This article discusses identification in continuous triangular systems without restrictions on heterogeneity or functional form. We do not assume separability of structural functions, restrictions on the dimensionality of unobservables, or monotonicity in unobservables. We do maintain monotonicity of the first stage relationship in the instrument and consider the case of real-valued treatment. Under these conditions alone, and given rich enough support of the data, potential outcome distributions, the average structural function, and quantile structural functions are point identified. If the support of the continuous instrument is not large enough, potential outcome distributions are partially identified. If the instrument is discrete, identification fails completely. If treatment is multi-dimensional, additional exclusion restrictions yield identification. The set-up discussed in this article covers important cases not covered by existing approaches such as conditional moment restrictions (cf. Newey and Powell, 2003) and control variables (cf. Imbens and Newey, 2009). It covers, in particular, random coefficient models, as well as systems of structural equations.

We discuss the tension between “what we can get” (identification) and “what we want” (parameters of interest) in models of policy choice (treatment assignment). Our nonstandard empirical object of interest is the ranking of counterfactual policies. Partial identification of treatment effects maps into a partial welfare ranking of treatment assignment policies. We characterize the identified ranking and show how the identifiability of the ranking depends on identifying assumptions, the feasible policy set, and distributional preferences. An application to the project STAR experiment illustrates this dependence. This paper connects the literatures on partial identification, robust statistics, and choice under Knightian uncertainty.

Many settings in empirical economics involve estimation of a large number of parameters. In such settings, methods that combine regularized estimation and data-driven choices of regularization parameters are useful. We provide guidance to applied researchers on the choice between regularized estimators and data-driven selection of regularization parameters. We characterize the risk and relative performance of regularized estimators as a function of the data-generating process and show that data-driven choices of regularization parameters yield estimators with risk uniformly close to the risk attained under the optimal (unfeasible) choice of regularization parameters. We illustrate using examples from empirical economics.

Some empirical results are more likely to be published than others. Selective publication leads to biased estimates and distorted inference. We propose two approaches for identifying the conditional probability of publication as a function of a study’s results, the first based on systematic replication studies and the second on meta-studies. For known conditional publication probabilities, we propose bias-corrected estimators and confidence sets. We apply our methods to recent replication studies in experimental economics and psychology, and to a meta-study on the effect of the minimum wage. When replication and meta-study data are available, we find similar results from both.

Standard experimental designs are geared toward point estimation and hypothesis testing, while bandit algorithms are geared toward in‐sample outcomes. Here, we instead consider treatment assignment in an experiment with several waves for choosing the best among a set of possible policies (treatments) at the end of the experiment. We propose a computationally tractable assignment algorithm that we call “exploration sampling,” where assignment probabilities in each wave are an increasing concave function of the posterior probabilities that each treatment is optimal. We prove an asymptotic optimality result for this algorithm and demonstrate improvements in welfare in calibrated simulations over both non‐adaptive designs and bandit algorithms. An application to selecting between six different recruitment strategies for an agricultural extension service in India demonstrates practical feasibility.

We propose to use economic theories to construct shrinkage estimators that perform well when the theories' empirical implications are approximately correct but perform no worse than unrestricted estimators when the theories' implications do not hold. We implement this construction in various settings, including labor demand and wage inequality, and estimation of consumer demand. We provide asymptotic and finite sample characterizations of the behavior of the proposed estimators. Our approach is an alternative to the use of theory as something to be tested or to be imposed on estimates. Our approach complements uses of theory for identification and extrapolation.

We consider the problem of repeatedly choosing policies to maximize social welfare. Welfare is a weighted sum of private utility and public revenue. Earlier outcomes inform later policies. Utility is not observed, but indirectly inferred. Response functions are learned through experimentation. We derive a lower bound on regret, and a matching adversarial upper bound for a variant of the Exp3 algorithm. Cumulative regret grows at a rate of T 2/3 . This implies that (i) welfare maximization is harder than the multiarmed bandit problem (with a rate of T 1/2 for finite policy sets), and (ii) our algorithm achieves the optimal rate. For the stochastic setting, if social welfare is concave, we can achieve a rate of T 1/2 (for continuous policy sets), using a dyadic search algorithm. We analyze an extension to nonlinear income taxation, and sketch an extension to commodity taxation. We compare our setting to monopoly pricing (which is easier), and price setting for bilateral trade (which is harder).