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The stationarity fallacy

I just met my new personal banker in LBBW, a German-owned Czech bank, and he looked like a rather competent investor and adviser relatively to many others.

Just to entertain some of the multi-millionaire TRF readers, the interesting discussion included an offer to have a regular checking account where I would save a whopping monthly $3 fee relatively to some other accounts where people are forced to pay this insanely huge fee. ;-)



Well, we have had some more general discussion about the LBBW products, the markets etc. Your humble correspondent is just a theorist. ;-) And the guy also reminded me that a deposit product has just matured. It's only interesting because it was previously discussed on this blog, I made a prediction, and we can now evaluate the prediction - which is an important feedback that most people who predict things usually avoid.

(That's why almost no one notices that most predictions about complex systems that "experts" do, especially the self-confident ones, are wrong.)

One year ago, I wrote about

betting on the stability of currency exchange rates
Let me just remind you about the rules of the game. The product is not terribly important - I just use it as a good example of a very incorrect mindset of most people that I have called "the stationarity fallacy". Before July 1st, you could make a deposit. The EUR/CZK exchange rate was 25.50 CZK per EUR.




You would get a zero - more precisely, 0.35% - interest if the EUR/CZK exchange rate weren't stable and 3.35% if it were "rather stable". What does it exactly mean "rather stable"? It means that between April 2011 and June 2011 (incl.), the exchange rate must stay between 24.70 and 26.50 CZK per EUR on each day - the interval was originally defined by percentage variations from the July 1st, 2010 rate which was about 25.70.

In my 2010 text on this issue, I calculated the probability that the condition is satisfied simply by looking how a similar contract would end up if it began on a random day between 1999 and 2010. The probability that the exchange rate remains "rather stable" in the next year was only 17%, so in average, the deposit only offered you the interest 0.35% + 0.17 * 3.35% = 0.85% which is safely below the interest rate of the saving accounts, around 1.75%.

(A 15% withholding tax is subtracted from all the interests.)

Needless to say, the result - which is known today - confirmed my "very likely" prediction. Despite the apparent recent "stability" of the crown/euro exchange rate, the stability condition wasn't satisfied. The number of crowns per euro had to be above 24.70 between April and June - but in fact, it was always lower than 24.70 and the minimum was as low as 24.05 or so.

No doubts about it, the condition was safely violated and the people who made the deposit got their 0.35%. I was right.

This is obviously not an important financial question but the belief of some investors that this product was a good offer is a textbook example of the "stationarity fallacy". I decided that this is the most natural term for what seems to be a flawed mindset of a vast majority of the people - climate alarmists but even many climate skeptics; left-wingers but even many right-wingers; and pretty much all other people in almost all groups. The TRF readers could be an exception as a group but you will surely find many examples in which TRF readers suffer from this fallacy, too. I had a feeling that someone else had to have discussed the very same point and someone had to give it the same name.

So I made a Google search and indeed, I found pages about the "stationarity fallacy"! Many of the pages about the fallacy were about water management. One of them, an article in the Newsweek, conveniently defined the stationarity fallacy as
the belief that natural systems fluctuate within a narrow, predictable range, even over long periods.
In the case of the water management, they say that the bad consequence is that decisions were effectively left to short-sighted people who can't really react to long-term changes.

But the stationarity fallacy as I see it is not only about the overlooking of inevitable long-term variations. It is about the overlooking of short-term variations, too.

The stationarity fallacy influences most people's expectations about the prices of arbitrary things; exchange rates; supply and demand of arbitrary products; global mean temperature and any local temperature; and pretty much all other quantities describing the world that are able to change but that may be thought of as continuous functions of time.

The EUR/CZK stability bet was a financial example. But the climate examples may be equally prominent.

Climate alarmists who "fight against climate change" suffer from the stationarity fallacy of the most severe form. They believe that the Earth is naturally obliged to stay constant. Well, the global mean temperature is constant, they think. In this form, the belief obviously contradicts even the most elementary observations of year-on-year temperature changes that have always been observed to be nonzero.

So these alarmists improve their belief just a little bit. The interannual temperature changes may be nonzero but those changes must surely average out - the variations of the temperatures are naturally some kind of a "white noise" - so if you study the average temperature over a decade, this number wouldn't be naturally changing from one decade to another.

Except that it is definitely changing and it has always been changing. There is absolutely no physical reason why the global mean temperature averaged over a decade should be equal to the global mean temperature averaged over another decade. The temperature is fluctuating because of weather and many chaotic, unpredictable effects. But even if you imagined that those average out, the temperature is also fluctuating because of changes of the cloud cover; the average albedo of the Earth's surface (including the changing vegetation), and so on, and so on.

