Motivated by a recent literature on the double-descent phenomenon in machine learning, we consider highly over-parameterized models in causal inference, including synthetic control with many control units. In such models, there may be so many free parameters that the model fits the training data perfectly. We first investigate high-dimensional linear regression for imputing wage data and estimating average treatment effects, where we find that models with many more covariates than sample size can outperform simple ones. We then document the performance of high-dimensional synthetic control estimators with many control units. We find that adding control units can help improve imputation performance even beyond the point where the pre-treatment fit is perfect. We provide a unified theoretical perspective on the performance of these high-dimensional models. Specifically, we show that more complex models can be interpreted as model-averaging estimators over simpler ones, which we link to an improvement in average performance. This perspective yields concrete insights into the use of synthetic control when control units are many relative to the number of pre-treatment periods.

One important application of algorithms is to turn complex data into simple predictions or recommendations that help decision-makers take better decisions. Examples of this include risk assessments presented to judges or doctors. We typically think of such algorithmic assessments as providing additional information about which choices will lead to better outcomes. But when a decision-maker obtains algorithmic assistance, they may not only react to the information. The decision-maker may view the input of the algorithm as recommending a default action, making it costly for them to deviate. In this article, we consider the effect and design of algorithmic recommendations when they affect choices not just by shifting beliefs, but also by altering preferences. We show that recommendation dependence creates inefficiencies where the decision-maker is overly responsive to the recommendation, and propose changes to the design of recommendation algorithms to counteract this response.

We show how to optimally regulate prediction algorithms in a world where (a) high-stakes decisions such as lending, medical testing or hiring are made by a complex 'black-box' prediction functions, (b) there is an incentive conflict between the agent who designs the prediction function and a principal who oversees the use of the algorithm, and (c) the principal is limited in how much she can learn about the agent's black-box model. We show that limiting agents to prediction functions that are simple enough to be fully transparent is inefficient as long as the bias induced by misalignment between principal's and agent's preferences is small relative to the uncertainty about the true state of the world. Algorithmic audits can improve welfare, but the gains depend on the design of the audit tools. Tools that focus on minimizing overall information loss, the focus of many post-hoc explainer tools, will generally be inefficient since they focus on explaining the average behavior of the prediction function rather than those aspects that are most indicative of a misaligned choice. Targeted tools that focus on the source of incentive misalignment, e.g., excess false positives or racial disparities, can provide first-best solutions. We provide empirical support for our theoretical findings using an application in consumer lending.

We investigate the optimal design of experimental studies that have pre-treatment outcome data available. The average treatment effect is estimated as the difference between the weighted average outcomes of the treated and control units. A number of commonly used approaches fit this formulation, including the difference-in-means estimator and a variety of synthetic-control techniques. We propose several methods for choosing the set of treated units in conjunction with the weights. Observing the NP-hardness of the problem, we introduce a mixed-integer programming formulation which selects both the treatment and control sets and unit weightings. We prove that these proposed approaches lead to qualitatively different experimental units being selected for treatment. We use simulations based on publicly available data from the US Bureau of Labor Statistics that show improvements in terms of mean squared error and statistical power when compared to simple and commonly used alternatives such as randomized trials.