arXiv subjects

Jeffrey Wooldridge

Publications and source records attributed to Jeffrey Wooldridge.

2 recordsLinked to original sources

arXiv2022

What Estimators Are Unbiased For Linear Models?

The recent thought-provoking paper by Hansen [2022, Econometrica] proved that the Gauss-Markov theorem continues to hold without the requirement that competing estimators are linear in the vector of outcomes. Despite the elegant proof, it was shown by the authors and other researchers that the main result in the earlier version of Hansen's paper does not extend the classic Gauss-Markov theorem because no nonlinear unbiased estimator exists under his conditions. To address the issue, Hansen [2022] added statements in the latest version with new conditions under which nonlinear unbiased estimators exist. Motivated by the lively discussion, we study a fundamental problem: what estimators are unbiased for a given class of linear models? We first review a line of highly relevant work dating back to the 1960s, which, unfortunately, have not drawn enough attention. Then, we introduce notation that allows us to restate and unify results from earlier work and Hansen [2022]. The new framework also allows us to highlight differences among previous conclusions. Lastly, we establish new representation theorems for unbiased estimators under different restrictions on the linear model, allowing the coefficients and covariance matrix to take only a finite number of values, the higher moments of the estimator and the dependent variable to exist, and the error distribution to be discrete, absolutely continuous, or dominated by another probability measure. Our results substantially generalize the claims of parallel commentaries on Hansen [2022] and a remarkable result by Koopmann [1982].

econ.EM↗

arXiv2022

When Should You Adjust Standard Errors for Clustering?

In empirical work it is common to estimate parameters of models and report associated standard errors that account for "clustering" of units, where clusters are defined by factors such as geography. Clustering adjustments are typically motivated by the concern that unobserved components of outcomes for units within clusters are correlated. However, this motivation does not provide guidance about questions such as: (i) Why should we adjust standard errors for clustering in some situations but not others? How can we justify the common practice of clustering in observational studies but not randomized experiments, or clustering by state but not by gender? (ii) Why is conventional clustering a potentially conservative "all-or-nothing" adjustment, and are there alternative methods that respond to data and are less conservative? (iii) In what settings does the choice of whether and how to cluster make a difference? We address these questions using a framework of sampling and design inference. We argue that clustering can be needed to address sampling issues if sampling follows a two stage process where in the first stage, a subset of clusters are sampled from a population of clusters, and in the second stage, units are sampled from the sampled clusters. Then, clustered standard errors account for the existence of clusters in the population that we do not see in the sample. Clustering can be needed to account for design issues if treatment assignment is correlated with membership in a cluster. We propose new variance estimators to deal with intermediate settings where conventional cluster standard errors are unnecessarily conservative and robust standard errors are too small.

math.ST↗