cointegration in a multivariate setting, i.e., extending the tests proposed by Johansen (1991). It is also related to Pesavento (2007) who analyzed the local asymptotic power functions of similar tests we suggested in an earlier version of this paper. She showed that our approach did lead to marked improvements in power. This paper o¤ers improved tests with non-centrality parameters to quasi-di¤erence the data that are better grounded in theory compared to what was available in the previous version. This paper is organized as follows. Section 2 presents the data-generating process considered including the speci…cations of the deterministic components for the regression and the variables. Section 3 presents a motivation for the use of GLS detrended data and outlines our strategy for the construction of the tests and the selection of the non-centrality parameter to quasi-di¤erence the data. Section 4 describes the various tests considered. Section 5 is concerned with asymptotic results, in particular the local asymptotic power functions of the tests, including the limit null distributions that allow tabulating critical values. Section 6 describes issues related to the implementation of the tests, in particular the evaluation of the relevant non-centrality parameter to quasi-di¤erence the data. It also provides asymptotic 3 critical values and a comparison of the local asymptotic power functions of the tests, both with OLS and GLS detrended data. Section 7 presents simulation results about the size and power of the tests in …nite samples. A more general data generating process is adopted that does not impose weakly exogenous regressors. The relative merits and drawbacks of each tests are analyzed. Section 8 o¤ers brief concluding remarks and recommendations for practical implementations. An appendix contains technical derivations. 2 The Data Generating Process We consider the following Data Generating Process (DGP): 0 yt = dyt + (1) xt + ut xt = dxt + uxt uxt = uxt ut = for t = 1; :::; T , where xt is a m Px i components: dxt = pi=0 ix t ut 1 + v1t 1 + v2t ; 1 vector, yt is a scalar, dxt and dyt are the deterministic Ppy y 0 0 x i x px 0 x mt , dyt = y mt , where mt = (1; t; :::; t ) i=0 iy t 0 and myt = (1; t; :::; tpy )0 . The vector vt = (v1t ; v2t )0 contains (potentially) serially correlated P1 P errors with vt = (L) t = 1 i t i and i=0 i det j i j < 1 ( 0 = In ) where t is a i=0 martingale di¤erence sequence with respect to the sigma-…eld Ft = -f ield f:::; with E( t jFt 1 ) = : The total number of variables in the system is n = m + 1. Assumption 1 f t g satis…es a functional central limit theorem, that is T 1=2 W(r) where W(r) is a standard n denotes weak convergence and 1=2 t 2; t 1g P[T r] t=1 t ) 1 vector of independent Wiener processes, ) is the variance-covariance matrix of t. As a matter of notation, let W = (W01 ; W2 )0 where W1 is an m-vector and W2 is a scalar. Also, Assumption 1 along with the conditions on T 1=2 [T r] X t=1 where (L) implies that vt ) B(r) = 1=2 (2) W(r) is the spectral density at frequency zero of vt scaled by 2 , that is, = lim T T !1 1 T T X X Ef[ vt ][ vt0 ]g = 2 fvv (0) = 2 t=1 t=1 4 (1) (1)0 : We de…ne the following partition of the n n matrix as 3 2 11 ! 12 5; =4 ! 21 ! 22 and de…ne R2 = 0 , where = 1=2 1=2 11 ! 12 ! 22 is a vector containing the bivariate zero frequency correlations of each element of v1t with v2t . The coe¢ cient R2 lies between zero and one and represents the long-run contribution of the right-hand side variables in the …rst equation of (1). It is zero when the variables xt are not correlated in the long run with the errors from the cointegration regression. Furthermore, under the assumption that xt are not individually cointegrated, The null hypothesis is 11 is non singular, an assumption that we maintain throughout. = 1 and the alternative hypothesis is j j < 1. Under the alternative hypothesis, the linear combination yt dyt 0 xt is stationary, yt and xt are cointegrated and the system (1) contains m unit roots. Under the null hypothesis, the two variables are not cointegrated and there are n unit roots in the system. Some comments about the speci…cations of the deterministic components are in order. First, we restrict the analysis to px = 0 (non-trending data) and px = 1 (trending data) as these are of most interest in practice. When px = 0, we set py = 0 (only a constant in the cointegrating regression). When px = 1, we consider both cases with py = 0 and py = 1. In the terminology of Campbell and Perron (1991), setting py = 0 amounts to testing for “deterministic cointegration”, meaning that the cointegrating vector eliminates the nonstationarity due to both the deterministic and stochastic trends. This is the case of most interest in practice. On the other hand, setting py = 1 amounts to testing for “stochastic cointegration”meaning that the cointegrating vector eliminates the non-stationarity due to the stochastic components only so that the cointegrating relationship is allowed to be a trendstationary process. This case is less