Leave-out Estimation of Variance Components

Abstract

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.

Fixed‐Effect Regressions on Network Data

“…If the functional of interest is the variance, an exact bias correction can be performed (see Andrews et al (2008) and Kline, Saggio, and Sølvsten (2018)). Simple regularity conditions on ϕ for this bias result to hold are that it is differentiable with E(ϕ (α i ) 2 ) < ∞ and bounded third derivative.…”

“…Therefore, for the bias to vanish andτ to be consistent, we need that the degrees of the individual vertices grow with n for an increasing fraction of the vertices. If the functional of interest is the variance, an exact bias correction can be performed (see Andrews et al (2008) and Kline, Saggio, and Sølvsten (2018)). For functionals like τ, a plug-in estimator of the leading-order bias b is easily formed and so an adjusted estimator is readily constructed.…”

“…We postpone a detailed analysis to future work. In a recent contribution, Kline, Saggio, and Sølvsten (2018) presented limit theory for quadratic forms in α.…”

A Distributional Framework for Matched Employer Employee Data

“… See Abowd, Kramarz, Lengermann, and Pérez-Duarte (2004),Andrews, Gill, Schank, andUpward (2008, 2012), and recentlyKline, Saggio, and Sølvsten (2018) for methods to address incidental parameter bias in fixed-effects regressions.4 Similarly as in most of the literature on discrete estimation, this result is derived under the assumption that the population of firms consists of a finite, known number of classes. InBonhomme, Lamadon, and Manresa (2017), we considered a setting where the discrete modeling is viewed as an approximation to an underlying, possibly continuous, distribution of firm unobserved heterogeneity, and we provided consistency results and rates of convergence.…”

“… See Abowd, Kramarz, Lengermann, and Pérez‐Duarte (), Andrews, Gill, Schank, and Upward (, ), and recently Kline, Saggio, and Sølvsten () for methods to address incidental parameter bias in fixed‐effects regressions. …”