Saturday, March 25, 2017

The mystery of Japan's inflation

Noah Smith has said that Japan is where macroeconomic theories go to die (well, except this one), and when I initially looked at Japan's price level it with the dynamic equilibrium model (more here) it looked grim. However, as the Dude says, my thinking about this case had become very uptight. While Japan would not be a good example of the dynamic equilibrium framework on its own, given the framework works for other inflation data (e.g. here or here) we can be a bit more assured that the giant ongoing shock required to fit the price level data isn't spurious. It still could be spurious, but at least it's not a "surely you're joking" ad hoc hypothesis.

Anyway, here are the model results using the same codes available here:

Note that I removed the effects of the VAT increases from the data. The shocks are centered at 1974.6 (positive) and 2006.5 (negative). What's also interesting is that the width (similar to the standard deviation of a normal distribution) of the latter shock is 108 months (9 years) which means it won't be over for many years to come. Instead of "Abenomics", the recent rising/stabilizing inflation is the ending of the giant "lost decade" shock.

Of course, this picture also makes a bold claim that the equilibrium rate of inflation is 4.3% per year (± 0.4 percentage points at 2σ). That seems like something that would be good to test, so here are the relevant forecasts through 2020 (and will be added to the forecast page):

PS For another macro mystery set in Japan, here's my old post on Japan's monetary base.

Thursday, March 23, 2017

Three centuries of dynamic equilibria in the UK

If the price level has a dynamic equilibrium (e.g. per the previous post or here), it should be interesting to look at some very long run historical data ‒ specifically this three century time series available for the UK. It turns out that there are two dynamic equilibria. The first hold from the 1600s to the early 1900s is associated with an inflation rate consistent with zero (call it the gold standard equilibrium); the second holding over the post WWII period is the one I've observed before (links at beginning of this paragraph) with an inflation rate of 2.6%:

The major shocks are centered at 1780.7, 1916.1, 1945.5, and 1976.1. My intuition is that these are associated with the industrial revolution, WWI, WWII, and women entering the workforce. However, I'm open to alternative theories. If you're looking for a demographic cause of the first shocks, there was a major change in population growth in the early 1800s associated with improved sanitation for example.

The "quantity theory of labor" and dynamic equilibrium

One of the earliest models I looked at with the information equilibrium (IE) framework was the relationship $P : N \rightleftarrows L$ where $P$ is the price level, $N$ is nominal output, and $L$ is the level of employment. The relationship effectively captures Okun's law (i.e. changes in real GDP are related to changes in employment), and is a component of what I called the "quantity theory of labor" (and capital). The IE notation is shorthand for the equation:

\text{(1) } \; P \equiv \frac{dN}{dL} = k \; \frac{N}{L}

with information transfer (IT) index (a free parameter) $k$. The price level is shown as the "detector" of information flow, and represents an abstract price in the information equilibrium framework. The solution to this differential equation is

\frac{N}{N_{0}} = & \left( \frac{L}{L_{0}}\right)^{k}\\
P = & \; k \frac{N_{0}}{L_{0}} \left( \frac{L}{L_{0}}\right)^{k-1}

Note that if $k = 2$, we have $P \sim L$ (hence the "quantity theory of labor" moniker), and we can obtain (one form of) Okun's law

\frac{d}{dt} \log L = \frac{d}{dt} \log R

by simply differentiating the equation (1) above.

Dynamic equilibrium

Now dynamic equilibrium (in the presence of shocks) is another way of looking at the same equation where we look at the growth of the (abstract) price on the left hand side and the growth of the ratio on the right hand side. Symbolically:

\frac{d}{dt} \log P = & \; (k - 1) \frac{d}{dt} \log L\\
\frac{d}{dt} \log \frac{N}{L} = & \; (k - 1) \frac{d}{dt} \log L

If we take $L \sim e^{\lambda t}$, $P \sim e^{\pi t}$, and $N/L \sim e^{\gamma t}$ then

\text{(2) } \; \pi = \gamma = (k - 1) \lambda

I have previously looked at the core PCE price level, and noticed that it is well-described by a single shock centered at 1978.7 (I hypothesized this was related to a demographic shock of women entering the workforce). My question to myself was: what about $N/L$? The answer turned out to tell us that the abstract price $P$ isn't the measured by PCE (i.e. equation (2) is wrong), but rather something in information equilibrium with PCE. We also gain some new perspective on shocks.

