Even though the original question was asking for a software, the "hard" part is in statistical theory. The answer is called the multivariate delta method, which states that, given a variance-covariance matrix
$$V = \begin{bmatrix}
\sigma^2_{A} & \sigma_{A, B} & \sigma_{A, C}\\
\sigma_{A, B} & \sigma^2_{B} & \sigma_{B, C} \\
\sigma_{A, C} & \sigma_{B, C} & \sigma^2_{C} &
\end{bmatrix} $$
and defining the gradient of $\hat{\beta}$
$$\nabla \hat{\beta}(A,B,C) = \begin{bmatrix}
\frac{\partial \beta}{\partial A}\\
\frac{\partial \beta}{\partial B}\\
\frac{\partial \beta}{\partial C}\\
\end{bmatrix}$$
The required variance is approximately
$$Var(\hat{\beta}) \approx \nabla \hat{\beta}^t V \nabla \hat{\beta}$$
Given the difficult formula for $\hat{\beta}$, finding the gradient is the "difficult" part, but indeed any symbolic software is able to carry out this symbolic derivation.
lincom
command. $\endgroup$ – AdamO Feb 20 '18 at 17:40nlcom
. However, that will only work after an actual estimation command, rather taking an immediate argument. $\endgroup$ – Dimitriy V. Masterov Feb 20 '18 at 17:54