Asymptotic 2SLS Inference
We can derive the limiting distribution of the 2SLS coefficient vector using an argument similar to that used
in Section 3.1.3 for OLS. In this case, let Vi = Xі s,■
stage, equation (4.1.9). The 2SLS estimator can then be written
where the second equality comes from the fact that the first-stage residuals, (Si — S/), are orthogonal to Vi in the sample. The limiting distribution of the 2SLS coefficient vector is therefore the limiting distribution of Ei V/V/] 1 ^0i Vp. This quantity is a little harder to work with than the corresponding OLS quantity, because the regressors in this case involve estimated fitted values, S/. A Slutsky-type argument shows, however, that we get the same limiting distribution replacing estimated fitted values with the corresponding population fitted values (i. e., replacing S/ with [Х/^1о + ^11Z/]). It therefore follows that Г2sls has an asymptotically normal distribution, with probability limit Г, and a covariance matrix estimated consistently by E i Vi Vi']-1 [E i ViV/^2] E i ViVi']-1. This is a sandwich formula like the one for OLS standard errors (White, 1982). As with OLS, if p/ is conditionally homoskedastic given covariates and instruments, the consistent covariance matrix estimator simplifies to Ei V/V/] 1 .
There is little new here, but there is one tricky point. It seems natural to construct 2SLS estimates manually by first estimating the first stage (4.1.4a) and then plugging the fitted values into equation (4.1.9) and estimating this by OLS. That’s fine as far as the coefficient estimates go, but the resulting standard errors will be incorrect. Conventional regression software does not know that you are trying to construct a 2SLS estimate. The residual variance estimator that goes into the standard formulas will therefore be incorrect. When constructing standard errors, the software will estimate the residual variance of the equation you estimate by OLS in the second stage:
Yi - [а'Х/ + psi] = [pi + p(Si - Si)],
replacing the coefficients with the corresponding estimates. The correct residual variance estimator, however, uses the original endogenous regressor to construct residuals and not the first-stage fitted values, S/. In other words, the residual you want is Y/ — [a'X/ + pS/] = p/, so as to consistently estimate a^, and not Pi + p(Si — Si). Although this problem is easy to fix (you can construct the appropriate residual variance estimator in a separate calculation), software designed for 2SLS gets this right automatically, and may help
you avoid other common 2SLS mistakes.