commonly encountered in practice but has nevertheless been useful in a variety of applications. Note …nally that the cases with (px = 0; py = 1) and (px = 1; py = 1) are asymptotically the same since when a time trend is included as regressors the tests are invariant to the value of the trend function of the regressors. Since the case (px = 0; py = 1) corresponds to one with an irrelevant trend regressor included, we shall not discuss it further. Some comments about the DPG adopted are in order. First, note that it is the same DGP adopted by, among others, Elliott, Jansson and Pesavento (2005) to analyze optimal tests with a known cointegrating vector, Pesavento (2004, 2007) to analyze the local asymptotic power functions of various tests for cointegration, and Zivot (2000) to analyze the power of 5 single equation ECM tests for cointegration when the cointegrating vector is known. It is, however, restrictive since it implies that the regressors xt are “weakly exogenous”. To see this note that we can write (1) as the following Error Correction Model (ECM): 1 1 0 1 0 1 0 0 y ( 1) yt A 1 @ t 1 A + @ 1t A ; A=@ @ xt 1 0 xt 2t where = ( 1t = v2t + v1t and 2t = v1t . Hence, xt is weakly exogenous for the parameter 1), which is the parameter of interest underlying the construction of the ECM- based tests. It is well known that in the presence of weakly exogenous regressors, ECM-based tests are more powerful than residual-based tests but are very sensitive (even potentially inconsistent) to the presence of endogenous regressors. On the other hand, the residualbased tests are more robust to non-weakly exogenous regressors. Ideally, one would like to adopt a DGP that is the most general possible, not imposing weakly exogenous regressors. The problem is that in this general case, the local asymptotic power functions of the tests are not only di¢ cult to obtain but very complex functions of many nuisance parameters (see, e.g., Zivot, 2000, p. 427). On the other hand, with weakly exogenous regressors, the local asymptotic power functions will depend on a single nuisance parameter, namely R2 (and m, of course). Also, as will be shown in the simulations reported later, the power functions of the residual-based tests are not much in‡uenced by nuisance parameters other than R2 and m, even though they are not invariant to them. However, the power functions of the ECM-based tests are very sensitive to the degree of endogeneity of the regressors xt , with power that can often be zero. Accordingly, the strategy we adopt is to work with the DGP (1) even if it implies weakly exogenous regressors for the relevant parameter in the ECM. This will allow obtaining the local asymptotic power functions in this base case, from which we can select the relevant non-centrality parameter to quasi-di¤erence the data. These may not correspond exactly to the values that yield 50% local asymptotic power but given the relative robustness of the residual-based tests to departures from the weakly exogenous regressors case, the di¤erences will have little practical e¤ects on the power of the tests. For example, selecting the noncentrality parameter that corresponds to a local asymptotic power function of, say, 40% instead of 50% will not have much impact in the end. As we shall show, even with this less than optimal approach to select the non-centrality parameter, the GLS residual-based tests will have much improved power over their OLS-based versions in all DGPs considered including those with non-weakly exogenous regressors. The GLS versions of the ECM based 6 tests will have higher power in the case of weakly exogenous regressors (higher than their OLS-based counterparts and also higher than the GLS versions of the residual-based tests) but will be very de…cient when the DGP involves non-weakly exogenous regressors. 3 Motivation Consider a special case of the DGP given by: yt = y 0 y mt 0 + (3) xt + ut ; where ut = ut with = 1+c=T and, for simplicity, "t + "t ; 1 2 " ). i:i:d: (0; In this setup, yt and xt are cointegrated for any …nite samples but not cointegrated in the limit as T ! 1. Hence, this is a case of a nearly non-cointegrated process similar to the nearly integrated framework used by ERS. Let zt = (yt ; x0t )0 and zt = (1 myt L)zt ; L)myt ; = (1 Ignoring the …rst observation, we can apply a GLS transformation and write (3) as yt = and the GLS estimates of for y and y 0 y mt 0 + xt + "t ; (4) are obtained by applying OLS to equation (4) which, , is equivalent to applying OLS to the regression: ytd = 0 d xt + "t ; 0 0 0 where ytd and xdt are the GLS detrended data, i.e. (ytd ; xd0 t ) = (yt ; xt ) x0 0 ^ is is given by (my0 t ; mt ) and ^ = (My0 My ) 1 My0 Z : ^ 0 mt where mt = (5) where Z = [z1 ; :::; zT ]0 and My = [my1 ; :::; myT ]0 . This implies that each series is detrended separately by an OLS regression on quasidi¤erenced data with di¤erencing parameter . Accordingly, the method to construct the test statistic is one that provides, via the GLS transform, better estimates of the parameters 7 of the cointegrating