Yukitoshi Matsushita, Taisuke Otsu and Keita Sunada

Published 9 October 2026

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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 with probability approaching one, 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.