54 K. Li and N.R. Prabhala
Virtually all studies routinely match on size, industry, the book-to-market ratio, and so
on. The “treatment effect” is the matched-pair difference in outcomes. There is nothing
inherently wrong with these methods. They involve the same economic assumptions
as other matching methods based on propensity scores used in recent applications. In
fact, dimension-by-dimension matching imposes less structure and probably represents
a reasonable first line of attack in typical corporate finance applications.
Matching on all dimensions and estimating the matched-pair differences in outcomes
poses two difficulties. One is that characteristics are not always exactly matched in cor-
porate finance applications. For instance, we often match firm size or book-to-market
ratios with 30% calipers. When matches are inexact, substantial biases could build up
as we traverse different characteristics being matched. A second issue that proponents
of matching methods frequently mention is dimensionality. When the number of di-
mensions to be matched goes up and the matching calipers become fine (e.g., size and
prior performance matched within 5% rather than 30%, and 4-digit rather than 2-digit
SIC matches), finding matches becomes difficult or even infeasible. When dimension-
by-dimension matching is not feasible, a convenient alternative is methods based on
propensity scores. We turn to these next.
4.3.2. Propensity score (PS) matching
Propensity score (PS) matching methods handle the problems caused by dimension-
by-dimension matching by reducing it to a problem of matching on a single one: the
probability of undergoing treatmentE. The probability of treatment is called thepropen-
sity score. Given a probability model such as equation(24), the treatment effect equals
the outcome for the treated firm minus the outcome for an untreated firm with equal
treatment probability. The simplicity of the estimator and its straightforward interpreta-
tion makes the propensity score estimator attractive.
It is useful to review the key assumptions underlying the propensity score method.
FollowingRosenbaum and Rubin (1983), suppose that the probability model in equa-
tion(24)satisfies
- PS1: 0<pr(E|Z) <1.
- PS2: GivenZ, outcomesYE,YNEdo not depend on whether the firm is in groupE
(NE).
Assumption (PS1) requires that at each level of the explanatory variableZ,some
firms should pickEand others pickNE. This constraint is frequently imposed in em-
pirical applications by requiring that treated and untreated firms have common support.
Assumption (PS2) is thestrong ignorabilityor conditional independence condition.
It requires that unobserved private information should not explain outcome differentials
between firms choosingEand those choosingNE. This is a crucial assumption. As
Heckman and Navarro-Lozano (2004)show, even fairly mild departures can trigger
substantial biases in treatment effect estimates.
Given assumptions (PS1) and (PS2),Rosenbaum and Rubin (1983)show that the
treatment effect is the difference between outcomes of treated and untreated firms hav-