regression, y and , and not better estimates of the coe¢ cients of the trend function of the variables of the system, e.g., x for the regressors xt . The role of the GLS or quasi-di¤erence device is not to obtain better estimates of the trend function as is the case when doing unit root tests. In the latter case, better estimates of the trend function indeed lead to more powerful unit root tests (see Canjels and Watson, 1997, and Phillips and Lee, 1996 for evidence on the latter assertion). This becomes even more transparent in the case for which the data are trending and only a constant is included in the set of deterministic components to detrend the data. Then, there is not even an attempt to estimate the deterministic components. Here, what will permit an increase in power is not a more precise estimate of the trend function but rather the fact that by applying a GLS transformation we get more precise estimates of the coe¢ cients in the static regression under the alternative hypothesis of cointegration. Xiao and Phillips (1999), in the context of tests within a full system with unit root processes, obtained simulation results showing that power was higher with the non-centrality parameter c set at -7.0 (px = py = 0) or -13.5 (px = py = 1) compared to using c = 0. Their explanation was that the estimates were numerically less reliable when c = 0 which caused a loss of power in …nite samples. We believe, instead, that the simulations show support for our interpretation that in the context of cointegration tests, the use of GLS detrended data does not arise from a concern to get better estimates of the coe¢ cients of the trend functions of the variables in the system but rather arises as a consequence of trying to obtain better estimates of the cointegrating regression when the errors are modelled as nearly-integrated. Following ERS, the next step would be to derive the Gaussian local power envelope of the likelihood ratio test for testing H0 : c = 0 versus the family of point alternatives H1 : c some …xed negative value. Unfortunately, this problem is intractable (at least to us) given the fact that the cointegrating vector is not identi…ed under the null hypothesis. It nevertheless seems sensible to argue that tests that perform best in the unit root case would also allow power gains in the cointegration context. This is the approach we follow. To motivate further, consider the case with a known cointegrating vector and R2 = 0, then the tests of ERS and those of Ng and Perron (2001) have power functions nearly identical to the local Gaussian power envelope. As shown by Elliott et al. (2005), this is no longer the case when R2 > 0 with a known cointegrating vector, though it make sense to argue that power gains are nevertheless possible even when R2 > 0. Extending the argument further, there is every reason to believe that power gains can also be obtained in the case of an unknown cointegrating vector even though the tests cannot be claimed to be optimal. Hence, the 8 approach taken is to consider nearly optimal tests for unit roots and apply them in the cointegration context. Consider the regression model: 0 d xt ytd = + udt (6) where udt = ut (^y 0 y y ) mt + y 0 ^0 x mt : (7) The tests are then based on the OLS residuals u^t , the residuals using GLS detrended data, de…ned by 0 ytd = ^ xdt + u^t (8) where ^ is the OLS estimate of ^ obtained from (6). As we shall show this will lead to considerable power gains over existing residual-based tests for cointegration (e.g., Phillips and Ouliaris, 1990). 4 The Tests The residual-based tests considered are constructed from u^t , the residuals obtained from the static cointegration regression (8) with GLS detrended variables, i.e., u^t = ytd ^ 0 xdt : (9) Consider …rst the M PTGLS proposed by Ng and Perron (2001). In our case, it is de…ned by GLS M PT; = GLS M PT; = c2 T PT u^2t s2 P T 2 c T 2 t=1 u^2t 2 t=1 1 cT 1 + (1 1 2 u^T ; c)T 1 2 u^T s2 ; for the demeaned ( ) and detrended ( ) cases, respectively. Another class of tests analyzed is the class of Z tests, proposed by Phillips (1987) and Phillips and Perron (1988) in the context of testing for a unit root. These statistics can be applied to test the null hypothesis of no-cointegration as showed by Phillips and Ouliaris (1990). In the present context, these are de…ned by ZbGLS = T (^ ZtGLS = b su t^ s (s2 s2u ) ; P (2T 2 Tt=1 u^2t 1 ) (s2 s2u ) ; P (4s2 T 2 Tt=1 u^2t 1 )1=2 1) 9 where ^ is the OLS estimate in the following regression: u^t = ^u^t 1 (10) +! ^ t; = 1, s2u = T and t^ is the corresponding t-statistic for testing 1 below. PT t=1 ! ^ 2t and s2 is described The class of M -tests, originally proposed by Stock (1999), and further analyzed by Perron and Ng (1996) and Ng and Perron (2001), exploit the feature that a series converges with di¤erent rates of normalization under the null and the alternative hypotheses. These were shown to have far less size distortions than the Z tests in the presence of important negative serial correlation in the …rst-di¤erences of the data. Constructed using