Evaluation of the Australian Wage Subsidy Special Youth ...

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occur by design, because the missing-ness usually does depend on recorded values
(Dillman et al. (2002): 17).
The possibility that the data are missing for reasons of self-selection raises problems in
econometric models of outcomes. In other words, attrition is most problematic when it is
related to the endogenous variable. The data are then not missing completely at random.
If the probability of attrition is related to the outcomes, then the estimates are biased and
inconsistent.
The 2 period model is as follows:
(1) y it = β ‘ x it + v it
(2) a i =1 if y i2 is observed,
a i =0 if y i2 is not observed
(3) a i =1 if a i * = γ y i2 + θ ‘x i2 + δ ‘ w i + ε i * ≥ 0
where time T=2, v it is the error term, y it is the endogenous variable, a i is the attrition
dummy, a i * is the latent variable for attrition, w i is the vector of variables that do not
enter the conditional expectation of y but affect the probability of observing y, θ and δ are
parameters, and ε i * are normally distributed. A common procedure for analysis where
there is attrition, is to use only those cases present in all surveys, also called a balanced
panel. Attrition in the second period would mean that in a balanced panel, the
observations for which y i2 is missing are discarded.
If the probability of observing y i2 varies with the value of attrition , as well as the value of
other variables, then the probability of observing y i2 depends on the error term v i2 . The
reduced form, where y i2 is substituted into equation (3), becomes:
(4) a i * = (γ β ‘ + θ ‘)x i2 + δ ‘ w i + γ v i2 + ε i *
This can be rewritten as:
(5) a i * = π ‘x i2 + δ ‘ w i + γ v i2 + ε i
and finally collecting terms in matrix notation: