This paper analyzes dynamic games with continuous states and controls. There are two main contributions. First, we give conditions under which the payoff function is nonparametrically identified by the observed distribution of states and controls. The identification conditions are fairly general and can be expected to hold in many potential applications. The key identifying restrictions include that one of the partial derivatives of the payoff function is known and that there is some component of the state space that enters the policy function, but not the payoff function directly. The latter of these restrictions is a standard exclusion restriction and is used to identify the payoff function off the equilibrium path. By manipulating the first order condition, we can show that the payoff function satisfies an integro-differential equation. Due to the presence of the value function in the first order condition, this integro-differential equation contains a Fredholm integral operator of the second kind. Invertibility of this operator, and knowledge of one of the partial derivatives of the payoff function is used to ensure that the integro-differential equation has a unique solution. The second contribution of this paper is to propose a two-step semiparametric estimator for the model. In the first step the transition densities and policy function are estimated nonparametrically. In the second step, the parameters of the payoff function are estimated from the optimality conditions of the model. Because the state and action space are continuous, there is a continuum of optimality conditions. The parameter estimates minimize the norm of the these conditions. Hence, the estimator is related to recent papers on GMM in Hilbert spaces and semiparametric estimators with conditional moment restrictions. We give high-level conditions on the first step nonparametric estimates for the parameter estimates to be consistent and parameters to be root-n asymptotically normal. Finally, we show that a kernel based estimator satisfies these conditions.

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We consider the estimation of the set of best linear approximations to a set identified function. We extend the partial identification literature by allowing our bounds to by any estimable functions, potentially even indexed by some parameter. Characterizing the identified set via its support function, we develop the limit theory for the support function and prove that the function approximately converges to a Gaussian process. Limit inference results and the validity of a Bayesian bootstrap is proved as well. The bounds may be estimated by either non-parametric or parametric means and may carry an index. This nests the canonical examples in the literature– interval valued outcome data and interval valued regressor data in mean regression– as special cases. Since the bounds may carry an index, our method covers applications beyond mean regression. These include quantile and distribution regression with interval valued data, sample selection problems, as well as mean, quantile, and distribution treatment effects. Moreover, our framework allows for the utilization of instruments. To illustrate our framework, we perform simulations for the quantile treatment effect in the selection model and, as an example, study female labor participation along the lines of Mulligan and Rubinstein (2008).

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We use employee-level panel data from a single firm to explore the possibility that individuals may select insurance coverage in part based on their anticipated behavioral (“moral hazard”) response to insurance, a phenomenon we label “selection on moral hazard.” Using a model of plan choice and medical utilization, we present evidence of heterogeneous moral hazard as well as selection on it, and explore some of its implications. For example, we show that, at least in our context, abstracting from selection on moral hazard could lead to over-estimates of the spending reduction associated with introducing a high-deductible health insurance option.

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Much of the extensive empirical literature on insurance markets has focused on whether adverse selection can be detected. Once detected, however, there has been little attempt to quantify its welfare cost or to assess whether and what potential government interventions may reduce these costs. To do so, we develop a model of annuity contract choice and estimate it using data from the U.K. annuity market. The model allows for private information about mortality risk as well as heterogeneity in preferences over different contract options. We focus on the choice of length of guarantee among individuals who are required to buy annuities. The results suggest that asymmetric information along the guarantee margin reduces welfare relative to a first-best symmetric information benchmark by about £127 million per year or about 2 percent of annuitized wealth. We also find that by requiring that individuals choose the longest guarantee period allowed, mandates could achieve the first-best allocation. However, we estimate that other mandated guarantee lengths would have detrimental effects on welfare. Since determining the optimal mandate is empirically difficult, our findings suggest that achieving welfare gains through mandatory social insurance may be harder in practice than simple theory may suggest.

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