Testing whether the coefficients of one model and another are statistically significantly different I am currently testing two models which have the exact same specification except for one is when z=0 and when z=1, where z could be something like male and female. Essentially, the effect of X on Y if Z=0 and X on Y if Z=1. 
I want to test whether the coefficients in these models are statistically significantly different from one another. 
In my mind I should do some form of likelihood ratio test, but I'm not certain. i was wondering if anybody could confirm my suspicion that I'm wrong. 
 A: This sounds like an interaction effect. You estimate the model:
$y = \beta_0 + \beta_1 x + \beta_2z + \beta_3 x z + \varepsilon $


*

*$\beta_0$ is the constant when $z=0$

*$\beta_0 + \beta_2$ is the constant when $z=1$

*$\beta_1$ is the effect of $x$ when $z=0$

*$\beta_1 + \beta_3$ is the effect of $x$ when $z=1$


So $\beta_3$ is the difference in effect of $x$ before and after the crisis. If it is 0 there is no difference in effect. This would be your null hypothesis, and the test statistic is (depending on your software) typically reported immediately next to that coefficient. 
If $\beta_3$ is positive then the effect of $x$ is larger after the crisis than before. If $\beta_3$ is negative then the effect of $x$ is smaller after the crisis than before.
The model with interactions is almost the same as estimating two separate models. You can see this by looking at the point estimates: these will be exactly the same. The only difference is that you constrain the variance of the error term to be the same before and after the crisis (homoscedasticity). You can easily work around that by asking for robust/Huber-White/sandwich standard errors. 
A: I agree with Maarten that it is an interaction and it should be solved as one model. But, as your variable $z$ is categorical and not continuous, the Maarten's model can be confusing. The model should look like this:
$y = \alpha_z + \beta_z x + \varepsilon $
So, $\alpha_z$ and $\beta_z$ are sex-specific coefficients. This model is called ANCOVA. See example at http://r-eco-evo.blogspot.cz/2011/08/comparing-two-regression-slopes-by.html - R will report the significance of the intercept slope being different for each category (citing the resource). Ugly things will happen if you try to fit $z$ as continuous variable.
