Correcting Small Sample Bias in Linear Models With Many Covariates

Estimations of quadratic forms in the parameters of linear models exhibit small-sample bias. The direct computation for a bias correction is not feasible when the number of covariates is large. We propose a boot-strap method for correcting this bias that accommodates different assumptions on the structure of the error term including general heteroscedasticity and serial correlation. Our approach is suited to correct variance decompositions and the bias of multiple quadratic forms of the same linear model without increasing the computational cost. We show with Monte Carlo simulations that our bootstrap procedure is effective in correcting the bias and find that is faster than other methods in the literature. Using administrative data for France, we apply our method by carrying out a variance decomposition of a linear model of log wages with person and firm fixed effects. We find that the person and firm effects are less important in explaining the variance of log wages after correcting for the bias and depending on the specification their correlation becomes positive after the correction.