# Regression: OLS: residual split

A friend of mine is estimating the following model, using ols:

$y=\alpha + \beta X + u$,

where $y$ and $X$ are continious variables, $\alpha$ and $\beta$ are parameters and $u$ is an error component. In a second step, he then splits the sample at the median of the residuals. He then estimates the following regression:

$y=\alpha + \beta X + \gamma Dummy(above\_Median=1) + \delta X \times Dummy(above\_Median=1)$ + v,

where $Dummy(.)$ is one if an observation is above the median and zero otherwise. My feeling is that this procedure is extremely strange and the estimates are probably biased. But I currently cannot depict/formalize the problem. What potential problems do you see? Can this be done? What do you think about this procedure?

• Thanks a lot. Just as a follow up. His problem is that residuals in his model have "a deeper interpretation" and he is interested in modelling heterogeneity in this componend (i.e. heterogeneity in $\beta$). I don't see how this can be done without using sort of a proxy for the residuals. Commented Nov 14, 2017 at 9:46