Test for one coefficient doubled another one? I want to run a regression in order to test ( thinking of the $t$-test ) the null hypothesis that one coefficient ($b_2$) is twice another coefficient ($b_3$), so $b_2= 2 b_3$, The variable for $b_3$ is a dummy variable. The $t$-test is $(b_2-2b_3)/se(b_2-2b_3)$. How can I find $se(b_2-2b_3)$?
 A: Most any statistics package will report, somewhere, the covariance matrix of your estimators. Let $S$ denote this covariance matrix. 
Let $w = \left[ \begin{array}{c}1\\-2\end{array} \right]$ and $b = \left[ \begin{array}{c}b_1\\b_2\end{array} \right]$
$\mathrm{Var}(w'b) = w'\mathrm{Var}(b)w = w'Sw$. The standard error of $w'b$ would then be $\sqrt{w'Sw}$.
A: The answer by MatthewGunn applies more generally. If you are working under the framework of linear regression (or generalized linear regression), then
$$se(b_2 - 2b_3) = \sqrt{\dfrac{Var(b_2 - 2b_3)}{n}},$$
where $n$ is the number of observations. The covariance matrix of regression coefficients is $\sigma^2 (X^TX)^{-1}$, which can be calculates for every problem.
$$Var(b_2 - 2b_3) = Var(b_2) + 4Var(b_3) - 4Cov(b_2, b_3).$$
These values can be picked from the estimate of $\sigma^2 (X^TX)^{-1}$. For example, in R you can use the vcov function on the lm object. I present the example on the dataset cars below.
> fit <- lm(cars)
> vcov(fit)
            (Intercept)          dist
(Intercept)  0.76454824 -0.0131543211
dist        -0.01315432  0.0003060568

Here $Var(dist - 2Intercept)$ = .0003 + 4(.76454) - 4(-.01315). 
