Kernel-based hidden Markov conditional densities

A natural way to obtain conditional density estimates for time series processes is to adopt a kernel-based (nonparametric) conditional density estimation (KCDE) method. To this end, the data generating process is commonly assumed to be Markovian of finite order. Markov processes, however, have limited memory range so that only the most recent observations are informative for estimating future observations, assuming the underlying model is known. Hidden Markov models (HMMs), on the other hand, can integrate information over arbitrary lengths of time and thus describe a wider variety of data generating processes. The KCDE and HMMs are combined into one method. The resulting KCDE-HMM method is described in detail, and an iterative algorithm is presented for estimating its transition probabilities, weights and bandwidths. Consistency and asymptotic normality of the resulting conditional density estimator are proved. The conditional forecast ability of the proposed conditional density method is examined and compared via a rolling forecasting window with three benchmark methods: HMM, autoregressive HMM, and KCDE-MM. Large-sample performance of the above conditional estimation methods as a function of training data size is explored. Finally, the methods are applied to the U.S. Industrial Production series and the S&P 500 index. The results indicate that KCDE-HMM outperforms the benchmark methods for moderate-to-large sample sizes, irrespective of the number of hidden states considered.