Sequential regression using first regression coefficient (not an IV)

[bagus wahyu]

Hi.
I am conducting a brand-level empirical study on tax pass-through in addictive markets with heterogenous product (3 product criteria). My goal is to examine how the extent of competition affects the degree of tax shifting of each product.
My research design involves two regressions:
I first estimate a brand-level output on rival's output to identify the extent of competition/collusion
output i,t=f(output j,t, controls). From this, I obtain the predicted output j,t (rival), which represents the equilibrium output level dictated by the competition environment.

I then use the predicted output as a regressor in the price equation to see how competition moderates tax shifting:
price i,t=f(tax, predicted rival, (tax*predicted rival), controls)

My question is: Is it appropriate to use the predicted value of competition from the first regression directly as a regressor in the second regression (not as a formal IV), as I need the extent of competition in predicting the tax shifting. Under what conditions would this approach be valid?

Additionally, I would appreciate any guidance on this.
Thank you very much for your advice.

Best regards,
Aloysius

[George Ford]

No problem using generated regressors, though you need to bootstrap the second stage.

The bigger problem is that I'm not sure the first stage is exogenous to the second, or whether the first stage is estimating what you think it is.

Might look at:

https://www.annualreviews.org/content/journals/10.1146/annurev-economics-080315-015050