Hello I am struggling a bit understanding the random effects model.
What I understood is that we fit a model but allow for sysematic differences of the variance of residuals per group.
I would incorporate such different variances in the expected negative Gaussian log likelihood loss to be $-\log \mathcal{L} = \frac{1}{2} \sum_{i} \left( \log(\sigma_{j(i)}^2) + \frac{(y_{i} - \mu_{j(i)})^2}{\sigma_{j(i)}^2} \right)+C$ (*), where the variance depends on the group $j(i)$ of sample $i$. This will allow better fit of the resulting model: when the variance of a group is high, then its mean squared error terms will have "less weight" and the model can fit the remaining data points better.
Such a model would deal with "group wise heteroscdasticity".
I understand that people might also define a random effects model in terms of a generating random process $Y_i =\sum_k A_k X_{k,i} + \epsilon +\eta_{j(i)}$... But in my mind this boils down to (*) when assuming that $Y_i$ and the error terms $\epsilon$ and $\eta_{j(i)}$ follow normal distributions and then doing maximum likelihood estimation...
Since I cannot find any literatur which uses the word heteroscdasticity and random effects model together, I guess my understanding/intuition is wrong. Could you please guide me to the essential logic piece behind random effects models?
