# What is the most appropriate way to test significance of effect difference across two groups?

Here is the question: After running two separate OLS regressions, one for each group (male vs. female), we get coefficient b1 for a specific IV in "male" regression, and coefficient b2 for the same IV in "female" regression. What would be the most appropriate way to test the hypothesis: Ho: b1>b2? As far as I know, Wald is a widely used test, but it only check whether the effect difference is different from zero (Ho: b1-b2=0)

Thanks!

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## 2 Answers

A one sided test like a t-test can be obtained by merging your two separate regressions into one that models both groups. If you are interested in the effect of x on y across groups then add a $x_i D_i$ regressor where $D_i$ is a dummy variable that codes your group:

$y_i = a x_i + b x_i D_i + u_i$

As you can see, your group effects are given by $a$ and $a+b$. Thus your hypothesis test becomes $H_0: b>0$ and its statistic would be based on a one sided t-statistic

ABOUT COLINEARITY. These 2 IV are not going to be colinear as there is no linear combination that makes its sum = 0 under standard conditions (and a OLS, ML estimator). to see why let's only take the sample for which $D_i=0$ and assume the following vector notation: $y = x a + z b + u$ where z is a vector with $x_iD_i$components.

Colinearity may airse as the linear combination $c_1x + c_2z$ approaches to zero (for $c_j \neq 0$ But just considering the block of observations where $D_i=0$ automatically leads to a linear combination block equal to $c_1 x\neq 0$ if $x \neq 0$.

I guess you are thinking in term of standard theory that tells you that higher IV corellation may lead to colinearity, but don't forget that standard theory refers to the IV which in our case are $x_i$ and $z_i(=x_iD_i)$ and not $x_i$ and $D_i$. Moreover, the mathematical issue behind the correlation issue is the fact of having linearly dependent variables, so linear dependence between $x_i$ and $z_i$ must be adressed.

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I agree with JDav.  I think we came up with the same answer at almost the same time. He is really confirming what I happened to get submitted first. I suspect that he was writing his before I finished mine and came up with it independently.  Ihope this will convince you to use this approach rather than work with two spearate regressions. – Michael Chernick Aug 4 '12 at 0:52
Thanks!!The joint model approach sounds rather straightforward and flexible indeed! My only concerns is that, in my dataset, the gender dummy is highly correlated with the IV, and for this reason I was a bit hesitant to pool the two groups together... – Bill718 Aug 4 '12 at 0:57
Haha, indeed, almost same answer at almost the same time! I think I am convinced what the best method is:) – Bill718 Aug 4 '12 at 1:14
It took me long to write these very few lines and it was certainly almost simultaneous answer. About colinearity, see the edited answer. – JDav Aug 4 '12 at 12:33
Many thanks for your time, really appreciated!I was stuck with the standard theory as you mentioned;worried about high IV correlation that could cause serious colinearity issues...it sounds clear now! – Bill718 Aug 5 '12 at 2:13

Why not do a more conventional thing and do one regression with gender as a covariate? Then a statistically significant gender effect would indicate that gender is important and more specifically include a gender interaction term with your IV. If the interaction is statistically significant that indicates that the male and female slopes differ. Make the test one sided and you have your answer.

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Thanks!!Please see comment above, high correlation between IV and dummy is a point of concern if doing one regression only. – Bill718 Aug 4 '12 at 0:59

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Correlation between covariates is not a problem unless it is so high as to create a near multicollinearity. In fact if you believe one slope is different from the other then the IV interacts with gender and there is necessarily a correlation. Also correlation between a binary variable and and a continuous one is different from correlation between two continuous variables. Certainly a pearson correlation cannot be defined. – Michael Chernick Aug 4 '12 at 3:03
I see, I was not aware that pearson correlation cannot be defined in case of correlation between binary and continuous variables, many thanks! – Bill718 Aug 5 '12 at 2:15
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