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will be biased downward relative to the “true” union effect ∆v(c). Assuming that potential wage outcomes are generated by equations (5a, 5b), it can be shown thatthe difference in the variance of wages in the presence of unions and in the counterfactual situation inwhich all workers are paid according to the nonunion wage structure is V- VN= Var[U(c)∆w(c)] + 2Cov[WN(c), U(c)∆w(c)] + E[U(c)∆v(c)] + E[U(c)(1!U(c)){ (θ(c)+∆w(c))2 !θ(c)2 }] .(6)Only the last term of this equation differs from equation (4), the expression that applies when θ(c)=0 forall groups.2In the presence of unobserved heterogeneity, however, ∆w(c) and ∆v(c) can no longer beestimated consistently from the observed differences in the means and variances of union and nonunionworkers in skill group c. By the same token, it is no longer possible to use a reweighting procedure basedon the fraction of union members in different observed skill groups to estimate VN.It is instructive to compare the estimated effect of unions under the “as good as random”assumption to the true effect, when potential wages are generated by equations (5a,5b). The estimatedeffect is given by equation (4), using the observed within-skill group union differences Dw(c) and Dv(c) asestimates of ∆w(c) and ∆v(c). The true effect is given by equation (6). The difference isBias = Var[U(c)Dw(c)] !Var[U(c)∆w(c)] + 2Cov[WN(c), U(c)( Dw(c) !∆w(c) ) ]