Monday, August 24, 2015

Entropy is working for the weekend

Nick Rowe employs his ability for distillation in the task of explaining a monetary coordination failure that results in a recession. In his recent post, he mentions the coordination occurring on the weekend:
Every Saturday Canadian output and employment drop. And they drop again every Sunday. Every weekend, output and employment drop for two successive days. Are weekends mini recessions? I would say "no".
If you've been following along this blog -- in particular this post [1] -- you might ask: I thought you said coordination causes recessions?

I did say that. The important thing to understand is that it is coordination relative to the equilibrium distribution. Let's say here's what (the probability density of) output looks like during a typical week (I made this data up):


Call this distribution P. It's the equilibrium distribution. Some people (by no means everybody) have the opportunity to take weekends off. Now week to week the information loss measured by the KL divergence is zero:

D(P||P) = 0

But if suddenly this happens (call this distribution Q):


We get a KL-divergence that results in an information loss of:

D(P||Q) = 0.18 bit

It turns out that is a loss of about 6.6% relative to the information entropy of P ~ 2.78 bits. That would be a mini recession (if weeks were all the same except that one, it would be a recession of about 0.1% of output).

What would really be happening? What is the story behind this mini-recession? With more people off on days that they used to have on, they might go shopping. But with fewer people to stock the shelves and no one expecting a Thursday rush of long weekend consumers, less gets sold than in the status quo. ATM's might not have enough money in them for Wednesday being the new Friday and cash only establishments would miss out. Restaurants unexpectedly fill up for Friday brunch and people can't get a table.

We seasonally adjust data; that's an admissible procedure only because the seasonality represents the equilibrium distribution.

Now you may ask: What about holidays? Or the Stanley Cup?

Well, the first graph was actually a simplification. The actual distribution would be more complex, taking into account your country's public holidays, major sporting events and vacations. That would be the real P.

If part way through the year, 5% more people became unemployed so that your distribution of output changed (i.e. more the output would have been at the beginning of the year relative to an equilibrium year), then you would probably have a recession. That's a bad coordination.

Now I actually think the real coordination comes in the form of pessimism about asset prices and future sales, so the rise in unemployment is a symptom, not the cause. The entropy loss manifests as a fall in employment (and a rise in the number of people with zero wage change) as seen in the post above [1].

Sunday, August 23, 2015

Rational expectations and information theory

KL Divergence using Gaussian distributions from Wikipedia.

Noah Smith once again deleted my comment on his blog, so I'll just have to preserve it (well, the gist of it) here.

He discussed an argument against rational expectations he'd never considered before. Since counterfactual universes are never realized, one can never explore the entire state space to learn the fundamental probability distribution from which macro observable are drawn. Let's call this probability distribution A. The best we can get is some approximation B.

Rational expectations is the assumption A ≈ B.

If this sounds familiar, it's exactly the way one would approach this with the information equilibrium model as I discussed several months ago.

In that post, I showed that the KL divergence measures information loss in the macroeconomy based on the difference between the distributions A and B.

D(A||B) = ΔI

That was the content of my comment on Noah's post. I go a bit further at the link and say that this information loss is measured by the difference between the price level and how much NGDP changes when the monetary base changes

ΔI ~ P - dN/dM = dN*/dM - dN/dM 

Which to me seems intuitive: it compares how much the economy should grow from an expansion of the money supply (ideally) to how much it actually does grow.

Just the aggregate ΔI is measured, however. Two different distributions BB' and B'' can have the same KL divergence so this doesn't give us a way to estimate A better.

Now rational expectations are clearly wrong at some given level of accuracy, but then so are Newton's laws. The question of whether you can apply rational expectations depends on the size of ΔI. Since ΔI is roughly proportional to nominal shocks (the difference between where the economy is and where it should be based on growth of M alone [1]) and these nominal shocks are basically the size of the business cycle, it means rational expectations are not a useful approximation to make when analyzing the business cycle.

As far as I know, this is the first macroeconomic estimate of the limits of the rational expectations assumption that doesn't compare it to a different model of expectations (e.g. bounded rationality, adaptive expectations). (There are lots of estimates for micro.)


Footnotes:
[1] In case anyone was curious, this also illustrates the apparent inconsistency between e.g. this post where nominal shocks are negative and e.g. this post where they are positive. It depends on whether you apply them before including the effect of inflation or after. Specifically

0 = dN*/dM - (dN/dM + σ) = (dN*/dM - σ) - dN/dM

Friday, August 21, 2015

Sticky wages?

I'm not sure I understand how economists (including Mark Thoma) can look at this data:


... and say:
Taken at face value, this analysis suggests the presence of some amount of wage rigidity.
24% of people are reporting nominal hourly wage declines. Only 20% are reporting the same wage. That means something like 80% of people are reporting wage changes. They should check out the section of my paper on entropic forces and nominal rigidity. Or this post on macro rigidity and micro flexibility.

