Can you multiply or divide standard errors? I'm running a regression in R, and the way it is set up, I get a series of coefficients and their terms.  As it happens, I know that one coefficient is the quotient of two terms (coef1 = r/k).  The output gives a standard error for coef1.  the term r also appears elsewhere in the output, thus I know the standard error of r.  Since coef1 = r/k, and I know the value and standard error of coef1 and r, and I also know the value of k, is it possible to calculate the standard error of k?
 A: From the way you've written the question I assume that you have two input variables $r$ and $k$ and you have estimated the linear model:
$$ E[Y|r, k] = \beta_0 + \beta_1 (r/k) + \beta_2 r $$
Ordinary least squares (and pretty much most regression models) give asymptotic normal distributions with estimable variances for the parameter estimates:
$$ \sqrt{n} \left[ \hat{\beta} - \beta \right] \rightarrow_d \mathcal{N} \left( 0, \text{var}(\beta) \right) $$ 
And in running the regression model, you come up with two estimates: $\hat{\beta}$ and $\hat{\text{var}}(\hat{\beta})$, both of which are consistent for the things they're estimating.
I understand further that now you're interested in what a regression parameter would have been for $k$ had you estimated it in a model like $E[Y | r, k] = \alpha_0 + \alpha_1 k$. You noticed mathematically that, technically speaking $\beta_1 * \beta_2 = \alpha_1$ so $\hat{\beta_1} * \hat{\beta}_2$ is an estimator of $\alpha_1$. But how can you use the regression model results to estimate the variance of $\hat{\beta_1}*\hat{\beta_2}$?
There is no exact result, however the $\delta$-method gives you an approximation. You simply need to calculate the Jacobian of $ J = \nabla f(x, y)$ where $f(x,y ) = x * y$ and multiply with the covariante $\text{var} \left( \hat{\beta_1} \hat{\beta_2} \right) \approx J^T \hat{\text{var}}(\hat{\beta}) J$.
A: It seems to me rather that @Larry is trying to estimate a model that looks like:
$$ E[y|x_1,x_2] = \alpha_0 + \alpha_1x_1 + \alpha_2x_2,$$
where it is a priori known that $\alpha_1=\beta_1/\beta_2$ and $\alpha_2=\beta_1$. By fitting an OLS regression, we get the estimates $\hat\alpha_1$ and $\hat\alpha_2$ and corresponding standard errors. And using these estimate we can calcualte, $\hat\beta_2=\hat\alpha_2/\hat\alpha_2$. The question seems to be "what is the standard error of $\hat\beta_2$?". 
A practical approach to estimating the standard error would be simply to bootstrap $\hat\beta_2$. If your sample has $n$ observations, resample $n$ of them with replacement (according to the design by which the sample was obtained). Call this resampled set of observations $S$. Then fit your model on $S$ and calculate your estimate of $\hat\beta_2^{(S)} = \hat\alpha_2^{(S)}/\hat\alpha_1^{(S)}$. Repeat this process many times, each time drawing a sample from your original $n$ observations with replacement. Say you have fitted your model to $k$ resampled datasets. This will give you $k$ different $\hat\beta_2^{(S)}$s. The standard deviation of the distribution of these estimates will give you an approximation to the the standard error of $\hat\beta_2$.
