Estimation and Hypothesis Testing of a Cointegrating Vector under a Possible Variance Break Nikolaos Kourogenis,
y
Ekaterini Panopoulouz
Nikitas Pittisx
Abstract This paper investigates the performance of standard estimators for estimating a cointegrating vector in the context of a triangular system that exhibits a break in the long-run covariance matrix of the errors. It is shown that the limiting distributions of OLS and the associated test statistics are signi…cantly a¤ected by the presence of such shifts. Based on this result, we obtain an estimator that corrects the OLS, not only for long-run correlation and endogeneity but also for possible breaks in the long-run covariance matrix of the errors, occured at a single, not a priori known, point in the sample. JEL Classi…cation: C22 Keywords: cointegration, variance breaks, limiting distributions, t-statistics.
Department of Banking and Financial Management, University of Piraeus. Correspondence to: Nikolaos Kourogenis, Department of Banking and Financial Management, University of Piraeus, 80 Karaoli and Dimitriou str., 18534 Piraeus, Greece. Email1:
[email protected] Email2:
[email protected] . Tel: 00302104142142. Fax: 00302104142341. z Department of Statistics and Insurance Science, University of Piraeus & IIIS, Trinity College Dublin. x Department of Banking and Financial Management, University of Piraeus. y
1 Introduction Some recent empirical studies have shown that the volatility of the innovations that drive many macroeconomic and …nancial series has experienced a sharp decline since the mid 1980s (see, for example, Kim and Nelson 1999, McConnell and Perez-Quiros 2000, Stock and Watson 2002, Sensier and van Dijk 2004). This phenomenon is particularly profound for the case of the U.S. economy and it is usually referred to as the "Great Moderation". This empirical evidence has motivated extensive theoretical research that investigates whether variance breaks a¤ect standard methods for analyzing univariate time series. More speci…cally, a great deal of research e¤orts have been devoted to study the e¤ects of such breaks on the usual unit root and stationarity tests (see, for example, Hamori and Tokihisa 1997, Kim, Leybourne and Newbold 2002, Burridge and Taylor 2001, Busetti and Taylor 2003, Cavaliere 2004a, 2004b, Cavaliere and Taylor 2005, 2008). The results of these studies have demonstrated that the presence of a permanent shift in the error variance, not accounted for by the testing procedure, seriously a¤ects inferences on the time-heterogeneity properties of the relevant series. More recently, Cavaliere and Taylor (2006) analyze the issue of shifts in the error variance in a multivariate framework. In particular, they assume a multivariate cointegrated time series model and analyze the e¤ects of several types of variance breaks that occur either in the common stochastic trends or in the cointegrating relations on the tests for the null hypothesis of cointegration. They show that such breaks a¤ect the limiting distributions of the relevant test statistics under both the null hypothesis of cointegration and the alternative of no cointegration, thus a¤ecting both the size and the power properties of these tests. Similar e¤ects are reported in Cavaliere, Rahbek and Taylor (2007) for the conventional trace and maximum eigenvalue statistics of Johansen (1988). Despite the problems mentioned above, an applied researcher is still likely to make the correct decision concerning the cointegration properties of his model. For example, if the null hypothesis of cointegration is true, the applied researcher faces a signi…cant probability of not rejecting the null, even by means of a relatively over-sized test. In such a case, his next step is to employ a standard cointegration estimator in order to estimate the cointegrating vector(s). The question that naturally arises at this point is the following: "what are the asymptotic properties of such an estimator and those of the associated test statistics in the presence of a variance break?" In other words, even if the applied researcher is fortunate enough to make the correct inferences
1
