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!
 A: 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 COLLINEARITY.
These 2 IV are not going to be collinear 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. 
Collinearity may airse as the linear combination $c_1x + c_2z$ approaches 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 correlation may lead to collinearity, 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 addressed. 
A: 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.
