# Clustering in Instrumental Variables Regression?

I am wondering whether clustering in IV estimation would mean I have a fixed effect for both error terms or just for the structural error. For example, in the model \begin{eqnarray} y = X \beta + \epsilon \\ X = Z \Pi + V \end{eqnarray} where say y and X are both endogenous and I would expect clustering of errors, would this clustering term have to carry over into the first-stage equation, as well?

The relevant reference would be Shore-Sheppard (1996) "The Precision of Instrumental Variables Estimates With Grouped Data". You can directly calculate by how much the standard errors in 2SLS are over-estimated by using the Moulton factor

$$\frac{Var(\widehat{\beta}^c)}{Var(\widehat{\beta}^{ols})} = 1 + \left(\frac{Var(n_g)}{\overline{n}} + \overline{n} -1 \right)\rho_z\rho$$ where $g$ are the groups, $\overline{n}$ is the average group size $$\rho_z = \frac{\sum_g \sum_{i\neq k}(z_{ig}-\overline{z})(z_{kg}-\overline{z})}{Var(z_{ig})\sum_g n_g (n_g - 1)}$$ is the intra-class correlation coefficient of the instrument $z$ and $\rho$ is the intra-class correlation coefficient of the second stage error - clustering in the first stage error does not matter for this.

From this you see that your 2SLS standard error depends on the number of groups and their average sizes, and the two intra-class correlation coefficients. If you need more information on this have a look at these lecture notes by Steve Pischke.

• Thanks so much @Andy this is an amazing reference. The thing is that a whole class of tests robust to weak instruments turn out to be robust against clustering and heteroskedastic errors, as well. At least that's what my proof argues. But this Princeton working paper is very good! – Hirek Feb 19 '15 at 21:30

I did some background research and found this here which characterizes the clustering issue in IV regression. Naturally, the clustering of errors will only appear in the covariance matrix of the structural errors. Therefore it is non-sensical to write down clustered first-stage errors. Hence \begin{eqnarray} Y_{i,g} = X'_{i,g} \beta + \eta_{g} + \epsilon_{i,g} \end{eqnarray} would be one line of the second stage regression while the other remains unchanged.

In the standard instrumental variable case with 2-SLS, you indeed not do need to take into account the errors in the first stage as you say.

However, if you were confronted with weak instruments, or want some more fancy endogeneity tests etc, then the usual weak instruments asymptotic need to be adjusted for the presence of cluster heteroskedasticity.

A good overview of this can be found in: . Colin Cameron and Douglas L. Miller, "A Practitioner's Guide to Cluster-Robust Inference", Journal of Human Resources, forthcoming, Spring 2015, page 33-34.

• Thanks @Mat! Yeah, I wrote down a LIML estimation problem and it seems to hold that the first-stage errors don't matter. Does that sound plausible? – Hirek Feb 19 '15 at 19:43