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High-Dimensional Semiparametric Selection Models: Estimation Theory with an Application to the Retail Gasoline Market

This paper proposes a multi-stage projection-based Lasso procedure for the semiparametric sample selection model in high-dimensional settings under a weak nonparametric restriction on the selection correction. In particular, the number of regressors in the main equation, p, and the number of regressors in the selection equation, d, can grow with and exceed the sample size n. The analysis considers the exact sparsity case and the approximate sparsity case. The main theoretical results are finite-sample bounds from which sufficient scaling conditions on the sample size for estimation consistency and variable-selection consistency are established. Statistical efficiency of the proposed estimators is studied via lower bounds on minimax risks and the result shows that, for a family of models with exactly sparse structure on the coefficient vector in the main equation, one of the proposed estimators attains the smallest estimation error up to the (n,d,p)-scaling among a class of procedures in worst-case scenarios. Inference procedures for the coefficients of the main equation, one based on a pivotal Dantzig selector to construct non-asymptotic confidence sets and one based on a post-selection strategy, are discussed. Other theoretical contributions include establishing the non-asymptotic counterpart of the familiar asymptotic oracle results from previous literature: the estimator of the coefficients in the main equation behaves as if the unknown nonparametric component were known, provided the nonparametric component is sufficiently smooth. Small-sample performance of the high-dimensional multi-stage estimation procedure is evaluated by Monte-Carlo simulations and illustrated with an empirical application to the retail gasoline market in the Greater Saint Louis area.