will be biased downward relative to the true union effect v c Assuming that

Will be biased downward relative to the true union

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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 that the difference in the variance of wages in the presence of unions and in the counterfactual situation in which all workers are paid according to the nonunion wage structure is V - V N = Var[ U ( c ) w ( c )] + 2Cov[ W N ( 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 for all groups. 2 In the presence of unobserved heterogeneity, however, w ( c ) and v ( c ) can no longer be estimated consistently from the observed differences in the means and variances of union and nonunion workers in skill group c . By the same token, it is no longer possible to use a reweighting procedure based on the fraction of union members in different observed skill groups to estimate V N . 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 estimated effect is given by equation (4), using the observed within-skill group union differences D w ( c ) and D v ( c ) as estimates of w ( c ) and v ( c ). The true effect is given by equation (6). The difference is Bias = Var[ U ( c ) D w ( c )] ! Var[ U ( c ) w ( c )] + 2Cov[ W N ( c ), U ( c )( D w ( c ) ! w ( c ) ) ]
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