Thursday, June 14, 2012

The "unit root" thing (arcane:)

Unit root? No, I don't understand. Wikipedia says:

In time series models in econometrics (the application of statistical methods to economics), a unit root is a feature of processes that evolve through time that can cause problems in statistical inference if it is not adequately dealt with.

If a "linear stochastic process" has a "unit root" the process is "non-stationary". Otherwise it is "stationary". Something like that (based on the Wikipedia article). Apparently it is something that can be calculated. Sometimes. Maybe.

This rang a bell. From the same article:

The diagram above depicts an example of a potential unit root. The red line represents an observed drop in output. Green shows the path of recovery if the series has a unit root. Blue shows the recovery if there is no unit root and the series is trend stationary. The blue line returns to meet and follow the dashed trend line while the green line remains permanently below the trend. The unit root hypothesis also holds that a spike in output will lead to levels of output higher than the past trend.

Okay: The dotted line is Potential GDP. The red line is actual GDP. The blue and green lines are two possible post-crisis paths. We seem to be on the green path.

This is exactly what James Bullard was talking about, what John Taylor worked out and John Cochrane showed, what David Andolfatto presented, and what I reviewed a few months back. (See especially Cochrane's presentation of Taylor's graphs.)

I was totally fascinated by Bullard's approach. I didn't know it had all this backstory.

Among the ending thoughts of the Wikipedia article:

The issue is particularly popular in the literature on business cycles.

Research on the subject began with Nelson and Plosser whose paper on GNP and other output aggregates failed to reject the unit root hypothesis for these series. Since then, a debate—entwined with technical disputes on statistical methods—has ensued.

While the literature on the unit root hypothesis may consist of arcane debate on statistical methods, the implications of the hypothesis can have concrete implications for economic forecasts and policies.

Arcane debate, indeed.

Paul Krugman (the Wikipedia article links to him) provides a relevant example, which I've highlighted:

March 3, 2009, 9:06 pm

Roots of evil (wonkish)

As Brad DeLong says, sigh. Greg Mankiw challenges the administration’s prediction of relatively fast growth a few years from now on the basis that real GDP may have a unit root — that is, there’s no tendency for bad years to be offset by good years later.

I always thought the unit root thing involved a bit of deliberate obtuseness — it involved pretending that you didn’t know the difference between, say, low GDP growth due to a productivity slowdown like the one that happened from 1973 to 1995, on one side, and low GDP growth due to a severe recession. For one thing is very clear: variables that measure the use of resources, like unemployment or capacity utilization, do NOT have unit roots: when unemployment is high, it tends to fall. And together with Okun’s law, this says that yes, it is right to expect high growth in future if the economy is depressed now.

But to invoke the unit root thing to disparage growth forecasts now involves more than a bit of deliberate obtuseness. How can you fail to acknowledge that there’s huge slack capacity in the economy right now? And yes, we can expect fast growth if and when that capacity comes back into use.

Kling responds:

But the unemployment rate is not trend-stationary either--the "natural rate" tends to wander around.

This all comes up because I came upon a link from besttrousers at Reddit (Reddit links to a PDF, 50+ pages), a paper dated 1989, written by Lawrence J. Christiano and Martin Eichenbaum. From a time before computers, by the look of it.

From the opening of the paper:

Macroeconomists have traditionally viewed movements in aggregate output as representing temporary fluctuations about a deterministic trend. According to this view, innovations [shocks] to real gross national product (GNP) should have no impact on long-run forecasts of aggregate output. Increasingly, however, this view of aggregate fluctuations has been challenged. Following the important work of Nelson and Plosser (1982), numerous economists have argued that real GDP is best characterized as a stochastic process that does not revert to a deterministic trend path.

Let me ask a question: What is this trend that may or may not exist?

It is always an unwavering path, a straight line on a log chart. This straight-line path is the universal, fundamental assumption, whether we think it exists or not.

The straight line is the wrong picture. The long-term trend looks like a sine curve. It looks like a business cycle, only longer than the normal business cycle. Longer than a Kondratieff wave. Longer than David Hackett Fischer's Great Wave. I'm looking at a really long business cycle, what we normally think of as civilization, the rise and fall of civilization.

The cycle of civilization. Change that dotted line to a sine wave with a 2000-year period, and all your assumptions about stationary, non-stationary, and unit root go out the window.