Tuesday, February 16, 2016

Recoveries do grow old; they just have an uncertain lifetime

There was an article out of the SF Fed that seems to suggest that post-WWII recoveries do not "grow old". They assume a Weibull distribution for the PDF for post-WWII recovery lengths (and I'm assuming for the pre-WWII recoveries as well) and then observe that the hazard function is "nearly flat" (has no time scale). The implication is that post-WWII recoveries don't "grow old" and end in recession.

I wasn't entirely convinced by that argument -- the reason for a hazard function to not be flat is that it has not only a well defined mean lifetime (scale), but additionally a very low variance. Those properties result in a strongly peaked PDF which leads to a sharply increasing hazard function over a short period of time. Because of the graph limits shown in the SF Fed article, it makes it look like the pre-WWII hazard function shoots up forever, while the post-WWII hazard function gradually increases. The pre-WWII hazard function actually has to level off after that initial increase -- which isn't shown.

The real argument that post-WWII recoveries don't grow old is that they are scale-free -- lack a well defined lifetime. However the use of a Weibull distribution for the post-WWII recoveries assumes they do have a lifetime (i.e. Weibull distribution has time scale).

But you can't assume a different type of distribution for the post-WWII recoveries than for the pre-WWII recoveries because you don't have very much post-WWII data. In a sense, you're kind of stuck. The data is mostly pre-WWII and a Weibull distribution looks appropriate for that. That means you're stuck with a Weibull distribution with different parameters for the post-WWII data [1] -- and therefore stuck with a lifetime.

I basically re-did the analysis using a gamma distribution to find out the scales (mean and variance). I fitted the empirical CDFs to regularized gamma functions and extracted the distributions for pre-WWII, post-WWII and used a sum of the distributions to represent all the data. Here are the resulting fits:


And here are the resulting PDFs:


The two distributions have means of 2.2 years (pre-WWII) and 4.8 years (post-WWII), but the big difference is in the variance. The square root of the variances are 1.0 years and 3.1 years. The post-WWII distributions are quite spread out. The result is a "flat" hazard function:


But post-WWII recoveries still "grow old" (after about 4.8 years), they just don't have a well-defined lifetime (± 3.1 years). As the current recovery is about 30 quarters long, so we'd expect about a 9% chance of a recession in the next quarter. But since the variance is so high, it takes 10 years (40 quarters) to reach a 99% chance of a recession (at least with this model):


...

Footnotes

[1] Statistical tests pretty definitively show that the post-WWII data is drawn from a different distribution than the pre-WWII data. We know that the distribution changed, but we don't really have enough data to pick a new kind of distribution -- even though that might be warranted.

19 comments:

  1. Interesting as always. But I wonder if the important difference between the datasets is not whether they were before or after WWII, but rather before or after New Deal (social security, widespread unemployment benefits, larger proportion of economy due to Federal government, etc.) Federal spending and transfers could be smoothing out the distributions, and thus resulting in longer recoveries. Thoughts?

    ReplyDelete
    Replies
    1. My opinion is that the fiscal expansion during WWII increased the information transfer index k to a value closer to 2, making monetary policy an effective tool of demand management.

      But before 1914, there wasn't a Fed, so about 60 years of pre-war data occur without a central bank.

      Delete
    2. Fair enough, but is that likely to have been caused by a temporary fiscal expansion of war and defense spending (along with price controls, etc.)? I am thinking that more likely it would have been caused by the permanent fiscal expansion of the new deal, elements of which have been estimated to have smoothed the great recession, for example. I recall Paul Krugman's argument that proxy Federal transfers (medicaid, medicare, unemployment insurance) represented something like 4% of Florida's GDP during the great recession, which certainly helped.

      Delete
    3. The permanent fiscal expansion (15% of NGDP) wasn't large enough to change the ratio of log NGDP to log M. You need something like the war effort (sustained 40% of NGDP) to move the needle:

      https://research.stlouisfed.org/fred2/series/FYONGDA188S

      That is just one theory though. It could be the hyperinflation associated with the war and the pegged interest rates.

      Or it could be both.

      I don't deny that in a liquidity trap government spending and automatic stabilizers can help cushion a demand shock.

      Delete
    4. There may well be three regimes.

      1) Before the New Deal
      2) After the New Deal
      3) Post New Deal

      Post New Deal being after the New Deal financial regulation had been gutted. That was accomplished by the end of Clinton's second term. Post New Deal looks a lot like Before New Deal, eh? But we did stave off another Great Depression in '08, I think. We just ended up with a Not So Great Depression.

      Delete
    5. As for the length of this recovery, what recovery? What heights do we have to fall from? It's like the old ethnic joke. Why don't more (fill in ethnic butt of joke) commit suicide? Because you can't kill yourself jumping out of a basement window.