the residuals from the cointegrating regression, they are de…ned by: M Z^GLS = M SB GLS = T 2T " T M ZtGLS = h ^ 1 2 u^T s2 ; PT 2 ^2t t=1 u 2 PT ^2t t=1 u s2 T 4s2 T s2 1 2 u^T 2 #1=2 PT ^2t t=1 u ; i1=2 : The …rst statistic is a modi…ed version of the Z^ test, the second is a modi…ed version of Bhargava’s (1986) R1 statistic which builds upon the work of Sargan and Bhargava (1983), and the third is a modi…ed version of the Zt^ test. In all tests de…ned above, the term s2 is an autoregressive estimate of (2 times) the spectral density at frequency zero of udt ; de…ned by: s2 k s2 = h (11) i2 ; ^ 1 b(1) P P where s2k = T 1 Tt=k+1 ^2tk ; ^b(1) = kj=1 ^bj , with ^bj and f^tk g obtained from the autoregression3 : u^t = ^t 1 0u + k X bj u^t j + tk : (12) j=1 Another test of interest is the so-called ADF test (Dickey and Fuller, 1979, Said and Dickey, 1984) which is the t-statistic for testing 0 = 0 in regression (12). We denote this test by ADF GLS . 3 The advantages of using this autoregressive-based spectral density estimator over the more traditional kernel-based methods are discussed in Perron and Ng (1998). 10 1=3 Note that we require that k be selected in such a way that T k ! 0 and k ! 1 as T ! 1 (see, e.g., Ng and Perron, 1995). This requirement will be established formally in Assumption 2 later. As mentioned above another framework in which to test for cointegration is the singleequation conditional error correction model initially proposed by Phillips (1954, 1957), and further developed by Sargan (1964). The …rst approach in this type of tests has been proposed by Banerjee et al. (1986), Kremers et al. (1992), Banerjee et al. (1993), Banerjee et al. (1998) and Zivot (2000) who considered the restrictive case of a known cointegration vector. Banerjee et al. (1986), Banerjee et al. (1993), and Banerjee et al. (1998) suggested adding redundant regressors to have tests valid with an estimated cointegrating vector. The tests are then labelled ECR instead of ECM tests. The equation to be estimated, using OLS detrended data is 4 : yt = dt + 0 1 xt + ^t 0u 1+ 0 1 xt 1 + k X ( 0 ix xt i + iy (13) yt i ) + t : i=1 If using GLS-detrended data allows residual-based tests to have higher power, it is also likely to yield improvements for this class of tests as well. Hence, we shall consider tests based on the following regression ytd = 0 1 xdt + ^t 0u 1+ 0 d 1 xt 1 + k X ( 0 ix xdt i + iy ytd i ) + t : (14) i=1 The corresponding statistic is then the t-ratio for testing H0 : 0 = 0, denoted by tGLS ECR . GLS . It is Also, the corresponding F-statistic for testing H0 : ( 0 ; 01 )0 = 0 is denoted by FECR equivalent to the Wald test suggested by Boswijk (1994) but using GLS detrended data. As in the case of the ADF test, we also require that k be selected in such a way that T and k ! 1 as T ! 1. 5 1=3 k!0 Asymptotic Distributions In order to derive the limit distributions of the tests under the null hypothesis of no cointegration and their local asymptotic power functions, the DGP is assumed to be given by (1) with the local to unity speci…cation: = 1 + c=T: 4 See Pesavento (2004) for an analysis of the asymptotic distributions and local asymptotic power functions of the ECR tests in this context using OLS detrended data. 11 We consider in turn the limit distribution of the estimate of the trend function under GLS detrending, the limit distribution of the estimate of the cointegrating vector and …nally the limit distributions of the various tests described in the previous Section 5 . 5.1 Preliminaries We start with a weak convergence result about the limit of u[T r] . Lemma 1 T 1=2 u[T r] ) ! 2:1 J12c (r), where ! 2:1 = ! 22 ! 21 111 ! 12 and J12c (r) = W12 (r) + Rs c 0 e(s r)c W12 (r)dr, where W12 (r) is a scalar Wiener process with variance ! 2:1 that has the equivalent representation W12 = R2 =(1 with W 1 (r) = m 1=2 Pm i=1 R2 ) 1=2 W 1 (r) + W2 (r); Wi (r). The following Lemma gives the asymptotic properties of the estimates of the coe¢ cients of the trend function using the GLS approach. Px 0 i x Lemma 2 Suppose that zt = [x0t ; yt ]0 is generated by (1) with dxt = pi=0 ix t = x mt , Ppy y 0 i dyt = i=0 iy t = y mt , and that each variable in the n 1 vector zt is detrended separately. Let ^ be the GLS estimates of the coe¢ cients of the trend function obtained using = 1+c=T . Then, 0 1. If py = 0, with px = 0 or px = 1: T vec( ^ x T = diag(T x ) ) 0m , where an m m matrix and 0m denotes an m 1 vector of zeros; also T 1=2 ( ^ y 2. If py = 1, then: T vec[ where T ^x = [diag(T x] 1=2 2 )4 ; :::; T 0m 1=2 B1 (1) + 3(1 ) R1 0 rB1 (r) ); diag(T 1=2 ; :::; T 1=2 )]; a 2m c + c2 =3). Also, 2 0 ^ 4 T vec[ y y] ) f J12c (1) + 3(1 3 5 c)=(1 where T = dia[T 1=2 ) ; T 1=2 ]; a diagonal 2 5 R1 0 rJ12c (r)g + 2 matrix. 0 Hx 2 4 0m Hx 5 2 4 ; :::; T y ) ) 0. 3 5 2m matrix, 3 1=2 = (1 0 Hy + 0 Hx 3 5 Many of our proofs are similar to those of Pesavento (2004) who use OLS detrended data. However, her proofs are only valid in the case with a single right-hand-side variable. The case with multiple regressors adds complications, which we address. 