A consistent nonparametric Bayesian procedure for estimating autoregressive conditional densities

This article proposes a Bayesian infinite mixture model for the estimation of the conditional density of an ergodic time series. A nonparametric prior on the conditional density is described through the Dirichlet process. In the mixture model, a kernel is used leading to a dynamic nonlinear autoregressive model. This model can approximate any linear autoregressive model arbitrarily closely while imposing no constraint on parameters to ensure stationarity. We establish sufficient conditions for posterior consistency in two different topologies. The proposed method is compared with the mixture of autoregressive model [Wong and Li, 2000. On a mixture autoregressive model. J. Roy. Statist. Soc. Ser. B 62(1), 91–115] and the double-kernel local linear approach [Fan et al., 1996. Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83, 189–206] by simulations and real examples. Our method shows excellent performances in these studies.