Efficient simulation and integrated likelihood estimation in state space models

Abstract

Abstract:We consider the problem of implementing simple and efficient Markov chain Monte Carlo (MCMC) estimation algorithms for state space models. A conceptually transparent derivation of the posterior distribution of the states is discussed, which also leads to an efficient simulation algorithm that is modular, scalable and widely applicable. We also discuss a simple approach for evaluating the integrated likelihood, defined as the density of the data given the parameters but marginal of the state vector. We show that this high-dimensional integral can be easily evaluated with minimal computational and conceptual difficulty. Two empirical applications in macroeconomics demonstrate that the methods are versatile and computationally undemanding. In one application, involving a time-varying parameter model, we show that the methods allow for efficient handling of large state vectors. In our second application, involving a dynamic factor model, we introduce a new blocking strategy which results in improved MCMC mixing at little cost. The results demonstrate that the framework is simple, flexible and efficient.

Time Varying Dimension Models

“…The fact that is integrated out analytically greatly improves the e¢ ciency of the algorithm. We draw (conditional on all the model parameters, including K) using the algorithm of Chan and Jeliazkov (2009), although any of the standard algorithms for drawing states in state space models (e.g. Kohn, 1994 or Durbin andKoopman, 2002) could be used.…”

Estimation in Non-Linear Non-Gaussian State Space Models with Precision-Based Methods

“…In this section we present a precision-based sampler developed independently in Chan and Jeliazkov (2009b) and McCausland, Millera, and Pelletier (2011) for simulating the states in linear Gaussian state space models. By exploiting the sparseness structure of the precision matrix for the conditional density of the states, this new simulation algorithm is more e¢ cient than Kalman …lter-based methods in general.…”

“…Due to the general applicability of the proposed approach, it will prove useful in a wide range of applications. We extend the recently developed precision-based samplers Jeliazkov, 2009b andMcCausland, Millera, andPelletier, 2011) and sparse matrix procedures to build fast, e¢ cient samplers for these nonlinear models. We develop a practical way to sample the model parameters and the states jointly to circumvent the problem of high autocorrelations in high-dimensional settings.…”

“…Substantial progress has been made in the last two decades in estimating linear Gaussian state space models. For example, Kalman …lter-based algorithms include Carter and Kohn (1994), Früwirth-Schnatter (1994), de Jong and Shephard (1995) and Durbin and Koopman (2002); more recently, precision-based algorithms are proposed in Rue, Martino, and Chopin (2009), Chan and Jeliazkov (2009b) and McCausland, Millera, and Pelletier (2011). E¢ cient simulation algorithms also exist for certain speci…c non-linear non-Gaussian state space models, the most notable example is the class of stochastic volatility models.…”

“…Speci…-cally, the contributions of this paper are three-fold. In the …rst contribution we build upon the recently proposed precision-based sampler in Chan and Jeliazkov (2009b) and McCausland, Millera, and Pelletier (2011) originally developed for linear Gaussian state space models, and we present a quick method to obtain a Gaussian or a student t approximation for the conditional density of the states p ( j ; y). By exploiting the sparseness structure of the precision matrix for p ( j ; y), the precision-based algorithm is more e¢ cient than Kalman …lter-based methods in general.…”