Team Obama on the Unit Root Hypothesis

All academics, to some degree, suffer from the infliction of seeing the world through the lens of their own research. I admit, I do it too. So when I read the CEA's forecast analysis, this sentence jumped out at me:

a key fact is that recessions are followed by rebounds. Indeed, if periods of lower-than-normal growth were not followed by periods of higher-than-normal growth, the unemployment rate would never return to normal.

That is, according to the CEA, because we are now experiencing below-average growth, we should raise our growth forecast in the future to put the economy back on trend in the long run. In the language of time-series econometrics, the CEA is premising its forecast on the economy being trend stationary.

Some years ago, I engaged in a small intellectual skirmish over this topic along with my coauthor John Campbell. Here is the abstract of our paper:

According to the conventional view of the business cycle, fluctuations in output represent temporary deviations from trend. The purpose of this paper is to question this conventional view. If fluctuations in output are dominated by temporary deviations from the natural rate of output, then an unexpected change in output today should not substantially change one's forecast of output in, say, five or ten years. Our examination of quarterly postwar United States data leads us to be skeptical about this implication. The data suggest that an unexpected change in real GNP of 1 percent should change one's forecast by over 1 percent over a long horizon.

The view that Campbell and I advocated is sometimes called the unit-root hypothesis (for technical reasons that I will not bother with here). It contrasts starkly with the trend-stationary hypothesis.

In the CEA document, Table 2 shows growth rates immediately after recessions end. It demonstrates that growth is higher than normal in most of the recoveries. Is this evidence against the hypothesis that Campbell and I advanced?

I don't think so. The problem is that those numbers start at the end of the recessions, and we do not know when the recession will end. In other words, if God came down and told us the exact date the current recession was going to end, my forecast subsequent to that date would be for higher than normal growth. But absent that divine intervention, there is always some chance the recession will linger (remember the Great Depression), and an optimal forecast has to give some positive probability weight to that scenario as well. The forecast should be an unconditional expectation, not an expectation conditional on a particular end date for the recession.

The CEA document also gives an intriguing picture:

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.*

There is another possible interpretation, however: Imagine that the shocks to the economy have time-varying variance. When the variance is high and the economy experience a negative shock, one gets a deep recession. But when recovery comes, it tends to be more robust.

For the econgeeks out there, let me put the point more formally. Suppose growth G is a random variable distributed N(M,V(t)), where M is the mean growth rate and V(t) is the time-varying variance. A recession is when G is negative. Now compute two conditional expectations: E[G / G less than 0] and E[G / G greater than 0]. You will find, I am pretty sure, that an increase in V(t) reduces the first conditional expectation and increases the second. That is, higher variance makes average recessions deeper and average recoveries more robust. But if you don't know whether a future date will occur in a recession or recovery, the best forecast is M, the unconditional mean.

Right now, we are facing a particularly high-variance economy. (Just look at the VIX index.) That means, under the conjecture I just described, that when recovery comes, it will probably be a robust one. But this logic is not necessarily a reason to raise the unconditional expectation of economic growth, because we don't know when that recovery will begin.

Finally, I should note that there is much to forecasting beyond the univariate models in my work with Campbell. And our paper, of course, was only one piece of a large literature. The CEA might well be right that we are in for a robust recovery over the next few years. I don't pretend to have as good a forecasting staff sitting in my Harvard office as the CEA has. (I miss you, Steve Braun.) I certainly hope they are right. We could all use some good economic news right now.

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* One odd feature of this figure is that it omits the 1980 recession. Perhaps the CEA left it out because that recession was followed quickly by another recession. As a result, in the two years after the 1980 recession ended, growth averaged less than one percent. That episode underscores my main point: when forecasting, you cannot be assured of being in a recovery state rather than in a recession state.