**Efficient Estimation of Directionally Differentiable Functionals** [revision in progress]

Job Market Paper

This paper studies estimation of parameters of the form , where is a known directionally differentiable function, and is an estimable feature of the observed distribution of the data. 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 show that the efficient (Locally Asymptotically Minimax) estimators for such parameters take the form , where is the efficient estimator for , and are suitable adjustment terms. I show that the optimal adjustment terms depend on the chosen loss function and develop a general procedure to compute them from the data. A simulation study demonstrates that the proposed estimator can have lower finite-sample bias and variance than the existing alternatives. As an application, I construct efficient estimators for 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.

**Social Interactions on Endogenous Networks with Positive Spillovers** [draft coming soon]

This paper studies a class of network games in which each player chooses both an activity level and a set of connections, and the activity levels exhibit positive spillovers. Examples include peer effects in education, labor market participation, criminal activity, and R&D collaboration. The contribution of this paper is twofold. First, I provide a detailed characterization of the equilibrium (Pairwise Nash Stable) networks, activity levels, and welfare distribution. Importantly, I allow for heterogeneous players with different types, defined by ex-ante ability level and linking costs, and do not impose strong separability restrictions on payoffs. I show that the equilibrium networks consist of several interlinked components with simple structures: each component contains central and peripheral players of the same type; central players of different types may be interlinked; within each type, central players exhibit higher activity levels and receive higher payoffs. The resulting network structures closely resemble the asymmetric and weakly connected networks observed in the data. Second, I discuss the implications of the above results for empirical analysis and construct identified sets for the payoff parameters that remain tractable in networks of moderate size. A simulation study suggests that the proposed identified sets can be sufficiently informative.

**Selecting Inequalities for Sharp Identification in Models with Set-Valued Predictions**

[draft coming soon]

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 Artstein’s inequalities. For a given parameter value, the inequalities verify that the observed conditional distribution of outcome given covariates belongs to the set of distributions admitted by the model (called the core). However, checking all inequalities is often computationally infeasible, so one must select a subset thereof (e.g., a core-determining class). This paper offers guidance for inequality selection in such settings. I propose a new criterion that dramatically reduces the number of inequalities without loss of information and applies both in the population and in finite samples. In settings where the outcome space is finite, I derive the smallest subset of the inequalities that guarantees sharpness, which can be efficiently computed using graph propagation techniques. I illustrate the utility of the proposed criterion in the context of market entry games, network formation models, and auctions.