Is there an analytic adjustment of the standard errors when running residualized regressions? Based on the Frisch-Waugh-Lovell theorem, the multivariate regression of $Y$ on $D$ with $X$ as covariate should give the same estimate as a bivariate regression of $Y$ on $\hat{r}$, where $\hat{r}$ is the residual of a regression of $X$ on $D$.
$ Y_i = \alpha_1 + \delta D_i + \gamma_1 X_i +\varepsilon_{1i} \\
    D_i = \beta_0 + \gamma_2 X_i + \varepsilon_{2i} \\
    Y_i = \alpha_2 + \delta \hat{r}_i +\varepsilon_{3i}
   $
When doing this in practice, the coefficient estimates are the same but standard errors of the residualized regression are different, in my case larger (I would have expected smaller...). When I bootstrap the standard errors the standard errors increase even more and the difference just becomes larger.
Is there an analytic adjustment of the standard errors when running residualized regressions?
Addendum
I noticed that for full-fledged FWL I should have residualized $Y$ too (even though that works too for the coefficient, see MHE). If I do that and run the residuals of $Y$ on the residuals of $D$
$\hat{R_Y} = \alpha_3 + \delta \hat{R}_D +\varepsilon_{4i}$
the standard errors is now below the multivariate one and further away than the single-residualized one.

Note that the Frisch-Waugh-Lovell theorem implies that the residual vectors $e\mathrel{:=}Y - \left(\hat\alpha_1 + \hat\delta D + \hat\gamma_1 X\right)$  and  $\tilde{e}\mathrel{:=}\hat{R}_Y - \left(\hat\alpha_3 + \hat\delta \hat{R}_D\right)$ coincide.
Therefore, an unbiased estimator $\hat\sigma^2$ for $\mathop{\mathbb{V}}\left(\varepsilon_{1i}\right)$ is given by
$$
\hat\sigma^2 = \frac{e^\top e}{n-3} = \frac{\tilde{e}^\top \tilde{e}}{n-3}=
\frac{n-2}{n-3} \cdot \tilde{\sigma}^2,
$$
where the realized value of $\tilde{\sigma}^2 = \tilde{e}^\top \tilde{e}/\left(n-2\right)$ is probably what your statistical software uses to compute the standard error of $\hat\delta$ in the last regression you describe. If so, you would have to multiply that standard error by a factor of $\sqrt{\frac{n-2}{n-3}}$ to make it match the one in your first regression.
