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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?

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This amounts to discretizing (specifically: dichotomizing) a continuous variable. Your friend fits a model that has a discrete jump as an observation moves from the "below median" to the "above median" group. This almost certainly doesn't make sense. Read more here: What is the benefit of breaking up a continuous predictor variable?

If there is remaining nonlinearity, which you could detect, e.g., in plotting residuals against actuals or against fitted values, then it's far better to refit a model with .

Friends don't let friends discretize continuous variables.

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  • $\begingroup$ 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. $\endgroup$
    – bachelor
    Nov 14, 2017 at 9:46
  • $\begingroup$ Hm. Is he looking for an interaction term? Or does he want to model heteroskedasticity? $\endgroup$ Nov 14, 2017 at 10:04
  • $\begingroup$ He is rather interested in interaction terms. $\endgroup$
    – bachelor
    Nov 14, 2017 at 10:15
  • $\begingroup$ Then I'd recommend that he explicitly models it as such. $\endgroup$ Nov 14, 2017 at 10:16
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    $\begingroup$ Well, to me it does, too. An interaction with residuals sounds very much like conditional heteroskedasticity to me. If you have a statistician available, I'd recommend your friend buy him a cup of coffee and explain his setup to him. $\endgroup$ Nov 14, 2017 at 11:06

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