Multivariate Lagrange Multiplier Tests for Fractional Integration
Cornell University
Address correspondence to Morten Ørregaard Nielsen, Department of Economics, 482 Uris Hall, Cornell University, Ithaca, NY 14853, or e-mail: mon2{at}cornell.edu.
Received April 2, 2004; revised January 24, 2005; accepted February 2, 2005.
We introduce a multivariate Lagrange multiplier (LM) test for fractional integration. We derive and analyze the LM statistic and show that it is asymptotically noncentral chi-squared distributed under local alternatives, and that, under Gaussianity, the LM test is asymptotically efficient against local alternatives. It is shown that the regression variant in Breitung and Hassler (2002, Journal of Econometrics 110, 167–185) is not equivalent to the LM test in the multivariate case, although it is in the univariate case. A generalization of the LM test that explicitly allows for different integration orders for each variable is also introduced. The finite sample properties of the LM test are evaluated by Monte Carlo experiments which demonstrate that it is superior to the Breitung and Hassler (2002) test. An application to multivariate time series of real interest rates for six countries is offered, demonstrating that more clear-cut evidence can be drawn from multivariate tests compared to conducting several univariate tests.
KEYWORDS: asymptotic local power, efficient test, fractional integration, Lagrange multiplier test, multivariate fractional unit root, nonstationarity