Average weekly hours in the UK

I came across this chart (via a re-tweet from Ian Wright) where Alfie Sterling extrapolated the 1946-1980 trend in average hours worked in the UK alongside an extrapolation from data post-1980:

There seems to be an entire industry in the UK built out of extrapolations like this (here's productivity). I've reproduced a version of this — it uses data for all employees, not just full time employees, so the level is a bit higher [click to enlarge]:

But the story is roughly the same — the trend was a steeper decline before 1980 and shallower after. However, plotting the graph on this scale (as well as cutting off the data at 1946) obscures some of the issues with extrapolating linearly willy-nilly. Zooming in a bit and taking that linear fit back to 1900 shows the 1946-1980 trend is unique to the period 1946-1980:

In fact, as I looked at a couple of years ago, this data is pretty well described by a dynamic information equilibrium model (DIEM):

The trend from World War II (WWII) to 1980 is almost certainly part of the demographic shift of women into the workforce in Anglophone countries that seems to govern so many things. The other major effects seem to be WWI and WWII ending in 1918 and 1945, respectively. Aside from those three events, average weekly hours is on a steady decline of 0.13% (consistent with what I found earlier for US data [1]).

This perspective is a lot less policy (or productivity, or wage) dependent, depending more on major social changes (war, women entering the workforce) — a recurring theme in my book. The trends differ across countries (with e.g. France and Germany's annual labor hours falling at closer to a 1% rate) implying that they may be set more by social norms.

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Footnotes:

[1] Here's the figure showing the -0.13% trend and the same 60s-70s demographic shock:

The S&P 500 since 2017

One of the forecasts I made when I first worked out the theory behind the dynamic information equilibrium model (DIEM) besides the unemployment rate was for the S&P 500. This forecast has worked out remarkably well — though one might ask how could it not with error bands on the order of 20%? I changed the color scheme a bit since the original forecast, but here's where we are (note that it's a log plot):

The black line is post-forecast data. The vertical blue bands are NBER recessions. The vertical red/pink bands are the non-equilibrium shocks to the S&P 500. The green error bands are the 90% confidence bands for the entire data series since the 1950s and the blue error band over the forecast data was the 90% confidence projection from estimating an AR process on the deviations from the dynamic equilibrium starting from the forecast date. That AR process was trained on the data since 2010.

The AR process error gets fairly close to the error bands for the whole series. My interpretation of this is that the AR process model has captured pretty much nearly the entire range of error except for recessions — that is to say the data from 2010 to 2017 gave us a decent estimate of non-recessionary deviations from the dynamic equilibrium due to news shocks, policy, foreign affairs, et cetera.

I think the past couple years has given us some additional evidence that hypothesis is correct as the current administration has effectively conducted some "natural experiments". I estimated a couple of non-equilibrium shocks to the post-forecast data — we can zoom in to see them better:

The first shock is a positive one taking place at the end of 2017 and the beginning of 2018 — almost certainly due to the TCJA (which incidentally appears to have had other effects). If that had been the only thing that happened since 2017, everyone with a 401(k) account invested in an S&P index fund (full disclosure: that's what I do) would have had 17% more in it today even without contributions.

Instead of being around 3650, we're closer to 3100 today. Why? It appears to be due to the "trade war" with China. The major announcements are shown with black arrows (↑). The first round of tariffs began before the ink on the TCJA had dried (green arrow) and basically cut what was estimated to be a sizable 20% gain back to zero. A second round of tariffs comes in August and September of 2018, accompanied by a subsequent shock.

The Fed's rate increase in December 2018 did produce a rapid drop in the S&P 500, but the effect seems to have since evaporated. I estimated the tariff shock with and without the data from from December 2018 and January 2019 and got nearly the exact same result in terms of the longer run level in both cases. It's of course not impossible that the effect of tariffs is what evaporated and what we're seeing is purely the effect of the Fed — but this is inconsistent with a) the fact that the tariffs seem to have had a lasting effect in 2018 and b) the December 2015 rate increase also largely evaporated [1].

So it seems that mismanagement of government policy does have sizable & quantifiable effects on the stock market. However, the key conclusion here is that these policy decisions appear to be within the range of the overall 20% error since the 1950s — and that policy changes are basically a 2nd order effect on the stock market after recessions with deviations on the order of δ log SP500 ~ 0.1 with recessions having an effect δ log SP500 ~ 0.6-0.8 or more [2].