Information equilibrium as an economic principle


I have finished the first public draft of the information equilibrium paper I started to write back in February. Here is a link (please let me know if it doesn't work -- I think I've set my Google Drive settings properly) and the outline below:

Information equilibrium as an economic principle [pdf]

1 Introduction
2 Information equilibrium
   2.1 Supply and demand
   2.2 Physical analogy
   2.3 Alternative motivation

3 Macroeconomics
   3.1 AD-AS model
   3.2 Labor market and Okun's law
   3.3 IS-LM model and interest rates
   3.4 Price level and inflation
   3.5 Solow-Swan growth model
   3.6 A note on constructing models
   3.7 Summary

4 Statistical economics
   4.1 Entropic forces and emergent properties
5 Summary and conclusion

Thursday, August 20, 2015

Don't forget the VAT

Update: Mark Sadowski (in comments below) correctly points out that there was no VAT change between 2008-2009. I have corrected my error in the figures and results. There is a tiny change in the conclusion (where the "VAT increase" is instead referred to as a "spike in inflation" in the second to last paragraph of the + 2 hour update)
Scott Sumner is linking to Mark Sadowski again and showing us that inflation as measured by the GDP deflator has risen in Japan since 'Abenomics' went into effect. However, the graph shown fails to take into account three two VAT increases -- which Scott Sumner explicitly pointed out might fool people -- that show up visibly in the GDP deflator data. Here's Sumner:
Japan will be hit by an adverse supply shock next year (higher sales tax rates) which will boost inflation–making it look like they will hit their 2% target. Don’t be fooled.
And here's the data and a version with the VAT increases filtered out ...


If you remove these VAT increases, the change is less dramatic and the change appears to start before the monetary policy changes were announced.


I make a pretty good case here that the change in direction of the price level (core CPI) is mostly due to the fiscal policy component. The information equilibrium model seems to get the data about right without any monetary policy effects.

...

Update + 2 hours:

If you reduce the size of the averaging window from 10 quarters to 4 (i.e. 1 year), you can see that the effect of the VAT doesn't stick around in 2015 ... much like the effect of the VAT increase in 2008 and 1997:


Also using a single year averaging means that the start of the increase in inflation is pushed all the way back to 2010 (that year is no longer impacted by the VAT increase spike in 2008). Heck, here's using only a single quarter 'average':


The trend towards increased inflation appears to start after the financial crisis; Abe appears in the middle of it.

Wednesday, August 19, 2015

The Chinese unemployment rate

I wrote awhile ago about Chinese economic statistics (CPI and NGDP) not seeming terribly suspect. Alex Tabarrok points to the stillness of the Chinese unemployment rate as a sign that the statistics can't be trusted. And I tentatively agree -- there appears to be something wrong with the reported Chinese unemployment rate.

Combining the China model linked above with this post on employment growth, I thought I'd try to estimate the Chinese unemployment rate.

Here are the nominal shocks (see the post on employment growth):


And here is an estimate of the resulting unemployment rate (making an assumption of 5% for the "natural rate" and a 50% employment-population ratio):


The yellow line is the 'official' rate (from here).

This result is consistent with the unemployment rate rising to 11% in 2002 in the estimate presented in Tabarrok's post. That analysis says that unemployment stays high through 2009. However, my model seems to think that the unemployment rate dropped to 4-5% before 2009. My model is more consistent with the burst of NGDP (and RGDP) growth between 2005 and 2010 (using e.g. Okun's law). And it puts unemployment nearer to 4% before the recent economic trouble.

This is not to say unemployment isn't high today in China (the last NGDP data I have is from 2014). And China does not appear to be reporting the unemployment spikes from recessions. This of  course could be a difference due to the economic systems. While it seems to operate a large capitalist economy nowadays, the country is officially communist. Involuntary unemployment during recessions may not be the same thing there as it is in the US.

Employment doesn't depend on inflation


Robert Waldmann and Simon Wren-Lewis, in discussing Paul Romer's history of macro 1977-1982, bring up the Phillips curve. I've also written about it on occasion.

I thought I'd have a look at the Phillips using the DSGE form of the information equilibrium model. Turns out it results in something really cool ... Here are the relevant equations from the DSGE form link:

$$
\text{(1) } n_{t} =  \sigma_{t} + \left( \frac{1}{\kappa} - 1 \right) (m_{t} - m_{t-1}) + n_{t-1}
$$
$$
\text{(2) } \pi_{t} = \left( \frac{1}{\kappa} - 1 \right) (m_{t} + m^{*}) + c_{m}
$$
$$
\text{(4) } \ell_{t} = n_{t} - \pi_{t} + c_{\ell}
$$

Here $n$ is nominal output, $m$ is base money (minus reserves), $\pi$ is the price level, and $\ell$ is the total employed. The symbol $\sigma$ represents 'nominal shocks'. They are the stochastic part of the model, and they're typically positive. They represent the difference between where $n$ is at time $t$ and where it should be based on the change in $m$ from time $t-1$ to time $t$ alone -- essentially equation (1). The rest of the symbols are constants, with $\kappa$ being the IT index (approximately constant over the short run).

I had hoped to show some kind of relationship between changes in the total employed ($\ell_{t} - \ell_{t-1}$) -- and thus changes in unemployment -- and inflation ($\pi_{t} - \pi_{t-1}$). But the math led me to something I didn't expect. With a bit of algebra, you can show that labor growth is given by:

$$
\ell_{t} - \ell_{t - 1} = \sigma_{t}
$$

regardless of the information transfer index $\kappa$. Those nominal shocks I've talked about since this post? They are basically changes in the number of employed. That's why they're typically positive and typically around a few percent.

Effectively, employment growth is the part of nominal output (NGDP) that is left over after accounting for inflation. Thus there shouldn't be any relationship between inflation and unemployment -- i.e. the Phillips curve isn't real. This even applies to a version of the model consistent with expectations, since we could easily write:

$$
E_{t} \ell_{t+1} - \ell_{t} = E_{t} \sigma_{t+1}
$$

That is to say expected changes in unemployment are the expected shocks to the economy after accounting for inflation.

How does this look empirically? Pretty good ($\sigma$ in blue, $\ell$ in yellow):