12 1=2 ), The following Lemma progvides the limit distribution of the residuals from the OLS regression involving GLS-detrended variables, i.e. udt de…ned by (7). 1=2 d u[T r] Lemma 3 T d d ) ! 2:1 J12c (r), where; a) if py = 0: J12c (r) = J12c (r) as de…ned in d = J12c Lemma 1; b) if py = 1: and J12c rHy . The following theorem gives the limit of the estimate ^ obtained from applying OLS to (6) with GLS-detrended variables. Theorem 1 Suppose that zt = [x0t ; yt ]0 is generated by (1) with = 1+c=T and Assumption 1 holding. Let ytd and xdt be GLS detrended variables with non-centrality parameter = 1+c=T . Let ^ be the OLS estimates of the cointegrating vector de…ned by (8). Then: (^ )) 1=2 ! 2:1 1=2 11 Z 1 1 W1d W1d0 0 Z 1 1=2 d W1d J12c 1=2 11 ! 2:1 d c; 0 where W = (W01 ; W2 )0 and d = J12c ; 1. If px = 0, py = 0: W1d = W1 and J12c R1 d = J12c ) 0 sW1 (s)ds]r, J12c 2. If px = 1, py = 1: W1d = W1 [ W1 (1) + 3(1 R1 2 [ J12c (1) + 3(1 ) 0 sJ12c (s)ds]r and = (1 c)=(1 c + c =3); 3. Also, if px = 1, py = 0: T 1=2 DT C ( ^ 1 )) 1=2 ! 2:1 Z 1 d d0 W1R W1R Z d 0 0 1=2 where W1R = [W1(m Im 1 ; T ) and 1) ; r] , DT = diag(T h i h 0 1=2 C = C1 C2 = 1x ( 1x 11 1x ) where 5.2 1x is the m (m 1=2 ! 2:1 e dc d W1R J12c 1x ( 0 1x 1x ) 1) matrix which spans the null space of 1 i ; 1x . The Asymptotic Distributions of the Tests We now consider the limit distributions of the tests. We start with results pertaining to the single-equation static regression or residual-based tests stated in the next Theorem. Before we state a required Assumption. Assumption 2. For regressions (12) and (14), we require that T T ! 1. 13 1=3 k ! 0 and k ! 1 as Theorem 2 Let the data be generated by (1) with = 1 + c=T , Assumptions 1-2 holding and the residuals u^t obtained from (9) with a non-centrality parameter = 1 + c=T . Then, as T ! 1: i h d d d d0 d (1) A A c2 d0 c c c c c c c GLS ) M PT; ; d d0 D c c h i d d d d0 d c2 d0 A + (1 c) A (1) c c c c c c GLS ) M PT; ; d d0 D c c ZtGLS ; ADF GLS ) c b ZbGLS ) c+ M SB GLS ) M ZbGLS ) M ZtGLS ) b d0 D c h d0 c R + [ f Wcd dW h d0 c R f d Wcd dW c i1=2 d0 D d 1=2 c c d0 Ad d ]1=2 c c c i1=2 d c ; ; d0 Ad d ]1=2 c c c d0 d d 1=2 c Ac c ; d0 D d 1=2 c c d d d0 d0 d c D c c Ac (1) c ; d d 2 d0 c Ac c d d d0 d0 d c D c c Ac (1) c ; d0 D d 1=2 d d ]1=2 2 [ d0 A c c c c c [ 1=2 1=2 11 ! 12 ! 22 , d d0 c ; 1 , Wc = 0 1=2 1=2 R2 = 0 , = ! 2:1 ! 21 11 , d0 c = R 1 d f0 = , Adc = 0 Wcd Wcd0 , W W1d0 ; J12c P R2 )]1=2 W 1 + W2 , where W 1 (r) = m 1=2 m i=1 Wi (r), 2 3 I 5; D=4 0 0 1+ where ! 2:1 = ! 22 ! 21 111 ! 12 , = R R d )( W1d W1d0 ) 1 ; 1] = [ ( W1d J12c [W10 ; W12 ]0 , W12 = [R2 =(1 d 1=2 c d 1=2 c d0 d c Ac 2 Adc (1) = 4 W1d (1)Wd1 (1)0 d W1d (1)J12c (1) d J12c (1)W1d (1)0 d d J12c (1)J12c (1) 3 5; J12c (r) is an Ornstein-Uhlenbeck process such that J12c (r) = W12 (r) + c and: d 1. If px = 0, py = 0: W1d = W1 and J12c = J12c ; 14 Rs 0 e(s r)c W12 (r)dr R1 d = J12c ) 0 sW1 (s)ds]r, J12c 2. If px = 1, py = 1: W1d = W1 [ W1 (1) + 3(1 R1 2 [ J12c (1) + 3(1 ) 0 sJ12c (s)ds]r and = (1 c)=(1 c + c =3); d 0 3. If px = 1, py = 0: W1d = W1R = [W1(m 1) ; r] 0 d and J12c = J12c . The next Theorem presents the asymptotic distributions of the ECR statistics based on GLS detrended variables. Theorem 3 Let the data be generated by (1) with = 1 + c=T , Assumptions 1-2 holding and regression (14) estimated with GLS-detrended variables with non-centrality parameter = 1 + c=T . Then, as T ! 1: tGLS ECR GLS FECR ) c Z d 2 (J12c ) R Z Z d W1d J12c ( W1d W1d0 ) 1 Z 1=2 d W1d J12c R R R d d J12c dW2 W1d J12c ( W1d W1d0 ) 1 W1d dW2 ; + R R R R 1=2 d d 2 d (J12c ) W1d J12c ( W1d W1d0 ) 1 W1d J12c Z Z Z Z 2 d 2 d d0 d 1 ) c (J12c ) + 2c J12c dW2 + Wc dW2 (Ac ) Wcd dW2 ; 1=2 1=2 0 1=2 1=2 ! 21 111 ! 12 , = 11 ! 12 ! 22 , R2 = 0 , = ! 2:1 ! 21 11 , Wcd = R1 d W1d0 ; J12c , Adc = 0 Wcd Wcd0 , J12c (r) is an Ornstein-Uhlenbeck process such that J12c (r) = Rs W12 (r) + c 0 e(s r)c W12 (r)dr and: where ! 2:1 = ! 22 d 1. If px = 0, py = 0: W1d = W1 and J12c = J12c ; R1 d = J12c ) 0 sW1 (s)ds]r, J12c 2. If px = 1, py = 1: W1d = W1 [ W1 (1) + 3(1 R1 [ J12c (1) + 3(1 ) 0 sJ12c (s)ds]r and = (1 c)=(1 c + c2 =3); d 0 3. If px = 1, py = 0: W1d = W1R = [W1(m 6 1) ; r] 0 d and J12c = J12c . Implementation and Power Results The theoretical results obtained in the previous section are used to address the following issues for the implementation and assessment of the properties of the tests: 1) the choice of the relevant non-centrality parameter c; 2) given a choice for c, the derivation of the critical values needed to implement the tests; 3) an analysis of the local asymptotic power functions of the tests to assess their relative merits. Throughout, the results are obtained from 10,000 replications using 1,000 steps to approximate the Wiener processes. 