Testing for homogeneity

Abstract

In this chapter, various tests for homogeneity will be discussed. In section 5.1, an exact test for the case of normal, homoskedastic errors which are independent in time is derived. This test corresponds to Anderson’s U test (see Anderson 1984, 298ff). In section 5.2, it will be shown that Anderson’s U statistic is functionally equivalent to Laitinen’s statistic (see Laitinen 1978). In section 5.3, it is assumed that the errors follow a vector autoregressive process. The likelihood ratio statistic for testing homogeneity will be defined. Since only the asymptotic distribution of this test statistic is known, a small sample version of the likelihood ratio test and a Monte Carlo test is proposed. In section 5.4, we study the consequences of testing for homogeneity if the error process is wrongly specified. It will be shown that Anderson’s U statistic derived in section 5.1 is asymptotically equivalent to a quadratic form in normal variables under quite general assumptions. In section 5.5, a robust Wald test is defined. This Wald test is based on the quasi-maximum likelihood estimation of the coefficients of the Rotterdam model and on the heteroskedasticity and autocorrelation consistent estimation of its variance-covariance matrix. This test is robust in the sense that the Wald statistic is asymptotically distributed as a χ² under the null hypothesis of homogeneity under fairly general assumptions.