All these things may be drifting and they have no reason to return where they started, not even after 10 years. In fact, just the opposite is true: the longer period of time you consider, the larger change you should expect. If you divide the overall change by the time interval, you get a slower trend for longer periods of time. But the overall change is of course larger if the time interval is larger.

The Brownian motion - random walk - is the most canonical example of such changes. A particle is moving back and forth randomly. After time \( t \), the average squared distance from the origin goes like \( t \), so the typical distance how far the particle gets goes like
\[ \Delta X \sim \sqrt{t} .\]
It's the square root of time. So the longer time \( t \) you consider, the larger deviation of \( X \) you will observe. On the other hand, if you try to evaluate the average velocity over the time interval, you will get
\[ \left|\langle \vec v \rangle\right | \sim \frac{\sqrt{t}}{t} = \frac{1}{\sqrt{t}} \]
which goes to zero if \( t \to \infty \). But you must distinguish the slope of the idealized line from the overall change. The overall change becomes larger if the time separation gets larger, too.

Aside from climate alarmists, all other prophets of catastrophes and dooms (Alexander Ač is a person who unifies all different sorts of doom prophets who have evolved so far) suffer from the stationarity fallacy because they talk about the "future" as if it were a single moment. In reality, the future consists of infinitely many different moments - much like the past (in fact, the future is longer because the past began 13.7 billion years ago while future of our de Sitter space will continue indefinitely) - and different things will occur at different moments in the future. I will explain this point on the example of an exchange rate in 2013 and 2014 momentarily.

The same error appears when people talk about the past, too. The very term "pre-industrial temperature" or "pre-industrial anything" is an incarnation of the fallacious reasoning that envisions the past as a single moment.

Contradictions in stationarity assumptions

Many people believe all kinds of things about stationarity of the economy. Many of those beliefs are totally incompatible with each other. For example, many people believe that the United States - or any other country - will have about the same federal expenses. And at the same moment, they also believe that the self-confidence of the sustainability of the U.S. debt will stay about the same, too.

These two things directly contradict one another.

Unless the federal expenses will be radically reduced in coming years, the budget deficit will stay comparable to 10% of the GDP, so the debt/GDP ratio will be increasing roughly by 10% per year, too. In five years, it will inevitably reach the Greek proportions. Note that the GDP growth as well as inflation are very small relatively to those 10% so I can neglect them - and in fact, these two "helpful" factors are almost entirely compensated by the increasing interests that the U.S. will have to pay.

Clearly, if the "velocity" \( dX / dt \) is nonzero, you can't expect both a stationary \( dX / dt \) as well as a stationary \( X \). The nonzero value of the velocity is what it means that \( X \) itself isn't stationary.

Not distinguishing different moments in the future

The stationarity fallacy is linked to the people's tendency to neglect the future dynamics of prices, temperatures, and everything else.

For example, it is nearly impossible predict the detailed moments when prices or temperatures will increase or drop in the future. So our predictions for e.g. the euro-dollar currency exchange rate \( R \), to pick a totally random example (the temperatures are a seemingly different example but the logic is identical) will be almost identical for January 2013 and January 2014. Schematically,
\[ \langle R_{2013} \rangle = \langle R_{2014} \rangle. \]
That's fine. This is what is typically true for our predictions. The mean value of a predicted quantity - sometimes it has the character of a coordinate \( X \) but sometimes it has the character of its velocity \( dX / dt \) - isn't too different for some points in the future.

The equation above may be rewritten as
\[ \langle R_{2013} - R_{2014} \rangle = 0. \]
However, most people misinterpret the equation above as
\[ \langle \left | R_{2013} - R_{2014} \right | \rangle = 0. \]
Note that I just added an innocently looking absolute value inside the expectation value. However, the second equation doesn't follow from the first one. In fact, you can be very sure that
\[ \langle \left | R_{2013} - R_{2014} \right | \rangle > 0. \]
Between 2013 and 2014, it's guaranteed that the euro-dollar will be changing in one direction or another, much like it was always changing in the past. The mean values of the exchange rate may be equal for 2013 and 2014. But the mean value of the change of the exchange rate isn't the same thing as the difference of the mean values of the exchange rates themselves. And the mean value of the change - with a stripped sign (as a physicist, I would normally square the change to make it natural, but the absolute value is more comprehensible to the laymen, so please forgive me the unnatural functions here) - is definitely nonzero, somewhat calculable, and independent from the mean values themselves.

Quite generally, people like to think that quantities that oscillate with time oscillate within a predetermined, narrow enough interval. Almost equivalently, they are assuming that the quantities behave as a kind of the white noise - the deviation of the quantity from the "holy central value" at one moment is almost independent from the variation of the quantity from the "holy central value" at another moment.

