An Integrated Treatment of Monte Carlo Numerical Integration Techniques

Using an unified framework, we integrate the two major Monte Carlo techniques currently available: Efficient Importance Sampling (EIS) and Markov Chain Monte Carlo (MCMC). We do so for two reasons. First, the two methods complement one another. Loosely, EIS is effective for integrating high-dimensional latent processes, such as stochastic volatility in financial series and unobserved heterogeneity in panel data, and all situations for which there are sequential factorizations. MCMC works better when such factorizations are not available, as when dealing with posterior densities of highly non-linear models. Second, and more fundamentally, both methods rely on efficient samplers for their low-dimensional (typically univariate) components. By embedding EIS auxiliary steps within MCMC, we can construct efficient and automated (therefore, also replicable) MCMC implementations that include the selection of critical calibrating constants and starting values. We also include tests essential for validating these low-dimensional samplers. EIS also offers an operational 'single block' alternative to MCMC for high-dimensional latent processes with high correlation. The excellent performance of our mixed EIS-MCMC approach is illustrated by two examples, a Bayesian analysis of stochastic volatility models and the Bayesian analysis of a stationary autoregressive processes with parametrization linked directly to its roots. In both cases the EIS-MCMC implementation is numerically efficient and converges much faster than conventional MCMC