Inference in semiparametric conditional moment models with partial identification

This paper develops inference methods for conditional moment models in which the unknown parameter is possibly partially identified and may contain infinite-dimensional components. For a conjectured restriction on the parameter, we consider testing the hypothesis that the restriction is satisfied by at least one element of the identified set. We propose using the sieve minimum of a Kolmogorov–Smirnov type statistic as the test statistic, derive its asymptotic distribution, and provide consistent bootstrap critical values. In this way a broad family of restrictions can be consistently tested, making the proposed procedure applicable to testing the model specification and constructing confidence set for any given component or some feature of the parameter. Our methods are robust to partial identification, and allow for the moment functions to be nonsmooth. As an illustration, we apply the proposed inference methods to study the quantile instrumental variable Engel curves for gasoline in Brazil. A Monte Carlo study demonstrates finite sample performance.