Interact dummy of dependent variable with independent variable

[高佳]

Hi all,

I have a resarch setting in which we have strong theoretical support that the effect of X (variable of interest) on Y (the dependent variable) is conditional on threshhold of Y. For instance, we speculate that the effect of X on Y depends on whether Y is positive or negative according to the prospect theory. Per my own knowledge, the most intuitive research design is the following:

Y ~ X + Z + X* Z + u

In which

Z = 1 if Y > 0, Z = 0 otherwise

.

The model will be estimated by OLS. I can mathmatically prove that when E[u|X] = 0 holds, then E[u|X*Z] = 0 holds (check the link here). I have two questions regarding this approach:

1, Can I use this research design judging from the causal inference perspective or more generally, the econometrics perspective?
2, Is there any research papers that use such a design?

[George Ford]

Y = XB + e, so you are conditioning on e. There will be endogeneity.

Do you have anything exogenous to work with? Maybe the lag of Y would work.

[高佳]

Y = XB + e, so you are conditioning on e. There will be endogeneity.

Do you have anything exogenous to work with? Maybe the lag of Y would work.

Hi George, thanks for your comments. I'm not quite following what you are saying. Are you suggesting that even if X itself is exogenous, X*Z would still be endegenous?

It is hard to find something exogenous in this case becaus Z is a demterministic function of Y.

[George Ford]

You need Y to be independent of e, which it is not.

[高佳]

You need Y to be independent of e, which it is not.

I doubt that wil ever be satisfied.

[Jeff Wooldridge]

What you're doing is the same as estimating two different linear models -- assuming you're including an overall intercept. One model is for Y > 0 and the other for Y < 0. It's unusual to do that because what sense does it make to hold Z fixed and then change X when changing X might make Y go from negative to positive? So I think you need to rethink what you want to learn.

[高佳]

What you're doing is the same as estimating two different linear models -- assuming you're including an overall intercept. One model is for Y > 0 and the other for Y < 0. It's unusual to do that because what sense does it make to hold Z fixed and then change X when changing X might make Y go from negative to positive? So I think you need to rethink what you want to learn.

Dear Dr. Wooldridge,

Thank you so much for your insights. This is surely a problem if X itself could explain the variance of Y or could predict the direction of Y. But in our specific research setting, we have strong theoretical argument that X should not affect the direction of Y. Instead, X should only explian Y depends on the direction of Y. That's why we are interested in this model specification in the first place. Shall there be any other model specification that could help us to investiagte this question, we are more than happy to look into it.

Nonetheless, we are aware of this issue you mentioned and test these possibilities by running the following two models:

Y= X + u

Pr(Y=1) = X + e

In either model, we find no statistically significant effect of X on Y.

Given the theoretical argument and empirical evidence that X could not predict the direction of Y, could we use the model specification in the original post to investigate the research question we have?

[George Ford]

If you have other Xs, I wonder if semipar would work.

[高佳]

If you have other Xs, I wonder if semipar would work.

Thanks, but I I'm not quite following, what is semipar?

[George Ford]

sysuse auto, clear
semipar mpg foreign, nonpar(weight)

at a minimum, this will give some idea about whether there's a difference across the range of YX.

[Rich Goldstein]

note that -semipar- is user-written and needs to be installed; use -search- to find and install

[George Ford]

A finite mixture model (fmm) might be a solution.