Simple and reliable estimators of coefficients of interest in a model with high-dimensional confounding effects

Often an investigator is interested in a single parameter or low-dimensional parameter vector (e.g. a treatment effect) in a regression model, rather than in the full set of regression coefficients. There may also be a relatively high-dimensional set of potential explanatory variables other than the effect of interest which, if omitted, could bias the estimate of the parameter of interest. However there may be too many such variables to include all of them without substantial efficiency loss. We suggest a simple, easily computed estimator for this case, using principal components to compute auxiliary regressors from the set of potential controls, and establish the limit properties of the estimator allowing for dependence and heterogeneity as well as increasing dimension of the set of controls. We also provide finite-sample evidence on the performance of the estimator where principal components are selected in a one-dimensional search using an appropriate information criterion as stopping rule for the number of components. The results suggest that the estimator has practical usefulness in small samples.