Residual-based inference on moment hypotheses, with an application to testing for constant correlation

Abstract

Often, inference on moment properties of unobserved processes are conducted on the basis of estimated counterparts obtained in a preliminary step. In some situations, the use of residuals instead of the true quantities affects inference even in the limit, while in others there is no asymptotic residual effect. For the case of statistics based on partial sums of nonlinear functions of the residuals, we give here a characterization of the conditions under which the residual effect does not vanish as the sample size goes to infinity (generic regularity conditions provided). An overview of methods to account for the residual effect is also provided. The analysis extends to models with change points in parameters at estimated time, in spite of the discontinuous manner in which the break time enters the model of interest. To illustrate the usefulness of the results, we propose a test for constant correlations allowing for breaks at unknown time in the marginal means and variances. We find, in Monte Carlo simulations and in an application to US and German stock returns, that not accounting for changes in the marginal moments has severe consequences.