15 6.1 The Choice of c and the Asymptotic Critical Values Of importance for the implementation of the tests is the choice of c. We performed very extensive simulations about the local asymptotic power functions of the tests for all cases with a wide range of values for R2 and up to 5 right-hand-side regressors (the complete set of results is available upon request). In each case, we computed the non-centrality parameter c that corresponds to a local asymptotic power of 50%. In order to provide the non-centrality parameters used to construct all tests, we calibrated the results using the M PTGLS test for the three cases analyzed for the deterministic components. This is motivated in part by the fact that the M PTGLS test has a local asymptotic power function closest to the Gaussian local power envelope in the case of unit root tests. In any event, the di¤rences in power are very minor across the residual-based tests and, hence, similar results would apply had we settled on any other test in this class. We did not calibrate the results to one of the ECR tests since, as we shall see, these have very bad properties in the case with non-weakly exogenous regressors; by selecting one of the residual-based tests, our results are therefore more robust to a wider range of DGPs. What transpired from these results are the following features. First, the local asymptotic power functions of the residual-based tests are very similar implying an “optimal c” that is nearly the same; of course, assuming the same number of I(1) regressors m). Second, variations in R2 do a¤ect the local power function and the corresponding “optimal c” but not as much as variations in m. In order not to overburden practitioners with endless tables of critical values, we opted to select a single representative value for R2 and for each m compute the associated “optimal c”. Since extreme values of R2 near 0 or 1 are unlikely in practice, it made sense to select a value in the mid-range. We settled upon using R2 = 0:4 which appears to be a sensible value often considered in the literature; see Pesavento (2007) for a similar argument. The “optimal c”does not vary much for small deviations, e.g., values of R2 between .25 and .55. So it indeed appears to be a sensible choice. Finally, since the power functions vary considerably with m, we report a separate “optimal c”for m = 1; :::; 5, presented in Table 1. As can be seen in the Table 1, the “optimal c”are very di¤erent from the unit root case. They also are larger in absolute value (more negative) with more deterministic components and with more regressors. Using these values of c, we simulated the limit distributions of the tests from Theorems 2 and 3, under the null hypothesis that c = 0. Notice that under the null hypothesis none of the statistics depends on R2 or any other nuisance parameters. Tables 2, 3 and 4 present the 16 asymptotic critical values for the three sets of possibilities for the deterministic components (px = 0, py = 0; px = 1, py = 1; and px = 1, py = 0). We present the critical values for tests with nominal sizes ranging from 1% to 20%. 6.2 Asymptotic Power Functions This section presents the local asymptotic power functions of the statistics proposed and compare them to those of tests constructed with OLS detrended data. We present results only for the case py = px = 0, that is, the demeaned case. The results for the other cases, px = 1, py = 0 and px = 1, py = 1, are available upon request. They are qualitatively similar with power being lower overall when more deterministic components are included. We also consider results for R2 = 0; :2; :4 and :8 and up to …ve right hand-side variables. The results for the OLS version of the tests were obtained using the limit distributions derived in Pesavento (2004). Figures 1 shows the asymptotic power functions of some tests based on OLS and GLS detrending: M PTOLS and M PTGLS in Figure 1.a, ADF OLS and ADF GLS in Figure 1.b, tOLS ECR and tGLS ECR in Figure 1.c. It is clear that the power of a test that uses GLS-detrended data is higher than its OLS-based counterpart for all cases, although two features are important. First, the advantages of GLS-based tests are clear when jcj 10. Second, for small R2 (0:0, 0:2 in Figure 1) the advantages of GLS-based over OLS-based tests are small when there are 5 regressors. Also, as expected, power decreases when R2 increases and when more righthand side variables are included. The increase in power when using GLS instead of OLS detrended data can be substantial. It also applies to all tests considered. We do not present detailed results to that e¤ect as they are similar. Instead, in what follows, we concentrate on comparing the various tests with GLS detrended data. Figure 2 shows the asymptotic power functions of three of the residual-based tests, the M PTGLS , M SB GLS and ADF GLS . What transpires is that the local asymptotic power functions are essentially the same unless the number of regressors is very large (see Figure 2c), in which case the ADF GLS is slightly more powerful. The similarity in the power functions of the residual-based tests extends to all tests considered in this paper. GLS Figures 3a-3c present a comparison of the ECR-based tests (tGLS ECR and FECR ) and the residual-based tests (M PTGLS , M SB GLS and ADF GLS ). The results show the ECR-based tests to dominate the residual-based tests in all cases, and more so as R2 increases and/or more regressors are included. Of the two ECR-based tests, tGLS ECR shows more power unless GLS GLS the R2 is very high (R2 = 0:8), in which case the FECR dominates. Also FECR dominates 17 when there are more right-hand regressors included in the cointegration equation. The …ve tests have similar power when R2 = 0 or R2 = 0:2 and m = 1. With larger values of R2 and number of regressors, the di¤erences become more pronounced. 