Identification and Efficiency Bounds for the Average Match Function Under Conditionally Exogenous Matching

ABSTRACT

Consider two heterogenous populations of agents who, when matched, jointly produce an output, Y. For example, teachers and classrooms of students together produce achievement, parents raise children, whose life outcomes vary in adulthood, assembly plant managers and workers produce a certain number of cars per month, and lieutenants and their platoons vary in unit effectiveness. Let $W\in \mathbb{W}=\{w_1,\dots ,w_J\}$ and $X\in \mathbb{X}=\{x_1,\dots ,x_K\}$ denote agent types in the two populations. Consider the following matching mechanism: take a random draw from the W = w_j subgroup of the first population and match her with an independent random draw from the X = x_k subgroup of the second population. Let β(w_j, x_k), the average match function (AMF), denote the expected output associated with this match. We show that (i) the AMF is identified when matching is conditionally exogenous, (ii) conditionally exogenous matching is compatible with a pairwise stable aggregate matching equilibrium under specific informational assumptions, and (iii) we calculate the AMF’s semiparametric efficiency bound.

KEY WORDS:

• Average match function
• Causal inference
• Choo–Siow model
• Matching
• Semiparametric efficiency.

ACKNOWLEDGMENTS

This article was presented at the 2012 European Summer Meetings of the Econometric Society, the December 2012 SFB 884 Research Conference on the Evaluation of Political reforms at the University of Mannheim, and at seminars hosted by the University of California - Berkeley, University of California - Davis, University of Wisconsin - Madison, Northwestern University, the University of Southern California, and Singapore Management University. The authors thank these seminar audiences for useful feedback. The authors are also grateful to Stephane Bonhomme, Richard Blundell, Konrad Menzel, and James Powell for useful discussions. The current draft reflects valuable suggests made by a co-editor, associate editor, and two referees. Financial support from the National Science Foundation (SES #0820361) for the first author’s contribution is gratefully acknowledged. All the usual disclaimers apply.

Funding

Division of Social and Economic Sciences [0820361].

Notes

1 In what follows random variables are denoted by capital Roman letters, specific realizations by lower case Roman letters, and their support by blackboard bold Roman letters. That is, Y, y, and $\mathbb{Y}$, respectively, denote a generic random draw of, a specific value of, and the support of, Y.

2 If we consider classroom h’s “treatment” to be the assignment to a specific teacher, then the fact that classroom h has a different potential outcome when assigned teacher i vs. teacher i′ is not a violation of SUTVA (see Imbens and Rubin 2015). However, there is a violation of SUTVA if, instead, we consider the type of i as the treatment (e.g., assignment to an inexperienced vs. experienced teacher). This follows, as explained in the main text, because teachers of the same observed type may vary in terms of unobserved, output-effecting, attributes. Another violation of SUTVA implicit in our setup is that of no treatment interference. Interference in our setting arises because if classroom h is assigned to teacher i, then classroom h′ cannot be assigned to teacher i; matching is one-to-one and rivalrous. Graham (2011a, Section 5) anticipated some of the discussion which follows.

3 The “inclusive definition of type” assumption imposes independence of, in the current notation, W_i and U_i and also of X^h and V^h. This assumption, which also features in Graham, Imbens, and Ridder (2010), is not imposed here. This difference is consequential when controlling for additional covariates. This can be seen by comparing the form of the identification result in Graham (2011a, Proposition 3.2) with the one outlined below. Here, identification involves averaging over the product of two conditional distributions, not two marginals as in Graham (2011a). Analog estimators based on the two results will numerically differ. Graham (2011a) did not derive a semiparametric efficiency bound for his estimand, but it too will differ from the one outlined here. We prefer the set of assumptions maintained here (which are weaker).