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Footnotes:

[1] The December 2015 rate increase was a shock on the order of δ log SP500 ~ 0.15, but in less than a quarter was only δ log SP500 ~ 0.05 and consistent with zero by 2017. The December 2018 rate increase has almost exactly the same structure: an initial drop by δ log SP500 ~ 0.15 and a quarter later the level was back up to within δ log SP500 ~ 0.05 of the pre-hike level.

[2] And per [1], Fed decisions having an effect on the order of δ log SP500 ~ 0.15 that quickly evaporate over the next quarter.

Projections, predictions, accountability, and accuracy

John Quiggin has an entertaining article up at The Conversation that looks at the persistently undershooting IEA "projections" of renewable energy production as a case study in the lack of accountability for statements about the future. This particular case comes up every two years because the IEA updates their "projections" every two years (Quiggin cites a 2017 critique from Paul Mainwood and David Roberts talked about it at Vox two years prior in 2015). It's 2019, so time for another look!

I personally have a soft spot in my heart for these "hedgehog" graphs where the future lines keep missing the data — I keep a gallery of macroeconomic "projections" (predictions? forecasts?) here. Often, the dynamic information equilibrium model (DIEM) is a much better model (such as for the unemployment rate), so I wanted to try it on this data.

Why should the DIEM be an appropriate model? Well, for one thing we can view the generation and consumption of electricity as a matching model — a megawatt-hour of production is matched with a megawatt-hour of consumption. Renewable energy is then like a manufacturing sector job or a retail sector job (... or an unemployment sector job a.k.a. being unemployed). But a more visually compelling reason is that technology adoption tends to follow (sums of) logistic curves per Dave Roberts' article at Vox:

These are the same logistic curves the DIEM's non-equilibrium component is built from. Logistic curves are also seriously problematic for "center predictions" — you really need to understand the error bands. The initial take-off is exponential, resulting in enormous error bands. The center is approximately linear, and only once you have reached that point do the error bands begin to calm down (see here for an explicit example of the unemployment rate during the Great Recession).

One issue was that I had the hardest time finding corroborating data that went back further than the "actual" data in Mainwood's graph. I eventually found this which is claimed to come from the IEA (via the World Bank). It largely matches up except for a single point in the 90s, allowing for error in digitizing the data from the plot. (Be careful about production versus consumption and world versus OECD if you try to find some data yourself.) That point in the 90s is inexplicably near zero in Mainwood's "actual" data. It's possible there are some definition issues (it's non-hydro renewables, which may or may not include biomass). But as this isn't a formal paper, the recent data seems fine, and the details of the fit aren't the main focus here we can just proceed.

I ran the DIEM model for the IEA data from 1971 to 2015, and this was the result:

Overall, the DIEM forecast is highly uncertain, but encompasses the 2012 and 2016 IEA forecasts for the near future. Mainwood's "corrected" forecast (not shown here) is well above any of these — it represents a typical problem with forecasts of logistic processes where people first see a lot of under-estimation, over-correct, and seriously over-estimate the result.

The best way to see the DIEM forecast is on a log scale:

There are three major events in this data — one centered in the early 80s (possibly to due to oil shocks and changes in energy policy such as the Carter administration), a sharp change in the late 80s, and then finally the current renewable revolution with wind and solar power generation due to a combination of policy and technology. The equilibrium growth rate (the "dynamic equilibrium") is consistent with zero — i.e. without policy or technology changes, renewables don't grow very fast if at all.

You can also see that it's likely we have seen the turnaround point in the data around 2010 — but it is also possible the global recession affected the data (causing renewables to fall as a fraction of global energy production). The global recession may be making it look like the turnaround has passed.

Quiggin's larger point, however, is something I've never really even considered. Do people really see projections as different from predictions or forecasts? If someone tried to hide and say their lines going into the future were "projections" and therefore not meant to be "predictions", I would just laugh. Does this really fool anyone?

I cannot come up with a serious rational argument that projections are different from predictions. We sometimes call predictions forecasts because that seems to move a step away from oracles and goat entrails. But any statement about the value of a variable in the future is a prediction. Sure, you can say "this line is just linear extrapolation" (a particular model of expected future data) and that it most certainly won't be right (a particular confidence interval). But it's still a prediction.

That's why the error bands (or fan charts, or whatever) are important! If you draw a line and say that we shouldn't take it seriously when we discover it's wrong, that just means the ex ante error bands were effectively infinite (or at least the range of the dependent variable). As such, there's literally zero information in the "projection" compared to a maximally uninformative prior — i.e. a uniform distribution over the range of the data. You can show that with information theory. Any claim that a projection that shouldn't be compared to future data yet has some kind of value is an informational paradox. It represents information and yet it doesn't!

