Instrumental variable regression with endogenous ratio variable

Question

I want to estimate the following regression:

$y_t=a+bx_t+e_t$

where $x_t$ is an endogenous ratio variable, i.e., continuous values bound between 0 and 1 and $y_t$ is a continuous variable.

I want to instrument for $x_t$ using $z_t$. The first stage regression will take the form:

$x_t=a+bz_t+e_t$

As $x_t$ is a ratio variable, are there issues estimating the first stage with OLS as the predicted values could plausibly be <0 or >1 ?

Answer 1

We implement Instrumental Variables estimation to obtain asymptotic consistency, paying the price of higher estimator variance. The issue you mention does not affect asymptotic consistency -it still obtains.

As regards its effect on small-sample behavior, it cannot be explicitly quantified separately from the estimation error from the first-stage regression.

Since you still obtain consistency, I believe you can proceed.