Friday, March 27, 2015

Potential RGDP and forecast RGDP


Brad DeLong had a post up a month ago on a study (well, blog post) of the various forecasts of potential RGDP and their changes over time. He quotes the study's (well, blog post's) authors Cecchetti and Schoenholtz:
We should all be wary of anyone who claims to be able to forecast trend growth accurately and reliably. Even after the fact, it takes some time to discern the underlying trend.
I (sort of) reproduce the authors' chart 2 (shown at DeLong's link), and add the information transfer model (ITM) trend in gray derived from the partition function approach:


Note that potential RGDP isn't some sort of speed limit, although there are some interpretations that are more like one. Like the Fed's estimate of potential RGDP, the ITM trend isn't a speed limit -- it's more of an equilibrium level above which there is a greater tendency to fall.

With the exception of 2000-2015 [1], the ITM lines up relatively well with the Fed's estimate. Of course, both of these measures are looking at trends in the roughly the same data so this overlap is not surprising. It's the differences that are interesting.

One way to interpret these two measures over the past 30 or so years in the graph is that both the ITM and the Fed say the 1990s dot-com boom was sustainable (it was recovery from low performance after the financial crises of the late 1980s and early 90s), but they differ on the housing boom of the 2000s. The ITM effectively says that boom was unsustainable [2], while the Fed's estimate shows potential RGDP decaying away -- as if something could have been done in the aftermath of the 2008 financial crisis.

One useful feature of the Fed's estimate is that the differences between potential and measured RGDP match up with the unemployment rate (modified by the "natural rate"):


This is not true in the ITM. However this is not much more than Okun's law (already a part of the ITM) -- changes of RGDP relative to any smooth baseline will result in a pattern that looks like the unemployment rate because changes in RGDP and changes in employment are correlated (i.e. Okun's law). The Fed's version does this without a derivative -- the absolute difference between potential RGDP and actual RGDP is proportional to the unemployment rate. Mathematically we have the Fed's

$$
u \sim RGDP - RGDP_{p}
$$

versus the ITM's

$$
\frac{d}{dt} \log u \sim \frac{d}{dt} \log (RGDP - RGDP_{p}) \sim \frac{d}{dt} \log RGDP + c
$$

where we've used the smoothness of potential RGDP to reduce it to a constant $c$.

Footnotes:

[1] And in the longer view, there's a discrepancy between the Fed's estimate and the ITM for 1960-1980. Here is the longer view:



[2] Actually, the ITM hints that the boom (and its inevitable bust) was caused by Fed policy.

Swiss update

It's been two months since Switzerland let their currency appreciate (although it seems to have rebounded from much of its original fall -- but that may just be because markets have the incorrect theory). In most monetary theories of the price level, this would imply that the price level should fall. There is no sign of that yet -- to be sure, it's still to early to tell. I am mostly just setting up my data ingest and graphics output so I can more easily produce updates to the information transfer model prediction that not much will happen.

Here is the graph [1]


The red lines aren't supposed to be exact model predictions -- they're just general trends to be expected based on a generic monetarist-view theory. Looser monetary policy should increase the price level (it actually decreased after the currency depreciation) and tighter policy should decrease the price level.

The dotted green is the data the information transfer model prediction is based on (I used data up through December of 2013). The solid green is data from after Dec 2013. And the black line is the data that has come out both since the model prediction and since the CHF appreciation (15 January 2015).

Footnotes:

[1] Here is the zoomed-out version of the whole data set:


Wednesday, March 25, 2015

Entropy and NGDP, take two

I guess I was a little over-zealous in the results in the previous post. I discovered an error in the programming -- effectively the price at which items were being bought and sold was constant, not fluctuating with the market.

In correcting the behavior and the error, I've discovered that it's not the entropy of the money distribution, but rather the goods distribution that is (approximately) proportional to NGDP. The reason I write "approximately" is that I've also turned up a bias in NGDP  that isn't the result of an immediately obvious programming error. It could be a real effect (the entropy changing impact of the shock has some effect) or it could be some error I haven't discovered (my initial intuition was that it had something to do with rounding errors in using whole numbers of money units, but it doesn't seem to go away with increased money resolution).

Anyway, here are the graphs again (each line is one of the 10 sectors on the Wicksellian roundabout, with the red line being the first sector). Here are the goods and money graphs:


You can see the effect that an increase demand for money by the first sector and the consequent fall in goods held. You can also see the "cyclic" fluctuations brought on by the initial shock as the excess supply of money makes its way back around the roundabout. Here are the prices:


As the first sector decides to hold more money, the price for its goods (now more scarce) shoots up. And finally is the problematic graph:


There is some discrepancy between the goods entropy (black) and the goods NGDP (blue). The entropy of money (green) is shown not to be the same as NGDP.

