Essays in Econometrics The chapters of this dissertation are devoted to three different topics. The first chapter studies estimation of parameters expressed via non-differentiable functions. Such parameters are abundant in econometric models and typically take the form of maxima or minima of some estimable objects. Examples include bounds on the average treatment effects in non-experimental settings, identified sets for the coefficients in regression models with interval-valued data, bounds on the distribution of wages accounting for selection into employment, and many others. I consider estimators of the form $\phi(\hat{\theta}_n + \hat{v}_{1, n}) + \hat{v}_{2, n}$, where $\hat{\theta}_n$ is the efficient estimator for $\theta_0$, and $\hat{v}_{1, n}, \hat{v}_{2, n}$ are suitable adjustment terms. I characterize the optimal adjustment terms and develop a general procedure to compute them from the data. A simulation study shows that the proposed estimator can have lower finite-sample bias and variance than the existing alternatives. As an application, I consider estimating the bounds on the distribution of valuations and the optimal reserve price in English auctions with independent private values. Empirically calibrated simulations show that the resulting estimates are substantially sharper than the previously available ones. The second chapter studies inequality selection in partially identified models. Many partially identified models have the following structure: given a parameter vector and covariates, the model produces a set of predictions while the researcher observes a single outcome. Examples include entry games with multiple equilibria, network formation models, discrete-choice models with endogenous explanatory variables or heterogeneous choice sets, and auctions. Sharp identified sets for structural parameters in such models can be characterized via a special kind of moment inequalities. For a given parameter value, the inequalities verify that the observed conditional distribution of the outcome given covariates belongs to the set of distributions admitted by the model. In practice, checking all of the inequalities is often computationally infeasible, and many of them may not even be informative. Therefore, some inequality selection is required. In this chapter, I propose a new analytical criterion that dramatically reduces the number of inequalities required to characterize the sharp identified set. In settings where the outcome space is finite, I characterize the smallest subset of inequalities that guarantees sharpness and show that it can be efficiently computed using graph propagation techniques. I apply the proposed criterion in the context of market entry games, network formation, auctions, and discrete-choice. The third chapter (coauthored with Liqiang Shi) is about model selection for policy learning. When treatment effects are heterogeneous, a decision-maker that has access to (quasi-)experimental data can attempt to find the optimal policy function, mapping observable characteristics into treatment choices, to maximize utilitarian welfare. When several different policy classes are available, the choice of the policy class poses a model selection problem. In this chapter, following Athey and Wager (2021) and Mbakop and Tabord-Meehan (2021), we propose a policy learning algorithm that leverages doubly-robust estimation and incorporates data-driven model selection. We show that the proposed algorithm automatically selects the best available class of policies and achieves the optimal $n^{-1/2}$ rate of convergence in terms of expected regret. We also refine some of the existing related results and derive a new finite-sample lower bound on expected regret.