Small Data': Efficient Inference with Occasionally Observed States

We study the estimation of dynamic economic models for which some of the state variables are observed only occasionally by the econometrician—a common problem in many fields, ranging from marketing to finance to industrial organization. If such occasional state observations are serially correlated, the likelihood function of the model becomes a potentially high-dimensional integral over a nonstandard domain. We propose a method that generalizes the recursive likelihood function integration procedure (RLI; Reich, 2018) to numerically approximate this integral. We prove consistency and asymptotic normality of our (approximate) estimator and demonstrate its high efficiency in several well-understood examples from finance and industrial organization. In extensive Monte Carlo studies, we compare the performance of our approach to a recently suggested method of simulated moments. In all our demonstrations, we identify all model parameters with high efficiency, and we find that the additional variance of our estimator when going from full to occasional state observations is small for the parameters of interest.