# Large $N$, small $T$ in SUR: workaround using system GMM

Consider a system of linear equations as in seemingly unrelated regression (SUR). If the number of equations $$N$$ is large relative to the sample size $$T$$, the weighting matrix in SUR (i.e. the error covariance matrix of equation-by-equation OLS) is hard to estimate precisely and the SUR estimator may have poor properties. E.g. hypothesis tests based on it may have poor size. We can read this in the help file of systemfit::systemfit (where $$N$$ is replaced by g):

It is important to realize the limitations on estimating the residuals covariance matrix imposed by the number of observations T in each equation. With g equations we estimate g ∗ (g + 1)/2 elements using T ∗ g observations total. Beck and Katz (1995,1993) discuss the issue and the resulting overconfidence when the ratio of T /g is small (e.g. 3). Even for T /g = 5 the estimate is unstable both numerically and statistically and the 95 approximately [0.5 ∗ σ 2 , 3 ∗ σ 2 ], which is inadequate precision if the covariance matrix will be used for simulation of asset return paths either for investment or risk management decisions. For a starter on models with large cross-sections see Reichlin (2002). [This paragraph has been provided by Stephen C. Bond -- Thanks!]

Instead of doing SUR estimation, what about using a system GMM estimator with a simpler-than-optimal (e.g. identity or diagonal) weighting matrix and heteroskedasticity-robust covariance matrix? Could this help, or would we just exchange a poorly estimated weighting matrix in SUR to a poorly estimated heteroskedasticity-robust covariance matrix in GMM, ending up in a similarly bad spot?

• It appears that the surff function in VGAM allows you to specify an $M \times M$ covariance matrix for the $M$ equations; this suggests a possible solution where you estimate the equations using OLS, form a heteroskedasticity-robust covariance matrix using whatever technique you want, then form the final estimate using surff (or even iterating between steps 2 and 3.) Not on point to your question, though. Commented Apr 26 at 16:13
• @jbowman, thank you! Let me see if I follow you. I mentioned HC robust error covariance matrix as the one to be estimated after a single-step GMM with a given weighting matrix (the point estimate of which could be equivalent to WLS or equation-by-equation OLS, depending on the weighting matrix). Are you suggesting to use the HC robust matrix as the weighting matrix? Would that not yield the same point estimate as SUR and efficient GMM? But I want to avoid having a flexible weighting matrix because of the problems mentioned above. Commented Apr 26 at 16:33
• Well the covariance matrix would be different, as it would be the HC robust one instead of the usual one, so the final estimates would be different, but whether they would be better or not I don't know. I've found this whole thread of questions very interesting, as I ran into a similar problem some years ago, which I never felt like I'd addressed in a completely satisfactory manner. I used a shrinkage estimator of the covariance matrix in a one-step FGLS SUR estimate at that time. I think this paper was one of the resources I used: ncbi.nlm.nih.gov/pmc/articles/PMC2748251 Commented Apr 26 at 17:46
• @jbowman, I am glad you are interested, as thus I receive your comments that are enlightening. I guess I was wrong to say SUR and efficient GMM would have the same point estimate, as the weighting matrix in the efficient GMM should account for HC while the one for SUR would not. Yes, shrinkage is quite intuitive in such a setting. For now (for a class) I would like to use something simpler, as I was not planning to cover shrinkage at the level I am teaching. Commented Apr 26 at 18:02
• Thanks for the compliment! I'll definitely think about this some more. Commented Apr 26 at 18:06