7 Size and Power of the Tests in Finite Samples In this section, we evaluate the size and power of the tests in …nite samples. The datagenerating process adopted is presented in Section 7.1. It generalizes the DGP used in the theoretical analysis of the limit distributions of the tests by allowing non-weakly exogenous regressors. The di¤erences between the DGPs and their implications are discussed further in Section 7.2. The results about the size and power of the tests are discussed in Section 7.3. 7.1 The Monte Carlo Design The data-generating process considered is a bivariate system (m = 1) given by: yt a1 y t a2 xt = u2t ; u2t = u2t 0 @ (15) xt = u1t ; v1t v2t u1t = 1 1 u1t + v1t ; 1 + v2t ; 20 A = i:i:d: N 4@ 0 0 1 0 A;@ ! 11 ! 12 ! 21 ! 22 13 A5 : This data-generating process is similar to that used by Banerjee et al. (1986), Engle and Granger (1987), Hansen and Phillips (1990), Gonzalo (1994), Haug (1993, 1996) and Ostermark and Hoglund (1999). Here, yt and xt are scalars. We use the following values: ! 11 = 2, ! 22 = 1, R2 = 0:0; 0:4; 0:8, = (R2 )1=2 , which implies that R2 = 2 for any values of ! ij (i; j = 1; 2). We consider c = 0; 5; 10; 15; 20. We present results for T = 200, in which case = 1; 0:975; 0:95; 0:925; 0:90, and for T = 500, in which case = 1; 0:99; 0:98; 0:97; 0:96. Without loss of generality, we set = 1 and we consider four values of a1 . The …rst is a1 = 0, in which case the regressor xt is weakly exogenous for the relevant parameter in the ECM. We also consider a1 = 1; 2; 3, in which case the regressor is not weakly exogenous. Note that when a1 = 0 and R2 = 0, the regressor xt is strongly exogenous. Finally, we set a2 = 1 in all experiments. We present results only for the demeaned case (px = py = 0) since the results are qualitatively similar for the other cases (available upon request). In all cases, the results are obtained using 5,000 replications. 18 7.2 Some Remarks about the DGP In order to better understand the results to be presented, it is useful to explain some features of the DGP. Following Gonzalo (1994), after some algebra, it can be shown that the DGP (15) can be written in the form of an ECM: 1 1 0 0 yt A= @ 1 A 1 @ xt 2 where 1 = ( 1)(a2 =d), 2 = ( 0 @ yt 1 xt 1 1)(a1 =d), d = (a1 1 0 A+@ a2 ), 1t 2t 1t 1 A; (16) = ( a2 v2t + v1t )=d and 2t = ( a1 v2t + v1t )=d. Note that the stability condition is 2 < 1 < 0 (e.g., Zivot, 2000), which is satis…ed for our parameter con…gurations. In this ECM representation, the parameters of importance are 1 and occurs when a1 = 0. When a1 6= 0, 2. When 2 = 0, xt is weakly exogenous for 1. This 6= 0 and weak exogeneity does not hold. In particular, 2 the value of 1 depends on a1 in such a way that an increase in a1 causes a reduction in the absolute value of 1 , which reduces the power of the ECM-based tests. Remark 1 The DGP can be represented in a form that is closer to equation (16) on which the ECR-tests are based. Following Gonzalo (1994), we have by diagonalizing the covariance matrix of 0 @ t: yt xt where e1t = 1 0 A=@ 1t E( yt j 1 0 1 2t , 1 0 A xt + @ e1t = 2t , 1 2 and 1 2 1 1 A 1 0 @ yt 1 xt 1 1)ut 1 is de…ned by the relation xt ; ut 1 ; lags( yt ; xt )) = 1 xt + ( 1 0 A+@ e1t e2t 1 A; (17) + lags( yt ; xt ): The same qualitative features discussed above about the in‡uence of the endogeneity parameter a1 remain valid with this representation. 7.3 Results about Size and Power We present results for the tests M PT , Z^ , ADF , tECR , FECR using both OLS and GLS detrended data. They are presented in Table 5 for T = 200 and Table 6 for T = 500. The lag length was selected using the BIC with kmin = 0 and kmax = (4 (T =100))1=4 , which for T = 200 and T = 500 imply kmax = 5 and 6, respectively. The critical values for the OLS residual-based tests are from Phillips and Ouliaris (1990) and those for the OLS ECR 19 tests are from Ericsson and Mackinnon (2002) and Pesavento (2004). The nominal size of the tests is set at 5%. Note …rst that the exact size of the tests is close to the nominal 5% level. There are some slight liberal distortions for some of the GLS-based tests when T = 200 but they disappear when T = 500. The test M PT is slightly conservative when T = 200 but again the distortions disappear when T = 500. These features hold for all values of a1 . Consider now, more importantly, the power of the tests. The …rst thing to note is that for all tests and all parameter con…gurations, the GLS versions are more powerful than their OLS counterparts. The di¤erence in