about the cointegrability of his model, the presence of a variance break is likely to distort his inferences on the cointegrating vector itself. To this end, Kourogenis and Pittis (2008) show that the asymptotic distribution of the OLS estimator of a cointegration parameter depends on the time, the size and the direction of the variance break. Hansen (2003) o¤ers an extensive analysis of the problem of estimation and hypothesis testing of cointegrating vectors under various types of structural breaks, within the context of Vector Error Correction (VEC) models. By extending the estimation method of Boswijk (1995), he develops the so called Generalized Reduced Rank Regression (GRRR) technique which enables him to deal with structural breaks in the adjustment coe¢ cients, the cointegrating vectors and the covariance matrix of the errors. Despite the generality of these results, however, their usefulness to applications is limited by the fact that the break dates are assumed to be known a priori. All the aforementioned studies deal with the e¤ets of variance breaks on the Johansen procedures (rank tests of cointegration and inference on the cointegrating vectors), which are based on the VEC model of cointegration. To the best of our knowledge, there are no studies analyzing these e¤ects for the case of other commonly used cointegration estimators, such as the single-equation ones. This study attempts to …ll this gap in the literature. In particular it analyzes the performance of various single-equation estimation methods for cointegration in the case of abrupt changes in the long-run covariance matrix of the errors of a cointegrated model. Concerning the latter, we adopt the triangular system, proposed by Phillips (1988a, 1991), and assume that the long-run covariance of the errors is subject to abrupt upward and/or downward shifts at a single point in the sample. These shifts may occur due to a change in the persistence of the errors, or in the covariance matrix of the innovations that drive the errors or both. Under this set of assumptions, we …rst derive the limiting distribution of the OLS estimator. By extending the results of Kourogenis and Pittis (2008), we show that the presence of a variance break seriously a¤ects the limiting distribution of OLS as well as those of the related test statistics. The usual corrections on OLS, which in the absence of any breaks result in the usual cointegration estimators, such as the Fully Modi…ed Least Squares (FMLS) estimator of Phillips and Hansen (1990), or the Dynamic OLS estimator, (DOLS) (see Saikonnen 1991, Phillips and Loretan 1991 and Stock and Watson 1993) are not su¢ cient to deliver standard asymptotics in the presence of a variance break. Indeed, it is shown that the FMLS and DOLS t-statistics su¤er from severe size distortions if the elements of the covariance matrix of the errors experience a single upward shift at some point in the sample. For these cases, we propose a new OLS2
based estimator, in the spirit of FMLS, which corrects the ordinary OLS estimator, not only for long-run correlation and endogeneity but also for possible variance shifts. Moreover, this robust version of FMLS is fully operational in the sense that the break date is not assumed to be known a priori but instead is estimated from the available data. Simulation evidence suggests that the new estimator performs very well even when the sample is small and the variance breaks are quite large. It is also shown that the performance of the new estimator is comparable to that of FMLS in the case of no variance break. The remainder of the paper is organized as follows. Section 2 introduces the model and derives the limiting distribution of OLS in the case of variance breaks. It is shown that the presence of a variance break seriously a¤ects the limiting distribution of OLS as well as those of the related test statistics. Based on these results, it suggests several modi…cations on OLS that account for the e¤ects of the variance break, thus producing test statistics with standard asymptotic distributions. Section 3 proposes a consistent estimator of the timing of the break, and derives the feasible versions of the aforementioned test statistics for testing hypotheses on the elements of the cointegration vector. Section 4 presents the results from a Monte Carlo study that compares the size performance of the t-statistics, which are based on standard cointegration estimators, with that of the new t-statistic, under various combinations of shifts in the long-run covariance matrix of the errors. Section 5 compares the results from testing the so-called "Fisher hypothesis" by means of standard cointegration methods to those produced by the newly proposed robust procedure. Section 5 concludes the paper. Proofs can be found in the Appendix.
2 The Model and Theoretical Results Let zt and ut be two (p + 1)-dimensional processes with z0t = [yt ; x0t ] and u0t = [u1t ; u02t ]. Obviously, yt and u1t are scalars. We further assume that the generating mechanism for yt is given by the system: yt =
0
xt + u1t
xt = xt where
1 + u2t
is a p-vector. We also de…ne
S[rT ] =
[rT ] X t=1
3
ut
(1)
and 1=2
XT (r) = T
S[rT ] :
Concerning the error process, ut , we assume that it experiences a unique structural break at time [sT ]; 0
s < 1; as described by the next assumption.
Assumption A: L
XT (r) ! B(r)
XT (r)
BM ( ); 0
L
XT (s) ! B (r
s)
r
