Hi,
I’m trying to use a Bayesian measurement error model. I am using R Stan to estimate this with brms() for help. My question is a modelling question though: concerning how mathematically to write out my multilevel (hierarchical / mixed effects) model in order to prove a point about what it does for my regressor, which is measured with measurement error. Importantly, this measurement error originates from the grouplevel of the model too (more on this below).
I have a dataset of firms across countries , across time , from 19902017. It is not a balanced panel, but could be made so.
Assume a situation of measurement error in a continuous predictor x^* (where x is the mismeasured version of x^* ). We are using x to predict the investment behavior of the firm (y). But now imagine that this measurement error in x^* has a ‘structural’ nature to it, with specific grouplevel components to the error, such that we can decompose the measurement error into three different error terms:
On source of the measurement error is _i firm specific; another _t is time specific; and finally another source of measurement error _{it} is crosssectional. (More generally we could use the grouplevel _j subscript and think about this being at the countrylevel too). Substituting for x in a regression equation we are left with the following equation, where attenuation bias in our \beta coefficient has three different sources of error due to the above:
Also note that we are using a standard Bayesian measurement error model:
Now: I’m trying to determine, based on a vague hunch: if using a hierarchical model to estimate equation 7.57 can reduce some of the measurement error arising at that the group level (say at the country or time level or both)? The intuition being that partial pooling minimizes the grouplevel error, between countries or between time periods; and so this partial pooling might ‘naturally’ counteract some of the attenuation bias arising from the measurement error of x at the grouplevel. So:

Does this hunch have any actual basis to it? Namely, that using a hierarchical model will help reduce the grouplevel measurement error in equation 7.57 above.

If by chance the hunch is correct, how to show it mathematically?