The purpose of this picture is to show that deeper-than-average recessions are followed by faster-than-average recoveries. (Similar evidence was compiled in the 2005 Economic Report of the President, Chapter 2.) This evidence might be taken as evidence in favor of the trend-stationary hypothesis over the unit-root hypothesis.
Later, he gives a shout-out to Steve Braun, a great economist and an old friend of mine from when we both worked at the Fed.
[UPDATE: Greg says more here. Apparently, he drew the wrath of Paul Krugman and Brad DeLong, the latter arguing that you can tell a permanent change from a transitory change by looking at the unemployment rate. But the unemployment rate is not trend-stationary either--the "natural rate" tends to wander around. So I am basically on Greg's side on this one. But see my original thoughts, below the fold.]
The conflict between trend-stationary and unit-root is basically about the statistical properties of macro data. Under the unit-root hypothesis, which I thought was well established, I see the attempt to do macroeconometrics as basically hopeless. The way I see it, with a unit root, every change is a structural change. As a result, there is no way to use macroeconomic data from different decades as if they allowed you to conduct a controlled experiment. See my lost history paper.
I am not sure that it is correct to attach economic meaning to the concepts of trend-stationary and unit-root, but it will help a non-technical reader to try to do so. Think of trend-stationary as an economy where the movements in output are driven by transitory factors. Eventually, the economy returns to a path determined by steady growth in capacity.
Think of a unit-root economy as one in which movements in output are driven by permanent factors. The economy does not return to some pre-determined path. Instead, starting from wherever you are, you grow at some average rate.
In terms of the current situation, suppose that the economy is down because people are postponing the purchases of cars and other goods out of fear. That would be a trend-stationary story, because it suggests a nice rebound when people get over their fear. On the other hand, suppose that the economy is down because we accumulated the wrong types of physical capital (houses) and human capital (skills in mortgage securitization). That is a unit-root type problem. Yes, the economy will start growing again at some point, but that misallocation of physical and human capital is a permanent loss. We are not going to make up for it with some above-normal growth.
Put in these terms, the trend-stationary story has a more Keynesian flavor, and the unit-root story has a more Austrian flavor. But Austrians don't typically do econometrics, so they would not be playing this game.
I style myself an Austro-Keynesian. I think that both Austrian and Keynesian factors are at work in the current economy. So, I think there is a potential for some catch-up demand down the road in consumer durables. However, there is also just a lot of lost output due to misallocation. In practical terms, I think that a bit of the pain could be alleviated by macroeconomic policy, but not all of it. My sense is that Tyler Cowen and I are in synch on this one.