Mankiw Fails to Understand Unit Roots
Matthew Martin 10/02/2012 04:00:00 PM
Recently Mankiw published this post which was a reference to the internet-famous Mankiw-Krugman bet (which don't think Krugman was ever made aware of). So this is a good opportunity to point out the error in Mankiw's reasoning.
So here's how it went down:
- 6:00AM, 10 January 2009: Obama administration releases estimates of the impact of the Recovery Act, which included forecasts of economic growth
- 7:25AM, 10 January 2009: Paul Krugman criticizes the forecasts in the Obama administration baseline estimates as being too optimistic, and pointing out that the stimulus plan was too weak.
- 3 March 2009: Mankiw tries to use the "unit root hypothesis" to argue that the Obama administration baseline estimates were too optimistic
- 9:06PM 3 March 2009: Krugman fires back with a bad pun in "Roots of evil", arguing that Mankiw was mischaracterizing the unit root hypothesis, which did not support the conclusion which he was using it to argue.
- 4 March 2009: Mankiw misreads Krugman's entire point, misses the pun (this is forgivable) and challenges Krugman to a bet that the estimates were too optimistic, even though this was a point on which they both agreed.
- 22 September 2012: Mankiw links us to an article by David O. Cushman, arguing that "Mankiw would likely win the bet."
- 9:49PM September 24 2012: Krugman responds by suggesting that Mankiw and Cushman must be thinking of a different Paul Krugman, since Mankiw and Krugman never disagreed on the point that the estimates were too optimistic.
What is a unit root? Consider a time series data like GDP. GDP fluctuates from period to period in response to a variety of shocks including technology, demand, and policy shocks. However, these shocks are not strictly iid--instead, they tend to persist across several quarters. This means that we can predict future values of GDP based on previous periods. There is a caveat though--in order to be able to estimate a predictive model, we need the shocks to be sufficiently short lived in nature. If we had, for example, a permanent structural shock, then it's effects on GDP cannot be estimated from the time-series. There are a couple of ways of thinking about why we cannot estimate such a shock--one is that if the shock is permanent, then we would need to include an infinite number of periods in the model in order to accurately estimate its effect--any missed period would bias the estimate. The other way of thinking about it is that with permanent shocks there simply isn't enough variation in the data to establish statistical significance--you can never claim that a policy has an effect because you cannot rule out the possibility that it is due to something else. This phenomenon--where we have shocks that last too long in time to be able to measure it accurately--is known as a unit root.
And that is exactly where Mankiw's argument goes off the rails. He argues that the Obama estimates were too optimistic because GDP follows a unit root. But unit roots are a problem that depends on the type of shocks, not the type of data. So what Mankiw was, perhaps unknowingly, arguing was that the shock that caused the recession was permanent in nature. That may or may not have been true--this is a topic for another post--but it has nothing to do with the empirical question of unit roots, which is all about whether or not we can accurately estimate a predictive model. By definition, you cannot use the unit root problem to argue a prediction--if we could make a prediction then it wouldn't be a unit root!
In conclusion, Krugman was entirely right to suggest that Mankiw's critique was entirely empty. He made an argument that requires as its premise that the recession was the result of a permanent shock, but not only failed to argue this point, but failed even to acknowledge that this was his implicit assumption. As Krugman points out, the fact that there were structural shocks back in the 1970s has no bearing on whether the shock during the recession was structural or temporary.