Many settings in empirical economics involve estimation of a large number of parameters. In such settings, methods that combine regularized estimation and data-driven choices of regularization parameters are useful. We provide guidance to applied researchers on the choice between regularized estimators and data-driven selection of regularization parameters. We characterize the risk and relative performance of regularized estimators as a function of the data-generating process and show that data-driven choices of regularization parameters yield estimators with risk uniformly close to the risk attained under the optimal (unfeasible) choice of regularization parameters. We illustrate using examples from empirical economics.

We propose to use economic theories to construct shrinkage estimators that perform well when the theories' empirical implications are approximately correct but perform no worse than unrestricted estimators when the theories' implications do not hold. We implement this construction in various settings, including labor demand and wage inequality, and estimation of consumer demand. We provide asymptotic and finite sample characterizations of the behavior of the proposed estimators. Our approach is an alternative to the use of theory as something to be tested or to be imposed on estimates. Our approach complements uses of theory for identification and extrapolation.

We discuss the tension between “what we can get” (identification) and “what we want” (parameters of interest) in models of policy choice (treatment assignment). Our nonstandard empirical object of interest is the ranking of counterfactual policies. Partial identification of treatment effects maps into a partial welfare ranking of treatment assignment policies. We characterize the identified ranking and show how the identifiability of the ranking depends on identifying assumptions, the feasible policy set, and distributional preferences. An application to the project STAR experiment illustrates this dependence. This paper connects the literatures on partial identification, robust statistics, and choice under Knightian uncertainty.