Pooled OLS with time-variant regressors Could someone please explain how a pooled OLS is set-up when the regressors are aggregate variables that are time-varying only (i.e. do not vary for the cross-section at each time t)?
If the pooled regression is expressed as:
$$\;Y_{i,t} = \;X_t \;\beta_i  + \;\epsilon_{i,t} $$
The response vector $Y$ is stacked: in other words, we stack the time-series of each $Y_i$ to form only one response vector of dimension (TN x 1).
The regressor X should be stacked similarly to obtain a vector of dimension (TN x 1). The issue here is that X is time-varying only, which means that the stacked vector would be rep(X, N): i.e. the same vector X is repeated N times.
Would it be valid to run a pooled OLS with such a stacked regressor?
Thank you,
 A: It would be valid but you must understand what you are estimating when the regressor matrix $X$ is identical for each cross-section. Denote $n$ the number of cross-sections.
Write the stacked regressor matrix and its transpose in block form
$$ \mathbf X   =\left [ \begin{matrix}  X \\ .\\.\\.\\ X  \end{matrix}\right ]\qquad \mathbf X'   =\left [   X' :\,X' :\, ...\,: X'  \right ]$$
and the dependent variable
$$ \mathbf Y   =\left [ \begin{matrix}  Y_1 \\ .\\.\\.\\ Y_n  \end{matrix}\right ]  $$
The pooled OLS estimator is just
$$\hat \beta_{POLS} = \Big ( \mathbf X' \mathbf X\Big)^{-1}\mathbf X'\mathbf Y  $$
Now
$$ \Big ( \mathbf X' \mathbf X\Big) = \left [   X' :\,X' :\, ...\,: X'  \right ]\left [ \begin{matrix}  X \\ .\\.\\.\\ X  \end{matrix}\right ]  = X'X + X'X +... +X'X = nX'X$$
while 
$$\mathbf X'\mathbf Y = \left [   X' :\,X' :\, ...\,: X'  \right ]\left [ \begin{matrix}  Y_1 \\ .\\.\\.\\ Y_n  \end{matrix}\right ] = X'Y_1 + X'Y_2 + ... + X'Y_n = X'\sum_i^nY_i$$
Inserting into the estimator
$$\hat \beta_{POLS} =\Big (nX'X\Big )^{-1}X'\sum_i^nY_i = \Big (X'X\Big )^{-1}X'\bar Y_n $$
...where $\bar Y_n$ is just the cross-sectional average of the series of the dependent variable,
$$\bar Y_n = \frac 1nY_1 + ...+\frac 1n Y_n =  \left [ \begin{matrix}  \frac 1n \sum_i^ny_{i1} \\ .\\.\\.\\ \frac 1n \sum_i^ny_{iT}  \end{matrix}\right ]  $$
So it boils down to averaging the dependent variable over cross-sections and then run a simple OLS regression using just once the identical regressor matrix.