Is this why a lot of economic and public policy forecasts leave off error bands? Is somehow not explicitly putting the bands down believed to keep the confidence in some kind of unmeasured quantum state such that it can't be wrong?

But as Quiggin mentions, this has ramifications for accountability. People year after year cite the IEA "projections" that continue to be wrong year after year. And year after year (or at least every two years) some rando on the internet takes them to task for getting it wrong, and the cycle begins again.

The thing is that it's not that difficult to explain why the IEA projections are wrong. Forecasting the course of a non-equilibrium shock (in the DIEM picture) is nigh impossible without accepting a great deal of uncertainty. Even if you don't believe the DIEM, a logistic picture of technology adoption is sufficient to understand the data. The only problem is that they'd have to show those enormous error bands.

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PS I am almost certain those error bands exist in their models; they just don't make it into the reports or executive summaries.

PPS The existence and form of "executive summaries" should be all the evidence we need that CEOs and other "executives" aren't super-genius Übermenschen.

World GDP growth and silly models

In my travels on the internet, I came across this paper (Koppl et al [1]) from almost exactly a year ago. It has the silliest model of the world economy I've ever seen. Here's the abstract:

We use a simple combinatorial model of technological change to explain the Industrial Revolution. The Industrial Revolution was a sudden large improvement in technology, which resulted in significant increases in human wealth and life spans. In our model, technological change is combining or modifying earlier goods to produce new goods. The underlying process, which has been the same for at least 200,000 years, was sure to produce a very long period of relatively slow change followed with probability one by a combinatorial explosion and sudden takeoff. Thus, in our model, after many millennia of relative quiescence in wealth and technology, a combinatorial explosion created the sudden takeoff of the Industrial Revolution.

Caveats about extrapolating that far back notwithstanding, the problem isn’t so much what is written in the abstract but rather that the model cannot support any of the statements in it. Overall, it’s a good lesson (cautionary tale?) in how to go about mathematical modeling.

Just so there are no complaints that I "didn't understand the model", I went and reproduced the results. There's something they kind of gloss over in their paper that I'll come back to later that accounts for the small discrepancies.

First, the population data is basically exactly their graph (I have two different sources that largely match up):

The black dashed line at the end will come back later. Constructing their recursive M function (that represents the combinatorial explosion) and putting it together with the Solow model/Cobb-Douglas production function in the paper allows us to reproduce their graph of world output (GDP in Geary–Khamis international dollars) since the dawn of the Common Era (CE):

Like the population graph above and the M-function graph below, it is also graphed on a linear axis for some reason. They zoom in to 1800-2000 because they want to talk about the Industrial Revolution:

We reproduce this down to the segmented lines drawn between points in the time steps. Although you can't really see it in this graph, this is really part of a continuous curve in the model that goes back to at least the 1600s — it's not the Industrial Revolution (for more on take-off growth, see here or here). A log-log graph helps illustrate it a bit better:

The authors then show their output points alongside a measure of GDP in international dollars. For some reason it’s now points instead of line segments. But at least we’re on a log scale!

I didn’t use the exact same time series for comparison; instead, I used GDP estimates from Brad Delong here [pdf] that I had on hand. However, they're reasonably close to the data they present in their paper. In fact, it’s a bit better fit! I'm doing my best to be charitable. There is an almost exact factor of 4 difference in the level of their data and Delong's, which I think is accounting for “seasonally adjusted annual rate” for quarterly data. Koppl et al actually have two other model fits in their paper with different parameters. I just reproduced the yellow one that was closest to the data  (see the others at the end of this post).

The one graph I’m not reproducing exactly here is their M function. I think they just plotted a version with different parameters than for their yellow model result. As I didn’t care that much, I just did the M function from the yellow model result since that’ll be most germane to our discussion. Like most combinatorial functions, it goes along fairly flat (in linear space) and then jumps up suddenly (again, in linear space):

It starts at M₀, which is 50 in the parameters for their yellow result. The last several numbers in the series are 117.1, 125.3, 136.1, 151.0, 173.7, 213.2, 303.1, 668.8, 9323.4, 326360625.7. The next number is 4.9 × 10^26. Five hundred octillion.

What is supposedly happening in this model is that a current inventory of products (stick, flint, feathers) are at random brought together to produce a new product (an arrow for a bow) with some probability. That new inventory then has elements brought together to craft a new output good. It's basically a Minecraft crafting economy with the number of products you discover increasing combinatorially (roughly on the order of e.g. the gamma function or factorial). The factorials enter through a binomial coefficient.

