Regression with an imputed dependent variable

Researchers are often interested in the relationship between two variables, with no single data set containing both. A common strategy is to use proxies for the dependent variable that are common to two surveys to impute the dependent variable into the data set containing the independent variable. We show that commonly employed regression or matching-based imputation procedures lead to inconsistent estimates. We offer an easily-implemented correction and correct asymptotic standard errors. We illustrate these with empirical examples using data from the US Consumer Expenditure Survey (CE) and the Panel Study of Income Dynamics (PSID).