Does anyone know of any good (theoretical or empirical) literature that presents evidence showing the merits of Bayesian measurement error models (where there’s uncertainty in the dependent / response variable) compared to the use of weighted OLS regression models?
To formalise things, consider the following model:
Y = \beta X,
where Y is measured with error, say s .
In a multi-level Bayesian (say brms) setting, one can directly model the measurement error, which intuitively seems to be the right way to go.
The problem I have is—in the field of (economic) literature that I’m working in—most previous studies use weighted OLS regression models, where observations are weighted by, say, 1 / s^2 , to estimate this model.
My co-authors and I have scanned the economic literature but haven’t yet found any good references to support our empirical approach.
Any pointers are appreciated, and apologies in advance if we’ve missed any obvious references.
I have no background in economics, so I’ll just tag a few people who I think work in at least somewhat related fields and hopefully someone will have a better answer. @RachaelMeager , @imadmali, @rtrangucci , @James_Savage .
I also don’t have any economics references for you; this paper on measurement error in covariates doesn’t consider weighted OLS but there are a couple conceptual points you could consider borrowing, or adapting, anyways [IJERPH | Free Full-Text | Modeling Community Health with Areal Data: Bayesian Inference with Survey Standard Errors and Spatial Structure]. From that paper, you could consider drawing out a couple things that Bayesian measurement error models do that weighed least squares doesn’t: 1) they differentiate between observational uncertainty and the actual amount dispersion around the fitted values; 2) they model the variable y itself (there’s a prior on y), which can be really important. This paper shows the importance for autocorrelated data (of having a prior and of thinking very carefully about that prior), but you may find something relevant for your context.
For example, if there’s an outlier that has been observed with high uncertainty, the weighted OLS would use the raw point estimate and assign low weight to it; the Bayesian model will produce a probability distribution for y_i, and may lead to the conclusion that values less extreme that the given point estimate are far more probable than values more extreme than it; i.e., the model will impose some ‘shrinkage’ on y_i. (That’s also why its so important to think carefully about the prior on y). The relevant references in that paper are mostly from spatial statistics, I don’t remember seeing any spatial econometric papers on this