I find myself puzzled by a passage about marginal effects of discrete variables in quantile regression. On p. 217 of Cameron and Trivedi's MUS book, the authors write:
For the $j$th (continuous) regressor, the ME is $$\frac{\partial Q_\tau(y\vert x)}{\partial x_j}=\beta_{\tau j}$$ As for linear least-squares regression, the ME is given by the slope coefficient and is invariant across individuals, simplifying analysis. The interpretation is somewhat delicate for discrete changes that are more than infinitessimal, however, because the partial derivative measures the impact of a change in $x_j$ under the assumption that the individual remains in the same quantile of the distribution after the change. For larger changes in a regressor, the individual may shift into a different quantile.
Above, $y$ is the outcome, $\tau$ is the quantile, $x$ is the vector of covariates.
I am not sure what the bold part means. Is that just a different way of saying that while we can estimate the average treatment effect for each individual, $\mathbb E(Y_1-Y_0)$, with $\mathbb E(Y_1)-\mathbb E(Y_0)$ with quality experimental data, the median treatment effect $$median(Y_1-Y_0)\ne median(Y_1)- median(Y_0)?$$
The only other passage I've found on this topic is on p. 48 of Roger Koenker's Quantile Regression:
The interpretation of the partial derivative itself, $\partial Q_\tau(y\vert x)/\partial x_j$, often requires considerable care. We emphasized earlier in the context of the two-sample problem that the Lehmann–Doksum quantile treatment effect is simply the response necessary to keep a respondent at the same quantile under both control and treatment regimes. Of course, this is not to say that a particular subject who happens to fall at the τth quantile initially, and then receives an increment $\Delta x_j$ , say, another year of education, will necessarily fall on the $\tau$th conditional quantile function following the increment. Indeed, as much of the recent literature on treatment effects has stressed (see, e.g., Angrist, Imbens, and Rubin, 1996), we are typically unable to identify features of the joint distribution of control and treatment responses because we do not observe responses under both regimes for the same subjects.