main equation
$y_1 = y_2 \beta + x_1 \gamma + u_i $
Instrumental equation
$y_2=x_1 \pi_1 + x_2 \pi_2$
I have a binary endogenous variable $y_2$ in my main estimation equation.
My instrument $x_2$ is binary as well and recently my supervisor suggested a structural equation approach to me. In STATA I ran
gsem (y1 <- y2 x1 L@a, oprobit) (y2 <- x1 x2 L@a)
Where the latent variable $L$ is part of the parametrisation (cf. link before). The main issue is that this procedure is that I do not account for the binary nature of the instrument and the instrumented variable and consequently the log-likelihood function breaks as it is not continous.
Is there a way around this discontinous function problem? Or do I need to estimate the first part with probit/logit? If so is that then still IV-regression or something else?
Best Answer
You can fit this sort of model very easily with David Roodman's conditional mixed process estimator
cmp
.Here's a toy example, where the "first stage" is a probit:
The repair record rating runs from Poor (1) to Excellent (5). We can see that foreign cars are less likely to have low repair ratings and more likely to have higher ones (though everything is insignificant). For example, Excellent is 5.4 percentage points more likely for foreign carts.
I think the
gsem
approach is also doable, but I think you will have to normalize the variance:As you can see, the estimates of the average marginal effects are very similar to
cmp
.