Conditional score residuals and diagnostic analysis of serial dependence in time series models

We introduce conditional score residuals and provide a general framework for the diagnostic analysis of time series models. A key feature of conditional score residuals is that they account for the shape of the conditional distribution. These residuals offer reliable and powerful diagnostic tools for testing residual autocorrelation. Furthermore, they can be employed in models of which it is not clear how to define residuals. The asymptotic properties of the empirical autocorrelation function for conditional score residuals are formally derived. The results yield a unified theory for the diagnostic analysis of a wide class of time series models. The practical relevance of the proposed framework is illustrated for heavy-tailed GARCH models. Monte Carlo and empirical results support the finding that conditional score residuals are more reliable in testing residual autocorrelation, when compared to squared GARCH residuals. We finally show how a diagnostic analysis can be designed for dynamic copula models.