Quantile regression is often advertised as a way of "predicting change in the dependent variable that is not the mean." It seems like one can do this with linear regression, however. Am I correct?
Suppose I am estimating a quantile regression (QR) with a constant and a single predictor X where I am interesting in predicting moving from the 25th percentile to the 75th percentile of a continuous dependent variable (Y): $Y = a + X$. My understanding is that the quantile regression models a different conditional mean: the mean for observations at those points in the distribution.
What is the difference between this approach and subsetting my data to observations in the 25th and 75th percentiles then adding a dummy to the treatment: $ Y = a + X + I + X*I$? My hunch is that it arises because the constant term is not allowed to vary with $I$ and presumably the error term is different for this subpopulation.