Standard errors of *partial* two stage least squares coefficients

Here partial 2SLS (I coined this term and found it descriptive) is the approach that SAS uses for 2SLS. Compared to "ordinary" 2SLS which in the first stage projects all explanatory variables $$X$$ onto the column space of the instrumental variable matrix $$Z$$, partial 2SLS will first allow some of the explanatory variables to be labelled as endogenous whereas the others are left as exogenous, and in the first stage only endogenous variables will be projected onto $$Z$$ and exogenous variables will remain unchanged. Below I'll post the details for how SAS computes the coefficients for the partial 2SLS approach, or you can just go to check SAS's syslin manual (p2144-2145):

The above gives an explicit formula for computing the coefficients. However, I couldn't find within the manual how to estimate the standard errors of the coefficients. Note that, since we're no longer using the "ordinary" 2SLS, the classical 2SLS estimator that appears in many places on the Internet or in the econometrics tutorials (e.g. Greene's book) is no longer suitable here.

Hence, what would be an appropriate estimator for $$\Bbb V(\widehat\delta_{2SLS})$$ under the partial 2SLS model? Or, is there any way to find out how SAS computes it in details? Thanks!

EDIT: it can be shown that, under the exogeneity assumptions that $$\Bbb E(\epsilon\mid \mathbf x) = 0,\quad \Bbb E(\epsilon\mid \mathbf z)=0$$ the partial 2SLS estimator $$\widehat\delta_{2SLS}$$ is still (asymptotically) consistent. But it's not clear to me whether $$\widehat\delta_{2SLS} - \text{vec}([\beta,\gamma])$$ still asymptotically follows a normal distribution.

If $$X$$ contains the exogenous regressors, then the variables in $$X$$ are also part of $$Z$$, the set of instruments. It is then superfluous to project the variables in $$X$$ on $$Z$$, as finding the predicted values of a regression of a variable onto itself and something else always returns the variable itself as fitted values, as nothing predicts a variable as well as the variable itself - in fact, it does so without error:

Say, you want to explain height of people via gender, age and...height. The fitted value for someone who is 1m81 will always be 1m81 given that s/he is male/female, of whatever age and 1m81.

There is then also no issue about other standard errors.

• Thanks but the issue is: if some of X are exogenous but I don't want them to also be instruments, then they won't be part of Z.
– Vim
Dec 6, 2018 at 0:10
• OK, but is that the question? I do not think that this is what SAS does. Dec 6, 2018 at 5:22
• yes SAS does allow for non-instrumental exogenous variables. This is exactly what the manual shows, and is also confirmed by my hard coded test examples (I got the same regression coefficients).
– Vim
Dec 6, 2018 at 5:43
• Of course it does allow for non-instrumental exogenous variables. The question is if the exogenous regressors are part of the regressors of the first stage regression. And your screenshot and the pages you link to in the manual are unclear (at least to me and as it stands) about that, as they do not show how $Z$ is defined. Let $\tilde Z$ denote the instruments that are not $X$. If $Z=(X\;\tilde Z)$ then there is no "partial 2SLS". If $Z=\tilde Z$, there is. Dec 6, 2018 at 10:25
• But of course it would be interesting to see the "hard coded test examples" - although I cannot promise I'll be able to make sense of them, as I do not use SAS. Dec 6, 2018 at 10:26