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Unformatted text preview: Detecting Discrimination in Audit and Correspondence Studies Audit David Neumark Audit and correspondence studies Audit Fictitious individuals who are identical except for race, sex, or Fictitious ethnicity apply for jobs ethnicity Audit studies – actual testers, observe job offers Correspondence studies – paper/on-line applications, observe callbacks Evidence of group differences in outcomes – for example, blacks Evidence getting fewer job offers than whites – is viewed as compelling evidence of discrimination (Pager, 2007; Riach and Rich, 2002) evidence A/C studies nearly unanimous in finding evidence of A/C discrimination discrimination Blacks, Hispanics, and women in the United States (Mincy, 1993; Blacks, Neumark, 1996; Bertrand and Mullainathan [BM], 2004) Neumark, Moroccans in Belgium and the Netherlands (Smeeters and Nayer, Moroccans 1998; Bovenkerk et al., 1995) 1998; Lower castes in India (Banerjee et al., 2008) Criticisms: controlling for mean differences differences Difficult to control for experimenter effects Difficult to control for all productivity-relevant differences Difficult employers can observe employers E.g., in Urban Institute study, white and minority testers were aware E.g., of the purpose of the test, and even – in their training – informed about “the pervasive problem of discrimination in the United States” Arguments sometimes a stretch, and surely better than regression Arguments studies (“residual approach” to discrimination) studies Controlling for subset of relevant characteristics can lead to more Controlling bias than not controlling for any, when characteristics controlled and not controlled are negative correlated, and can exacerbate effects of unimportant differences (but is negative correlation likely?) unimportant These criticisms (Heckman and Siegelman [HS], 1993) have been These addressed in many ways, including switch from audit to correspondence studies correspondence Criticisms: distributional differences Criticisms: Best case scenario: no mean differences in observables or Best unobservables unobservables More likely to hold in correspondence study No average observable differences between groups, and information No rich enough that no reason to assume unobservable group differences differences HS show that even this case is problematic Differences in variance of unobservable can lead to over- or underestimate of discrimination (equivalently, effect of discrimination on estimate hiring is unidentified) hiring Potentially devastating criticism that has been ignored (most recently, Potentially Pager’s 2007 review of the contributions and critiques of these studies) studies) Assumption of different variances of unobservables is common in Assumption models of statistical discrimination (Aigner and Cain, 1977; Lundberg and Startz, 1983) and Goals of paper Goals Develop method to (1) test for difference in variance of Develop unobservables and (2) identify discrimination even when variances differ, in correspondence study variances Requires data that are not usually collected in correspondence Requires study, but can be easily added so that this method can be easily implemented implemented Implement method using existing data set from one Implement correspondence study (BM) that has such data correspondence Assess estimator Intuition behind problem of differences in variance of unobservables Ignore any mean differences Correspondence study controls for one characteristic XI, and and employers care about sum of XI and XIIII employers XI and XIIII uncorrelated (not required), and XIIII not observed uncorrelated not Hire only if expected sum exceeds some critical value (hiring Hire standard) with sufficiently high probability standard) XI low relative to hiring standard – employer will favor group with high variance of unobservable, since there is higher probability that this group will high expected sum XI + XII that XI high relative to hiring standard – employer less likely to hire from group with high variance of unobservable, since this group is more likely to have low sum XI + XII more Statistical discrimination models assume higher variance for Statistical blacks