Imposing Moment Restrictions from Auxiliary Data by Weighting

In this paper we analyze estimation of coefficients in regression models under moment restrictions where the moment restrictions are derived from auxiliary data. Our approach is similar to those that have been used in statistics for analyzing contingency tables with known marginals. These methods are useful in cases where data from a small, potentially non-representative data set can be supplemented with auxiliary information from another data set which may be larger and/or more representative of the target population. The moment restrictions yield weights for each observation that can subsequently be used in weighted regression analysis. We discuss the interpretation of these weights both under the assumption that the target population and the sampled population are the same, as well as under the assumption that these popula- tions differ. We present an application based on omitted ability bias in estimation of wage regressions. The National Longitudinal Survey Young Men's Cohort (NLS), as well as containing information for each observation on earn- ings, education and experience, records data on two test scores that may be considered proxies for ability. The NLS is a small data set, however, with a high attrition rate. We investigate how to mitigate these problems in the NLS by forming moments from the joint distribution of education, experience and earnings in the 1% sample of the 1980 U.S. Census and using these moments to construct weights for weighted regression analysis of the NLS. We analyze the impacts of our weighted regression techniques on the estimated coefficients and standard errors on returns to education and experience in the NLS control- ling for ability, with and without assuming that the NLS and the Census samples are random samples from the same population.