The language of conditional quantiles is tricky. Sometimes we talk about "quantile regression coefficients at the median," or "effects on those at the lower decile." But it’s important to remember that quantile coefficients tell us about effects on distributions and not on individuals. If we discover, for example, that a training program raises the lower decile of the wage distribution, this does not necessarily mean that someone who would have been poor (i. e. at the lower decile without training) is now less poor. It only means that those who are poor in the regime with training are less poor than the poor would be in a regime without training.
The distinction between making a given set of poor people richer and changing what it means to be poor is subtle. This distinction has to do with whether we think an intervention preserves an individual’s rank in the wage (or other dependent variable) distribution. If an intervention is rank-preserving, then an increase in the lower decile indeed makes those who would have been poor richer since rank preservations means relative status is unchanged. Otherwise, we can only say that the poor - defined as the group in the bottom 10 percent of the wage distribution, whoever they may be - are better off. We elaborate on this point briefly in Section 7.2, below.
A second tricky point is the transition from conditional quantiles to marginal quantiles. A link from conditional to marginal quantiles allows us to investigate the impact of changes in quantile regression coefficients on overall inequality. Suppose, for example, that quantile coefficients fan out even further with schooling, beyond what’s observed in the 2000 Census. What does this imply for the ratio of upper-decile to lower-decile wages? Alternately, we can ask: how much of the overall increase in inequality (say, as measured by the ratio of upper - to lower-deciles) is explained by the fanning out of quantile regression coefficients? These sorts of questions turn out to be surprisingly difficult to answer. The difficulty has to do with the fact that all conditional quantiles are needed to pin down a particular marginal quantile (Machado and Mata, 2005). In particular, QT(Yj|Xj) =Xj^T does not imply QT(Yj) = QT(Xj)'PT. This contrast this with the much more tractable expectations operator, where if E(Yj|Xj) =Х'ф, then by iterating expectations, we have E(yj) = E(Xi)'p.
To show the link between conditional quantiles and marginal distributions more formally, suppose the CQF is indeed linear, so that QT(Yj|Xj) =Xj/3T. Let Fy(y|Xj) = P[Yj < y|Xj] with marginal distribution Fy(y) = P[Yj < y]. By definition of a conditional quantile,
J l[Fy1 (t|Xj) < y]dr = Fy(y|Xj). (7.1.9)
In other words, the proportion of the population below y conditional on Xj is the same as the proportion of conditional quantiles that are below y. Substituting for the CQF inside the integral,
Fy(y|Xj) = J 1[Xj3r <y]dr.
Next, we use the CDF of Xj, Fx (x), to integrate and get the marginal distribution function, Fy(y):
Fy(у)=/ J 1[X'd3r <y]drdFx(x). (7.1.10)
Finally, marginal quantiles, say, Qr(Yj) for r 2 (0,1), come from inverting Fy(y):
Qr(Yj) = inf fy : Fy(y) > r}.
An estimator of the marginal distribution replaces integrals with sums in (7.1.10), where the integral over quantiles comes from quantile regression estimates at, say, every.01 quantile. In a sample of size n, this is:
Fy(y) = n-^2(1/100) E 1[Xj3r < У]. j r=0
The corresponding marginal quantile estimator inverts Fy (y).
There are a number of difficulties with this approach in practice. For one thing, you have to estimate lots of quantile regressions. Another is that the distribution theory is messy (though not insurmountable; see, e. g., Melly, 2005). Simplifying the conditional-to-marginal quantile transition is an area of active research. Gosling, Machin, and Meghir (2000) and Machado and Mata (2005) are among the first empirical studies to go from conditional to marginal quantiles. When the variable of primary interest in a quantile regression model is a dummy variable and the other regressors are seen as controls, a propensity-score type weighting scheme can be used to produce the difference in quantiles conditional on the dummy. See Firpo (2007) for the exogenous case and Frolich and Melly (2007) for a marginalization scheme that works for quantile treatment effects of the sort discussed in the next section.