standard error of ratio of coefficients

#1
hi all,

I am interested in the ratio of two regression coefficients. Is it possible to obtain a standard error for such a ratio?

eg: I estimate the following OLS regression:

y=a+bx1+error.

but what interests me is a/b. How can I get the standard error of (a/b)?

Thanks
 

ricky

New Member
#2
hi all,

I am interested in the ratio of two regression coefficients. Is it possible to obtain a standard error for such a ratio?

eg: I estimate the following OLS regression:

y=a+bx1+error.

but what interests me is a/b. How can I get the standard error of (a/b)?

Thanks
I believe this depends on in what are you interested? If you are just regarding the random uncertainty, you could performed the uncertainty analysis of simple linear regression by the method found here:
http://www.physics.upenn.edu/~uglabs/Least-squares-fitting-with-Excel.pdf
You could compute the uncertainty associated with a and b seperately and propagate the uncertainty to obatain the uncertainty of a/b. In this simple case of division, it is just the sqrt(sum(u(a)^2+u(b)^2)) where u(a) is the relative uncertainty of a.

Correct me if I am wrong.
 

BioStatMatt

TS Contributor
#3
Well, but consider (b/a) for a moment. The variance of (b/a) is:

var(b/a)

however, a = ybar - b*xbar so

var(b/a) = var(b / (ybar - b*xbar))

Which will not be very fun to try and solve. However, it can be approximated using the delta method, which is common when the exact SE is difficult.

var(a/b) can be found also in a similar manner.

~Matt