Combinatorial explosion is building all along, but it really doesn't explain the Industrial Revolution. In fact, you can’t really say this “starts” anywhere with any kind of objective criteria. It starts at M₀, if anything, which is assigned to “year 1” (t = 0) — the beginning of the CE. The location of the super-exponential “take off” point (viewed on a linear scale) is then 60 or so time steps from year 1. But what is a time step? That’s what the authors gloss over. The time is just “scaled” so that the combinatorial series fits in the period from about “year 1” to about the present year.

The time steps turn out to be about 31 years (at least that's what I used), which is remarkably close to a “generation”. But this time scale is a fundamental parameter of their model — telling us where and when the combinatorial explosion occurs. If it had instead been on the order of a quarter, we could go from subsistence to the modern age in about 16 years. Instead whatever combination of current output goods that produces a new product with some probability happens only once every 30 years. You could of course adjust the probability to compensate a change in the time scale — making the probability parameters smaller increases the number of time steps it takes to cover the dynamic range of GDP values. However, since none of these parameters are estimated from some underlying data, the exact location and span of the model result in time is completely arbitrary.

I will pause to note that leaving out time scales like this is a general failing in economics (see here or here), making it impossible to understand the scope of their theoretical models.

The real  problem is when you go to the next time steps (I've also started adding graph labels to the graphs themselves). Combinatorial explosion doesn’t stop once you’ve explained as much of the data you want to explain. It keeps going, and going, and going, and going ...

Of course the GDP data ends so we can't see just how realistic this model is. Remember — their M function is heading towards 10^26 when it's about 600 around the year 2000.

This made me want to use the  the dynamic equilibrium model to extrapolate the data a bit further. In it, we have general exponential growth interrupted by periods of much higher (or lower) growth (“shocks”).

I wrote about population growth and how you might go about modeling it with the dynamic equilibrium model about two years ago with a follow up referencing the well-known 1970s report titled Limits To Growth. The general result there is that the recent population data is consistent with a saturation level of about 12-13 billion people by 2300. That most recent surge in population growth is associated with the advent of modern medicine (others seem to be associated with e.g. the Neolithic revolution in farming or sanitation). Maybe that’s right, maybe that’s wrong. But at least it’s a realistic extrapolation based on a slow decline in world population growth.

I used the world GDP data and population data to create a GDP per capita measure. I then extrapolated that data using another dynamic equilibrium model — one that’s remarkably consistent with the widespread phenomenon of women entering the workforce in larger numbers in the 1950s, 60s and 70s in the world’s largest economies. Again, it’s possible GDP per capita will continue to expand at its current rate for much longer than the next 25 to 50 years, but with growth slowing in most Western countries and even China, it’s entirely possible we’ll see a decline to a rate of growth more consistent with the 1800s than the 1900s.

We can combine our extrapolation of GDP per capita with population to form an extrapolation of world GDP over the next hundred years. The new picture of the longer term output growth shows how silly the combinatorial model is unless we arbitrarily restrict it to the most recent 2000 years.

In 2077 [2], world GDP by this extrapolation is about 513 trillion 1990 Geary–Khamis international dollars instead of the combinatorial version which gives 8.2 duodecillion (10^39) international dollars. We can compare this to world GDP in 2000 which was about 96 trillion international dollars in this data.

An increase by a factor of 5 from the year 2000 is not entirely unreasonable given slowing global growth, but an increase by a factor of a duodecillion (which I had to look up) [3] seems ... um, improbable.

US real GDP grew by a factor of about 10 over 70 years from 1950 to today, but that also includes the period in the middle of the last century where growth was much higher. Plus, the data in the GDP extrapolation also grows by a factor of about 10 over 70 years from 1950 to 2020.

The main take away is that this combinatorial model is both arbitrary in its timing — it's set up to have growth explode after the industrial revolution — but also its scope, being limited to the period from about 1 CE to about 2000 CE [4]. Going a single time step too far gives not just unrealistic but absolutely silly results. The model seems very much like someone (maybe Koppl) had this combinatorial idea (maybe after someone mentioned Minecraft to him) and it was given to a bunch of grad students to figure out how to make it fit the data. Odd parameters, large time steps that result in segmented data graphs, arbitrarily setting terms in sums to zero — it's not a natural evolution of a model towards the data. I saw this in their figure 4 and laughed:

Of course, the default color scheme for Mathematica is instantly recognizable to me (and in part why I tried to reproduce the figures exactly down to the dotted grid lines). But these line segments are all supposed to be aiming for that blue line. None of them are remotely close to even qualitatively explaining the data.

