Evaluation of the Australian Wage Subsidy Special Youth ...

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4.3 Theory underlying propensity score matching methods
Matching methods are steeped in experimental and evaluation terminology, and the
following exposition which draws on the literature of Rosenbaum and Rubin (1983),
Dehijia and Wahba (1998), Heckman, Ichimura and Todd (1997), Heckman, Ichimura
and Todd (1998) reflects this.
The impact of participating in the programme is defined as
(1) ∆ = Y t – Y c
where Y t is the potential outcome conditional on participation and Y c is the potential
outcome conditional on non-participation. Following Rubin (1974), treatment assignment
may be random given a set of characteristics or variables X that are unaffected by the
treatment. The potential values of X for the different treatment states coincide, and are
thus exogenous. 72
The parameter most commonly estimated is the mean impact of treatment on the treated:
(2) θ = E(∆ | D = 1, X=x)
= E(Y t – Y c | D = 1, X=x)
= E(Y t | D = 1, X=x) - E(Y c | D = 1, X=x)
where D=1 denotes treatment, D=0 denotes non-treatment and X is a set of conditioning
variables. 73 The evaluation problem arises from the term E(Y c | D = 1, X=x). This is the
mean of the counterfactual which, since it is unobservable, must be identified and
estimated. This issue in evaluation was earlier discussed in Chapter 1.
The construction of a valid comparison group for matching techniques is based on the
Conditional Independence Assumption (CIA). Under CIA, potential outcomes Y t ,Y c are
independent of the assignment to treatment, when conditional on all pre-treatment
72 Hujer and Caliendo (2000) p9 explain that this is not exogeneity in the econometric sense. Heckman
Ichimura and Todd (1997) p610 note that it is not necessary to assume that the expected value of the errors
in the model of Y conditional on X are zero, so X can fail to be exogenous in the conventional sense of the
term.
73 Population quantities are denoted by capital letters and sample quantities by lower-case letters.