      Delete
    6. I think another reason macroeconomics is in such a terrible state is that way too many things changed at roughly the same time ...

      Increased government spending
      Leaving the gold standard
      Central bank independence/pegged interest rates
      Wartime hyperinflation

      It's a bit like someone having a cold and:

      Giving them tea
      Feeding them lots of food
      Wrapping them in blankets**
      Giving them fever reducers

      If there is only one cold, then there is really no way to tell which treatment helped ... if any at all did.

      And regarding the recovery -- that's just the term for the inter-recession period of growth. If there had been sustained contraction, that would be labeled a new recession.


      **This is how Joseph Fourier died -- falling down the stairs because he wrapped himself in blankets.

      Delete
  2. And as for recoveries of indefinite length, aren't post WWII slow recoveries a phenomenon of the 1990s and 2000s? I. e., of the Post New Deal era?

    ReplyDelete
    Replies
    1. There could be a connection between slow recoveries and long inter-recession periods ... it takes more time to build up more snow before an avalanche.

      But in any case, there isn't enough data to say much conclusively about post-WWII expansions. The distribution is actually indistinguishable from a uniform distribution. If you didn't have the prior of the pre-WWII expansions following a Poisson-esque distribution, you'd really have no place to start.

      Delete
    2. Indeed. Macroeconomics suffers from a paucity of data. It also suffers from relatively rapid political changes, causing rapid change of potentially relevant variables. For the Great Depression, the data suggests that for advanced economies, going off of the gold standard was important. You can see immediate positive effects on growth. But now nobody is on the gold standard, so nobody can go off it. After the high inflation of the 1970s, central banks around the globe have focused on keeping inflation low. If both of these factors are important, we have only 35 years or so when both have been in effect. We have only 15-20 years without the New Deal regulations, something else that may be important, and the only time in history, I think, when all three conditions have held.

      Delete
  3. Jason, minor point: but either I've gone color blind, or the curves you call green are actually red.

    ReplyDelete
    Replies
    1. I changed the color of the graphs at the last minute to conform to the scheme at the link, but forgot to change the labels for the plots themselves. I'll fix it.

      Delete
  4. Hey, this is unrelated to your post. I'm wondering if there is a non technical summary of the main principles of your information equilibrium paper? Thanks!

    ReplyDelete
    Replies
    1. Ok, here's my quick one. Information equilibrium (IE) might apply if you have two aggregate variables (say X and Y) with a plausible communication channel between them. The resulting relation is dX/dY = k*X/Y or equivalently dY/dX = (1/k)*Y/X. k is a constant that you typically determine by fitting the "general equilibrium" (GE) solution:

      X = c*Y^k

      to data. c is another constant. Using ~ for "goes as" it's even simpler:

      X ~ Y^k

      and

      dX/dY ~ Y^(k-1)

      Power laws. If they don't fit for any k, then it's not a case of IE, or you've framed the problem incorrectly, or the GE solution doesn't apply (perhaps partial equilibrium applies).

      dX/dY might have a special meaning, like price. How much X per unit of Y?

      I recommend this link (also located in the right hand column):

      How money transfers information

      Delete
    2. Hello Britonomist,

      Here is a link to a less technical overview:

      http://informationtransfereconomics.blogspot.com/2015/04/information-theory-and-economics-primer.html

      And here is some flavor of how the info eq principle changes how you look at e.g. incentives:

      http://informationtransfereconomics.blogspot.com/2015/04/incentives-are-entropic-force.html

      And here I've restated the "principles of economics" that appear at the beginning of many intro economics books in terms of info eq:

      http://informationtransfereconomics.blogspot.com/2015/05/information-equilibrium-as-economic.html

      Delete
  5. Jason, I'm inspired to start a companion blog called "Information Transfer for Morons" in which I feebly carry on Socratic dialogs with myself trying to understand various aspects of information transfer you present here.

    My tag line will be

    "What we have here is a failure to transfer information... ...ideally."

    A la "Cool Hand Luke."

    ...and meaning from your brain to mine. Or from my brain to any happless soul who has the misfortune to stumble upon my blog and try to understand what in the hell I'm talking about.

    ReplyDelete
    Replies
    1. ... Actually a la the 70's SNL send up of "Cool Hand Luke" called "French Camp" in which the instructor/warden says at one point

      "What we have here is a failure to communicate... ...bilingually"

      Delete
    2. Hi, Tom. :)

      Recently overheard in a move on TV:

      "My son may be a moron, but he's not an idiot."

      ;)

      Delete
    3. Bill, that's another possibility for my tag line!

      (and I guess it was called "Camp Beau Soleil")

      Delete

Comments are welcome. Please see the Moderation and comment policy.

Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead.