Neural network based approximations to posterior densities: a class of flexible sampling methods with applications to reduced rank models

Likelihoods and posteriors of econometric models with strong endogeneity and weak instruments may exhibit rather non-elliptical contours in the parameter space. This feature also holds for cointegration models when near non-stationarity occurs and determining the number of cointegrating relations is a nontrivial issue, and in mixture processes where the modes are relatively far apart. The performance of Monte Carlo integration methods like importance sampling or Markov Chain Monte Carlo procedures greatly depends in all these cases on the choice of the importance or candidate density. Such a density has to be `close' to the target density in order to yield numerically accurate results with efficient sampling. Neural networks seem to be natural importance or candidate densities, as they have a universal approximation property and are easy to sample from. That is, conditionally upon the specification of the neural network, sampling can be done either directly or using a Gibbs sampling technique, possibly using auxiliary variables. A key step in the proposed class of methods is the construction of a neural network that approximates the target density accurately. The methods are tested on a set of illustrative models which include a mixture of normal distributions, a Bayesian instrumental variable regression problem with weak instruments and near non-identification, a cointegration model with near non-stationarity and a two-regime growth model for US recessions and expansions. These examples involve experiments with non-standard, non-elliptical posterior distributions. The results indicate the feasibility of the neural network approach.