Okun's law and the labor force

I was curious about how the dynamic information equilibrium model of RGDP (described in a presentation/Twitter talk available here) matched up with an equivalent model of employment L (FRED PAYEMS) — they should to some degree because of Okun's law (for a more formal version in terms of information equilibrium "quantity theory of labor" see here). However a naive application doesn't work very well for basically the same reason that the "quantity theory of labor and capital" outperforms the "quantity theory of labor": there is effectively a dynamic equilibrium shock that is different between labor and NGDP that is compensated by the use of capital in the former of the two models. Here's that naive version:

So I tried to correct for this by combining the dynamic equilibrium model for the civilian labor force (CLF) and another one for the "employment rate", i.e. L/CLF. Here is the L/CLF model:

Multiplying by the dynamic equilibrium model for CLF (see e.g. here), we get a decent model of the employment level:

One big deviation is due to the fact that I am treating the 1980s recessions as a single recession (and there is a concomitant step response [1]). This won't be terribly relevant to the analysis here. The next thing to do is put the RGDP growth rate model (red) and the PAYEMS growth rate model (green) on a graph together:

These should be the identical models if Okun's law is a perfect description. As you can see, RGDP growth over estimates PAYEMS growth, specifically in the 90s and 2000s booms (dot-com and housing "bubbles") [2]. The thing is that the late 90s and 2000s is precisely where the RGDP and PAYEMS models are working best, so that deviations there imply that Okun's law is at best an approximation. It makes sense — increased real output during asset bubbles shouldn't be as closely linked to labor market booms.

The models in the "Phillips curve era" from the 60s through the 90s shouldn't exactly match up either because the oscillations due to RGDP are due to oscillations in the price level that precede the shocks to employment as can be seen in the graph above as well as in a chart from my presentation on macro trends:

All of this points to Okun's law being an approximation due to the fact that RGDP and PAYEMS are going to be highly correlated because a) recessions are where employment and output fall, and b) between recessions you usually have growth. In the past, when the Phillips curve was in full effect, the correlation was even better (the Phillips curve is a direct link between employment and inflation, the latter being essential to real output). This link persists through the entire business cycle in that era. More recently when recessions and output seem to be driven by exogenous factors to the labor market (e.g. commodity booms in Australia, asset booms in the US), the connection between the two variables is primarily via the recession.

I'm still trying to make sense of this myself, so I apologize if this comes across as a word salad. There does seem to be an effective macro theory consisting of Okun's law and the Phillips curve valid from the 60s through the 90s. More recently, a different — and less understood —effective theory has taken over.

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Footnotes:

[1] Speculating, but maybe the fading of the step response is linked to the fading of the Phillips curve I mention in my presentation?

[2] There are also significant deviations between the RGDP model and the RGDP data (faint red on the graph) in the case of the "Phillips curve" recessions of the 70s, 80s, and 90s. These could potentially be connected to the step response noted in footnote [1].