OLS, Frisch-Waugh-Lovell Theorem and few observations 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?
We know that $M_X\mathbf{u}= \mathbf{\hat u}$, where $\mathbf{y}=\mathbf{x\beta}+\mathbf{u}$. So even if the error terms are independent, the residuals will not be independent of the other residuals. This is makes the usual assumption of no autocorrelation invalid. 

Of course, if we use a HAC estimator then we can circumvent the existence of autocorrelation. However, the HAC estimators that I know of are just of asymptotic behaviour. So, it wouldn't be useful for a small sample size.