A specification test for dynamic conditional distribution models with function-valued parameters

This paper proposes a practical and consistent specification test of conditional distribution models for dependent data in a general setting. Our approach covers conditional distribution models indexed by function-valued parameters, allowing for a wide range of useful models for risk management and forecasting, such as the quantile autoregressive model, the CAViaR model, and the distributional regression model. The new specification test (i) is valid for general linear and nonlinear conditional quantile models under dependent data, (ii) allows for dynamic misspecification of the past information set, (iii) is consistent against fixed alternatives, and (iv) has nontrivial power against Pitman deviations from the null hypothesis. As the test statistic is non-pivotal, we propose and theoretically justify a subsampling approach to obtain valid inference. Finally, we illustrate the applicability of our approach by analyzing models of the returns distribution and Value-at-Risk (VaR) of two major stock indexes.