Consistent Testing of Cointegrating Relationships

Abstract

In this paper we investigate methods for testing the existence of a cointegration relationship among the components of a nonstationary fractionally integrated (NFI) vector time series. Our framework generalizes previous studies restricted to unit root integrated processes and permits simultaneous analysis of spurious and cointegrated NFI vectors. We propose a modified F-statistic, based on a particular studentization, which converges weakly under both hypotheses, despite the fact that OLS estimates are only consistent under cointegration. This statistic leads to a Wald-type test of cointegration when combined with a narrow band GLS-type estimate. Our semiparametric methodology allows consistent testing of the spurious regression hypothesis against the alternative of fractional cointegration without prior knowledge on the memory of the original series, their short run properties, the cointegrating vector, or the degree of cointegration. This semiparametric aspect of the modelization does not lead to an asymptotic loss of power, permitting the Wald statistic to diverge faster under the alternative of cointegration than when testing for a hypothesized cointegration vector. In our simulations we show that the method has comparable power to customary procedures under the unit root cointegration setup, and maintains good properties in a general framework where other methods may fail. We illustrate our method testing the cointegration hypothesis of nominal GNP and simple-sum (M1, M2, M3) monetary aggregates. Copyright The Econometric Society 2004.

A Wald test for the cointegration rank in nonstationary fractional systems

“…If r > 1 and the observables and cointegrating residuals are purely nonstationary, Marmol and Velasco (2005) showed that, in contrast to the standard C(1, 1) case (see Wooldrige (1991), Johansen (2002), the OLS estimate of the cointegrating vector (1, Γ ) does not provide a consistent estimate of a suitable linear combination of the cointegrating relations, though remains bounded in probability. Despite of that, in our setting of common error memory of cointegrating residuals, it was shown that the OLS residualsê t still approximate an I(1 − b) process as in the single equation set-up.…”

“…Despite of that, in our setting of common error memory of cointegrating residuals, it was shown that the OLS residualsê t still approximate an I(1 − b) process as in the single equation set-up. See also Marmol and Velasco (2004) for a discussion.…”

“…A similar idea was used by Hualde and Velasco (2007), employing the GLS estimates of Robinson and Hualde (2003). The chi-squared distribution of the GLS Wald statistic is inherited by a parallel cointegration test, hence avoiding the nonstandard asymptotic distribution of Marmol and Velasco's (2004) test and allowing for vector series with components of different integration orders.…”

“…Recently Robinson (2007) provided rigorous theoretical support to this idea. Marmol and Velasco (2004) proposed a Wald test of the null of spurious relationships against the alternative of a single cointegration relationship among the components of a nonstationary vector process. Their approach relies upon comparing OLS and narrow band GLS-type estimates of the cointegrating vector, with different properties under the competitive hypothesis.…”

Likelihood Based Testing for No Fractional Cointegration

“…Velasco (2003) considers semiparametric consistent estimation of the memory parameters of a nonstationary fractionally cointegrated vector time series. Marmol and Velasco (2004) propose tests of the null of cointegration, without information on the degree of integration, based on Wald statistics for OLS coe¢ cients. Hualde and Velasco (2008) employ GLS-type of estimator as in Robinson and Hualde (2003) and obtain a chi-squared distribution for the Wald test under the null of no cointegration.…”