To test whether it is appropriate to even use certain types of fixed effects, would it ever be appropriate to use a cross-validation methodology? Let's say that I have this type of mental model:
$$\text{wage} = \text{years of schooling} + \text{household FE} + \text{industry FE} + \text{year FE}$$
Suppose, however, that I use cross-validation to test this time series model out of sample, and somehow find out that if I don't include household FE, that my predictions of wage are more accurate (as measured by, say, lower out of sample RMSE). In that case, would it be more appropriate to use this model instead?
$$\text{wage} = \text{years of schooling} + \text{industry FE} + \text{year FE}$$
Why or why not?
My own intuition suggests that if there truly is household-level effects that are confounding estimation, then adding household FE should increase the accuracy of out-of-sample predictions. At the same time, those predictions seem to rely on accurate estimation of each household's fixed effects, which may not be accurate for an individual household. On the other hand, I know that one would never naively throw variables to predict an outcome to get a causal model.
As another way of putting it, are the following statements true or false:
'if there truly is a causal relationship with one variable, then adding that variable should help improve prediction.'
'if there is a causal relationship, AND my prediction is more inaccurate when using those specific causal variables, then there is no way I can accurately identify the causal parameter of interest' (in this case the slope on years of schooling)
