That is John Cochrane’s advice on the unit root issue.
A unit root means a random walk component. A random walk will eventually pass any upper and lower limit. Look at it [the unemployment rate]. That’s as stationary a series as you’re going to find in economics. (“Look at it” and “think about it” are the Cochrane unit root tests.)
Yes, unemployment like other stationary ratios in macro (consumption/GDP, hours/day, etc.) have important and frequently overlooked low-frequency movements. But they are far from random walks, and they like unemployment have a very large transitory component at business cycle frequencies. When unemployment is above 8%, it is a good bet that it will decline over the next 5 years.
I always wondered how a bounded sequence could have a unit root but I was too embarrassed to ask. So how do I reconcile Cochrane’s point above with a 2014 Journal of Econometrics Paper called “Testing for unit roots in bounded time series” (without carefully reading the paper)?