Testing for moderators of the difference between two regression coefficients

[Maoliang Ling]

Dear all,

Using cross-sectional survey data, I want to compare the effects of two independent variables X1 and X2 on Y (denoted as b1 and b2), and analyze whether b1 becomes significantly stronger than b2 as the variable S increases. In other words, does S moderate Δb = b1 - b2? All variables X1, X2, Y, and S are continuous. Here is what I have considered:

For the sample with S = s1, obtain the effects of X1 and X2 on Y denoted as b11 and b12, with a gap Δb1.
For the sample with S = s2, obtain the effects of X1 and X2 on Y denoted as b21 and b22, with a gap Δb2.
...
For the samples with S = sn, obtain the effects of X1 and X2 on Y denoted as bn1 and bn2, with a gap Δbn.

However, this approach does not allow for directly testing the differences between Δb1...Δbn.

Can anybody help me address the issue of heterogeneity in effect differences across variables? Specifically, how should I test whether Δb = b1 - b2 systematically varies with S?

Thanks in advance.

[Clyde Schechter]

Interesting problem.

Assuming your model is

regress Y c.S##c.(X1 X2) // AND PERHAPS SOME COVARIATES
// Y = b0 + b1*X1 + b2*X2 + b3*S + b31*S*X1 + b32*S*X2 + ... + error

then the marginal effect of X1 conditional on S will always be b1 + S*b31, and the marginal effect of X2 conditional on S will be b2 + S*b32. So the difference in the marginal effects, conditional on S, will be (b2-b1) + S*(b32 - b31). Your question, whether S moderates this difference then is nothing more or less than the question of whether b32-b31 = 0. So all you need to do is -lincom c.S#c.X2 - c.S#c.X1- and you will get from that an estimate of the slope of the linear Δb:S relationship, along with its standard error, a t-test of whether it is 0, and a 95% confidence interval for it.

[Maoliang Ling]

Dear Clyde, thank you very much for your help.

[Richard Williams]

I assume X1 and X2 are both measured the same way, right, e.g. both are measured in dollars?

[Maoliang Ling]

Yes, Richard, they are measured the same way.