might be polynomials or B splines and β y is an unknown vector of function

Might be polynomials or b splines and β y is an

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might be polynomials or B-splines), and β ( y ) is an unknown vector of function- valued parameters. Thus, if P ( x ) is a polynomial of order 1, and Λ( . ) is a logit link function, this model reduces to a standard logit model where the dependent variable is ( y i < y ) when y is fixed. Then by spanning the entire values of Y (or at least some sufficiently fine grid of values), we are able to estimate completely the conditional distribution and the impact of the covariates on the whole conditional distribution. 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 only if your set of covariates P ( X ) is not rich enough. 38 We also do not need to make specific assumptions about the smoothness of the conditional distribution since we are estimating it potentially at each values of Y. However, on the cons, it is computationally heavy since it implies running a lot of regressions. There might also be some problems with crossing of predictions therefore leading to distribu- tions that might be decreasing. However, you might overcome this problem with rearrangements. 39 Alternatively, we could have used quantile regressions to estimate the condi- tional distribution. However, this would have implied to inverse back the quantile function into the conditional distribution. But this is not necessary since distri- bution regressions provide directly such estimates. Ultimately, we are interested in segregation curves which are functionals of the conditional distributions. Then we do not need to go through the estimation of the quantile function. Finally, as mentioned earlier, quantile regressions require Y to be sufficiently smooth while 38 See Chernozhukov et al.[15] for more details on this point. 39 See Chernozhukov et al.[13][14] for more details about rearrangements. 14
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distribution regressions do not. Then, in our set up, there will probably some mass points 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 link function since the distribution of individuals across space is a discrete location choice problem. The logit function has been the dominant link function in this literature following McFadden.[36] We will then evaluate the counterfactual distri- butions at all values of our dependent variable, namely the number of individuals of a group in a location. 3.4 Detailed decomposition Our primary interest in this paper is to quantify each determinant of residential segregation. Thus, the aggregate decomposition is not enough to our purpose. We need to get a detailed decomposition of the gap between segregation curves. In the case of the standard Oaxaca-Blinder decomposition for the mean, because ev- erything is linear, the aggregate composition and structure effects are respectively the sum of the contribution of each factor: ( ¯ X W - ¯ X B ) β W = K X k =1 ( ¯ X kW - ¯ X kB ) β k W (6) and ¯
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