Still investigating.

Monday, March 23, 2015

Entropy and NGDP

I had previously shown analytically that the change in economic entropy is approximately given by the change in NGDP (times a constant).
UPDATE: 25 March 2015: There are some issues with this post. See the next post for the issues. The results generally seem to hold, but ΔS ~ ΔNGDP is only approximate.
This time I've taken the Wicksellian roundabout (with the first sector suddenly increasing its demand for money -- and I'm using 10 sectors here), added goods flowing in the opposite direction of money, and used the information transfer model definition of the price to calculate the change in "NGDP" both directly (i.e. from price × amount of goods exchanging hands) and from the entropy of the distribution of money. They turn out to be proportional to each other, explicitly verifying that ΔS ∝ ΔNGDP (in this model).

Here is the money demand shock in the first sector (red):


 Here are the goods flowing in the opposite direction:


Here are the market prices for the different goods:


And here is the change in entropy (- ∑ p log p, in black) and the change in NGDP (prices × amount of goods, in blue):


The Wicksellian roundabout probably isn't necessary -- it is simply a device to have the output of one sector dependent on the output of the others. In fact, looking at the price figure, you can see that the actions of the first sector have diminishing impact on the following sectors (orange is the most, then less on through green, blue and purple).

Supply and demand as entropy


Continuing in a series with the previous posts, here I'd like to show the forces of supply and demand as entropy. At the moment of the shock, we either add or remove points from the supply or demand. This produces shifts in the supply and demand curves (shocks), and the system returns to equilibrium. I used the differential equation:

$$
P = \frac{dD}{dS} = k \; \frac{D}{S}
$$

to determine the price. The model for partial equilibrium (i.e. supply and demand curves) is here for reference. Here are the four cases ... (demand is in blue on the left, supply in red on the right)

Increase in demand, leading to an increase in price:



Increase in supply, leading to a fall in price:



Fall in demand, leading to a fall in price:



Fall in supply, leading to an increase in price:



Saturday, March 21, 2015

Entropy and unemployment

Here's the same model from this post, except without the periodic boundary conditions and transitions can happen from any cell to any cell. In this case, I called the first cell the "unemployed sector" and during the recession a point (i.e. a member of the labor force) in the first cell can't leave the unemployment sector. The results are basically identical to the previous post, except now the fraction of people in the cell gives us the unemployment rate. It falls at a fairly linear rate (which I noted in this post from awhile ago), whereas output rises faster immediately after the recession and slower later. Here are the results:


The fraction of points in the first cell (the unemployment rate) is given in red. Here is the animation as well:


The Wicksellian roundabout and entropy

Nick Rowe wrote a post a year ago about a model of an economy as a "Wicksellian roundabout":
Imagine a large number of cars forever circling around a very large roundabout. Initially they are all going the same speed, and are evenly spaced. What happens if one car slows down temporarily?
This morning I remembered the post (mostly because of this and related discussion) and was inspired to look at the story as an entropy problem. I considered 5 cells where points (think of them as money) were randomly allowed to move counterclockwise one cell -- also, the fifth cell is next to both the fourth and first cell (i.e. periodic boundary conditions). Update 3:50pm: here is a picture of the 5-cell roundabout ...


If we start with all the money in the first cell (above the 3 o'clock position), the average configuration moves toward the maximum entropy distribution (equal amounts of money in each cell):


Note that the blue paths represent the entropy of a particular configuration (in the Monte Carlo simulation), while the black line represents the entropy of the average configuration. You can watch this in the following animation:


Rowe wanted to consider a pile-up (traffic jam) on this Wicksellian roundabout. I started with a maximum entropy configuration and added a period where the first cell (the one that starts with all the points in the picture above) increased its demand for money. During that period, that cell doesn't let a point leave its cell. Here is the entropy of the average configuration in that case:


The entropy takes a hit, and then starts heading back towards the maximum entropy distribution. Here is an animation for that case:


What does a loss in entropy mean for an economy? A fall in entropy is a fall in output. What the picture above shows is a recession.

What is interesting is that a decrease in the demand for money in the first cell also causes a recession. I modeled this by making it more likely that the first cell gives up one of its points. The result is similar:


And here is the animation:

The effect is less pronounced because in this case the first cell only affects the distribution using points that are in its own cell. In the first recession case, the first cell is interrupting the flow from all five cells. What a decrease in the demand for money in the first cell does is create an effective increase in demand in the second cell (it ends up with more money than maximum entropy would indicate).