Conditional densities in econometrics
Statistical inference in nested linear models that result from linear restrictions on the parameters of encompassing linear models can be considered to result from the conditional distribution under the encompassing model. We extend this reasoning to nested models that result from general (nonlinear) restrictions by defining sufficient conditions that, when satisfied by the random variables and the restrictions, ensure the existence of an unique expression of the conditional density. Statistical inference in these nested models can then be considered to result from such a conditional density. This novel manner of conducting statistical analyzes leads both to some new results and allows one to obtain some already known results in a different manner. In Bayesian statistics, the conditional densities show how to construct specific classes of priors for the parameters of nested models, priors on the parameters of encompassing models that imply an already specified prior on the parameters of the nested model, Bayes factors using (generalized) Savage-Dickey density ratios and Bayesian score statistics. In classical statistical analysis, the conditional densities offer an alternative approach for constructing small sample and limiting distributions of maximum likelihood estimators and classical score statistics.