Testing for structural change in conditional models

In the past decade, we have seen the development of a new set of tests for structural change of unknown timing in regression models, most notably the SupF statistic of Andrews (1993, Econometrica 61, 825–856), the ExpF and AveF statistics of Andrews-Ploberger (1994, Econometrica 62, 1383–1414), and the L statistic of Nyblom (1989, Journal of American Statistical Association 84, 223–230). The distribution theory used for these tests is primarily asymptotic, and has been derived under the maintained assumption that the regressors are stationary. This excludes structural change in the marginal distribution of the regressors. As a result, these tests technically cannot discriminate between structural change in the conditional and marginal distributions. This paper attempts to remedy this deficiency by deriving the large sample distributions of the test statistics allowing for structural change in the marginal distribution of the regressors. We find that the asymptotic distributions of the SupF, ExpF, AveF and L statistics are not invariant to structural change in the regressors. To solve the size problem, we introduce a ‘fixed regressor bootstrap’ which achieves the first-order asymptotic distribution, and appears to possess reasonable size properties in small samples. Our bootstrap theory allows for arbitrary structural change in the regressors, including structural shifts, polynomial trends, and exogenous stochastic trends. It allows for lagged dependent variables and heteroskedastic error processes.