Monday, April 27, 2015

Unit roots in English and Pictures

After my unit roots redux post, a few people have asked for a nontechnical explanation of what this is all about.

Suppose there is an unexpected movement in any of the data we look at -- inflation, unemployment, GDP, prices, etc.  Now, how does this "shock" affect our best estimate of where this variable will be in the future? The graph shows three possibilities.

First, green or "stationary."  There may be some short lived dynamics, the little hump shape I drew here. Then, given enough time, the variable will return to where we thought it was going all along. For unemployment, suppose your best guess of unemployment in 2050 was 5%. Then you see an upward unexpected 1% spike in today's unemployment. Ouch, that means that we're going back to a recession. But perhaps this news does not change your view of 2050 unemployment at all.

Second, blue or "pure random walk." That's more plausible (though no longer thought to be true) of stock prices. If the price goes up unexpectedly, your expectation of where the (log) price will be in the future goes up one-for-one, for all time.

Third, black, "unit root." This option recognizes the possibility that a shock may give rise to transitory dynamics, and may come back towards, but not all the way towards your previous estimate. As you can see the "unit root" is the same as a combination of a stationary component and a bit of a random walk. Perhaps seeing unemployment rise 1%, you think most of it will work itself out, but that even in the long run labor markets will be sticky and we'll never quite get back.

The "unit root" is most plausible and verified in the data for log GDP. Recessions and expansions have a lot of transitory component that will come back. But there are permanent movements too. Unemployment, being a ratio, strikes me as one that eventually must come back. But it can take a longer time than we usually think, which is interesting.

This is very simplified. A few of the issues:

For GDP the question is whether it will come back to a linear trend extrapolated from past data, not back to a level as I have shown.

Most of the issue is how standard statistical procedures work in these circumstances.

As you can see from the graph, the pure question whether the series will come back in an infinite time period is not really knowable. It could be that the series will come back eventually, but take a very long time. It could be stationary plus a second very slow moving stationary component. This is a statistical problem but not really an economic problem. The appearance of unit roots are economically interesting as they show a lot of "low frequency" movement, series that are coming back slowly -- even if they do come back eventually. The economics of "slumps" and (we hope, someday) "booms" is hot on the agenda, and this is one indication of the fact.

This is all much more interesting if you look at multiple series together. For the canonical example, if you just look at stock prices, they are very very close to a random walk. A price rise or decline are permanent. However, if you see stock prices rise relative to dividends, that's almost entirely stationary. GDP and consumption have a similar relationship. As in the latest recession, if GDP declines with a big consumption decline, that looks pretty darn permanent. GDP declining and people still consuming is much more likely to go away.

I hope this helps.

1. Thanks, I had a general idea of what this meant but this makes more sense now. Doesn't this also highlight an issue with things like the Taylor Rule? How well can we really know what the output gap and potential GDP are at a given time, especially after a major downturn?

1. The Taylor rule is a play to the crowd. You have no idea of what the Potential GDP is, what the long run equilibrium interest rate is, or what the relative weights of the GDP and inglation components should be. Too many unknowns.

2. Is "log GDP" the logarithm to some base of the GDP? Is the base Euler's number, two, or ten? Similarly, is log stock price the logarithm to some base. Which base. And why does taking the logarithm make a difference?

1. Almost always it is natural logs. It's very common in econometrics because it can an index with exponential growth look linear. Also, log returns have some nice properties.

2. Usually (if not always) it's the natural logaritm. I'm in finance, so I'm certain for stock prices, and pretty certain they use the natural log in macro as well.

3. ln because it's derivative is 1/x, and therefore a change in natural log is approximately a % change, especially for small changes..\$s grow and shrink multiplicatively rather than additively

3. Why wait 10 years for the green (or black) line to return to full employment? Why not create another shock in the opposite direction with fiscal stimulus? Unemployment has long term social costs that should not be tolerated.
This is not an economics question. It is a question of fairness and the values of our society.
- jonny bakho

1. The question is if that can even be fine

2. Sorry I meant done not fine....auto correct...

4. A helpful and generous post, thanks!

5. Nicely done. As a follow up, John should discuss some of the different ways economists and statisticians try to deal with nonstationary series. I suspect a big reason why time frequency analysis is still a hot a field in macro even though people have been writing papers on it since the 90s.

6. The random walk also has a unit root, no?

1. I was thinking the same thing, which made this post a little bit confusing for me :) I guess the point is this:

random walk => unit root
random walk + stationary stuff => unit root
stationary stuff => no unit root

2. Yes, Kurt. I think you have got it right there.

7. It is vert clear, but I cannot see any utility for economics here.

1. You mean apart from the fact that all economic variables, on some level, require some kind of statistical analysis? :-)

8. Almost all macroeconomic time series are mean-reverting processes, but few are (covariance) stationary.

KP

1. If by mean-reversion you mean a trend stationary process that is very much questionable. It is quite difficult to discern highly persistent trend-stationary processes from a unit root with limited data.

9. I wonder if the words of Sam Ervin come to mind: 20th-century witchcraft.

Odd how 100% of economic models and constructs support the political biases of the modelers....

1. Odd how blog post commenters can find politics where there are absolutely none.

2. You need medication Benjamin. Or, more likely, new and better medication.

3. Professor Cochrane how did Anon 10.14 pass your abuse screening?

Cole, it is unlikely that a model will be, intentionally or unintentionally, unreflective of the preferences of the modeller, in that sense, all models are subjective. But that of course is a major debate in philosophy. Best to consult Karl Popper and Habermas on stuff like that.

In this case, however, this is less of an issue. That is in evidence by the fact that Professor Cochrane and Professor Farmer draw different conclusions - with very different political implications - from what the results that the statistical model presents. This is likely to reflect their interpretation of the historical record/and or their theorising and the theories they base their conjecture on.

Personally I think we should be wary of reading too much into random walk tests.

10. Dear John, thanks for the post!

As an (undergrad econ) rookie to the field of time series econometrics, I found the equation "unit root = random walk + stationary" to be mildly unsettling. To my understanding, every RW has a unit root, as mentioned by commentator Kurt Verstegen.

What do you think would be a more precise description for the process depicted by the black line? Something like a RWM with drift and deterministic trend?

1. Find "How big is the random walk in GNP" "Permanent and Transitory Components" and "Time Series for Macroeconomics and Finance" on my webpage. They explain these issues as clearly as I can.

11. Doesn't this assume a conventional linear equilibrium system where a perturbation is eventually dealt with and equilibrium is restored - either back to baseline or at a new equilibrium point? The line with the "hump" shows the behavior of a damped system after a single perturbation.

If the system is nonlinear a perturbation could knock the system completely off the tracks in an unanticipated (black swan) way and the graph could go parabolic upwards. For example, an avalanche, where a small input e.g., an acorn falling on a snow pack, triggers a massive response.

I wish I could express this better but my higher math skills are 4 decades past their expiration date and I can only think about this qualitatively.

12. Fantastic! A simple graph that tells the whole story. Thank you! A

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