Miscellanea. Combining parametric and empirical likelihoods

Imbens & Lancaster (1994) pointed out that census reports can be interpreted as providing nearly exact knowledge of moments of the marginal distribution of economic variables. In this paper we show that empirical likelihood can effectively incorporate auxiliary information like this as long as it can be summarised as unbiased estimating equations. By combining empirical and parametric likelihoods, we show that the combined likelihood can produce valid inferences for the underlying parameters. A Wilks' type theorem is proved for the combined likelihood ratio statistic. Simulation results demonstrate that the performance of the combined likelihood ratio confidence intervals is better than conventional confidence intervals that use a normal approximation.