# relation between OLS regressions using different data transformations

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?