Good point, Mladen.
I checked, and auto-correlation is not a problem. If we remove the denominator from each series we still get very high correlation. Here is the total employment of the top 200 compustat corporations (left, in thousands) and the total energy consumption of the US (right, in Exajoules):
Actually, I've recently become aware of a problem with least squares regressions. Any smooth, monotonic curve will be highly correlated with any other curve with the same properties.
This puts a great deal of econometrics in jeopardy, including much of economic growth theory. Real GDP growth is almost perfectly exponential in the long-run, meaning any other smoothly growing curve will give extremely high correlation. This is why neoclassical production functions work. We could just as easily assert that GDP is an exponential function of time: Y= a*exp(bt). Both will yield high correlation, both are non-nonsensical.