Bootstrap LR tests of stationarity, common trends and cointegration

Abstract

This paper considers the likelihood ratio (LR) tests of stationarity, common trends and cointegration for multivariate time series. As the distribution of these tests is not known, a bootstrap version is proposed via a state- space representation. The bootstrap samples are obtained from the Kalman filter innovations under the null hypothesis. Monte Carlo simulations for the Gaussian univariate random walk plus noise model show that the bootstrap LR test achieves higher power for medium-sized deviations from the null hypothesis than a locally optimal and one-sided Lagrange Multiplier (LM) test that has a known asymptotic distribution. The power gains of the bootstrap LR test are significantly larger for testing the hypothesis of common trends and cointegration in multivariate time series, as the alternative asymptotic procedure – obtained as an extension of the LM test of stationarity – does not possess properties of optimality. Finally, it is shown that the (pseudo-)LR tests maintain good size and power properties also for the non-Gaussian series. An empirical illustration is provided.

Keywords:

• bootstrap
• cointegration
• computer-intensive methods
• econometric models
• econometrics
• maximum likelihood
• Monte Carlo simulation
• resampling
• space–time models
• statistical inference
• time series

JEL Classification :

• C12
• C22

Notes

As regards the related literature on testing for unit roots, bootstrap methods have been investigated in 14–16 among others; a common feature with our tests is that resampling is done under the null hypothesis.

As the LBI test maximizes the slope of the power function at the null hypothesis, no gains were expected for the bootstrap LR test when c is small.

To be precise, we adopt the slightly different transform where is the unconditional variance of i=1, 2, 3, 4 (see [pp. 494–497]26).