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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:

$ols(y,x)$

$2.$ regression of cross sectional means (so mean of all columns of y and x at each point of time):

$ols(y.mean(axis=1), x.mean(axis=1)$

$3.$ 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?

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