standard error derivation using FWL theorem: Is this correct? I'm reading a paper (https://scholar.google.com.br/scholar?oi=bibs&cluster=986729284887040990&btnI=1&hl=en), and there's this derivation in the article which I'm not sure is correct, so I want to check.
Consider a linear regression $$ Y = D\tau + X \beta +\epsilon  $$
By the FWL theorem, $$\hat{\tau} =\frac{ cov(D^{\perp X}, Y^{\perp X} )}{var( D^{\perp X} )},$$ where $Y^{\perp X}$ is the variable $Y$ after removing the component linearly explained by $X$, and other variables using $\perp$ is similarly defined. I'm okay until this part. But three pages later, the paper states that $$ se(\hat{\tau}) = \frac{ sd( Y^{\perp X, D })}{sd(D^{\perp X})} \sqrt{ \frac{1}{df}},$$ which I don't get. Specifically, why is there suddenly $D$ appearing in the sd of $Y$?

In more conventional notation (with $Z$ here including all regressors), the standard error is the diagonal element of
$$
s^2(Z'Z)^{-1},
$$
where $s^2$ the sample variance of the residuals. In "your" notation, the residuals are what you get after projecting $Y$ on $X$ and $D$, so $Y^{\perp X, D}$.
