Ch. 2: Self-Selection Models in Corporate Finance 55
ing identical treatment probabilities (or propensity scores). Averaging across different
treatment probabilities gives the average treatment effect across the population.^15
4.3.3. Implementation of PS methods
In light ofRosenbaum and Rubin (1983), the treatment effect is the difference between
outcomes of treated and untreated firms with identical propensity scores. One issue
in implementing matching is that we need to know propensity scores, i.e., the treat-
ment probabilitypr(E|Z). This quantity is not ex-ante known but it must be estimated
from the data, using, for instance, probit, logit, or other less parametrically specified
approaches. The corresponding treatment effects are also estimated with error and the
literature develops standard error estimates (e.g.,Heckman, Ichimura and Todd, 1998;
Dehejia and Wahba, 1999; Wooldridge, 2002, Chapter 18).
A second implementation issue immediately follows. What variables do we include
in estimating the probability of treatment? While self-selection models differentiate be-
tween variables determining outcomes and variables determining probability of being
treated (XandZ, respectively, in equations(24)–(26)), matching models make no such
distinction. Roughly speaking, either a variable determines the treatment probability, in
which case it should be used in estimating treatment probability, or it does not, in which
case it should be randomly distributed across treated and untreated firms and is aver-
aged out in computing treatment effects. Thus, for matching models, the prescription is
to use all relevant variables in estimating propensity scores.^16
A third issue is estimation error. In principle, matching demands that treated firms
be compared to untreated firms with the same treatment probability. However, treat-
ment probabilities must be estimated, so exact matching based on the true treatment
probability is usually infeasible. A popular approach, followingDehejia and Wahba
(1999), divides the data into several probability bins. The treatment effect is estimated
as the average difference between the outcomes ofEandNEfirms within each bin.
Heckman, Ichimura and Todd (1998)suggest taking the weighted average of untreated
firms, with weights declining inversely in proportion to the distance between the treated
and untreated firms. For statistical reasons,Abadie and Imbens (2004)suggest that the
counterfactual outcomes should be estimated not as the actual outcomes for a matched
untreated firm, but as the fitted value in a regression of outcomes on explanatory vari-
ables.^17
(^15) This discussion points to another distinction between PS and selection methods. The finest level to which
PS methods can go is the propensity score or the probability of treatment. Because many firms can have the
same propensity score, PS methods do not estimate treatment effects at the level of the individual firm, while
selection methods can do so.
(^16) This statement is not, of course, a recommendation to engage in data snooping. For instance, in fitting
models to estimate propensity scores, using quality of fit as a model selection criterion leads to difficulties, as
pointed out byHeckman and Navarro-Lozano (2004).
(^17) The statistical properties of different estimators has been extensively discussed in the econometrics lit-
erature, most recently in a review issue devoted to the topic (Symposium on the Econometrics of Matching,
Review of Economics and Statistics 86 (1), 2004).