vs. whites, but we don’t know how this will bias results (if true) since we don’t know whether XI is high or low relative to true) Intuition behind solution Intuition In hiring/callback probit we only identify ratios of coefficients (in In latent variable model) to standard deviation of unobservable latent In typical study, latent variable model coefficients of “controls” like In resume characteristics should be zero, since applicants designed to be equally qualified to If data also include applicants with different levels of If qualifications, coefficients should be non-zero, so we can learn something from these data something If we rule out differences in latent variable coefficients of these If variables for blacks and whites (e.g.), then differences in probit coefficients on these variables are informative about differences in standard deviation on unobservables standard Formal setup Formal Let productivity depend on two individual characteristics X’ = (XI,XII) Let R be a dummy for race, with R = equal to 1 for blacks and 0 Let for whites Assume productivity is additive P(X’,F) = XI + XII + F Treatment of a worker (valuation of productivity, implicitly or Treatment explicitly) depends on P and possibly R (if there is discrimination), assumed also additive, so assumed T(P(X’,F),R) = P + γ ’R T(P(X’,F),R) Discrimination against blacks implies γ ’ < 0 Discrimination Simple example would be determination of wages in Becker’s Simple employer discrimination model employer Audit or correspondence study Audit Two testers (one with R = 1 and one with R = 0 in each pair) Two are sent to firms to apply for jobs are Researcher attempts to standardize productivity based on Researcher observable characteristics Denote expected productivity for blacks and whites, based on Denote what the firm observes, as PB* and PW* what Study tries to set PB* = PW* Outcome T is observed for each tester, so each “test” yields Outcome observation on T(PB*,1) − T(PW*,0) = PB* + γ ’ − PW* observation ,1) If PB* = PW*, then averaging across tests yields an estimate of *, γ’ Alternatively, we can estimate γ ’ from regression of T on R Alternatively, from Unbiased test requires equal means for uncontrolled variables uncontrolled Audit study controls only one component of productivity, e.g., Audit setting XBI = XWI = XI*, and XII is unobserved setting In this case each test yields an observation on T(PB*,1) − T(PW*,0) = PB* + γ ’ − PW* ,1) = XBI + E(XBII) + γ ’ − (XWI + E(XWII)) = γ ’ + E(XBII) − E(XWII) )) Study yields an unbiased estimate only if E(XBII) = E(XWII) Audit studies can control for some observable qualifications, Audit but not possible to control for all observable differences but Correspondence studies instead send resumes, typically Correspondence randomized by group, so there are no observable differences, and perhaps also no unobserved differences and But equal means is necessary, not sufficient for unbiasedness Taste and statistical discrimination Taste In correspondence study, still can’t rule out differences in group In means of unobservables (uncontrolled), in which case we estimate γ + E(XBII) − E(XWII), since E(XBII) ≠ E(XWII) In this case, estimated difference reflects taste discrimination plus In mean difference in unobservable assumed by employer (statistical discrimination) With rich controls, though, one might be skeptical about With unobserved mean differences unobserved EEOC describes as illegal “employment decisions based on EEOC stereotypes or assumptions about the abilities, traits, or performance of individuals of a certain sex, race, age, religion, or ethnic group” So correspondence study may estimate “illegal” discrimination, So without isolating taste discrimination without Some recent correspondence studies try to distinguish taste from Some statistical discrimination (difficult) statistical Distributional differences in best-case scenario scenario Consensus that correspondence studies meet higher standard Consensus of validity of But HS show that even in best case, when group means equal But for uncontrolled variables (conditional on observed, or simply all variables), results can be biased all Key reason is that outcome is hiring, dependent on expected Key productivity exceeding a threshold