Multivariate Lagrange Multiplier Tests for Fractional Integration

We introduce a multivariate Lagrange Multiplier (LM) test for fractional integration. We derive and analyze the LM statistic and show that it is asymptotically 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 & 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 variate is also introduced. The finite sample properties of the LM test are evaluated and compared to the Breitung & Hassler (2002) test by Monte Carlo experiments. 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.