11. When an endogenous variable enters the regression nonlinearly, the obvious r

Question

11. When an endogenous variable enters the regression nonlinearly, the obvious r estimator is inconsistent and a modification is needed. Specifically, suppose Y! ßyi + u, and the irst-stage equation for y2 is y2-2z + v, where the zero- mean errors u and v are correlated. Here the endogenous regressor appears in the structural equation as US rather than y2. The IV estimator is = ziyaJ-1Li y on yz with the instrument z: regress y? on z and then regress yi on the first- stage prediction If instead we regress Y2 on z at the first stage, giving 12. and then regress yi on (72)2, an inconsistent estimate is obtained. Generate a simula- tion sample to demonstrate these points. Consider whether this example can be generalized to other nonlinear models where the nonlinearity isin regressors only, so that Y1-g(Y2)3 + u, where g(y2) is a nonlinear function of y2. i Ziyii. This can be implemented by a regular IV regression of

Explanation / Answer

We can generate the simlulation by the following log.

clear

set seed 10101

set obs 10000

generate double z = 5*rnormal(0) /* instrument */

generate double x = 5*rnormal(0)

matrix C = (1, -0.5 -0.5, 1) /* correlation structure */

corr2data u v, corr(C) /* correlated errors */

generate double y2 = 3*z + v /* endogenous variable due to correlation

with error, u */

generate y2sq = y2^2

generate double y1 = 5 + 2*y2sq + x + u

reg y2sq z x

predict y2sq_hat, xb

reg y1 y2sq_hat x, robust

reg y2 z x

predict y2_hat, xb

generate y2_hat_sq = y2_hat^2

reg y1 y2_hat_sq x, robust

Results-The coefficient estimates on both and are near the actual coefficient of 2. The standard error for the estimate deemed inconsistent is remarkably small while the standard error for the estimate deemed consistent is slightly larger than the parameter estimate, yielding t-value of less than 1.

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