productivity Bias from distributional differences (I) Bias HS: even with group means equal for uncontrolled variables HS: results can be biased because hiring outcome depends on expected productivity exceeding a threshold (c) Hiring rules are Hiring T(P(X’,F)|R = 1) = 1 if βI’XBI + XBII + γ ’ > c’ T(P(X’,F)|R = 0) = 1 if βI’XWI + XWII > c’ T(P(X’,F)|R Audit study controls for XI, with XBI = XWI = XI* XBII and XWII are normally distributed, with mean zero (without loss of generality) and standard deviations σBII and σWII Bias from distributional differences (II) Bias Probabilities that blacks and whites are hired are Probabilities Pr[T(P(XI*,XBII)|R = 1) = 1] = Φ[( βI’XI* + γ ’ − c’)/σBII] )|R [( Pr[T(P(XI*,XWII) |R = 0) = 1] = Φ[( βI’ XI* − c’)/σWII] |R [( Difference between two expressions (or ratio) is supposed to Difference be informative about discrimination, but we can’t distinguish influence of nonzero γ from unequal standard deviations σWII ≠ σBI E.g., σWII > σBII, and XI* set at low level E.g., With βI’XI* < c’, σWII > σBII, first probability (black callback) lower With first than second even if γ ’ = 0 Different outcomes possible depending on relative magnitudes of Different σWII vs. σBII and βI’XI* (or XI*) vs. c’ So with different variances of unobservables, effect of So discrimination on callbacks is unidentified discrimination Bias from distributional differences (III) differences Heckman (1998) X2 is unobservable, superscript “1” denotes blacks, and c1, c0 are hiring thresholds So despite absence of discrimination, can get evidence in either direction E.g., with higher var. for whites (0), and low level of standardization, we “find” discrimination against blacks, because employers want high X2 when X1 low Similar spurious evidence can occur in cases when there is discrimination Solution(I) Solution(I) A higher variance for one group, cet. par., implies a smaller higher effect of observed characteristics on employment for that group group Intuition: in the limit, if the variance of unobserved XII were infinite for a group, then XI (the observed productivity-related infinite variable) would have no effect on the evaluation of whether an applicant from that group meets the standard for hiring applicant So information from a correspondence study on how variation So in observable qualifications is related to employment outcomes can be informative about the relative variances of the unobservables the And since we saw that identification problem comes down to And distinguishing between relative variance of unobservable and γ ’, this can identify discrimination Solution (II) Solution We are interested in We Φ[( βI’XI* + γ ’ − c’)/σBII] − Φ[( βI’XI* − c’)/σWII] [( [( Simplify by normalizing one variance, and writing the other in Simplify terms of the relative variance σBRII = σBII/σWII, so we are now so estimating Φ[( β1XI* + γ − c)/σBRII] − Φ[βIXI* − c] [( where now all the coefficients are normalized with respect to where σWII So identification of γ now gives us the marginal effect of So race race With meaningful variation in XI* we can identify βI/σBRIIII and γ /σBRII from the probit for blacks, and βI from the probit for from whites, in which case the ratio identifies σBRII, and therefore γ and With joint estimation, we can do inference on the parameter σ With , II Identifying assumption Identifying Assumption that βI is the same for blacks and whites is Assumption necessary for the ratio of the two coefficients to identify σBRII Untestable with data on only one productivity-related Untestable characteristic characteristic Multiple productivity-related characteristics yield testable Multiple restriction E.g., if there are two such characteristics we estimate E.g., * Φ[( βIBXI* + βIIBZI* + γ − c)/σBRII] − Φ[( βIWXII* + βIIWZI* − c)] [( [( allowing different coefficients for blacks and whites If the only reason coefficients for blacks and whites differ is If because σBRII ≠ 1, then we must have βIB/βIW = βIIB/βIIW So I start by estimating a probit model with a full set of race So interactions, and testing this constraint interactions, In application, constraint is not rejected Implementation Implementation Estimation of βI/σBRII and βI can be done via a heteroskedastic Estimation