Regularization and variable selection in Heckman selection model

Sample selection arises when the outcome of interest is partially observed in a study. A common challenge is the requirement for exclusion restrictions. That is, some of the covariates affecting missingness mechanism do not affect the outcome. The drive to establish this requirement often leads to the inclusion of irrelevant variables in the model. A suboptimal solution is the use of classical variable selection criteria such as AIC and BIC, and traditional variable selection procedures such as stepwise selection. These methods are unstable when there is limited expert knowledge about the variables to include in the model. To address this, we propose the use of adaptive Lasso for variable selection and parameter estimation in both the selection and outcome submodels simultaneously in the absence of exclusion restrictions. By using the maximum likelihood estimator of the sample selection model, we constructed a loss function similar to the least squares regression problem up to a constant, and minimized its penalized version using an efficient algorithm. We show that the estimator, with proper choice of regularization parameter, is consistent and possesses the oracle properties. The method is compared to Lasso and adaptively weighted L1 penalized Two-step method. We applied the methods to the well-known Ambulatory Expenditure Dat