# Standard error of a ratio

I have a linear regression and two estimates, say A and B and their standard errors. I need to find the standard error of a ratio A/B [or A/(1-B)].

I guess the main problem is that I don't know the correlation between A and B. I also guess that makes this unsolvable, since the correlation plays a major role. Is this correct?

If I knew it, what would be the way to calculate the standard error?

1) The variance of a ratio is approximately:

$Var(x/y) \approx \left(\frac{E(x)}{E(y)}\right)^2 \left(\frac{Var(x)}{E(x)^2} + \frac{Var(y)}{E(y)^2} - 2 \frac{Cov(x,y)}{E(x)E(y)}\right)$

With the Bayesian approach it is easy to simulate from the posterior distributions of $A$ and $B$ and then to get simulations of the posterior distribution of $A/B$. Using the standard noninformative prior for the Gaussian linear model, we do not need MCMC techniques and we probably obtain a good frequentist-matching property: a $95\%$-posterior credibility interval is approximately a $95\%$-confidence interval in the frequentist sense.