Model Selection for Treatment Choice: Penalized Welfare Maximization

Abstract

This paper studies a new statistical decision rule for the treatment assignment problem.Consider a utilitarian policy maker who must use sample data to allocate one of two treatments to members of a population, based on their observable characteristics. In practice, it is often the case that policy makers do not have full discretion on how these covariates can be used, for legal, ethical or political reasons. We treat this constrained problem as a statistical decision problem, where we evaluate the performance of decision rules by their maximum regret. We focus on settings in which the policy maker may want to select amongst a collection of such constrained classes: examples we consider include choosing the number of covariates over which to perform best-subset selection, and model selection when approximating a complicated class via a sieve. We adapt and extend results from statistical learning to develop a decision rule which we call the Penalized Welfare Maximization (PWM) rule. We establish an oracle inequality for the regret of the PWM rule which shows that it is able to perform model selection over the collection of available classes. We then use this oracle inequality to derive relevant bounds on maximum regret for PWM. We illustrate the model-selection capabilities of our method with a small simulation exercise, and conclude by applying our rule to data from the Job Training Partnership Act (JTPA) study.

Empirical Welfare Maximization with Constraints

“…Dhailiwal et al (2013), and to the recent literature on EWM, e.g. Kitagawa and Tetenov (2018), Mbakop and Tabord-Meehan (2019), Rai (2019) and Athey and Wager (2020).…”

“…Athey and Wager (2020) propose doubly-robust estimation of the average benefit, which leads to an optimal rule even with quasi-experimental data. Mbakop and Tabord-Meehan (2019) propose a Penalized Welfare Maximization assignment rule which relaxes restrictions of the criterion class.…”

Policy Transforms and Learning Optimal Policies

“…However, there is now a large and growing literature on statistical treatment rules in econometrics, including papers by Manski (2004), Hirano and Porter (2009), Stoye (2009), Stoye (2012), Chamberlain (2011), Tetenov (2012, Kasy (2016), Kitagawa and Tetenov (2018) and Mbakop and Tabord-Meehan (2019). In general these papers can be divided according to (i) whether they are frequentist/bayesian, (ii) whether they take a finitesample or asymptotic approach, and (iii) whether they consider decision problems under uncertainty or ambiguity (or "Knightian uncertainty").…”

“…This has the benefit of allowing the policymaker to avoid relying on any specific properties of the underlying function class, which are typically difficult to verify, and thus are applicable whether or not the associated policy space is learnable. Furthermore, the use of data-dependent complexity measures like the empirical Rademacher complexity ensures our finite sample 5 Kitagawa and Tetenov (2018) and Mbakop and Tabord-Meehan (2019) make some connections with the statistical learning literature. However, their method of evaluating statistical treatment rules is different from that considered by the PAC model.…”

Fair Policy Targeting

“…This paper relates to a growing literature on statistical treatment rules (Armstrong and Shen, 2015;Athey and Wager, 2017;Bhattacharya and Dupas, 2012;Dehejia, 2005;Hirano and Porter, 2009;Tetenov, 2018, 2019;Mbakop and Tabord-Meehan, 2016;Stoye, 2012;Tetenov, 2012;Viviano, 2019;Zhou et al, 2018). Further research on optimal treatment allocations also includes estimation of individualized optimal treatments via residuals weighting (Zhou et al, 2017), penalized methods (Qian and Murphy, 2011), inference on the welfare for optimal treatment strategies (Andrews et al, 2019;Labe et al, 2014;Luedtke and Van Der Laan, 2016;Rai, 2018), doubly robust methods for treatment allocations (Dudik et al, 2011;Zhang et al, 2012), reinforcement learning (Kallus, 2017;Lu et al, 2018), and dynamic treatment regimes (Murphy, 2003;Nie et al, 2019).…”