Evaluation of the Australian Wage Subsidy Special Youth ...

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The treatment parameter, θ, cannot be correctly identified for values of X which do not
satisfy this support condition. Matching operates by constructing, for those participants
with support, a counterfactual from the non-participants. There are a number of ways of
defining this counterfactual. 74 Once the counterfactuals are identified, the mean impact of
the programme can be estimated as the mean difference in the outcomes of the matched
pairs.
A refinement to the matching approach was introduced by Rosenbaum and Rubin (1983).
Rosenbaum and Rubin (1983) suggest the use of balancing scores which are functions of
the relevant observed covariates (here termed X). If assumptions (3) and (4) hold then
Rosenbaum and Rubin (1983) show that the conditional independence assumption (CIA)
extends to the use of the propensity score:
(5) Y c ╨ D | P(X) = p(x), ∀x∈ χ
where ( ╨ ) symbolises independence, P(X) is the propensity score more fully termed
P(D=1| X), the probability of participating in the programme. The important advantage of
Rosenbaum and Rubin’s (1983) innovation is that the dimensionality of the match can be
reduced to one. 75 Rather than matching on a vector of characteristics, it is possible to
match on just the propensity score. The information from the vector of characteristics is
condensed within the estimated probability representing the propensity score. Having
done so, the mean impact of the programme is again estimated as the mean difference in
the outcomes of the matched pairs:
(6) E[Y c | D=1, P(X)=P(x)] = E[Y c | D=0, P(X)=P(x)] = E(Y c | P(X)=P(x)]
So then the average treatment effect can be written:
74 See, for example, Heckman et al. (1997) for a comparison of alternative matching schemes.
75 Matching on the propensity score removes the dimensionality problem if the propensity scores are
estimated parametrically, and relies on parametric estimation of the probability of participation providing
an advantage over parametric estimation of the outcome.