This project will use asymptotic statistical decision theory to develop new procedures and optimality results for two areas of current interest in econometrics: estimation and inference for partially identified parameters; and optimal treatment assignment rules. Partially identified models have received considerable recent attention in economics. In partially identified statistical economic models, not all quantities of interest can be perfectly recovered even with an idealized data set, but one can obtain bounds on the quantities of interest. Although such models can increase the robustness of empirical analysis by relaxing auxiliary assumptions, they are nonstandard from a statistical viewpoint. By using tools from asymptotic statistical decision theory to analyze these models, we can obtain sharp restrictions on the properties of statistical procedures, compare alternative procedures simply, and obtain optimality results. The results of this research will provide economists with new tools, and methods for selecting the best tools, for conducting bounds analyses.
The second component of our project will develop decision-theoretic approaches to treatment and policy analysis. In this project, we consider optimal treatment assignment problems. A major goal of treatment evaluation in the social and medical sciences is to provide guidance on how to assign individuals to treatments. For example, a number of studies have examined the problem of profiling individuals to identify those likely to benefit from a social program. These empirical studies typically focus on estimation, or inference on the size of the treatment effect. Our research takes a decision-theoretic approach, which connects the statistical analysis of the data to a formal policy decision. In recent work, we show how such an approach can be used to develop optimal procedures for treatment assignment in a wide range of binary, static cases. In the next phase of our work, we will broaden our analysis to a number of situations of practical relevance: settings with multi-valued or continuous treatments; and dynamic treatment assignment problems, where decisions can be made sequentially in response to intermediate outcomes.
Broader Impact: Models with partial identification arise throughout the social and life sciences. Our research will provide estimation and inference tools for researchers in other social sciences, survey analysis, biostatistics, and other fields. Treatment assignment problems and related dynamic programming problems also have broad application. Our research will provide researchers in medicine, biostatistics, and many other fields with procedures to make treatment and policy recommendations optimally in light of past data.
This project considered a number of challenges in econometric data analysis in which the parameter of interest is related in a non-smooth fashion to the distribution of the observed data. Such challenges arise in bounds analysis for missing data problems, estimation of game-theoretic models of strategic interaction, instrumental variables estimation of causal effects from nonexperimental data, and other related empirical settings which have seen wide use in economics, other social sciences, and biostatistics. Using the powerful approximation tools of asymptotic statistical decision theory, we developed new theoretical results that show how the nonsmoothness of the relationship between the observable data distribution and the parameter of interest affects the possibilities for estimation and inference. The project resulted in the paper "Impossibility Results for Nondifferentiable Functionals," (Hirano and Porter, Econometrica), which considers the problem of estimating coefficients that are nondifferentiable functionals of the underlying data distribution. This situation arises in recent work in economics and biostatistics on estimating bounds for treatment effects, estimating parameters of moment inequality models, and estimating optimal dynamic treatment regimes. Existing estimators in these models have nonstandard distributional properties, complicating interpretation and inference using such estimators. We show that this problem is, to some extent, unavoidable: there do not exist locally unbiased, regular, or quantile unbiased estimators for such coefficients. This places sharp limits on bias-reduction methods and conventional inference procedures, and provides a theoretical basis for developing alternative methods of estimation and inference. The project also led to the paper "Location Properties of Point Estimators in Linear Instrumental Variables and Related Models" (Hirano and Porter, forthcoming, Econometric Reviews). This paper considers the standard linear instrumental variables model, which is widely used in economics and other social sciences to estimate causal effects in the presence of nonrandom treatment assignment. We show that the mapping from reduced form distribution to parameters is key to understanding the properties of estimators in these models. This mapping is singular, which rules out the possibility of constructing estimators that are unbiased or have other desirable location properties.
