Improved Errors-in-Variables Estimators for Grouped Data

In many economic applications, observations are naturally categorized into mutually exclusive and exhaustive groups. For example, individuals can be classified into cohorts and workers are employees of a particular firm. Grouping models are widely used in economics -- for example, cohort models have been used to study labour supply, wage inequality, consumption, and intergenerational transfer of human capital. The simplest grouping estimator involves taking the means of all variables for each group and then carrying out a group-level regression by OLS or weighted least squares. This estimator is biased in finite samples. I show that the standard errors in variables estimator (EVE) designed to correct for small sample bias is exactly equivalent to the Jack-knife Instrumental Variables Estimator (JIVE). Also EVE is closely related to the k-class of instrumental variables estimators. I then use results from the instrumental variables literature to develop an estimator (UEVE) with better finite-sample properties than existing errors in variables estimators. The theoretical results are demonstrated using Monte Carlo experiments. Finally, I use the estimators to implement a model of inter-temporal male labour supply using micro data from the United States Census. There are sizeable differences in the wage elasticity across estimators, showing the practical importance of the theoretical issues discussed in this paper even in circumstances where the sample size is quite large.