Compare Regression Coefficients I want to test if two regression coefficients from two separate regressions are significantly different from each other. In particular I want to test from the model
$$
\begin{aligned}
x1_{(t)} &= \mu1 + a*x1_{(t-1)} + b*x2_{(t-1)} + e1  \\
x2_{(t)} &= \mu2 + c*x1_{(t-1)} + d*x2_{(t-1)} + e2
\end{aligned}
$$
if $H0: b=c$ against $H1: b>c$. Can I do this with a simple t-test or do I require another test statistic?

Just wanted to give a brief update. In my reference paper, they claim that they can test H0:b=c against H1:b>c in the model stated above with a simple Z-test. Does this mean that my test statistic is simply (b-c)/se(b)?
 A: A Z-test doesn't pass the smell test. Assuming your sample size is large enough, and since your predictor variables are the same in both equations, a multivariate multiple regression model (as opposed to multiple regression with a single outcome variable) is able to estimate the bivariate normal relationship between $x1_{(t)}$ and $x2_{(t)}$ conditional on your predictors. Your planned test of the null $H_0: b \leq c$ will then be testable with an F(1,n-p) test, which is really the same as a t-test squared when df1=1. Just make sure to adjust your p-value and rejection region to account for the one-tailed test. This test is not appropriate if you estimate the regressions separately.
A more detailed explanation is here.
However, as @Peter Ellis points out, it can be misleading to compare betas if you're actually interested in predictive power. My first thought would be to bootstrap the partial $R^2$ of both variables of interest given the others and then see if the bootstrap CI overlap, but this seems to be a contentious subject: How to get confidence interval on population r-square change. I would love to know if there is an accepted method out there.
A: Since these are returns on two different stock portfolios, they might be also contemporaneously correlated in which case you might want to use simultaneous equations to address endogenity issue.
