In-sample confidence bands and out-of-sample forecast bands for time-varying parameters in observation-driven models

We study the performances of alternative methods for calculating in-sample confidence and out-of-sample forecast bands for time-varying parameters. The in-sample bands reflect parameter uncertainty, while the out-of-sample bands reflect not only parameter uncertainty, but also innovation uncertainty. The bands are applicable to a wide range of estimation procedures and a large class of observation driven models with differentiable transition functions. A Monte Carlo study is conducted to investigate time-varying parameter models such as generalized autoregressive conditional heteroskedasticity and autoregressive conditional duration models. Our results show convincing differences between the actual coverages provided by the different methods. We illustrate our findings in a volatility analysis for monthly Standard & Poor's 500 index returns.