State smoothing in Markov-switching time-frequency GARCH models

In this letter, we propose a state smoothing algorithm for path-dependent Markov-switching generalized autoregressive conditional heteroscedasticity (GARCH) processes. Our smoothing technique extends the forward-backward recursions of Chang and Hancock and the stable backward recursion of Lindgren, Askar and Derin. We derive two recursive steps for the evaluation of conditional densities of future observations. The first step is an upward recursion that manipulates the future observations for the evaluation of their conditional densities, and the second step is a backward recursion that integrates over the possible future paths. Experimental results demonstrate the improvement in performance, compared to using causal estimation.