This "theory" how to describe the varying quantities looks good enough to a superficial observer. The white noise makes it fluctuating so everything is fine, they think.

However, nothing is fine. This "theory" is easily falsified as well. In fact, there is no "holy central value" around which the quantities fluctuate. One may show theoretically as well as experimentally that there isn't anything such as a "holy central value". The Brownian motion is a good example once again. Most of the \( N \) moves in a random walk get averaged out but it is statistically inevitable that some of them - approximately \( \sqrt{N} \) of them - won't. So in the future, the Brownian particle will be oscillating around a different point than it did in the past. There's nothing that would guarantee that all the things that influenced the particle's position will return to their previous state.

Now, for the global mean temperature, there exist some regulating mechanisms that prevent the Earth from deviating by more than 10 °C or so from some normal value - like 15 °C. It's plausible that the global mean temperature hasn't left the 5-25 °C interval for billions of years. Moreover, most of the changes of the temperature may be "explained" or "attributed" to some causes.

However, it's clearly true that we can't attribute everything and there's still a lot of room for variations that are essentially random and that may accumulate sizable deviations from a would-be "holy central value" after a sufficiently long time period. I think it's obvious that this "irremovable" error margin is at least comparable to 0.5 °C if not much larger than that. We only know that it's much smaller than 6 °C because the temperature in the glaciation cycles agrees with the Milankovitch theory well. If you subtract the predicted and observed temperatures in the Milankovitch setup, you will obtain a much less fluctuating difference curve which will tell you that the "irremovable noise" may accumulate at most up to one or two or three degrees Celsius.

But this result is still fully compatible with the hypothesis that the overall change of the global mean temperature between 1900 and 2011 may be attributed to "random noise".

Predictions are hard, especially the predictions of the future. But the point I want to make is that predicting a value of \( \langle X \rangle \) or \( \langle dX/dt \rangle \) in the future is not the only kind of a prediction we can make. We may also predict other things such as
\[ \langle [X(year+1)-X(year)]^2 \rangle \]
and these quantities, telling us something about the variability, are independent of \( \langle X \rangle \) or \( \langle dX/dt \rangle \). In particular, the expectation value of the squared difference above is guaranteed to be strictly different from zero (and positive). Unless we have a really convincing theory, the most sensible way to have an idea about the magnitude is to compute the average of the value from the past data.

The character of the variability of \( X \) in the past may change as well but the simplest hypothesis is that it won't change too much - and this theory is certainly much better than the assumption that \( X \) will no longer change in the future! It will definitely change even though we can't predict whether it will go up, down, and when.

So please, try to avoid the stationarity fallacy. The fact that we can't predict the precise shape of the "noise" in the future doesn't mean that there will be no noise. And to admit "white noise" that allows to return a quantity to its initial state after a long enough time is not enough to explain the observed variability of a quantity. In fact, it's very obvious and inevitable that most quantities contain kinds of noise that is not white and whose changes tend to accumulate over time - much like in the case of the Brownian motion (although the optimum exponents to describe the color of the noise may differ from the Brownian motion).

Change is an inevitable feature of our world (including climate change) and it will remain so as long as there exists something called the "time". People who fight against the very existence of a change or against time are lunatics. But even people who are not lunatics often make wrong assumptions that certain things won't change. After all, even Einstein used to believe in the Einstein static Universe, to mention a prominent bright guy whose wrong theories were based on a similar misconception.



Off-topic: jazz

I spend some hours on the Bohemia Jazz Fest that came to Pilsen - and saw Inger Marie Gundersen and her group (from Norway) and Danilo Perez and his trio (from Panama).

It was a pleasant surprise that this non-mainstream music attracted a thousand or a few thousand people to the main square of Pilsen. Gundersen's music is kind of slow and romantic - I wouldn't even classify it as jazz if you asked me. ;-) Danilo Perez is real jazz and you can see it's the kind of real and fancy jazz music that I don't always understand.

They also showed a movie about the beauties of Norway - plus the EUR 100 million "Norwegian funds projects" in Czechia that Norway has paid for. Quite amazing given the fact that Norway isn't even an EU member. This only reinforces my sentiment that being or not being an EU member makes no impact on the emotional (and even financial) proximity to another European nation.

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reader netudiant said...

An excellent post.
This writeup should be part of the basic science curriculum in grade school. It would gradually immunize our people against a lot of fallacies.


reader Kamil said...

Panta rhei. Wise mans thought about it from ancient times.
Heraclitus: "Ever-newer waters flow on those who step into the same rivers ."


reader Stan said...

One example would be a reference to the "pre-industrial age temperature"