Estimating Regression Models in Which the Dependent Variable Is Based on Estimates

Abstract

Researchers often use as dependent variables quantities estimated from auxiliary data sets. Estimated dependent variable (EDV) models arise, for example, in studies where counties or states are the units of analysis and the dependent variable is an estimated mean, proportion, or regression coefficient. Scholars fitting EDV models have generally recognized that variation in the sampling variance of the observations on the dependent variable will induce heteroscedasticity. We show that the most common approach to this problem, weighted least squares, will usually lead to inefficient estimates and underestimated standard errors. In many cases, OLS with White's or Efron heteroscedastic consistent standard errors yields better results. We also suggest two simple alternative FGLS approaches that are more efficient and yield consistent standard error estimates. Finally, we apply the various alternative estimators to a replication of Cohen's (2004) cross-national study of presidential approval.

Unemployed and alone? Unemployment and social participation in Europe

“…These analyses allow us to test which macro-level variables are relevant in mediating social participation disparities between the unemployed and the employed, and also whether their relative importance varies with type of social participation. We use HC3 robust standard errors (Efron standard errors) in all our macro-level analyses to control for possible heteroskedasticities (Lewis and Linzer, 2005).…”

“…The usual random error present in all models as well as error due to the dependent variables being estimated (as opposed to observed). If sampling variance differs across observation levels there is a risk that the error component will be heteroscedastic (Lewis and Linzer 2005). As our estimated variables are based on very different samples we apply Lewis and Linzer's feasible generalized least squares (FGLS) procedure which allows us to address the problem of 'heteroscedasticity in the first level error component without assuming that the second level error component is similarly heteroscedastic ' (2005: 347).…”

The Democratic Effect of Direct Democracy

“…The slope falls because the negative effect of the preference gap on policy congruence dominates. In other words, the costs of using direct democracy become too 19 A potential objection against the hypothesis tests is that we underestimated the full uncertainty in Models 1 to 4 because some explanatory variables are measured with uncertainty (the preference variables) (see, e.g., Lewis and Linzer, 2005). To account for the full uncertainty, we estimate the same models relying on 1,000 posterior draws of the preference predictions (first stage), rerun the models for each draw, and save 30 draws of the secondstage posterior.…”

Examining cost variation across hospital departments–a two-stage multi-level approach using patient-level data

“…vi Note also that the potential gains in efficiency from estimating a two-stage model in a single stage are modest when considerable information is available at the bottom level (Lewis and Linzer, 2005). In our study we have almost one million observations at patient level, with each department having no less than one thousand observations.…”

“…Jusko and Shively and Lewis and Linzer discuss extensively the hypothesis under which EDV models involving a two stage approach are consistent and efficient (Jusko andShively, 2005, Lewis andLinzer, 2005). In particular, heteroscedastic sampling errors in the estimated dependent variables might result in biased standard errors in the second stage analysis.…”