probit model (e.g., Williams, 2009) that allows the variance of the unobservable to vary with race the We pool the data for blacks and whites, and estimate a probit We model with Var(εij) = [exp(µ + ωRi)]2 model Normalize µ = 0 (equivalent to standard normalization in probit, but for just whites in this case) but Estimate via MLE Estimate of exp(ω) iis exactly the estimate of σBRII s Marginal effects (I) Marginal Typically, to translate probit coefficient estimates into Typically, magnitudes that can be interpreted as the marginal effects of a variable (Zk, generically, with coefficient βk, when Z is the vector variable when of controls with coefficients β), we use ∂P(hire)/∂Zk = βkφ(Zβ) φ(.) is the standard normal density, standard deviation of the (.) unobservable is normalized to 1 unobservable Evaluated at the means of Z When Zk is a dummy variable – such as race – the difference in the cumulative normal distribution functions is often used instead, although the difference is usually trivial although In heteroskedastic probit model, if variances of unobservables In differ by race, then when race “changes” both the variance and the level of the latent variable (the valuation of a worker’s productivity) that determines hires can shift productivity) Marginal effects (II) Marginal I want to isolate effect on latent variable, since differential want treatment of blacks and whites based only on differences in variances of unobservables should not be interpreted as discrimination (more later) discrimination With continuous version of partial derivative, can decompose the With effect of a change in Zk into these two components effect For HP model with Var(εij) = [exp(Wω)], overall partial derivative [exp(Wω)], of P(hire) with respect to Zk is of ∂P(hire)/∂ Zk = φ{Zβ/exp(Wω)}∙{(βk – Zβ∙ωk)}/exp(Wω) First term (before minus sign) is partial derivative with respect to First changes in Zk affecting only the level of the latent variable – changes corresponding to the counterfactual of Zk changing the valuation corresponding of the worker without changing the variance of the unobservable of Second term is partial derivative with respect to changes via the Second Application/prior evidence (I) Application/prior Bertrand and Mullainathan (2004) do correspondence study Bertrand of race discrimination, based on black-sounding names of Explicitly designed resumes of varying quality to test whether Explicitly returns to higher qualifications were lower for blacks returns Evidence of discrimination against blacks The quality variation does matter Evidence generally consistent with lower returns to higher Evidence qualifications for blacks qualifications Replication (marginal effects) Replication Black Female Selected individual resume controls Bachelor’s degree Males and females (1) (2) (3) ­.033 ­.030 ­.030 (.006) .009 (.012) (6) ­.030 (.008) … (.007) … (.007) … Academic honors Special skills Other controls: Individual resume characteristics (.006) .001 (.011) .009 (.009) .076 (.028) ­.021 (.010) .040 (.015) .055 (.009) .019 (.010) .080 (.034) ­.019 (.013) .026 (.017) .060 (.010) .019 (.010) .076 (.033) ­.018 (.012) 028 (.017) .059 (.010) X Experience2 ∙10­2 (.006) ­.001 (.011) .009 (.009) .080 (.029) ­.022 (.011) .039 (.015) .056 (.009) Experience ∙10­1 X X X Neighborhood characteristics Mean callback rate (4) ­.033 Females (5) ­.030 X .080 4,784 .080 4,784 .080 4,784 X .082 3,670 .082 3,670 .082 3,670 Application/prior evidence (I) Application/prior Bertrand and Mullainathan (2004) do correspondence study Bertrand of race discrimination, based on black-sounding names of Explicitly designed resumes of varying quality to test whether Explicitly returns to higher qualifications were lower for blacks returns Evidence of discrimination against blacks The quality variation does matter Evidence generally consistent with lower returns to higher Evidence qualifications for blacks qualifications Replication Replication Black Female Selected individual resume controls Bachelor’s degree Males and females (1) (2) ­.033 ­.030 (.006) (.006) .009 ­.001 (.012) (.011) (3) ­.030 (.006) .001 (.011) (4) ­.033 (.008) … Females (5) ­.030 (.007) … (6) ­.030 (.007) … .009 .009 .019 .019 Experience ∙10­1 (.009) .080 (.009) .076 (.010) .080 (.010) .076 Experience2 ∙10­2 (.029) ­.022 (.028) ­.021 (.034) ­.019 (.033) ­.018 Academic honors (.011) .039 (.010) .040 (.013) .026 (.012) 028 Special skills (.015) .056 (.015) .055 (.017) .060 (.017) .059 (.009) (.009) (.010) (.010) X X X X Other controls: Individual resume characteristics Neighborhood characteristics Mean callback rate X .080 4,784 .080 4,784 .080 4,784 X .082 3,670 .082 3,670 .082 3,670 Application/prior evidence (I) Application/prior Bertrand and Mullainathan (2004) do correspondence study of Bertrand race discrimination, based on black-sounding names race Explicitly designed resumes of varying quality to test whether Explicitly returns to higher qualifications were lower for blacks returns Evidence of discrimination against blacks The quality variation does matter Evidence generally consistent with lower returns to higher Evidence qualifications for blacks qualifications Application/prior evidence (II) Application/prior Lower coefficients for blacks are consistent with a larger Lower variance for blacks, i.e., σBRII > 1 If BM study has chosen low levels of the control variables on If which to standardize applicants – and BM explicitly state that they tried to avoid overqualification even of the higher-quality resumes (p. 995) – then the HS analysis would imply that there is a bias towards finding discrimination in favor of blacks blacks With low level of standardization, employers want higher With probability of high value of unobservable probability In this case, evidence of discrimination would be stronger In absent the bias from differences in the distribution of unobservables unobservables We don’t really know whether level of controls is low or high We relative to hiring standard relative Fully interactive specifications Fully A. Estimates from basic probit (Table 1) Black B. Heteroskedastic probit model Black (unbiased estimates) Effect of race through level Effect of race through variance Standard deviation of unobservables, black/white Wald test statistic, null hypothesis that ratio of standard deviations = 1 (p­value) Wald test statistic, null hypothesis that ratios of coefficients for whites relative to blacks are constant, fully interactive probit model (p­value) Test overidentifying restrictions: include in heteroskedastic probit model interactions for variables with |white coefficient| < |black coefficient|, Wald test for joint significance of interactions (p­value) Number of overidentifying restrictions Other controls: Individual resume characteristics Neighborhood characteristics Males and females (1) (2) (3) Females (4) ­.030 (.006) ­.030 (.006) ­.030 (.007) ­.030 (.007) ­.024 (.007) ­.086 (.038) .062 (.042) 1.37 ­.026 (.007) ­.070 (.040) .044 (.043) 1.26 ­.026 (.008) ­.072 (.040) .046 (.045) 1.26 ­.027 (.008) ­.054 (.040) .028 (.044) 1.15 .22 .37 .37 .56 .62 .42 .17 .35 .83 .33 .34 .56 3 6 2 6 X X X X X X Fully interactive specifications Fully A. Estimates from basic probit (Table 1) Black B. Heteroskedastic probit model Black (unbiased estimates) Effect of race through level Effect of race through variance Standard deviation of unobservables, black/white Wald test statistic, null hypothesis that ratio of standard deviations = 1 (p­value) Wald test statistic, null hypothesis that ratios of coefficients for whites relative to blacks are constant, fully interactive probit model (p­value) Test overidentifying restrictions: include in heteroskedastic probit model interactions for variables with |white coefficient| < |black coefficient|, Wald test for joint significance of interactions (p­value) Number of overidentifying restrictions Other controls: Individual resume characteristics Neighborhood characteristics Males and females (1) (2) (3) Females (4) ­.030 (.006) ­.030 (.006) ­.030 (.007) ­.030 (.007) ­.024 (.007) ­.086 (.038) .062 (.042) 1.37 ­.026 (.007) ­.070 (.040) .044 (.043) 1.26 ­.026 (.008) ­.072 (.040) .046 (.045) 1.26 ­.027 (.008) ­.054 (.040) .028 (.044) 1.15 .22 .37 .37 .56 3 6 2 6 X X X X X X Based on ratio > 1, if standardization was at low level of .62 qualifications (which was BM’s .42 .17 .35 intention), we should expect larger estimate of discrimination, which is .83 exactly what marginal effects show .33 .34 .56 Fully interactive