I have a $(n \times d)$ panel $y$ of $n$ different variables , and a $(n \times d)$ panel $x$ of their forecasts.
$d=$ time length of data
$n=$ cross section width/ no. of variables
I run a pooled (panel) regression in three ways:
$1.$ simple pooled regression which simply aggregates the data across time and variables:
$2.$ regression of cross sectional means (so mean of all columns of y and x at each point of time):
$3.$ pooled regression of cross-sectionally demeaned variables:
$ols( (y-y.mean(axis=1) , (x-x.mean(axis=1)) )$
Can I expect to see some sort of relation between betas/t-stats/r-square of these regressions?
e.g. Should it hold that $R_1^2$ lies between $R_2^2$ and $R_3^2$ etc?
How about the case when I enforce the intercept to be 0?