Arnold Kling's askblog and Roger Farmer have a little exchange on GDP and unit roots. My two cents here.
I did a lot of work on this topic a long time ago, in How Big is the Random Walk in GNP? (the first one) Permanent and Transitory Components of GNP and Stock Prices” (The last, and I think best one) "Multivariate estimates" with Argia Sbordone, and "A critique of the application of unit root tests", particularly appropriate to Roger's battery of tests.
The conclusions, which I still think hold up today:
Log GDP has both random walk and stationary components. Consumption is a pretty good indicator of the random walk component. This is also what the standard stochastic growth model predicts: a random walk technology shock induces a random walk component in output but there are transitory dynamics around that value.
A linear trend in GDP is only visible ex-post, like a "bull" or "bear" market. It's not "wrong" to detrend GDP, but it is wrong to forecast that GDP will return to the linear trend or to take too seriously correlations of linearly detrended series, as Arnold mentions. Treating macro series as cointegrated with one common trend is a better idea.
Log stock prices have random walk and stationary components. Dividends are a pretty good indicator of the random walk component. (Most recently, here.)
Arnold asks "In stock market returns, econometricians have been able to identify long-term mean reversion even though the short run is a random walk. Can something similar be done with GDP data?" Answer: Yes, and Permanent and Transitory Components is it.
A critique", that it is a bad idea it is to blindly run unit root tests and then impose that structure.
A unit root means a random walk component. A random walk will eventually pass any upper and lower limit. Look at it. That's as stationary a series as you're going to find in economics. ("Look at the plot" and "think about the units" 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.
If you apply unit root tests to an hour of second by second temperature data from 9 to 10 AM you will think it has both a linear trend and a unit root. Millisecond data will not help you to detect climate change. That's why unit root tests are a problem. You have to think, and consider the span of data you have and the frequency of mean reversion that makes economic sense in your data.
The tests are about infinite horizon behavior which you can never tell with finite horizons. However, they can alert you to low-frequency movement in your data, which can make ordinary distribution theory a bad guide. So can looking at a plot.
As far as I can tell, "Potential GDP" is equivalent to a two sided filter. It looks great ex-post. None of this is inconsistent with Arnold's view that standard calculations of potential GDP gaps do little to forecast GDP growth, especially in real time.