Category Archives: explosive processes

An Update on Bitcoin

Fairly hum-drum days of articles on testing for unit roots in time series led to discovery of an extraordinary new forecasting approach – using the future to predict the present.

Since virtually the only empirical application of the new technique is predicting bubbles in Bitcoin values, I include some of the recent news about Bitcoins at the end of the post.

Noncausal Autoregressive Models

I think you have to describe the forecasting approach recently considered by Lanne and Saikkonen, as well as Hencic, Gouriéroux and others, as “exciting,” even “sexy” in a Saturday Night Live sort of way.

Here is a brief description from a 2015 article in the Econometrics of Risk called Noncausal Autoregressive Model in Application to Bitcoin/USD Exchange Rates


I’ve always been a little behind the curve on lag operators, but basically Ψ(L-1) is a function of the standard lagged operators, while Φ(L) is a second function of offsets to future time periods.

To give an example, consider,

yt = k1yt-1+s1yt+1 + et

where subscripts t indicate time period.

In other words, the current value of the variable y is related to its immediately past value, and also to its future value, with an error term e being included.

This is what I mean by the future being used to predict the present.

Ordinarily in forecasting, one would consider such models rather fruitless. After all, you are trying to forecast y for period t+1, so how can you include this variable in the drivers for the forecasting setup?

But the surprising thing is that it is possible to estimate a relationship like this on historic data, and then take the estimated parameters and develop simulations which lead to predictions at the event horizon, of, say, the next period’s value of y.

This is explained in the paragraph following the one cited above –


In other words, because et in equation (1) can have infinite variance, it is definitely not normally distributed, or distributed according to a Gaussian probability distribution.

This is fascinating, since many financial time series are associated with nonGaussian error generating processes – distributions with fat tails that often are platykurtotic.

I recommend the Hencic and Gouriéroux article as a good read, as well as interesting analytics.

The authors proposed that a stationary time series is overlaid by explosive speculative periods, and that something can be abstracted in common from the structure of these speculative excesses.

Mt. Gox, of course, mentioned in this article, was raided in 2013 by Japanese authorities, after losses of more than $465 million from Bitcoin holders.

Now, two years later, the financial industry is showing increasing interest in the underlying Bitcoin technology and Bitcoin prices are on the rise once again.


Anyway, the bottom line is that I really, really like a forecast methodology based on recognition that data come from nonGaussian processes, and am intrigued by the fact that the ability to forecast with noncausal AR models depends on the error process being nonGaussian.

Wither the Price of Oil?

Crude oil futures continue their descent, as the chart below from January 2, 2015 shows.


What is going to happen here?

Discussions organize around several issues, pretty well nailed recently by IMF bloggers (including Chief Economist Oliver Blanchard) in Seven Questions About The Recent Oil Price Slump.

  2. How persistent is this supply shift likely to be?
  6. What are the financial implications?

The first point to note is the drop in oil prices involves both supply and demand – and is not just the result of increased pumping by Saudi Arabia.

The IMF discussion includes this interesting comparison between oil and other commodity price indices.


So over 2014, there have been drops in other commodity prices – probably due to weakened global demand – but not nearly much as oil.

Overall, the IMF counts lower oil prices as a net positive to the global economy, resulting in a gain for world GDP between 0.3 and 0.7 percent in 2015, compared to a scenario without the drop in oil prices.

There are big losers, of course. These include oil exporters with higher production costs, such as Russia, Iran, and Venezuela.

To take some examples, energy accounts for 25 percent of Russia’s GDP, 70 percent of its exports, and 50 percent of federal revenues. In the Middle East, the share of oil in federal government revenue is 22.5 percent of GDP and 63.6 percent of exports for the Gulf Cooperation Council countries. In Africa, oil exports accounts for 40-50 percent of GDP for Gabon, Angola and the Republic of Congo, and 80 percent of GDP for Equatorial Guinea. Oil also accounts for 75 percent of government revenues in Angola, Republic of Congo and Equatorial Guinea. In Latin America, oil contributes respectively about 30 percent and 46.6 percent to public sector revenues, and about 55 percent and 94 percent of exports for Ecuador and Venezuela.[8] This shows the dimension of the challenge facing these countries.

How long the Saudi’s can hold the line, maintaining higher production? Is it true, for example, that Saudi Arabia has a $750bn war chest of foreign currency reserves that will be burned through quickly by propping up the shortfall in export revenues? There are speculations that the health of King Abdullah, a strong supporter of the current Saudi Oil Minister, could come into play in coming months.