It's not an a priori bad insight for a model — it makes sense! It's kind of a Gary Becker irrational agents meets a Minecraft opportunity set. But combinatorial explosion is just too big to explain GDP, which is much more in the realm of the exponential with varying growth rates. So instead of mathematical modeling, you start building a Rube Goldberg device to make the model output kind of look like the data ... if you squint ... from across the room.

And yet instead of languishing on a grad student's file share or hard drive where it should be, this model ended up LaTeX'd up on the arXiv.

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Footnotes

[1] It should be noted that Roger Koppl, the lead author, is associated with Mercatus and George Mason University (like one of the other co-authors) with lots of references to Hayek and Austrian economic in any description. Additionally, the paper came up on Marginal Revolution this past week. It should be a huge grain of salt, and in fact this paper is pretty typical of the quality of the work product from GMU-related activities [5].

[2] Chosen due to the time step scale.

[3] This made me think of Graham's number — for a time the largest number that has ever been used for anything practical (in this case it's an upper bound for a graph coloring problem). In part because the Koppl et al GDP is so high itself, but also because like the suspicion of mathematicians that the real answer for Graham's number is about 20, a more realistic estimate of GDP is much, much lower.

[4] There are other choices, such as limiting ourselves to only about 4 items in the combinations that I believe was more a computational limit (my computer has overflow problems if you increase that number or add too many time steps), that basically turn this "model" into a ~10 parameter fit.

[5] The paper goes on a tangent about "grabbing" which is basically a right wing rant:

Our explanation might seem to neglect the important fact of predation, whereby some persons seize (perhaps violently) goods made by others without offering anything in exchange for them. Such “grabbing,” as we may call it, discourages technological change. 

The model put forward has absolutely nothing to do with this and can't explain technological change well enough to warrant speculation about secondary effects like this.

In addition, this is completely ahistorical. Violently seizing others' goods is in fact a major driver of innovation in history — a huge amount of innovation comes in the form of weapons. The silicon chips you're using right now to read this? Needed to make the computations fast enough to accurately guide a nuclear weapon to its target. The basics of computers with vacuum tubes were built to better aim artillery — even physics itself came from this.

Wage growth and belated GDP updates

Wage growth data from the Atlanta Fed came out yesterday and the dynamic information equilibrium model (DIEM) has been doing fairly well for awhile — coming up on two years in a couple months!

J.W. Mason will have to continue to be puzzled. Black is post-forecast data, and click to enlarge. We can also say the DIEM forecast was better than Jan Hatzius' (Goldman Sachs) forecast from this same week one year ago. Orange is the actual average over that time period with one standard deviation errors, compared with Hatzius' range in purple:

Other forecasts where the DIEM is outperforming are RGDP growth and PCE inflation (those belated GDP updates). These cases aren't so much about the center of the prediction, but rather the error band being smaller for the DIEM model and still being accurate.

There's also the FOMC forecast, which is fine but claims a lot more precision in their "central tendency" that must be reflecting something other than RMS error [1]:

And here's the forecast of nominal GDP (NGDP) over employment (PAYEMS or here L) that forms the basis of (the information equilibrium view of) Okun's law and the "quantity theory of labor":

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Footnotes:

[1] A "central tendency" in the opinions of a group of people is sometimes called "groupthink".

Jobs day: October 2019

The Employment Situation data from BLS was released on FRED today (a.k.a. "Jobs Day") which includes the latest  unemployment rate and the "prime age" (25-54) labor force participation rate data among many other measures. I've emphasized those two particular measures since I've been tracking the performance of the Dynamic Information Equilibrium Model forecast since 2017. And now, almost three years later, they're as accurate as ever (black is the post-forecast data):

For a bit of context, here's a rogues' gallery of forecasts from the Federal Reserve Board of San Francisco (FRBSF), Ray Fair, Nobel laureate Paul Romer (a prediction from 2017 [1]), and Jan Hatzius (Goldman-Sachs) [click to enlarge]:

Additionally, PCE inflation data came out yesterday — it was also in line with the (very boring) DIEM forecast:

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Footnotes:

[1] Also in the tweet with the unemployment prediction is a horribly wrong labor force participation forecast (the DIEM model of CIVPART was based on the dynamic equilibrium for the prime age participation rate forecast above). Click to enlarge: