Rank based tests for testing the constancy of the regression coefficients against random walk alternatives

Abstract

A class of approximately locally most powerful type tests based on ranks of residuals is suggested for testing the hypothesis that the regression coefficient is constant in a standard regression model against the alternatives that a random walk process generates the successive regression coefficients. We derive the asymptotic null distribution of such a rank test. This distribution can be described as a generalization of the asymptotic distribution of the Cramer-von Mises test statistic. However, this distribution is quite complex and involves eigen values and eigen functions of a known positive definite kernel, as well as the unknown density function of the error term. It is then natural to apply bootstrap procedures. Extending a result due to Shorack in [25], we have shown that the weighted empirical process of residuals can be bootstrapped, which solves the problem of finding the null distribution of a rank test statistic. A simulation study is reported in order to judge performance of the suggested test statistic and the bootstrap procedure.