So the basic dynamic equilibrium model gave us this description of the PCE price level:

The same procedure comes up with a similar result for $N/L$:

The biggest shock is centered at 1977.5, which is equal to the location of the shock to PCE inflation within the error. However, the first issue is that there is an additional shock associated with the aftermath of the great recession (centered at 2014.7). This shock is not obvious in the PCE data. This can be cleared up a bit by forcing the model to have an additional shock:

The model places a tiny shock (also centered at 2014.7 [2]) that is basically buried in the noise. Note that this shock may explain part of the low post-recession inflation and the more recent rise (the equilibrium inflation rate is 1.7%). More on this below in the forecasting section. In any case, this shows the small post-recession shock is not a serious issue.

What is a serious issue is that $\pi =$ 1.7% equilibrium inflation rate compared with the ratio $N/L$ growth rate of $\gamma =$ 3.8%. This tells us that the IE relationship $PCE : NGDP \rightleftarrows L$ is at best an approximate effective theory. However, it's not a serious issue that can't be dealt with via a simple fix; much like the interest rate model [1] instead of PCE being the abstract price $P$, we just assume PCE is in information equilibrium with the abstract price $P$:

PCE & \rightleftarrows P\\
P : NGDP & \rightleftarrows L

This introduces a second IT index we'll call $c$ and take $P \sim e^{\gamma t}$ (required by the theory above) and $PCE \sim e^{\pi t}$ so that

\text{(2') } \; \frac{\pi}{c} = \gamma = (k - 1) \lambda

Now $\pi = $ 1.7% per year and $\gamma = $ 3.8% so $c = $ 0.45. Note that $\lambda \sim $ 2% meaning that $k \sim 2$ (i.e. the "quantity theory of labor").

What does this mean? Well, if we thought this model was accessing some fundamental "truth", we could say that the growth rate of $P$ (i.e. $\gamma$) is the "true" inflation rate that we only measure crudely with measures like PCE (actually CPI would be a slightly "better" measure with $c$ closer to 1). 

But such a simple model is unlikely to be the best possible description of a macroeconomy, so it's probably best to take an agnostic "effective theory" approach. In that view, we just take the model above to be a decent approximation for some scope (e.g. average values over time scales of 10s of years, with errors for e.g. oil shocks).

Addendum: forecasting

The dynamic equilibrium results do lend themselves to (conditional [3]) forecasting, so here is the recent data for $N/L$ and core PCE (and their derivatives) along with a forecast through 2020:

This forecast predicts growth will rise back towards the equilibrium growth rate of 3.8%. Note that this is the growth rate of $N/L$, not NGDP growth.

Here is PCE including the tiny post-recession shock:

The horizontal line is at 1.7% inflation; we can just make out the tiny dip centered at 2014.7 accounting for a 0.25 percentage point drop below 1.7% at the peak of the shock (i.e. about 1.4% inflation during 2014, which is in fact the average during that time ... 1.36%).

For completeness, here is the forecast without this shock:


[1] Instead of being the price of money, the interest rate is assumed to be in information equilibrium with the price of money.

[2] I previously associated the positive inflation shock with a demographic shift (women entering the workforce). The second shock occurs nearly 40 years later (potentially associated with the cohort that entered the workforce in the 1970s leaving the workforce). It could also be that we are seeing the demographic effect of people affected by the great recession leaving the workforce (the drop in the employment-population ratio), or of the baby boomer cohort retiring (the peak of which was centered in in the mid-1950s, meaning the mid-2010s shock is 60 years later).

[3] The condition is that no additional shock occurs during the forecast period.

Tuesday, March 21, 2017

India's demonetization and model scope

Srinivas [1] has been requesting that I look into India's demonetization using the information equilibrium (IE) framework for awhile now. One of the reasons I haven't done so is that I can't seem to find decent NGDP data from before 1996 or so. I'm going to proceed using this limited data set because there are several results that I think are a) illustrative of how to deal with different models with different scope, b) show that monetary models are not very useful.

Previously, the only "experiment" with currency in circulation I had encountered was the Fed's stockpiling of currency before the year 2000 in preparation for any issues:

This temporary spike had no impact on inflation or interest rates. Economists would say that the spike was expected to be taken away, and therefore there would be no impact. Scientifically, all we can say is that rapid changes in M0 do not necessarily cause rapid changes in other variables. This makes sense if it is entropic forces maintaining these relationships between macroeconomic aggregates. Another example is the Fed's recent increases in short term interest rates. The adjustment of the monetary base to the new equilibrium appears to be a process with a time scale on the order of years.

If either interest rates or monetary aggregates are changed, it takes time for agents to find or explore the corresponding change in state space.

India recently removed a bunch of currency in circulation (M0). If the historical relationship between nominal output (NGDP) and M0 were to hold, we'd get a massive fall in output and the price level. However, the change in M0 appears to be quick:

So, what do the various models have to say about this?

