Optimal Tests for Reduced Rank Time Variation in Regression Coefficients and Level Variation in the Multivariate Local Level Model

This paper constructs tests for martingale time variation in regression coefficients in the regression model yt = xt′βt + ut, where βt is k×1, and Σβ is the covariance matrix of Δβt. Under the null there is no time variation, so Ho: Σβ = 0, under the alternative there is time variation in r linear combinations of the coefficients, so Ha: rank(Σβ ) = r, where r may be less than k. The Gaussian point optimal invariant test for this reduced rank testing problem is derived, and the test’s asymptotic behavior is studied under local alternatives. The paper also considers the analogous testing problem in the multivariate local level model Zt = μt + at, where Zt is a k×1 vector, μt is a level process that is constant under the null but is subject to reduced rank martingale variation under the alternative, and at is an I(0) process. The test is used to investigate possible common trend variation in the growth rate of per-capita GDP in France, Germany and Italy.