2SLS with multiple endogenous variables

* I am wondering if when using 2SLS you must use a multivariate OLS in the reduced form or if you can just do each individual endogenous variable.

* Let's see!

clear
set obs 10000

* First generate the instruments
gen z1 = rnormal()
gen z2 = rnormal()

* Now the error.  u1 and u2 are the parts of w1 and w2 correlated with the error u.
gen u1 = rnormal()
gen u2 = rnormal()
gen u3 = rnormal()*3

gen w1 = 1*z1 + .5*z2 + rnormal() + 2*u1
gen w2 = -.5*z1 + 1.5*z2 + rnormal() - 2*u2 + .2*u1

gen u = u1 + u2 + u3
gen y = 4 + w1*1 + w2*1 + u

* We can see because u is correlated with w1 and w2, OLS will be biased and inconsistent.

reg y w1 w2

* Instrumental variable regression could solve this problem

ivreg y (w1 w2 = z1 z2)

* Let's see about our 2SLS (which should be the same as IV)

reg w1 z1 z2
predict w1_hat

reg w2 z1 z2
predict w2_hat

reg y w1_hat w2_hat
* It seems that 2SLS using separate regressors is producing the same results.

* This is probably because in SOLS if you use the same regressors then I think the coefficients are the same but the standard errors may be adjusted for cross equation correlations (I think I recall).