User:Stevelihn/sandbox/Unemployment momentum

The unemployment momentum $$U_{1y}(t)$$ is defined in Lihn (2019) as the 12-month difference of logarithm of the unemployment rate in the U.S.:


 * $$U_{1y}(t) = \log \text{UNRATE}(t) - \log \text{UNRATE}(t-1),$$

where $$\text{UNRATE}(t)$$ is the U.S. civilian unemployment rate published by the Federal Reserve, and $$t$$ is measured by year.

The unemployment acceleration $$A_{6m}(t)$$ is defined as the 6-month rate of change of $$U_{1y}(t)$$:


 * $$A_{6m}(t) = 2(U_{1y}(t)-U_{1y}(t-0.5)).$$

Applications
In the real-time recession probability model developed in Lihn (2019), $$U_{1y}(t)$$ has two states - normal state and crash state - in the Hidden Markov Model (HMM), and $$A_{6m}(t)$$ has three states - acceleration, middle (or normal), deceleration. The composite HMM can decode the NBER recession indicator, USREC, since 1960 with reasonable precision. It shows that positive momentum in unemployment kicks off a recession. The momentum accelerates during the recession. And eventually the rapid deceleration marks the end of it.


 * Illustration of unemployment momentum in Hidden Markov Model.png
 * Illustration of unemployment acceleration in Hidden Markov Model.png
 * Illustration of unemployment acceleration in Hidden Markov Model.png