Choosing the Dictionary and Penalty for IV-LASSO

Abstract

Estimating the first stage of an instrumental variables (IV) model with the least absolute shrinkage and selection operator (LASSO) requires choosing a dictionary of technical instruments and a penalty level. First-order asymptotic theory offers no guidance on these choices, as any consistent implementation yields a structural parameter estimator with the same limiting distribution. In finite samples, however, these choices can have a substantial impact on the resulting structural parameter estimate. Working in a model with a single endogenous regressor and homoskedastic Gaussian errors, we use first- and second-order Stein identities to derive the approximate mean squared error (AMSE) of the instrumental-variables LASSO (IV-LASSO) estimator, which can be consistently estimated and used to rank a prespecified list of dictionary-penalty candidates. The AMSE reveals a bias-variance trade-off: more complex first-stage fits better approximate the conditional mean of the endogenous variable but are also more correlated with the structural errors, with complexity measured by the degrees of freedom of the LASSO fit. The weight on this bias rises with the endogeneity of the regressor, a quantity that neither plug-in nor cross-validation penalty rules take into account. Despite the AMSE being derived in a Gaussian model, penalty selection by minimizing the feasible AMSE criterion delivers up to a one-third lower mean squared error compared to cross-validation and plug-in penalty rules in Gaussian and non-Gaussian simulation designs calibrated to the data of Gilchrist and Sands (2016).