Interest rates

The interest rate model says that a drop in M0 should raise interest rates ceteris paribus. However this IE relationship only holds on average over several years. Were the drop in M0 to remain, we should expect higher long term interest rates in India:

However, if M0 continues to rise as quickly as it has in Dec 2016, Jan 2017, and Feb 2017, then we probably won't see any effect at all (much like the year 2000 effect described above). M0 needs to maintain a lower level for an extended period for rates to rise appreciably.

This is to say that the model scope is long time periods (on the order of years to decades), and therefore sharp changes are out of scope.

Monetary model

Previously, like many other countries, India has shown an information equilibrium relationship (described at the end of these slides) between M0 and NGDP with an information transfer index (k) on the order of 1.5. A value of k = 2 means a quantity theory of money economy, while a lower value means that prices and output respond much less to changes in M0.

In fact, as I mentioned in a post from yesterday monetary models only appear to be good effective theories when inflation is above 10%, and in that case we should find k = 2. That k < 2 implies the monetary theory is out of scope and we have something more complex happening.

The quantity theory of labor

The monetary models don't appear to be very useful in this situation. However one model that does do well for countries with k < 2 is the quantity theory of labor (and capital). This is basically the information equilibrium version of the Solow model (but deals with nominal values, doesn't have varying factor productivity, and doesn't have constant returns to scale). unfortunately the time series data doesn't go back very far and there aren't a lot of major fluctuations. Even so, the model does provide a decent description of output and inflation:

The exponents are 1.6 for capital and 0.9 for labor meaning India is a great place to get return on capital investment (the US has 0.7 and 0.8, and the UK has 1.0 and 0.5, respectively).

This model tells us that inflation is primarily due to an expanding labor force, and therefore demonetization should have little to no effect on it.

Dynamic equilibrium

The dynamic equilibrium approach to prices (price indices) and ratios of quantities has shown remarkable descriptive power as I've shown in several recent posts (e.g. here). India is no different and inflation over the past 15 years can be pretty well described by a single shock centered in late 2010 continuing over a time scale on the order of one and a half years:

This model doesn't tell us the source of the shock, but unless another shocks hits we should expect inflation to continue at the same rate as it has over the past 2 years (averaging 4.7% inflation). This also means that the demonetization should have little to no effect.


The preponderance of model evidence tells us that the demonetization should have little to no effect on inflation or output. The speed at which it was enacted means that the monetary models are out of scope and tell us nothing; we can only rely on other models that are in scope and those have no dependence on M0.



[1] Srinivas also sent me much of the data used in this post.

Monday, March 20, 2017

Using PCA to remove cyclical effects

One potential use of the principal component analysis I did a couple days ago is to subtract the cyclical component of the various sectors. I thought I'd take it a step further and use a dynamic equilibrium model to describe the cyclical principal component and then subtract the estimated model. What should be left over are the non-cyclical pieces.

First, here's the model and the principal component data; the description is pretty good:

I won't bore you with listing the results for every sector (you can ask for an update with your favorite sector in comments and I will oblige). Let me just focus on the interesting sectors with regard to the economic "boom" of 2013-2014. There are three different behaviors. The first is a temporary bump that seems to be concentrated in retail trade:

The bump begins in mid-2013 (vertical line) and ends in mid-2016.

The second behavior is job growth. For example, here is health care and social assistance; the rise begins around the date the ACA goes into effect (vertical line):

The third behavior is unique to government hiring, specifically at the state and local level. It drops precipitously at the 2016 election (vertical line):

Note that this doesn't mean hiring dropped to zero, it just mean state and local government hiring dropped back to it's cyclical level after being above it (e.g. because of the ACA, for example).

Belarus and effective theories

Scott Sumner makes a good point that inflation is not only demographics using Belarus as an example. However I think this example is a great teachable moment about effective theories. The data on inflation versus monetary base growth shows two distinct regimes; the graph depicting this above is from a diagram in David Romer's Advanced Macroeconomics. One regime is high inflation and because it is pretty well described by the quantity theory of money (the blue line) I'll call it the quantity theory of money regime. The second regime is low inflation. It is much more complex and is probably related to multiple factors at least partially including demographics (or e.g. price controls).

The scale that separates the two regimes (and that defines the scope of the quantity theory of money theory) is on the order of 10% inflation (gray horizontal line). For inflation rates ~ 10% or greater, the quantity theory is a really good effective theory. What's also interesting is that the theory of inflation seems to simplify greatly (becoming a single-factor model). It is also important to point out that there is no accepted theory that covers the entire data set ‒ that is to say there is no theory with global scope.

In physics, we'd say that the quantity theory of money has a scale of τ₀ ~ 10 years (i.e. 10% per annum). For base growth scales shorter than this time scale like, say, β₀ ~ 5 years (i.e. 20% per annum), we can use quantity theory.