specifications Fully A. Estimates from basic probit (Table 1) Black Males and females (1) (2) (3) Females (4) ­.030 (.006) ­.030 (.006) ­.030 (.007) ­.030 (.007) ­.024 (.007) ­.086 (.038) .062 (.042) 1.37 ­.026 (.007) ­.070 (.040) .044 (.043) 1.26 ­.026 (.008) ­.072 (.040) .046 (.045) 1.26 ­.027 (.008) ­.054 (.040) .028 (.044) 1.15 Wald test statistic, null hypothesis that ratio of standard deviations = 1 (p­value) .22 .37 .37 .56 Wald test statistic, null hypothesis that ratios of coefficients for whites relative to blacks are constant, .62 .42 .17 .35 Test overidentifying restrictions: include in heteroskedastic probit model interactions for variables .83 .33 .34 .56 B. Heteroskedastic probit model Black (unbiased estimates) Effect of race through level Effect of race through variance Standard deviation of unobservables, black/white fully interactive probit model (p­value) with |white coefficient| < |black coefficient|, Wald test for joint significance of interactions (p­value) Number of overidentifying restrictions Other controls: Individual resume characteristics Neighborhood characteristics 3 X Only need to exclude one 6 2 variable for het. probit, but this is less arbitrary X X X 6 X X Drop controls with Drop |estimated effect| smaller for whites A. Estimates from basic probit Black Males and females (1) (2) (3) Females (4) ­.030 (.006) ­.030 (.006) ­.030 (.007) ­.030 (.006) ­.024 (.007) ­.090 (.037) .066 (.041) ­.025 (.007) ­.080 (.036) .056 (.039) ­.024 (.009) ­.086 (.040) .062 (.044) ­.025 (.008) ­.077 (.038) .052 (.042) 1.41 1.33 1.37 1.30 Wald test statistic, null hypothesis that ratio of standard deviations = 1 (p­value) .19 .23 .25 .29 Wald test statistic, null hypothesis that ratios of coefficients for whites relative to blacks are .84 .92 .68 .74 X X X X X X B. Heteroskedastic probit model Black (unbiased estimates) Effect of race through level Effect of race through variance Standard deviation of unobservables, black/white constant, fully interactive probit model (p­value) Other controls: Individual resume characteristics Neighborhood characteristics Insensitivity not surprising, Drop controls with Drop since control variables |estimated effect| smaller randomly assigned w.r.t. race for whites A. Estimates from basic probit Black B. Heteroskedastic probit model Black (unbiased estimates) Effect of race through level Effect of race through variance Standard deviation of unobservables, black/white Wald test statistic, null hypothesis that ratio of standard deviations = 1 (p­value) Wald test statistic, null hypothesis that ratios of coefficients for whites relative to blacks are constant, fully interactive probit model (p­value) Other controls: Individual resume characteristics Neighborhood characteristics Males and females (1) (2) (3) Females (4) ­.030 (.006) ­.030 (.006) ­.030 (.007) ­.030 (.006) ­.024 (.007) ­.090 (.037) .066 (.041) ­.025 (.007) ­.080 (.036) .056 (.039) ­.024 (.009) ­.086 (.040) .062 (.044) ­.025 (.008) ­.077 (.038) .052 (.042) 1.41 1.33 1.37 1.30 .19 .23 .25 .29 .84 .92 .68 .74 X X X X X X How well does approach work? How Start with same parameters as in earlier example from Start Heckman Heckman Generate data consistent with those parameters, and Generate assumptions under which this method should work assumptions Observable productivity measure affects probability of hiring Observable equally for blacks and whites equally Replicate Heckman result, using probit estimates on generated Replicate data instead of analytical solution (2000 blacks, 2000 whites, 100 simulations) 100 Implement approach using same simulated data, to see if we Implement identify marginal effects correctly identify Replication of results from Heckman (1998), estimates of γ Var(X2W)/Var(X2B) = 2.25 2.25 No discrimination No (γ =0) Coef. of X1 = 1 Dashed line analytic Solid line from 100 Solid simulations of probit estimates, drawing data from truncated normal at each value of XI* (by .1) value + .1 SD(XI) (=1) Replication of results from Heckman (1998), marginals (1998), Var(X2W)/Var(X2B) = 2.25 2.25 No discrimination No (γ =0) Coef. of X1 = 1 Dashed line analytic Solid line from 100 Solid simulations of probit estimates, drawing data from truncated normal at each value of XI* (by .1) value + .1 SD(XI) (=1) Replication of results from Heckman (1998), marginals (1998), Var(X2W)/Var(X2B) = 2.25 2.25 With With discrimination (γ =discrimination 0.5) Coef. of X1 = 1 Dashed line analytic Solid line from 100 Solid simulations of probit estimates, drawing data from truncated normal at each value of XI* (by .1) value + .1 SD(XI) (=1) Now mimic data with variation in productivity-related characteristics— productivity-related probit estimates No discrimination No (γ =0) All is the same, except dgp XI* is now sampled from two truncated normal distributions, one using XI*in steps of 0.1 + 0.1∙SD(XI*), as before, and the second using instead XI* + 0.5, again in steps of 0.1 + 0.1∙SD(XI*) Bias reduced slightly, as expected because Now mimic data with variation in productivity-related characteristics— productivity-related heteroskedastic probit estimates, no heteroskedastic disc. disc. Generated samples of 2,000 blacks, 2,000 whites, 5,000 simulations Instead, assume there is discrimination (γ = -0.5)—heteroskedastic probit estimates of γ Model mispecification masking higher unobserved variance for whites (I) unobserved Mild violation of identifying assumption in data generating process ( βI for whites = 1.1) γ = 0 γ = ­.5 Model mispecification masking higher unobserved variance for whites (II) unobserved Strong violation of identifying assumption in data generating process ( βI for whites = 2) γ = 0 γ = ­.5 Implications of violating identifying assumption assumption Ratio of the probit coefficient (white/black) identifies σBRIIII , so when Ratio true value of βI is larger for whites than for blacks, but it is assumed that they are equal, σBRII is overestimated Probit for blacks identifies (–c + γ )/exp(φ) = (–c + γ )/σBRII Probit (–c c = 0 iin the simulations, so we identify γ by multiplying the estimate of n by this expression by the estimate of σBRII Upward bias in estimate of σBRII biases estimate of γ away from zero Upward Irrelevant when γ = 0 Irrelevant When true γ iis non-zero (and negative), bias leads to larger negative s When estimate, generating the “bending” of the estimated marginal effects In standard marginal effect – γ φ(Zβ) – nearer the center of the In (Zβ) distribution the larger estimate of γ dominates the marginal effect, dominates whereas nearer the tails the larger estimate of γ lowers φ(Zβ) (Zβ) enough that the product γ φ(Zβ) is closer to zero But bias is multiplicative, so never generates wrong sign for the estimate But sign The meaning of discrimination The Heckman analysis shows that blacks and whites can be treated Heckman differently solely because employers perceive differences in variance of unobservables variance Semantic issue (?) whether to interpret this discrimination, whereas Semantic interpretation is clear for difference in treatment that arises via latent variable (à la Becker) variable Not aware of any legal consideration of this as statistical Not discrimination discrimination Differential treatment via variance is an artifact of correspondence Differential study in which applicants have narrow range of qualifications study In real economy applicants would be representative of the In range of observables of actual applicants to a job, in which case differential treatment depending on level of qualifications and the variance of the unobservable would balance out the So it is the effect through γ that we are interested in So Conclusions/summary Conclusions/summary Even in ideal case with no group differences in means, Even correspondence studies subject to bias in any direction from any differences in variances of unobservables, which are common to models of discrimination to Paper shows how to test for difference in variances of Paper unobservables, and recover unbiased estimate of discrimination when variance differs discrimination Implementation shows that method provides informative Implementation estimates estimates Using BM data provides stronger evidence of discrimination Using Required data typically not available in correspondence study Required But such data can easily be generated in any future But correspondence study, and this test/correction implemented correspondence ...
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