## 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.

clear
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.

#### 1 comment:

1. From memory all the Wooldridge text book says is that the test of the coefficient equal to zero is an endogeneity test. Also as you probably know you can see the square of this coefficient's t-statistic is approx the Wu-Hausman chi-squared test statistic from:

ivregress 2sls y (w=z)
estat endog

Best wishes,
Tom Palmer