Wednesday, February 20, 2013
Interpreting the Control Function Coefficient
* Is the control function coefficient a measure of the direction and size of the bias caused by endogeneity?
* Imagine the endogenous variable w being composed of three components: 1 endogenous portion, 2 exogenous portion correlated with z, 3 exogenous portion uncorrelated with z.
set obs 10000
gen z = rnormal()
gen v = rnormal()
gen endogenous = rnormal()
gen exog_with_z = z
gen exog_without_z = rnormal()
gen w = endogenous + exog_with_z + exog_without_z
* Likewise we can think of the error u as composed of both an exogenous portion and an endogenous portion (correlated with part of w)
gen u = endogenous + rnormal()*3
gen y = 1*w + 3*u
reg y w
* We can see that OLS is clearly upward biased
ivreg y (w=z)
* Instrumental variables seems to be working well
* Now for the control function
reg w z
predict v_hat, resid
reg y w v_hat
* I was thinking that the control function coefficient could generally not be interpretted directly the sign of the bias but looking at this simulation it appears I was wrong.
* I will have to do some more thinking on this.