by Salil Mehta, Statistical Ideas
Nature may reach the same result in many ways.
This was the quote repeatedly given by a Serbian physicist who, in the late 19th century, tried to make his mark in the U.S. Criss-crossing the eastern part of the country, he delivered these hard-to-believe lectures at various professional society meetings. Then 120 years later, another foreign-born businessman and revolutionary would himself try to make an indelible mark on the U.S., from the other side of the country. Producing electric vehicles bearing the name of that same physicist inspiration: Nikola Tesla.
The idea that Tesla had, about the random nature governing physical processes and the innovative applications that they could have on society, were built upon ideas created 75 years before his time. Near Paris, in the early 19th century, a bright young mathematician, Siméon Poisson, revolutionized branches of physics, wave theory in particular. For example, he corrected a second-order, partial differential equation in his field. An equation that just happens to have been developed by one of his two famous doctoral advisors, Pierre-Simon Laplace (the other was Joseph-Louis Lagrange). In the field of probability, Poisson also set an important mark in his narrow development of what has sprung to be one of the most popular discrete probability models we use today, to explain the frequency of random physical processes. And as we’ll see in this note, it has important applications far afield from physics, but only when used correctly.
Poisson was able to constrict a probability model in an unusual way, where we see that there is a mathematical relationship between the exponential process times between events and the entire frequency distribution of those events. Sure we can imagine that this would be an inverse relation, where there are a small number of events that can have a large time length between them. But to have the frequency distribution be mostly mound-shaped (with a median generally just less than the average) reveals the hidden beauty of the Poisson solution.
We have seen a number of popular Poisson applications for more than a century. From the accident mortality of Prussian soldiers, to yeast cells used for brewing Guinness beer, to call centre and postal volume traffic, to the discovery rate of an industrial R&D department, to the hourly delivery of babies at a modern hospital.
And given the “memoryless” property of the exponential distribution (i.e., the future probability of events are not impacted by the current amount of time that has elapsed), surely this has at least partial appeal for efficient-market, economists. These theoretical economists assume that all available valuation-related information is currently embedded into asset allocation decisions. And so it is generally easy to explain short-term changes that can only be due to unconditional, random chance. In order to use the exponential distribution though, we see that some modifications must be made, namely the limited range of cycle durations on both the high-end and particularly on the low-end.
It is important to distinguish this Poisson approach from that of the tail convolution mathematical theory, as the former assume the events are the same, even as we know from convolution examples that they are not. For example, we can scientifically count the number of snowstorms this winter, versus the typical number. But this only tells part of our story, since each snowstorm severity is inherently random in a number of respects. They are different in its intensity, total volume, damage to vehicles and property, and temperature.
In general society, outside of science, this concept has also been understood. The 20th century Argentinean fictional writer, Jorge Borges, keenly noted we must appreciate the times in which we live, since in each one they are all created different:
Time can’t be measured in days the way money is measured in pesos and centavos, because all pesos are equal, while every day, perhaps every hour, is different.
Now given this background, let’s here explore the exponential distribution as a model for the survival function of U.S. economic cycles going back 150 years, starting with just after the Civil War. The survival function is the inverse of the cumulative probability distribution, so for a given cycle length it is the portion of cycles whose duration is even lengthier than that given length.
Economic cycles alternates between peaks and troughs. And which came first? This is difficult to ascertain tracing economic cycles to its first origin, due to a lack of reliable data, and the absence of knowledge whether time started near an expansionary peak or a recessionary trough. Probability models therefore must provide an initial starting point, which we then judge for reasonableness in a later time-series analysis.
On the illustration above, the red peak-to-peak cycle would last from one economic peak, past the subsequent economic trough, through to the next economic peak. And the blue trough-to-trough cycle would last from one economic trough, past the subsequent economic peak, through until the next economic trough. Since these cycles weave together (e.g., one trough must reside between two peaks, and one peak must reside between two troughs), these random distributions are shown to be equal in aggregate.
Both cycles have the same average duration of 56 months (~4.5 years), and per the illustration they both have roughly 1/3 of the cycles enduring (or surviving) beyond this time. But here are a couple of wrinkles. First is that half of this average 56 months is 28 months, and there are virtually no economic cycles less than 28 months! This is a failure in the academic literature to model this survival function as if it richly covers the lower range of cycle times. And the literature multiplies this impact by applying this same impracticable survival pattern, further applied to many recent international economic cycles.
The gap on the lower-end implicitly denies very high frequency events (i.e., a large number of these brief cycles filled into a fixed time interval). Conversely, there is another critical point about the more rapid decay that we see on the on right side of the chart, implying that there are economic limits preventing economic cycles to be measured so large, that virtually none exist for a fixed time duration. We must keep this in mind as we explore relatively recent (e.g., past 25 years) economic history, and the small number of both lengthy and brief economic cycles. The proposed solution here is to maintain an exponential process, but with the possible embedded shift to the right. In other words, economic cycles that last at least two years, but then decay at a very rapid speed.
