Fixed effect model I'm currently working on modelling how house prices have changed in the last 5 years, with dummy variables for property type and region. I know that there is likely to be unobserved heterogeneity as property type and location aren't the only factors that impact house prices, therefore I have ruled out the pooled OLS regression. However, I'm trying to understand the differences between the fixed effect and random effect models? And if there are any statistical pre-estimation tests I can run to determine which is better.
Any help or direction would be appreciated.
 A: In pooled OLS regression, cross-sectional features like region and type are disregarded and are not used as input to the model. Of course, you can also apply models that do use those features, hoping for better predictions. And in such a situation people then often use one of those two approaches: fixed and random effect models.
The difference between fixed and random effect models, applied to your situation, is usually the following:
With fixed effect models, you fit a new regression model for each combination of values of your features type and region. And for each of those models, you only use the subset of your data that belongs to the pertinent combination of your cross-sectional features. (Note that you don't really have to consider all combinations, you might also consider only subsets.)
Random effects models, however, can be understood as doing the same as the fixed effect models, but trying to make all those separate models similar, because, after all, they are all models for house prices so they should behave similarly to a certain extent. This sometimes helps to improve the prediction performance.
There are no "statistical pre-estimation tests" to decide which model to apply. You have to try both and then compare the results, using methods of model selection.
