The importance of centering variables
So, maybe this is part of the learning experience of developing applied intuition, but I never really appreciated how important it is to center your variables, especially when dealing with finicky models like simultaneous autoregressive models common in spatial econometrics.
The following are three traces from a spatial econometric model, the spatial lag model, on some example data.
The first one hasn’t centered its variables. The lag coefficient is trying very hard to escape the (-1,1) bounding on stable autoregressive coefficients. While the other parameters look alright, you’d clearly look into misspecification checks here.
Removing the bounding, you find that Rho gravitates towards ~1.01, and the rest of the parameters still hit their mark… this is hard to square. The movement of that Rho parameter still is pretty bad, and there’s no accompanying movement in the other substantive parameters.
But, if you center the response, everything looks sane. Here, since I know the true mean and the MLE estimate, Rho looks okay. While it’s underestimating the spatial effect, the MLE is centered right at the mode of Rho, too. And, since I’m using a flat prior, that’s to be expected with this model.
It’s pretty weird to me. While people recommend you center responses and covariates sometimes when dealing with models in a maximum likelihood context, I’ve never found it to be that profound. But, in this context, the model goes from something is really wrong to this looks alright. Anecdotally, MCMC techniques seem to be sensitive to different kinds of ill-conditioning than what I’m used to in the classical econometric case.
imported from: yetanothergeographer