power can indeed be quite substantial. For example, for the ADF test with T = 200, a1 = 0 and R2 = 0, the power of the OLS version is .204 and that of the GLS version is .791 when c = 15; when c = 20 the corresponding …gures are :428 and :937. Hence, there are clear gains in using our GLS-based tests. Consider, more speci…cally, the case a1 = 0 with weakly exogenous regressors. When 2 R = 0, the power functions of the residual-based tests are comparable to those of the ECR tests. When R2 = :4, the di¤erences are minor with the ECR tests being slightly more powerful. When R2 = :8, the ECR tests are clearly more powerful than the residualbased tests. Hence, if the practitioner is con…dent that the regressors are weakly exogenous and strongly correlated with the regression errors, then the ECR tests are recommended. Otherwise, the di¤erences are minor. We now turn to cases with a1 6= 0. The …rst thing to note is that the power of the ECR tests is very much a¤ected. When a1 = 1, the residual-based tests are more powerful unless R2 = 0:8. More importantly when a1 = 2; 3, the power of tECR is very low, essentially zero, especially when R2 is high. The power of the test FECR is also very small except when a1 = 3 and R2 = 0:8. Hence, the ECR tests are unreliable when the regressors are not weakly exogenous. The power of the residual-based tests is much more robust to the presence of non-weakly exogenous regressors, even though the power is not invariant to the value of a1 . For instance, consider the ADF test with c = 15, R2 = 0:4 and T = 200. The power values for a1 = 0; 1; 2; 3 are .692, .765, .578 and .364, respectively. Hence, the power function is not monotonically increasing or decreasing as a1 increases, though for large values of a1 power is indeed lower. The results are broadly similar for T = 200 and T = 500 as one would expect given that the alternative values are speci…ed in a local fashion. The only notable di¤erence is that the superiority of the GLS-based tests is in general more pronounced with T = 200 than with T = 500, though it remains substantial. 20 8 Conclusions We analyzed two approaches to test whether or not non-stationary economic time series are cointegrated: single-equation static regression or residual-based tests and tests based on single-equation conditional error correction models. Using the GLS versions of the tests, we derived the limiting distributions of the test statistics for three di¤erent cases pertaining to the nature of the deterministic components. We derived the local asymptotic power function of the tests using a DGP with weakly exogenous regressors. Using these power functions, we calibrated useful values of the non-centrality parameters c to quasi-di¤erence the data. The asymptotic distributions depend of the number (m) of right-hand side variables, the type of deterministic components used in the cointegration equation and present in the data, and a nuisance parameter R2 which measures the long-run correlation between xt and yt . The theoretical results showed that important power gains can indeed be achieved by using GLS detrended data. With such weakly exogenous regressors, the local asymptotic power functions of the ECR tests are indeed higher than those of the residual-based tests, especially if the value of R2 is high. This was con…rmed by simulations for common samples sizes. Our simulations, however, highlighted the high sensitivity of the ECR tests on the assumption of weakly exogenous regressors. When this condition is not satis…ed, the power of the ECR tests can drop dramatically and even be near zero. This is not the case for the residual-based tests whose power functions are more robust to departures from weakly exogenous regressors. Our theoretical analysis does not o¤er a so-called local Gaussian power envelope. So we cannot claim that any of the test achieves the best power possible. Given the approach taken, we also cannot claim that the non-centrality parameters c suggested correspond exactly to the values that would yield a 50% local asymptotic power for the tests. While we agree that it would be of theoretical interest to be able to remedy these caveats, we believe that, for all practical purposes, it would make little di¤erences. Our GLS versions of the tests do indeed provide substantial improvements over their OLS counterparts and further gains in power for the classes of tests considered are likely to be minor. The end-products do provide improvements over currently available methods. With respect to which tests to suggest we o¤er the following recommendations. First, within the classes of residual-based or ECR tests, it makes little di¤erence which test to adopt. They have similar size and power. If one is con…dent that the regressors are weakly exogenous and the R2 is high, then the ECR tests are clearly superior. With weakly exogenous regressors and a small to medium value of the R2 both classes of tests perform similarly. 21 When the regressors are potentially non-weakly exogenous the residual-based tests are clearly preferred. 22