At 10% annual inflation, Belarus should be decently described by the quantity theory of money with other factors; indeed base growth has been on the order of 10%.

The problem is that then Scott says:
So why do demographics cause deflation in Japan but not Belarus?  Simple, demographics don’t cause deflation in Japan, or anywhere else.
Let me translate this into a statement about physics:
So why does quantum mechanics make paths probabilistic for electrons but not for baseballs? Simple, quantum mechanics doesn’t make paths probabilistic for electrons, or anything else.
As you can see this framing of the question completely ignores the fact that there are different regimes where different effective theories can operate (quantum mechanics on scales set by de Broglie wavelengths; when the de Broglie wavelength is small you have a Newtonian effective theory).

Wednesday, March 15, 2017

Washington's unemployment rate, Seattle's minimum wage, and dynamic equilibrium

I live in Seattle and the big thing in the national news about us is that we raised our minimum wage to $15, which just went into effect for large businesses in January of this year. According to many people who oppose the minimum wage, this should have lead to disaster. Did it have an effect? Let's try and see what the dynamic equilibrium model says. I added a couple of extra potential shocks after the 2009 big one:

People who believe the minimum wage did negatively impact could probably see the negative shock centered at 2015.8 as evidence in their favor. However, that could also be the end of whatever positive shock centered at 2013.0 (which I think was hiring associated with the ACA/Obamacare). I showed what the path would look like if those shocks (positive and negative) were left out using a dotted line. If it was the minimum wage, it would have to be based entirely on expectations because it is being phased in (not reaching $15/hour for all businesses until 2020):

However those expectations did not kick in when the original vote happened in June of 2014, so it must be some very complex expectations model. In this second graph I show what the graph looks like in the absence of both the 2013 and 2015.8 shocks (shorter dashes) as well as just the absence of the 2015.8 shock (longer dashes). Various theories welcome!

The Fed raised interest rates today, oh boy

The Fed raised its interest rate target to a band between 0.75 to 1.0 percent at today's meeting, so I have to update this graph with a new equilibrium level C'':

We might be able to see whether the interest rate indicator of a potential recession has any use:

This indicator is directly related to yield curve inversion (the green curve needs to be above the gray curve in order for yield curve inversion to become probable). Here are the 3-month and 10-year rates over the past 20 years showing these inversions preceding recessions (both in linear and log scales):

Principal component analysis of jobs data

Narayana Kocherlakota tweeted about employment making a random claim about the "steady state" employment growth that seems to come from nowhere which inspired me to do something I've been meaning to do for awhile: a principal component analysis of the Job Openings and Labor Turnover time series data ('JOLTS'):

I used a pretty basic Karhunen–Loève decomposition (Mathematica function here) on several seasonally adjusted hires time series from FRED (e.g. here). For those interested (apparently no one, but alas I'll do it anyway) the source code can be found in the Dynamic Equilibrium GitHub repository I set up. Here's the result (after normalizing the data):

The major components are the blue and yellow one (the rest are mostly noise, constant over time). I called these two components the "cyclical" (blue) and the "growth/decline" (yellow) for fairly obvious reasons (the growth/decline is strongest after 2011). It's growth or decline because the component can be added with a positive (growth) or negative (decline) coefficient. Here are how those two components match up with the original basis:

The story these components tell is consistent with the common narrative and some conventional wisdom:

  • Health care and education are not very cyclical
  • Health care and education are growing 
  • Manufacturing and construction are declining  

Here's health care on its own (which looks pretty much like the growth/decline component):

And here's manufacturing (durable goods):

To get back to Kocherlakota's claim, the "steady state" of jobs growth then might seem to depend on the exact mix of industries (because some are growing and some are declining) and where you are in the business cycle. However, as I showed back in January, total hires can be described by constant relative growth compared to the unemployment level and the number of vacancies ‒ except during a recession. This is all to say: it's complicated [1]. 

PS Here are all of the components (here's a link to my Google Drive which shows a higher quality picture):


Update 16 March 2017

I added government hiring, normalized data to the mean, and standardized the output of the algorithm. The results don't change, but it looks a bit better:

Here are the two main vectors (standardizing flipped the sign of the growth/decline vector):

And here's the original data, updated with the new data points released today (I subtracted the census peak by interpolating between the two adjacent entries):



[1] Although I'm still not sure where the 1.2 million jobs per year comes from; here's the employment change year over year:
The bottom line is Kocherlakota's 1.2 million figure. The second from the bottom is the 1.48 million rate that comes from averaging the growth rate including recessions (it's almost 1.6 million for just post 1960 data). The upper line is my "guesstimate" for the average excluding recessions (2.5 million). Maybe it was a typo and he meant 2.2 million.