How to instrument for dummy endogenous variable

[Chengmou Lei]

Hi all,

Suppose I’m regressing a continuous variable y on two dummies. The dummies comes from a continuous variable X, I split the continuous variable into three group (low, medium, high) and use low as the baseline group. I know the continuous variable X itself is endogenous, and I have find an instrument Z. However, if I run the regression with two dummies, then since the dummies are derived from X, they are also endogenous, so I need two instrument variables.
Is it possible to still use Z again in ivreghdfe?
Is there a way to transform Z somehow to get another IV?

Some explanation: the reason why I do not regress Y on X directly is because the relationship changes when X has different range. It is not linear, it could be any function form. This is why I want to first see: compared to low X group, what is the differential impact on Y from medium/high X group.

thank you

[Maxence Morlet]

The immediate fix is to derive two dummies from X, and use e.g. low+medium as base group. Then you need to instrument the high category, and your IV equation is identified. That would work for identification, but whether it is a good idea or not is another question.

Regarding the second part of your statement: you should make the functional form as flexible as possible, and you can test it using Ramsey RESET tests. Generally speaking, dichotomising or discretising continuous variables removes variation, and is not great...

[Chengmou Lei]

The immediate fix is to derive two dummies from X, and use e.g. low+medium as base group. Then you need to instrument the high category, and your IV equation is identified. That would work for identification, but whether it is a good idea or not is another question.

Regarding the second part of your statement: you should make the functional form as flexible as possible, and you can test it using Ramsey RESET tests. Generally speaking, dichotomising or discretising continuous variables removes variation, and is not great...

Hi Maxence,

Thank you for the reply. I totally agree with the your second part. Does it make sense to do this: make my X into N groups with different interval so that within each interval I can see the functional form (decide the functional form relatively precise), so for different levels of x, I fit using different functional form? For instance x from 0-0.4 is a quadratic function, x from 0.4-1.2 is a linear function, x from 1.2-5 is a cubic function.

Thank you