I’m currently working on an analysis of the latest state election in Rhineland-Palatinate using aggregate data alone, i.e. electoral returns and structural information, which is available at the level of the state’s roughly 2300 municipalities. The state’s Green party (historically very weak) has roughly tripled their share of the vote since the last election in 2006, and I want to know were all these additional votes come from. And yes, I’m treading very careful around the very large potential ecological fallacy that lurks at the centre of my analysis, regressing Green gains on factors such as tax receipts and distance from next university town, but never claiming that the rich or the students or both turned to the Greens.

One common problem with this type of analysis is that not all municipalities are created equal. There is a surprisingly large number of flyspeck villages with only a few dozen voters on, whereas the state’s capital boasts more than 140,000 registered voters. Most places are somewhere in between. Having many small municipalities in the regression feels wrong for at least two reasons. First, small-scale changes of political preferences in tiny electorates will result in relatively large percentage changes. Second, the behaviour of a relatively large number of voters who happen to live in a small number of relatively large municipalities will be grossly underrepresented, i.e. the countryside will drive the results.

My PhD supervisor, who did a lot of this stuff in his time, used to weigh municipalities by the size of their electorates to deal with these problems. But this would lead to pretty extreme weights in my case. Moreover, while voters bring about electoral results, I really don’t want to introduce claims about individual behaviour through the back door.

My next idea was to weigh municipalities by the square root of the size their electorates. Why? In a sense, the observed behaviour is like a sample from the underlying distribution of preferences, and the reliability of this estimate is proportional to the square root of the number of people in a given community. But even taking the square root left me with weights that were quite extreme, and the concern regarding the level of analysis still applied.

Then I realised that instead of weighing by size, I could simply include the size of the electorate as an additional independent variable to correct for potential bias. But this still left me exposed to the danger of extreme outliers (think small, poor, rural communities where the number of Green voters goes up from one to four, a whopping 300 per cent increase) playing havoc with my analysis. So I began reading up on robust regression and its various implementations in Stata.

The basic idea of robust regression is that real data are more likely than not a mixture of (at least) two mechanisms: the “true model” whose coefficients we want to estimate one the one hand, and some other process(es) that contaminate the data on the other. If these contaminating data points are far away from the multivariate mean of the x-Variables (outliers) and deviate substantially from the true regression line, they will bias the estimates.

Robust regression estimators are able to deal with a high degree of contamination, i.e. they can recover the true parameters even if there are many outliers amongst the data points. The downside is that the older generation of robust estimators also have a low efficiency (the estimates are unbiased but have a much higher variance than regular OLS-estimates).

A number of newer (post-1980) estimators, however, are less affected by this problem. One particular promising approach is the MM estimator, that has been implemented in Stata ados by Veradi/Croux (MMregress) and by Ben Jann (robreg mm). Jann’s ado seems to be faster and plays nicely with his esttab/estout package, so I went with that.

The MM estimator works basically by identifying outliers and weighing them down, so it amounts to a particularly sophisticated case of weighted least squares. Using the defaults, MM claims to have 85 per cent of the efficiency of OLS while being able to deal with up to 50 per cent contamination. As you can see in the table, the MM estimates deviate somewhat from their OLS counterparts. The difference is most pronounced for the effect of tax receipts (hekst).

robreg mm has an option to store the optimal weights. I ran OLS again using these weights (column 3), thereby recovering the MM estimates and demonstrating that MM is really just weighted least squares (standard errors (which are not very relevant here) differ, because robreg uses the robust variance estimator). This is fascinating stuff, and I’m looking forward to a forthcoming book by Jann and Veradi on robust regression in Stata (to be published by Stata Press in 2012).

OLS MM WLS greenpct2006 0.193*** 0.329*** 0.329*** (0.0349) (0.0592) (0.0278) hekst 0.311*** 0.634*** 0.634*** (0.0894) (0.124) (0.0688) senioren -0.0744*** -0.100*** -0.100*** (0.0131) (0.0149) (0.00994) kregvoters11 -0.0125 -0.00844 -0.00844 (0.0146) (0.00669) (0.00982) kbevdichte -0.433 -0.00750 -0.00750 (0.464) (0.330) (0.326) uni 1.258 0.816 0.816 (1.695) (0.765) (1.137) lnunidist -0.418** -0.372** -0.372*** (0.127) (0.113) (0.0918) _cons 8.232*** 7.078*** 7.078*** (0.627) (0.663) (0.461)

Dear Professor Arzheimer,

I have been playing around with mmregress and robreg mm and I get different results for both standard errors and coefficients. Do you know the reason?

Best

Roberto

Roberto, are you sure that you are calling them with equivalent options? If so, it’s best to get in touch with the authors of these two programs (see http://ideas.repec.org/c/boc/bocode/s457057.html and http://ideas.repec.org/c/boc/bocode/s457114.html for their contact data). They will need a minimal example that demonstrates the difference, i.e. a stripped down version of your dataset and the two commands that reproduce your findings.

Would it also make sense to directly model the variance/regression error as a function of municipality size along the lines discussed by Braumoeller in “Explaining variance”?

@NilsSteiner : thanks for that. The article you are referring to is Braumoeller 2006 http://pan.oxfordjournals.org/content/14/3/268.abstract

His point is valid and interesting, but I think my problem is slightly different: From my point of view, the additional variance in Y (and one of the Xs) is a nuisance, not a causal effect I’m trying to understand. Or am I missing something here?

@kai.arzheimer@NilsSteiner I agree that it would be a pure nuissance from your point of view (as opposed to Braumoellers’ substantial interest in actually explaining variance). Still, I think that it could make sense to deal with this nuissance by modeling the variance as being a function of municipality size as your discussion seems to suggest that the variance actually is dependent on municipality size. I know of no particular application (other than the vague reference to Braumoeller) nor how to implement that in Stata. Gelman/Hill’s 2007 multilevel book briefly mentions this as a possible extension in section 13.7. In the end, this might just give similar results to WLS as you would effectively down-weight observations that have a higher estimated variances…