Topics in density forecast in stationary parametric univariate time series models

Resumen

In this thesis we study the computation and evaluation of density forecasts under model uncertainty in time series univariate models. First, we analyze the effects of uncertainty on density forecasts of linear univariate ARMA models. We consider three specific sources of uncertainty: parameter estimation, error distribution and lag order. For moderate sample sizes, as those usually encountered in practice, the most important source of uncertainty is the error distribution. We consider alternative procedures proposed to deal with each of these sources of uncertainty and compare their finite sample properties by Monte Carlo experiments. In particular, we analyze asymptotic, Bayesian and bootstrap procedures, including some very recent procedures which have not been previously compared in the literature. Second, we propose an extension of the Generalized Autocontour (G-ACR) tests of González-Rivera and Sun (2015) for one-step-ahead dynamic specifications of conditional densities in-sample and of forecast densities out-of-sample. The new tests are based on probability integral transforms (PITs) computed from bootstrap conditional densities that incorporate the parameter uncertainty without assuming any particular forecast error density. Consequently, the parametric specification of the conditional moments can be tested without relying on any particular error distribution. We show that the asymptotic distributions of the bootstrapped G-ACR (BG-ACR) tests are well approximated using standard asymptotic distributions. Furthermore, the proposed tests are easy to implement and are accompanied by graphical tools which provide suggestions about the potential misspecification. The results are illustrated by testing the dynamic specification of the Heterogenous autoregressive (HAR) model when fitted to the popular U.S. volatility index VIX.