Moments of Random Coefficients in Short Panels

Abstract

We study identification and estimation of moments of random coefficients in short linear panels, allowing the number of heterogeneous coefficients to exceed the number of equations observed for each unit. Under moment homogeneity, different regressor histories impose restrictions on the same moment vector. We give necessary and sufficient conditions for these restrictions to identify moments of a given order, stated in terms of the row spaces generated by the regressor support. The results show that moments may be identified even when the coefficients cannot be recovered for any individual, and, for two full-row-rank histories, give the exact loss of independent restrictions caused by overlap of their row spaces. When the support condition fails, we establish nonidentification in the maintained model and characterize the sharp identified set implied by these conditional moments. At second order, the identified set is determined by positive-semidefinite covariance restrictions and is also sharp relative to the full joint distribution of outcomes and regressors. Under the support condition, weighted minimum-distance estimators are root-$N$ asymptotically normal; under conditional nondegeneracy, oracle generalized-inverse weighting attains the Chamberlain (1987) efficiency bound for the maintained conditional-moment model.