The latest data for July for the dynamic equilibrium model of the (prime age) civilian labor force participation rate is now available. No big news — just continuing to be a decent model:

The second graph is a zoomed-in version of the first.

...

There is a scope condition on this model: it won't work for CLF participation rates close to 100%. The model operates on a log-linear transform of the data which is fine for ratios that can go above 1. However, CLF is limited to being being between 0% and 100% (a labor force participation rate, like an unemployment rate above 100%, doesn't make sense). For these kinds of measures, instead of a log-linear transformation $x \rightarrow \log x$ we need to use

x \rightarrow \log \frac{x}{1-x}

$$

For $x \ll 1$ (i.e. sufficiently below 100%), we have

$$

\log \frac{x}{1-x} \sim \log x + o(x)

$$

As you can see, it is a decent approximation until you get close to 100%:

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