I was talking to a colleague a few days ago and the following question came up:
Why can't we just use projection matrices and the FWL Theorem to "gain" a few degrees of freedom in case of very few observations?
Imagine a case where you want to estimate
$$y_t = \beta_0 + z\beta_1 + x_2\beta_2 + x_3\beta_3 + u_t$$
but unfortunately, you just have 5 observations (extreme example). Luckily, you are just interested in $\beta_1$. Now you could just create some projection matrices in the form
$$ M_x = I - x'(x'x)^{-1}x'$$
and project off your "unwanted" $x_2$ and $x_3$ (following Davidson and MacKinnon). You would be left with the regression
$$M_x y = M_x z \beta_1 + residuals$$
where $\beta_1$ is numerically the same as in the regression above. But now you are able to estimate just one (beta-)parameter with 5 observations compared to three parameters before. So, obviously, the estimates improve, don't they?
Where is our mistake?