Finite Sample Inference Methods for Simultaneous Equations and Models with Unobserved and Generated Regressors

We propose finite sample tests and confidence sets for models with unobserved and generated regressors as well as various models estimated by instrumental variables method. We study two distinct approaches for various models considered by Pagan (1984). The first one is an instrument substitution method which generalizes an approach proposed by Anderson and Rubin (1949) and Fuller (1987) for different (although related) problems, while the second one is based on splitting the sample. The instrument substitution method uses the instruments directly, instead of generated regressors, in order to test hypotheses about the ``structural parameters'' of interest and build confidence sets. The second approach relies on ``generated regressors'', which allows a gain in degrees of freedom, and a sample-split technique. A distributional theory is obtained under the assumptions of Gaussian errors and strictly exogenous regressors. We show that the various tests and confidence sets proposed are (locally) ``asymptotically valid'' under much weaker assumptions. The properties of the tests proposed are examined in simulation experiments. In general, they outperform the usual asymptotic inference methods in terms of both reliability and power. Finally, the techniques suggested are applied to a model of Tobin's $q$ and to a model of academic performance.