The latest GDP numbers came out today and many people in the blogosphere are declaring the results are consistent with their priors; I will be no different, but I hope to add to the discussion by showing that RGDP growth does not have a unit root. This was inspired by a post by Scott Sumner.

What does unit root mean? Well, basically if a random process does not have a stationary trend, then it has a unit root and if it does, then it doesn't. That definition hinges rather precariously on knowing the trend, which is judged by a pretty squishy metric. I previously extracted a trend in a couple of different [1] ways [2] from the information transfer model (ITM) and it captures the empirical observation that RGDP growth has fallen over the post-war period. Various economic explanations are given for this fall: a productivity slowdown, or that GDP is hit by both supply and demand shocks with the former being more persistent. In the information transfer model, it follows from the diminishing marginal utility of monetary expansion. The most efficient way to increase economic growth in an economy with little money is to add money which allows the market to transfer more information from the demand to the supply. However, the bang for the (literal) buck tends to fall once there's enough money around which results in a slowly falling RGDP growth rate. There's no productivity slowdown or persistent supply shocks; your economy just ran out of low hanging fruit that could be grabbed by printing money.

And now, the unit root test!

Using the updated GDP data, I extracted the RGDP trend as I did in the links [1, 2] above. The RGDP growth data is shown in blue, the trend is shown as a dark blue line and the ITM de-trended data is in red:

I performed a unit root test on the two data sets (a Dickey-Fuller test using annual averages) and the results are on the graph. The de-trended data rejects a unit root at the

*p = 0.01*level while the RGDP growth data appears to have a unit root. Essentially, RGDP growth is consistent with random fluctuations around the ITM trend in blue on the graph such that there is a tendency to return to the trend -- there is a "bounce back" after a recession. However, due to the fall in RGDP growth trend, the "bounce back" appears to get smaller and smaller over the post-war period.
In the linked post, Sumner quotes Krugman as saying that Mankiw thinks that, "real GDP may have a unit root — that is, there’s no tendency for bad years to be offset by good years later."

ReplyDeleteSumner replies: "One point where I do agree with Krugman is the labor market, which does show clear trend reversion. . . . For those who don’t know, a unit root in a time series like RGDP implies that even after a change in RGDP, the optimal forecast of future RGDP is roughly the trend rate of growth. No “bounceback” can be expected."

Sorry, sportsfans, but that is not what having a unit root means. For instance, this difference equation has a unit root (somewhat loosely speaking):

Y(t) = Y(t-1) - Y(t-3) + Y(t-4)

That is, it has an associated (characteristic) equation,

y^4 - y^3 + y - 1 = 0

with a root that is an nth root of 1, for some n.

We can factor that equation to get

(y^3 + 1)*(y - 1) = 0

which obviously has a root at 1 but also has roots which are 6th roots of 1, i. e., cube roots of -1. All four of the roots of this equation are roots of 1. :)

Does this difference equation exhibit no tendency for bad years to be followed by good years, or is the optimal forecast the trend? Here is an example time series, starting with Y(0) = 1 up to Y(3) = 4, followed by values according to the equation above.

1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 . . .

Y(4) = Y(3) - Y(1) + Y(0) = 4 - 2 + 1 = 3

Y(5) = 3 - 3 + 2 = 2

Y(6) = 2 - 4 + 3 = 1

Y(7) = 1 - 3 + 4 = 2

Etc.

Note that the "bad" years, with Y = 1 or Y = 2, are followed by "good" years, with Y = 3 or Y = 4, and vice versa. And the best forecast is not the trend, which is Y = 2.5.

The sequence has a cycle of 6, which is indicated by the 6th roots of 1 for the characteristic equation. Also, the cycle neither converges (which, I suppose, "reversion to the mean" means), nor diverges, which would indicate a runaway (positive) feedback cycle. The unit root indicates such a stable cycle.

What about a sequence with a trend? Well, consider this equation,

Y(t) = Y(t-1) - Y(t-3) + Y(t-4) + 1

Given the same initial four values we have

1 2 3 4 4 4 4 5 6 7 7 7 7 8 . . .

which has a linear trend that gains 3 points in each cycle.

So why the misstatements about unit roots by both Krugman and Sumner? Neither indicates any awareness of the relation between stable cycles and unit roots.

