Cointegration versus Spurious Regression and Heterogeneity in Large Panels

This paper provides an estimation and testing framework to identify the source(s) of spuriousness in a large nonstationary panel. This can be determined by two non mutually exclusive causes: pooling units neglecting the presence of heterogeneity and genuine presence of I (1) errors in some of the units. The paper proposes two tests that complement a test for the null of cointegration: one test for the null of homogeneity (and thus presence of spuriousness due to some of the units being genuinely spurious regressions) and one for the null of genuine cointegration in all units of the panel (and thus spuriousness arising only from neglected heterogeneity). The results are derived using a linear combination of two estimators (one consistent, one inconsistent) for the variance of the estimated pooled parameter. The paper also derives two estimators for the degree of heterogeneity and for the fraction of spurious regressions; consistency is achieved as long as (n; T ) ! 1, with no need for special restriction on the rate of expansion between n and T as they pass to in…nity. JEL Codes: C23. Keywords: Large Panels, Heterogeneity, Spurious Regression. Cass Business School, Faculty of Finance, 106 Bunhill Row, London EC1Y 8TZ, Tel.: +44 (0) 207 040 5260; email: [email protected] 1 1 Introduction Consider the heterogeneous panel regression model yit = i + ixit + uit; (1) where i = 1; :::; n, t = 1; :::; T and the variables yit and xit are both I (1) for each i. As far as estimation is concerned, sometimes the pooled version of (1) is employed, i.e. yit = i + xit + vit; (2) either because the assumption of homogeneity ( i = for all i) is not rejected by the data or because the object of interest are not the unit-speci…c slopes but the long-run average parameter see also the comments in Temple (1999). However, pooling introduces a further component in the error term, ( i )xit, which is I (1) and thus makes the panel spurious, unless i = ; this was noted by Phillips and Moon (1999), who proved that under heterogeneity, (2) is equivalent to a spurious regression and the estimate of is p n-consistent as opposed to p nT -consistent which would be the case in a cointegrated panel. On the other hand, in (1), for each i the error term uit can be either I (0), and thus the unit is a cointegration relationship, or I (1), and therefore unit i is genuinely a spurious regression, irrespective of heterogeneity. This situation could e.g. correspond to the case (often found in empirical applications) where uit is truly stationary in accord with some theory, but it is observationally equivalent to an I (1) process due to mis-speci…cation. A comprehensive review of the literature on the possible causes of this is in a recent contribution by Fuertes (2008). Thus, letting 2 [0; 1] be the proportion of units that are spurious regressions, model (1) could be a cointegrated panel ( = 0), a panel where all units are described by spurious regressions ( = 1), or any situation in between. In light of these considerations, a pooled panel model can thus be spurious due to two (not mutually exclusive) reasons: neglected heterogeneity or genuine spuriousness of some units. Therefore, a test for the null of panel cointegration applied to equation (2) is in fact a test for both homogeneity/poolability and = 0. 2 The question then arises as to how to disentangle the two possible sources of spuriousness. The purpose of this paper is to provide a step further after rejecting the null of panel cointegration, by providing two tests, one for the null of homogeneity and one for the null of panel cointegration. These two tests make it possible to identify the source of spuriousness in the panel. As a by-product, this paper also proposes two consistent estimators, for the degree of heterogeneity across units and for the fraction of spurious regressions, . The two estimators are consistent and use the model with variables in levels directly, (2), as opposed to models using …rst di¤erences where the risk of overdi¤erencing is present. Particularly, the two estimators are based on a linear combination of two estimators, one consistent and one inconsistent, of the variance of the estimate ̂ in (2). Note that in a mixed panel context (where some units are cointegrated and some are not), estimating the degree of heterogeneity using data in levels is a nontrivial exercise as some of the unit speci…c estimates ̂is will be inconsistent and therefore cannot be used. Estimation of the fraction of spurious regressions has been recently addressed by Ng (2008), where a di¤erent estimator is proposed to estimate 2 (0; 1]. In this paper, a consistent estimator for is proposed and consistency is shown in all the interval [0; 1], including the boundary = 0; no special assumptions, such as unit long run variances in the uits are required. Based on this estimator, a test for the null of cointegration H0 : = 0 can be constructed, which has various advantages, since often cointegration is the working hypothesis of interest. Results are derived jointly for (n; T ) ! 1, and all the asymptotics is derived allowing for cross dependence of various strength, including strong cross dependence that could arise from a factor structure in the error term. As we show in the paper, consistency of the estimates is not hampered by the degree of cross sectional dependence. The paper is organised as follows. Section 2 discusses the model and the main assumptions; consistent estimation of the degree of heterogeneity and of the fraction of spurious regressions is discussed in Section 3, and the results concerning testing are in Section 4. Section 5 concludes. Notation is fairly standard. Throughout the paper, ! denotes the ordinary limit, d ! weak convergence and d ! H0 weak convergence under the null 3 hypothesis of a test, and p ! convergence in probability. Stochastic processes such asW (r) on [0; 1] are usually written asW , integrals such as R 1 0 W (r) dr as R W ; W1, W2 etc. denote independent demeaned standard Brownian motions. We let M1, M2 etc. such that Mj <1 be generic positive constants, not depending on T or n. 2 Model and assumptions Recall model (1), where for the sake of simplicity we consider only one regressor, xit, and the pooled model (2) yit = i + ixit + uit; yit = i + xit + vit: Let uit = u ( ) it d + u (1 ) it (1 d ) + u (f) it ; (3) where u it I (0) and d = 1 for i = 1; :::; bn c and zero otherwise. We let u ) it = " u it and u (1 ) it = " u it, where " u it is a stationary process for all i. Thus, we assume that u ) it is non-stationary and u (1 ) it is stationary. In (2), it holds that vit = uit + ( i )xit. The regressor xit is assumed to be i.i.d. across i have the following DGP xit = xit 1 + e x it: Let !it = [eit; " u it] , ! it = h !it; u (f) it i0 and consider the following assumptions. Assumption 1: [cross sectional properties] (i) !it is i.i.d. across i with E (!it); (ii) there exists an invariant -…eld C independent of !it such that E u (f) it C = 0 and u it C is i.i.d. across i. Assumption 2: [time series properties] (i) E k! itjCk 8+ < 1 for some > 0; (ii) and an invariance principle holds for the partial sums of ! itjC