Is the significance of difference in slopes equivalent to the significance of the slope of the difference of two series? Say you have an independent variable, $x$, and two dependent variables $y_1$ and $y_2$. I want to calculate whether these two variables have a significantly different slope. I can do it by calculating separate regressions, and then using he method mentioned at Test a significant difference between two slope values.
I was wondering, would be equivalent to testing for the significance of the slope of the series $(y_1-y_2)$ [e.g. null hypothesis slope=0]? I tried it with a couple of hundred independent series, and the results look very similar (similar distribution of p-values), but I'd think if it were that easy, more people would use it, as the computation is simpler. So, why is it not the same?
 A: If the x's are the same for both, you're effectively making the values paired (on $x$). 
More generally - as in the case where you have two separate fitted lines - you don't have that.
When you do have the two sets of $y$ paired on $x$, then you could test for a difference in slope by taking differences $y_{2i}-y_{1i}$ and regressing that on $x$.
If the y-values for a given $x$ are independent (conditional independence), it should make no difference which you do.
$\hat{\beta_2}-\hat{\beta_1}=(X'X)^{-1}X'y_2-(X'X)^{-1}X'y_1=(X'X)^{-1}X'(y_2-y_1)$
showing the estimates are the same, and (assuming conditional independence):
$\text{Var}((X'X)^{-1}X'(y_2-y_1))=(X'X)^{-1}X'\text{Var}(y_2-y_1)X'(X'X)^{-1}$
$=(X'X)^{-1}X'((\sigma_1^2+\sigma_2^2)I)X'(X'X)^{-1}$
$=(\sigma_1^2+\sigma_2^2)(X'X)^{-1}X'X'(X'X)^{-1}$
$=(\sigma_1^2+\sigma_2^2)(X'X)^{-1}$
$=\text{Var}(\hat{\beta_2})+\text{Var}(\hat{\beta_1})$
(though the sample values won't be perfectly uncorrelated). So I think the two are at least asymptotically equal, and should be equally efficient at smaller samples. I don't think it matters which you use.
However, if the y-values are positively correlated, it would be important to work with the differences. 
