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 data sets. 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 χ 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.

We investigate the time-series properties of firm effects in the AKM models popularized by Abowd et al. (1999). We consider two approaches. The first approach-labelled as the rolling approach-estimates AKM models separately in each T = 2 adjacent time interval. The second approach is based on an extension of the original AKM-labelled as the Time Varying AKM Model (TV-AKM)-in which we allow for unrestricted interactions of year and firm dummies. We correct for biases in the resulting variance decompositions using the leave out correction of Kline et al. (2019). These approaches allow us to examine how firm effects evolve stochastically, their relation to the business cycle, and their contribution to changes in the wage structure at a higher frequency than previously possible. Using data from Washington State, we find that firm effects in earnings and hourly wages are highly persistent. The autocorrelation coefficient between firm effects for wage rates in 2002 and 2014 is 0.74, and between firm effects for earnings in 2002 and 2014 is 0.82. The rolling approach uncovers a significant degree of cyclicality in firm effects. Variability in firm premiums tended to increase during the great recession while the degree of worker and firm assortativity decreased. Time-varying firm effects explains 13% of the variance of log wages and 21% of the variance of log earnings in the Washington state over 2002-2014. Between 2002-2003 and 2013-2014 the variance of firm wage premia decreased by 10%, but this decline was offset by increases in the variance in individual premia and increases in assortative matching that resulted in an overall increase in the variance of wages. Auxiliary evidence suggests that misspecification in AKM models due to the drifting of firm effects is a second-order concern.

We propose a framework for unbiased estimation of quadratic forms in the parameters of linear models with many regressors and unrestricted heteroscedasticity. Applications include variance component estimation and tests of linear restrictions inhierarchical and panel models. 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 jackknife bias corrections exhibit first-order biases. The estimator's limiting distribution can be represented by a linear combination of normal and non-central χ 2 random variables. Consistent variance estimators are proposed along with a procedure for constructing uniformly valid confidence intervals. Applying a two-way fixed effects model of wage determination to Italian social security records, we find that ignoring heteroscedasticity substantially biases conclusions regarding the relative contribution of workers, firms, and workerfirm sorting to wage inequality. Monte Carlo exercises corroborate the accuracy of our asymptotic approximations, with clear evidence of non-normality emerging when worker mobility between groups of firms is limited.

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