What can we say about the variance of the error term in a standard linear regression?

At its most basic we can consider the case with two regression lines as a special case of the single regression which contains an extra additive or multiplicative term. It's like putting in a dummy variable in which changes the slope and/or the intercept term.  Now they're on the same playing field we're just asking the question if adding an additional variable like we just did would increase or decrease the regression variance. The answer is pretty straight forward. Adding a new variable will NEVER increase the regression variance. It's ALWAYS monotone decreasing. That's because a new variable cannot make the model explain less than what's already been explained without it being present. So the regression variance stays the same or reduces. I can understand the motivation for the statistic you mention but it seems pretty redundant to me. Such a statistic would be monotone decreasing, as you have reasoned, but what new information does it contain that you can't get in a simpler, more intuitive way? There is a small flaw in your logic though. The single regression variance in the denominator would be constant. It's the numerators which are changing. And strictly speaking it would be better to use the standard deviations in the numerator since, if the data did not support two regression lines the statistic would approach 1 from below. Do you have a link to the paper? I'd like to see why the author would put forward the notion you mention. Maybe it's logically consistent but worded in a confusing way.