2SLS: (ivregress) having an interaction variable made with an endogenous variable and an instrumental variable

[Eli Chen]

Hi All,

I am conducting two related studies and desperately need your input to solve an issue.

In study 1 I look at

1) Y = f(X, controls) : How city-level income (X) influences the degree of a strategy of firms located in the focal city (Y)

In study 2, I test

2) Z = f(Y, Y*X, controls): How a firms strategy (Y) influences firms performance (Z) and how this effect of strategy on firm performance differs depending on the level of income of the city where the firm is located (Y*X).

To solve the endogeniety problem of Y, I used 2SLS.

First, I should get a fitted value of Y by regressing it on X and other explanatory variables. But here, should I include Y*X to get the fitted value of Y? If so, how? And if not, what should I do??


And would be there other alternative approaches than 2SLS to address this issue??

I would really really appreciate your advice.

Thanks in advance,
Eli

[Sergio Correia]

If Y is endogenous, then so is X*Y. This means that you need an excluded instrument for both Y and one for X*Y. AFAIK, if you find yourself an instrument (W1), you could then do W2 = W1*X in order to get the second instrument, and then regress it all on a 2SLS:

ivreg2 Z controls (Y YX = W1 W2)

Of course, finding a good instrument is nontrivial, and this also assumes you had the right functional form for how (Y,X) affect Z.

Hope this helps,
Sergio

[Oded Mcdossi]

Since you have firms within cities I would try mixed models (xtmixed) and address the two research questions (how city characteristics affect firms' performance) by interaction between the levels.

For research question 1:
You should begin by estimating how much the variance of degree_of_a_strategy is related to city characteristics (changes across cities).

xtmixed degree_of_a_strategy || cities:, var //this is a random intercept model

Then you can add your independent at the city level and estimate how much city income explain the variance of a firm's strategy between cities

​xtmixed degree_of_a_strategy city_income || cities:, var
xtmixed degree_of_a_strategy city_income (here you can add more controls for firms and cities)|| cities:, var

For research question 2:
Try to model a random slope for the firms' strategy with interaction with city characteristics, in mixed model language this is something like:

xtmixed firm_performance firms_strategy city_income (here you can add more controls for firms and cities)|| cities:firms_strategy city_income (here you can add more controls for cities), var

in this model the effect of firms_strategy is not constant between cities and its variance is explained by city characteristics.

[Eli Chen]

Thank you so much Sergio and Oded!

I will take a closer look at your comments and work through the steps. Thank you so much again!