Residual Empirical Processes and Weighted Sums for Time‐Varying Processes with Applications to Testing for Homoscedasticity

In the context of heteroscedastic time‐varying autoregressive (AR)‐process we study the estimation of the error/innovation distributions. Our study reveals that the non‐parametric estimation of the AR parameter functions has a negligible asymptotic effect on the estimation of the empirical distribution of the residuals even though the AR parameter functions are estimated non‐parametrically. The derivation of these results involves the study of both function‐indexed sequential residual empirical processes and weighted sum processes. Exponential inequalities and weak convergence results are derived. As an application of our results we discuss testing for the constancy of the variance function, which in special cases corresponds to testing for stationarity.