  We propose leave-out estimators of quadratic forms designed for the study of
linear models with unrestricted heteroscedasticity. Applications include
analysis of variance and tests of linear restrictions in models with many
regressors. An approximation algorithm is provided that enables accurate
computation of the estimator in very large datasets. We study the large sample
properties of our estimator allowing the number of regressors to grow in
proportion to the number of observations. Consistency is established in a
variety of settings where plug-in methods and estimators predicated on
homoscedasticity exhibit first-order biases. For quadratic forms of increasing
rank, the limiting distribution can be represented by a linear combination of
normal and non-central $\chi^2$ random variables, with normality ensuing under
strong identification. Standard error estimators are proposed that enable tests
of linear restrictions and the construction of uniformly valid confidence
intervals for quadratic forms of interest. We find in Italian social security
records that leave-out estimates of a variance decomposition in a two-way fixed
effects model of wage determination yield substantially different conclusions
regarding the relative contribution of workers, firms, and worker-firm sorting
to wage inequality than conventional methods. Monte Carlo exercises corroborate
the accuracy of our asymptotic approximations, with clear evidence of
non-normality emerging when worker mobility between blocks of firms is limited.
