Markov chain Monte Carlo estimation of spatial dynamic panel models for large samples

Focus is on efficient estimation of a dynamic space-time panel data model that incorporates spatial dependence, temporal dependence, as well as space-time covariance and can be implemented in large N and T situations, where N is the number of spatial units and T the number of time periods. Quasi-maximum likelihood (QML) estimation in cases involving large N and T poses computational challenges because optimizing the (log) likelihood requires: 1) evaluating the log-determinant of an NT x NT matrix that appears in the likelihood, 2) imposing stability restrictions on parameters reflecting space-time dynamics, as well as 3) simulations to produce an empirical distribution of the partial derivatives used to interpret model estimates that require numerous inversions of large matrices. We set forth a Markov Chain Monte Carlo (MCMC) estimation procedure capable of handling large problems, which we illustrate using a sample of T = 487 daily fuel prices for N = 12, 435 German gas stations, resulting in N x T over 6 million. The procedure produces estimates equivalent to those from QML and has the additional advantage of producing a Monte Carlo integrated estimate of the log-marginal likelihood, useful for purposes of model comparison. Our MCMC estimation procedure uses: 1) a Taylor series approximation to the logdeterminant based on traces of matrix products calculated prior to MCMC sampling, 2) block sampling of the spatiotemporal parameters, which allows imposition of the stability restrictions, and 3) a Metropolis-Hastings guided Monte Carlo integration of the logmarginal likelihood. We also provide an efficient approach to simulations needed to produce the empirical distribution of the partial derivatives for model interpretation.