Abstract
Abstract This study examines the consequences of using an estimated aggregate measure as an explanatory variable in linear regression. We show that neglecting the seemingly small sampling error in the estimated regressor could severely contaminate the estimates. We propose a simple statistical framework to account for the error. In particular, we apply our analysis to two aggregate indicators of economic development, the Gini coefficient and sex ratio. Our findings suggest that the impact of the estimated regressor could be substantially underestimated, when the sampling error is not accounted for.
Highlights
Empirical studies often encounter the following situation: A regressor in the linear regression needs to be estimated before it is included in the regression analysis
We propose an adjusted version of the ordinary least squares (OLS) estimator, which accounts for sampling error in the estimated regressor
We find that the OLS estimator without accounting for sampling error could severely underestimate the effect of the estimated regressor
Summary
Empirical studies often encounter the following situation: A regressor in the linear regression needs to be estimated before it is included in the regression analysis. By contrast, generated regressors typically result from a common functional form that holds across observations These subtle differences imply that existing methods, such as the classical errors-in-variables estimator and the adjustment in Murphy and Topel (1985), are no longer suitable to correct the sampling error in an estimated regressor. The underlying reason is that if the variation of the estimated regressor itself is small, the seemingly small sampling error could lead to a large difference in the estimates We illustrate this difference by comparing the OLS estimator with its adjusted version that accounts for the sampling error. Using the same data as in Jin et al (2011) and Wei and Zhang (2011), we find evidence that the OLS estimator is substantially different from its adjusted version that takes sampling error into account. Further details and the Monte Carlo evidence are presented in the Appendix
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