Computation of Gaussian orthant probabilities in high dimension

We study the computation of Gaussian orthant probabilities, i.e. the probability that a Gaussian falls inside a quadrant. The Geweke-Hajivassiliou-Keane (GHK) algorithm [Genz, 1992; Geweke, 1991; Hajivassiliou et al., 1996; Keane, 1993], is currently used for integrals of dimension greater than 10. In this paper we show that for Markovian covariances GHK can be interpreted as the estimator of the normalizing constant of a state space model using sequential importance sampling (SIS). We show for an AR(1) the variance of the GHK, properly normalized, diverges exponentially fast with the dimension. As an improvement we propose using a particle filter (PF). We then generalize this idea to arbitrary covariance matrices using Sequential Monte Carlo (SMC) with properly tailored MCMC moves. We show empirically that this can lead to drastic improvements on currently used algorithms. We also extend the framework to orthants of mixture of Gaussians (Student, Cauchy etc.), and to the simulation of truncated Gaussians

Misspecified heteroskedasticity in the panel probit model: A small sample comparison of GMM and SML estimators

This paper compares generalized method of moments (GMM) and simulated maximum likelihood (SML) approaches to the estimation of the panel probit model. Both techniques circumvent multiple integration of joint density functions without the need to restrict the error term variance- covariance matrix of the latent normal regression model. Particular attention is paid to a three-stage GMM estimator based on nonparametric estimation of the optimal instruments for given conditional moment functions. Monte Carlo experiments are carried out which focus on the small sample consequences of misspecification of the error term variance-covariance matrix. The correctly specified experiment reveals the asymptotic efficiency advantages of SML. The GMM estimators outperform SML in the presence of misspecification in terms of multiplicative heteroskedasticity. This holds in particular for the three-stage GMM estimator. Allowing for heteroskedasticity over time increases the robustness with respect to misspecification in terms of ultiplicative heteroskedasticity. An application to the product innovation activities of German manufacturing firms is presented.

Approximating multivariate posterior distribution functions from Monte Carlo samples for sequential Bayesian inference

An important feature of Bayesian statistics is the opportunity to do sequential inference: the posterior distribution obtained after seeing a dataset can be used as prior for a second inference. However, when Monte Carlo sampling methods are used for inference, we only have a set of samples from the posterior distribution. To do sequential inference, we then either have to evaluate the second posterior at only these locations and reweight the samples accordingly, or we can estimate a functional description of the posterior probability distribution from the samples and use that as prior for the second inference. Here, we investigated to what extent we can obtain an accurate joint posterior from two datasets if the inference is done sequentially rather than jointly, under the condition that each inference step is done using Monte Carlo sampling. To test this, we evaluated the accuracy of kernel density estimates, Gaussian mixtures, vine copulas and Gaussian processes in approximating posterior distributions, and then tested whether these approximations can be used in sequential inference. In low dimensionality, Gaussian processes are more accurate, whereas in higher dimensionality Gaussian mixtures or vine copulas perform better. In our test cases, posterior approximations are preferable over direct sample reweighting, although joint inference is still preferable over sequential inference. Since the performance is case-specific, we provide an R package mvdens with a unified interface for the density approximation methods