Regularized Regression When Boundary Values Are Possible

Regularized estimators have demonstrated outstanding performance across a wide range of applications. In this dissertation, I study regularized estimators when the true value of parameters possibly lies on the boundary of some parameter space. Due to the boundary, the asymptotic distribution of an estimator may no longer be normal. Moreover, regularization can introduce bias to the estimator, which can also interact with the boundary and make the distribution unclear. To understand these issues, in the first part of this dissertation, I derive the asymptotic distributions of the ridge regressor, the least absolute shrinkage and selection operator, and general regularized M-estimators under constraints and fixed-dimensional settings. Since regularization is often applied in high-dimensional contexts, in the second part, I propose a constraint debiased machine learning estimator and a quasi-likelihood ratio test for high-dimensional linear models. In the third part, I focus on the high-dimensional random-coefficient multinomial logit model, commonly used to study discrete choices in economics. I develop a regularized maximum likelihood estimator for simultaneous variable selection, and construct a constrained debiased machine learning estimator to account for both regularization bias and boundary. Finally, I illustrate the impacts of high-dimensional parameters and boundary in an empirical application to soft-drink markets in North Carolina.