Tests of Common Stochastic Trends

Abstract

This paper is concerned with tests in multivariate time series models made up of random walk (with drift) and stationary components. When the stationary component is white noise, a Lagrange multiplier test of the hypothesis that the covariance matrix of the disturbances driving the multivariate random walk is null is shown to be locally best invariant, something that does not automatically follow in the multivariate case. The asymptotic distribution of the test statistic is derived for the general model. The test is then extended to deal with a serially correlated stationary component. The main contribution of the paper is to propose a test of the validity of a specified value for the rank of the covariance matrix of the disturbances driving the multivariate random walk. This rank is equal to the number of common trends, or levels, in the series. The test is very simple insofar as it does not require any models to be estimated, even if serial correlation is present. Its use with real data is illustrated in the context of a stochastic volatility model, and the relationship with tests in the cointegration literature is discussed.

Current account patterns and national real estate markets

“…8 To examine further in details the stationarity of the real estate and the current account series, Table 2 reports a summary of unit root tests, one on the individual series for each country, and another across series in the panels. In the top panel, we can see that under the null hypothesis of a unit root the rejection rates of these Nyblom-Harvey (2000) test, the test statistic can be considered as the generalization of the Kwiatkowski-Phillips-Schmidt-Shin test, and a failure to reject the null hypothesis of zero common stochastic trends is an indication that the series do not form a cointegrated combination. Applying to the panel of Real Estate/(GDP deflator) appreciation, the Levin-Lin-Chu and the Im-Pesaran-Shin tests reject the null of non-stationarity.…”

“…The null hypothesis is non-stationarity for the Levin-Lin-Chu (2002) test and the Im-Pesaran-Shin (2003) test. For the Nyblom-Harvey (2000) test, the test statistic can be considered as the generalization of the Kwiatkowski-Phillips-Schmidt-Shin test, and a failure to reject the null hypothesis of zero common stochastic trends is an indication that the series do not form a cointegrated combination. The test statistics correspond to specifications with time trend.…”

The Stability of the Adjusted and Unadjusted Environmental Kuznets Curve

“…In particular, Roberts and Grimes (1997) Nyblom and Harvey (2000). 3 The test does not require any models to be estimated, even if serial correlation is present, and therefore it is particularly convenient in presence of nonlinear specifications such as that of the EKC.…”

“…When individual time series are short these tests should be accompanied by bootstrap critical values (Müller-Furstenberger, Wagner and Müller, 2004) 2 . An additional test for cointegration is the one developed by Nyblom and Harvey (2000). 3 The test does not require any models to be estimated, even if serial correlation is present, and therefore it is particularly convenient in presence of nonlinear specifications such as that of the EKC.…”

“…share a common stochastic trend) in order to obtain meaningful regression results with estimates in levels. The test, we perform, is based on Nyblom-Harvey (2000) and may be considered as a generalization of the Nyblom and Makelainen (1983) and Kwiatkowski et al (1992) univariate tests for stationary of a series. The null hypothesis of those tests is that the series is stationary, or stationary around a deterministic trend, against the alternative of the presence of a random walk component.…”