might be polynomials or B-splines), andβ(y) is an unknown vector of function-valued parameters. Thus, ifP(x) is a polynomial of order 1, and Λ(.) is a logitlink function, this model reduces to a standard logit model where the dependentvariable is(yi< y) when y is fixed. Then by spanning the entire values of Y (orat least some sufficiently fine grid of values), we are able to estimate completely theconditional distribution and the impact of the covariates on the whole conditionaldistribution.On the pros, this approach implies estimating well-known discrete choice mod-els depending on the link function chosen. It can be a probit, logit, linear, log-log,or Gosset/Student link function. But the choice of the link function is crucial onlyif your set of covariatesP(X) is not rich enough.38We also do not need to makespecific assumptions about the smoothness of the conditional distribution sincewe are estimating it potentially at each values of Y. However, on the cons, it iscomputationally heavy since it implies running a lot of regressions. There mightalso be some problems with crossing of predictions therefore leading to distribu-tions that might be decreasing. However, you might overcome this problem withrearrangements.39Alternatively, we could have used quantile regressions to estimate the condi-tional distribution. However, this would have implied to inverse back the quantilefunction into the conditional distribution. But this is not necessary since distri-bution regressions provide directly such estimates. Ultimately, we are interestedin segregation curves which are functionals of the conditional distributions. Thenwe do not need to go through the estimation of the quantile function. Finally, asmentioned earlier, quantile regressions require Y to be sufficiently smooth while38See Chernozhukov et al. for more details on this point.39See Chernozhukov et al. for more details about rearrangements.14
distribution regressions do not. Then, in our set up, there will probably some masspoints for Black and White ghettos which may violate this smoothness require-ment.Finally, we have to make a choice of the link function. We will use a logit linkfunction since the distribution of individuals across space is a discrete locationchoice problem.The logit function has been the dominant link function in thisliterature following McFadden. We will then evaluate the counterfactual distri-butions at all values of our dependent variable, namely the number of individualsof a group in a location.3.4Detailed decompositionOur primary interest in this paper is to quantify each determinant of residentialsegregation. Thus, the aggregate decomposition is not enough to our purpose. Weneed to get a detailed decomposition of the gap between segregation curves. Inthe case of the standard Oaxaca-Blinder decomposition for the mean, because ev-erything is linear, the aggregate composition and structure effects are respectivelythe sum of the contribution of each factor:(¯XW-¯XB)βW=KXk=1(¯XkW-¯XkB)βkW(6)and¯
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