What is Okun's law about, anyway?

In my iconoclastic way, I've referred to any relationship between employment and output as "Okun's law" (e.g. here and here). In the first of those links, I noted that "Okun's law" understood that way was at best an approximation. However, a few weeks ago Britonomist on Twitter piqued my interest in the original form by saying:

I don’t remember being taught any explanation of why c in Okum’s law was 2 for instance, but I would have found the subject very dull if it was just memorising various estimated parameters without any deeper explanation.

I set off to try and show why c = 2 using the dynamic information equilibrium model. What I learned was that the interesting part of Okun's law is almost entirely about the one or two quarters spent in recession. If we take the form ΔY/Y = k − c Δu where the change is over a year, and the typical coefficients where k = 3% and c = 2, we can show a pretty good "explanation" of k

k ≈ (e^γ − 1)  − c (e^α − 1) ū

where γ ≈  2.4% is the dynamic equilibrium for RGDP growth (i.e. the difference between the dynamic equilibria for NGDP = 3.8% and the GDP deflator = 1.4%), α ≈ − 8.3% is the dynamic equilibrium for the unemployment rate, and ū is the average unemployment rate over the period (~ 5.9%). This gives us

k ≈ 2.4% − 2 ( − 8.3%) × 5.9% = 3.4%

k ≈ 2.4% − 2 ( − 0.5%) = 3.4%

That ū shows up because the RHS of Okun's law is in terms of the change in the unemployment rate Δu rather than Δu/u ≈ Δu/ū (otherwise, we'd have a direct relationship between dynamic equilibria). The exponentials convert a logarithmic growth rate (dynamic equilibria) to a yearly change.

The truth is that we get k approximately right for c = 1, 2, or 1.4 (per the best fit later, and these values give 2.9%, 3.4% and 3.1%, respectively) as that term is about a 10-20% correction. The main driver behind the value of c is the recessions:

In the figure, I show year over year RGDP growth (gray) along with the DIEM model for the unemployment rate (purple). Okun's law is a transformation of the latter, and I show both the traditional coefficients (red) and the best fit over the data shown (dashed red). I also show the dynamic equilibria (dashed purple for the unemployment rate and dashed gray for RGDP). You can see that the k-value is mostly about the constant levels between non-equilibrium shocks (vertical lines) — it turns out the c-value is mostly about the shocks themselves.

The peak of shock n in (d/dt) log u(t) is aₙ/(4 bₙ) above the dynamic equilibrium α using the logistic function ansatz of step height aₙ and width bₙ. But again, we have that factor of ū to account for in relating Δu rather than Δu/u ≈ Δu/ū, which means that the shocks to unemployment are about 1/ū = 16.9 times bigger than the shocks to real GDP if c = 1. If c = 2, we have about a factor of 8.4. This turns out to be a better empirical fit (at least for more recent years):

So we end up with c = 2 because the shocks to the unemployment rate are about 8 times bigger than the shocks to real GDP and the average unemployment rate is about 5-6%. This is all to say that c = 2 is essentially a fluke of scaling two correlated relative rates of change. If we use that scaling factor of 8.4 to try and match (the log of) the unemployment rate and RGDP with a dynamic equilibrium subtracted (the correct frame), we can match the size of the steps without changing the scaling factor, but we have to move around the dynamic equilibrium and the offset (click to enlarge):

I had to change the dynamic equilibrium for Y = RGDP (γ) from 5% to 4% and eventually down to 2.4% to get these to line up (this is due to the demographic shift of the 1960s and 70s). Taking the rates of change ΔY and Δu removes the offset, but leaves the changing dynamic equilibrium which shows up in the graphic above as the size of the shocks not exactly matching.

Again, this makes Okun's law seem more like scaling two correlated series to match each other than any structural relationship between variables. Effectively, big recessions have big drops in RGDP and big spikes in unemployment while small recessions have small drops in RGDP and small spikes in unemployment. The fact that c = 2 means this proportional relationship hasn't changed very much over the past few decades (it's actually risen from about 1 in the 50s and 60s to about 2 in the 90s), but the actual value of c is an empirical point estimate. Mainly, c is based on the relative size of recession shocks to real output compared to recession shocks to the unemployment rate.