Now let’s see now how the implications of this modeling flaw contort our impression of economic cycles. Look below at this probability distribution illustration of economic cycle times and try identify the typical length from it. Our minds are trained to imagine a full theoretical chart (particularly asymptotically to the right) instead of the deeply truncated empirical one we see. As a result we see this bias in the press (be it from the Federal Reserve, or some outspoken, fear-mongering, Wall Street credit strategists) that cycles can now generally last at least 60 months (more than 5 years) due mostly to recent history. And many times we know in the zealous quest to avoid false positive research (not forewarning a downturn), they too often incur much false negative signals (ill-timed recession calls).
On the chart below, this amount of cycle time estimates come to the right of the beige bar above the green circle. But a hard look shows that the empirical and economic reality is we must plan for 5-10 months less, which is to the left of the bar above the green triangle.
This is not even taking into account the more complicated matter of median versus averages, but rather the proper model to take into account the narrow finite limits on both the left and right of the distribution as we have discussed. And, not withstanding the wide research distraction of decomposing the cycle into the random split between expansionary and contractionary periods (recall there is one trough jammed between two expansionary peaks), the break-down in the modeling range as we have seen in both of the above charts, there is still a desire to make the duration-dependent modeling leap with the Weibull distribution. We’ll discuss the Weibull continuous probability model a bit later.
Now let’s see the beige Poisson dotted-line in the illustration below (notice the right skew, with slight greater probability on the right tails versus the left tails). It accompanies the cycle survival models we have just seen. We once again note the large void at the high-end and low-end of length cycles. Also we use a fixed, 25-year time interval stretching past over the postbellum history, up to this current economic recovery.
Some cycles appear quick, but not enough given the empirical right shift of the cycle survival function. Equally dangerous is the recent 25-year window that includes one of the largest expansionary cycles in history (just after one of the briefest and most ignored cycles in history!), forcing out many other cycles in the lower-end frequency of 3. Note the rarity of this event, and equally, the lack of any frequencies of 2 or fewer cycles.
Then why do some suggest an even lengthier current economic cycle era? One that would take this record low frequency of 3 and create a possible scenarios where we are in an even lower one of 2? While we may see a small one-off effect in recent years from global liquidity pumped into the global economic system, the general cycle length expansion improbably breaks from the empirical data.
Also this empirical difference from the theoretical Poisson shows that there is a current empirical model approximation of economic cycle times that has medium dispersion in the interval times in-between cycles. This further dispels the academic literature where we stated forces a Weibull distribution to be used in the mortality function of predicting a conditional cycle duration.
This mortality function is also not as exotic as the empirical ones we develop in insurance literature. For example, here we use a custom mortality function to analyze the innovative battery warrants that in 2013 fueled the accounting earnings for Tesla vehicle company. But perhaps the mortality function we need for cycle durations should be customized as the Poisson probability function (e-1/λ*x1/λ/x!) and the corresponding meager, exponential survival function (e-λx) are not difficult to differentiate for modern quantitative professionals. In the probability functions just shown, the 11th Greek letter lambda (λ) represents the average cycle length and x represents the measured time (in months or 25 year increments, for the exponential or Poisson, respectively.)
We also see how the exponential process shows the empirical frequency distribution can’t help but to underestimate how many cycle events occur. Or this is better explained as it overstates the cycle times, within any fixed time period.
With additional analysis there is an extended NBER cycle history can augment (on an annual-basis) an additional 50% of cycles, going back at least an additional 50 more years to nearly Poisson’s time. This is versus the mostly postbellum (monthly-basis) data we have thus far analyzed above. The results from the augmented data differs from the empirical frequencies not in the average, but in how much less disperse they are at about the level of 5. This further erodes the ability to argue that we should seek lower frequency estimate according to the standards of an imaginary Poisson distribution that wouldn’t account for the exponential data shift.
This data augment of course converges our estimate that we have a dense set of brief economic cycles, consisting of about 5 to 6 cycles, per 25 years. And thus avoiding the idea of the typical economic cycle of 5 or more years in favor of about 4.5 years. Given the effort to precisely understand the duration of the economic cycle, this ½ year or so reduction (but not significantly more) in economic cycle estimate would be an important planning, lead time for many employees, and public and private sector asset allocators.
To summarize this note, we learn see this analysis that the exponential cycle time survival distribution, or the more binomial approximation from the frequency data, are clunky but powerful tools significantly bias economic cycle times to the upside. With modification they can work better. And from this analysis the typical economic peak, after the end of 2007, should be perhaps in the near future, with subsequent averages for future cycles returning to about 4.5 years, after this liquidity stimulus eventually runs its course.
Again, we shouldn’t lazily assume economic cycles are generally larger by structure, but rather that our current briefer economic cycle is running a little lengthier than normal. This is a duration we could expect about 25% of the time, or within 5% of what most are acknowledging. With each passing quarter we are getting closer to reaching this next economic peak, however.
The Poisson process, which even Nikola Tesla had appreciated, was a great starting point for this sort of economic analysis. But only after careful “modification”, an expression we use in probability and statistics vocabulary, to contrast with the empirical economic cycle data that we have historically experienced in the U.S., and globally.
We know from our statistical modeling that growth and risks can change per unit of time. But nonetheless, it is important to model them as correct stationary processes, from the current large, empirical experiences that we can model. We should adjust these cycle estimates to be less lengthy, or that they occur more frequently, then we generally see acknowledged while enjoying whatever benefits accrue from this economic recovery.