# Testing whether two regression coefficients are significantly different

I'm hoping somebody can help me out with this question. For a study I did a path analysis, which looks like this:

IV -->
Mediatior  --> DV
IV -->


So I have two IV's leading to my mediator. With SPSS I did two regression analysisses from the IV's to the Mediator.

Now I want to test whether the two regression coefficients are signifficantly different from each other using SPSS, but I have no idea how to do this. Except that I assume that I have to use a T-test, but don't know how.

I have the two regression coefficients, their standard deviation and my N (this n is the same for both regressions, because it's all from one sample).

I suspect that this is pretty basic for some, but my skills in statistics are not that great. So help would be greatly appreciated!

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Why do you want to do this? –  Peter Flom Jun 26 '12 at 11:01
Because according to my regression coefficients IV(a) has more influence on the mediator than IV(b) does. And now I would want to know whether this difference between the regression coefficients is significant –  user12208 Jun 26 '12 at 13:28

If you have the estimated covariance matrix for the coefficients, then you can construct the t-test as follows. Let the hypothesis, in its general form, be $R^T\beta = b$, and $\widehat{\Sigma} = \hat{\sigma}^2(X^TX)^{-1}$ be the estimated covariance matrix of the coefficients. In your case, assuming the test is that $\beta_2 = \beta_3$ and you have $K=3$ coefficients, $R^T = [0, 1, -1]$ and $b=0$. Then:

$T^* = \frac{R^T\beta - b}{\sqrt{R^T\widehat{\Sigma}R}}$

is distributed $t(N-K)$.

Source: Principles of Econometrics, Theil.

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Thanks for all of the answers! –  user12208 Jun 30 '12 at 13:36
I have the same question as user1 but jbowman's response isn't clear to me. Is it possible to clarify?? –  Candace White May 3 at 13:00
@Candace White I think you will need to be more specific than "isn't clear" if you would like a focused answer to your comment. –  whuber May 3 at 13:17