🤖 AI Summary
This study addresses the power loss of conventional variance estimators under alternative hypotheses, where residual drift causes test power to vanish at distant alternatives. To overcome this limitation, it proposes a cross-fitting variance estimation framework that leverages auxiliary linear combinations to eliminate bias induced by residual drift, while precisely characterizing conditional bias to establish estimator consistency under the alternative. This framework generalizes to variance estimation in econometric testing scenarios involving nonparametric specifications, overidentification, and multiple restrictions. Both theoretical analysis and empirical application to the Oregon Health Insurance Experiment demonstrate that the proposed approach substantially enhances the power of quadratic-form test statistics.
📝 Abstract
Quadratic-form test statistics are widely used in econometrics, and their performance depends on accurate variance estimation. Conventional plug-in estimators are consistent under the null hypothesis, but under alternatives the drift in the residuals inflates them and the test loses power. We develop a general framework for variance estimation in such statistics, replacing one of the two squared-residual factors by an auxiliary linear combination of the residuals (``cross-fitting'') chosen to annihilate the drift. We characterize the conditional bias of each estimator exactly. The drift enters the plug-in estimator squared, multiplied by quantities bounded away from zero, so its bias is positive whenever the drift is non-degenerate. It reaches the cross-fit estimator only through the part that survives the cross-fitting, and then only through off-diagonal entries of a residual-maker matrix. From this calculation we obtain conditions under which the cross-fit estimator remains consistent under alternatives while the plug-in estimator does not. At a common critical value the cross-fit test therefore rejects whenever the plug-in test does, and against distant alternatives the plug-in statistic converges to a finite limit, small when few observations carry the departure: its power can tend to zero where the cross-fit test's tends to one. We verify the conditions under primitive assumptions in nonparametric specification testing, overidentification testing, and testing many linear restrictions, and illustrate the procedure on the Oregon Health Insurance Experiment.