DeleteHere I am speculating, but perhaps that aspect of unit roots is not taught or is deemphasized in econometrics classes. Taking a cue from Sumner's remarks about "reversion to the mean" and prediction, let us consider the simple case where Y = 1, with random deviations. In that case the best predictor is E(Y(t)) = 1. But suppose that instead we have the predictor, E(Y(t)) = Y(t-1), which obviously has a unit root. There is no cycle, or a cycle of length 1.

Now suppose that Y(t-1) = 1.2. Then E(Y(t)) = 1.2, which is obviously wrong, as it perpetuates the error in Y(t-1). That's not so good.

However, while there is a better predictor, things are not so bad. For one thing, suppose that we predict Y(t+9). We still predict 1.2, but the yearly error is only 0.02. And the actual Y(t) will be close to 1, so that the predictions cluster around the true value.

One value of the predictor, E(Y(t)) = Y(t-1), is that it is better when there actually is no trend, or, more likely, when the trend changes. If the new value of Y is 1.2 or 1.15, then it gives a better prediction than E(Y) = 1. It may also give a better prediction when the trend is changing to a higher value.

In any event, if the best predictor is E(Y) = 1, based upon the evidence, then why are we even talking about E(Y(t)) = Y(t-1)?

If my speculation is correct, then when Krugman and Sumner say that Mankiw thinks that RGDP has a unit root, they probably mean that he thinks that RGDP has no trend, or, perhaps more lilkely, that its trend has changed.

Oops! That part about the yearly error being 0.02 is wrong. I forgot that the error was already 0.2. :( The error stays constant.

DeleteHi Bill,

DeleteSorry for the delay in getting back to you; I just got back from vacation where I had very limited connectivity. I think I follow what you are saying. In Sumner's quote, I think the "roughly" is covering the issues due to stable cycles (or autoregressive processes in general). The best guess for the future value of an autoregressive process is not necessarily the trend or previous value, but rather some other value -- specifically some function of the previous values (depending on the AR process) .

I think the thing that clears up the lack of mathematical rigor is that Sumner, Krugman, and Mankiw are talking about the long run -- a period of time much longer than the business cycle. In your first example, they would be talking about the average over many periods of the stable cycle.

So your final conclusion that what Mankiw and company mean by "unit root" is that they think the long term RGDP trend changes -- in particular after recessions. There is no "catch up growth" -- in the long term.

In my model above RGDP does have a changing trend, so RGDP growth techincally does have a unit root on its own. Once corrected for this trend, it becomes a stationary process (doesn't have a unit root) on timescales greater than a year.

Hi, Jason.

DeleteI was not expecting a response before August, if at all. :)

I think that there is a real disagreement between Mankiw and Krugman that would show up in the short term.

Speculating again, but with some background: I think that economists in general do not believe in stable economic cycles, but perhaps in damped or convergent cycles, those that revert to trend. There are those who believe in long term cycles, such as the Kondratieff wave. Ravi Batra wrote a book about the coming depression of 1990, based upon a 60 year cycle. He was only 18 years off. One trouble with long cycles is that their difference equations look approximately like Y(t) = 2*Y(t-1) - Y(t-2), which has a unit root. Such long cycles are easily perturbed.

I am kind of inclined to believe in long cycles myself. For instance, policy makers in the 1950s or 1960s would never have dismantled Glass-Steagle, because they remembered the Great Depression. You have to wait a generation or two for cultural forgetting.

Another thing about unit root cycles is that they are on the edge of chaos, the cusp between divergence and convergence. Again, I am inclined to thing that human systems tend toward the edge of chaos. But maybe that is because I grew up in modern times. For millenia civilization depended upon slavery and other caste systems, with little economic dynamism. I would like to think that we will turn away from the new caste system that we are building, but who knows?

P. S. Adding a constant to difference equations is my idea. The way I learned about time series, the equations did not include one. That makes the math more tractable, but it means that linear trends are represented by factors with unit roots, instead of, as in my examples, a constant.

Sorry, I misspoke. The problem with long cycles with characteristic equations close to y^2 - 2*y + 1 = (y - 1)^2 = 0 is not that the latter has a unit root. So does the equation of the long cycle. It is that Y(t) = 2*Y(t-1) - Y(t-2) will follow a linear trend.

DeleteThat is an interesting comment about unit root cycles being on the edge of chaos. And I completely agree that "cultural forgetting" is a major factor in not just economics, but human history in general!

DeleteSpeaking of the edge of chaos, how about Europe right now? It looks like the creditors have overplayed their hand. Spain, Portugal, and Ireland did not join with Greece, but if a rift develops between France and Germany, they could ally with France. Old yang becoming new yin?

Delete