Efficient high-dimensional importance sampling

Abstract

The paper describes a simple, generic and yet highly accurate efficient importance sampling (EIS) Monte Carlo (MC) procedure for the evaluation of high-dimensional numerical integrals. EIS is based upon a sequence of auxiliary weighted regressions which actually are linear under appropriate conditions. It can be used to evaluate likelihood functions and byproducts thereof, such as ML estimators, for models which depend upon unobservable variables. A dynamic stochastic volatility model and a logit panel data model with unobserved heterogeneity (random effects) in both dimensions are used to provide illustrations of EIS high numerical accuracy, even under small number of MC draws. MC simulations are used to characterize the finite sample numerical and statistical properties of EIS-based ML estimators. r 2007 Elsevier B.V. All rights reserved.JEL classification: C15

Predicting Time-Varying Parameters with Parameter-Driven and Observation-Driven Models

“…Following Richard and Zhang (2007) and Koopman, Lucas, and Scharth (2011), we obtain an efficient set of importance parameters χ t = {b t , C t } via the (approximate) variance minimisation problem min χt λ 2 (α t , y t ; ψ)ω(α t , y t ; ψ)g(α t |y; ψ) dα t (A.5)…”

“…Shephard and Pitt (1997) and Durbin and Koopman (1997) develop simulation-based methods for the estimation of ψ, α t and θ t . Liesenfeld and Richard (2003), Richard and Zhang (2007), Jungbacker and Koopman (2007) and Koopman, Lucas, and Scharth (2011) report recent developments on Monte Carlo methods for the analysis of general nonlinear non-Gaussian state space models.…”

“…The choice of the importance sampling density partly determines the accuracy of the likelihood estimate (24). We follow the approach of Richard and Zhang (2007) and choose an importance sampling density that (approximately) minimises the variance of (24). Koopman, Lucas, and Scharth (2011) develop a new method to obtain such an efficient importance sampler using a combination of numerical integration techniques and approximating linear state space methods.…”

Efficient simulated maximum likelihood estimation through explicitly parameter dependent importance sampling

“…It is important that research on simulated maximum likelihood recognizes the huge performance difference between simple and EPD importance sampling. As an example, a recent contribution, Richard and Zhang (2007), develops a new technique for importance sampling of likelihood functions containing high dimensional integrals. Their importance sampler is EPD in the sense defined here, with impressive results.…”

The Links between Inflation and Inflation Uncertainty at the Longer Horizon

“…A good proposal distribution can be very important for obtaining good Monte Carlo approximations to filtering and forecasting distributions (see below). Richard and Zhang (2007) propose a piecemeal approach to optimal fitting of a proposal distribution to be used in Monte Carlo integration in highdimensional models. The method is called the efficient importance sampling (EIS).…”