Sunday, April 28, 2013


Friday (like most days), Brad DeLong said some amazing stuff.

In particular he claimed that, "There Are Two Unit Roots and Strong Mean Reversion in U.S. GDP per Capita".


Phone call for Clive Granger.

First off, unit roots and mean reversion are incompatible concepts. A unit root is essentially a stochastic trend. There is no fixed mean of the series to revert to.

Second, "two unit roots" means that real gdp per capita is I(2). Which means in English that shocks to the growth rate are permanent, that the variance of the growth rate continually increases over time, and that there is no mean reversion in the growth rate.

We actually have a lot of statistical tests for unit roots. Sure they are not so great, especially in the power department. So if we fail to reject the null, we are not thrilled about rolling with it.

But if we can reject the null, the size of the tests (probability of rejecting a true null) are not far from accurate, especially over longer time periods, and we have over 200 years of data to work with!

So here's the augmented Dickey-Fuller test for a unit root in the growth rate of real GDP per capita(the null is that there is such a unit root):


Null Hypothesis: D(LRYPC) has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on Modified SIC, MAXLAG=14)

                                                                t-Statistic                     Prob.

Augmented Dickey-Fuller test statistic          -11.43155                  0.0000
Test critical values:  1% level                        -3.461630
                             5% level                        -2.875195
                           10% level                        -2.574125

*MacKinnon (1996) one-sided p-values.


We are rejecting (crushing) the null of a second unit root in the series at the 0.01 level. You can click through all the options in EVIEWS on lag length selection and get exactly the same rejection.

There are not two unit roots in real US GPD per capita.

Here's a graph of the growth rate of real GDP per capita in the US since 1800:

The data run from 1801 to 2010 (I'm pretty sure it's the same data Brad used).

The mean of the series is around 0.015, or a 1.5% growth rate. As you can see, while there is evidence of volatility clustering, the series is strongly mean reverting and shocks to the growth rate are decidedly not "highly persistent"  (i.e. it does not have a unit root).

There is even a big debate about whether real GDP per capita has even one unit root, because there are a lot of processes (long memory, Markov switching, structural breaks, breaking trends) that are not unit root processes but typical tests will fail to reject the null of a unit root anyway.


David O. Cushman said...

I certainly agree real GDP per person is unlikely to be I(2). But suppose it is. That's what DeLong's first two bullets assert (assuming "highly persistent" means "permanent"). Then for the 3rd bullet he says to add short-run transitory shocks to the level of real GDP per person. Finally he claims "univariate ARIMA just does not cut it as a description." But the model he has laid out is described (exactly, if there's no other serial correlation) by a univariate ARIMA: an ARIMA(0,2,2). For example, see Hyndman et al. (2008), Forecasting with Exponential Smoothing, Springer, p. 169. I guess the problem is Brad does not know much about "mean reversion" or "ARIMA." This may not be the first time ignorance hindered commentary by Brad.

Gerardo said...

I wonder if this is going to go "smackdown watch" or "stupidest person in the world"?

On a possibly related note, how come there aren't more "Augmented Dickey Fuller" jokes?

StatsJunkie said...

Maybe not an "Augmented Dickey Fuller" joke but check out this one about the 'null hypothesis.