Interestingly, low oil prices maintained long enough could be self-correcting. This is probably the bet the Saudi’s are making – that their policy can eventually trigger faster growth and enable them to maintain or increase their market share.

As I’ve said before, I think it’s a game changer. The trick is to figure out the linkages and connections, backwards and forwards along the supply chains.

Image of King Abdullah from Telegraph.

Hyperinflation and Asset Bubbles

According to Mizuno et al, the worst inflation in recent history occurred in Hungary after World War II. The exchange rate for the Hungarian Pengo to the US dollar rose from 100 in July 1945 to 6 x 1024 Pengos per dollar by July 1946.

Hyperinflations are triggered by inflationary expectations.  Past increases in prices influence expectations about future prices. These expectations trigger market behavior which accelerate price increases in the current period, in a positive feedback loop. Bounds on inflationary expectations are loosened by legitimacy issues relating to the state or social organization.

Hyperinflation can become important for applied forecasting in view of the possibility of smaller countries withdrawing from the Euro.

However, that is not the primary reason I want to address this topic at this point in time.

Rather, episodes of hyperinflation share broad and interesting similarities to the movement of prices in asset bubbles – like the dotcom bubble of the late 1990’s, the Hong Kong Hang Seng Stock Market Index from 1970 to 2000, and housing price bubbles in the US, Spain, Ireland, and elsewhere more recently.

Hyperinflations exhibit faster than exponential growth in prices to some point, at which time the regime shifts. This is analogous to the faster than exponential growth of asset prices in asset bubbles, and has a similar basis. Thus, in an asset bubble, the growth of prices becomes the focus of action. Noise or momentum traders become active, buying speculatively, often financing with Ponzi-like schemes. In a hyperinflation, inflation and its acceleration gets written into to the pricing equation. People stockpile and place advance orders  and, on the supply side, markup prices assuming rising factor costs.

The following is a logarithmic chart of inflation indexes in four South and Central American countries from 1970 to 1996 based on archives of the IMF World Economic Outlook (click to enlarge).

The straight line indicates an exponential growth of prices of 20 percent per year, underlining the faster than exponential growth in the other curves.


After an initial period, each of these hyperinflation curves exhibit similar mathematical properties. Mizuno et al fit negative double exponentials of the following form to the price data.


Sornette, Takayasu, and Zhouargue that the double exponential is “nothing but a discrete-time approximation of a general power law growth endowed with a finite-time singularity at some critical time tc.”

This enables the authors to develop an analysis which not only fits the ramping prices in each country, but also to predicts the end of the hyperinflation with varying success.

The rationale for this is simply that unleashing inflationary expectations, beyond a certain point, follows a common mathematical theme, and ends at a predictable point.

It is true that simple transformations render these hyperinflation curves very similar, as shown in the following chart.


Here I scale the logs of the cumulative price growth for Bolivia, Nicaragua, and Peru, adjusting them to the same time period (22 years). Clearly, the hyperinflation becomes more predictable after several years, and the takeoff rate to collapse seems to be approximately the same.

The same type of simple transformations would appear to regularize the market bubbles in the Macrotrends chart, although I have not yet collected all the data.

In reading the literature on asset bubbles, there is a split between so-called rational bubbles, and asset bubbles triggered, in some measure, by “bounded rationality” or what economists are prone to call “irrationality.”

Examples of how “irrational” agents might proceed to fuel an asset bubble are given in a selective review of the asset bubble literature developed recently by Anna Scherbina from which I take several extracts below.

For example, there is “feedback trading” involving traders who react solely to past price movements (momentum traders?). Scherbina writes,

In response to positive news, an asset experiences a high initial return. This is noticed by a group of feedback traders who assume that the high return will continue and, therefore, buy the asset, pushing prices above fundamentals. The further price increase attracts additional feedback traders, who also buy the asset and push prices even higher, thereby attracting subsequent feedback traders, and so on. The price will keep rising as long as more capital is being invested. Once the rate of new capital inflow slows down, so does the rate of price growth; at this point, capital might start flowing out, causing the bubble to deflate.

Other mechanisms are biased self-attribution and the representativeness heuristic. In biased self-attribution,

..people to take into account signals that confirm their beliefs and dismiss as noise signals that contradict their beliefs…. Investors form their initial beliefs by receiving a noisy private signal about the value of a security.. for example, by researching the security. Subsequently, investors receive a noisy public signal…..[can be]  assumed to be almost pure noise and therefore should be ignored. However, since investors suffer from biased self-attribution, they grow overconfident in their belief after the public signal confirms their private information and further revise their valuation in the direction of their private signal. When the public signal contradicts the investors’ private information, it is appropriately ignored and the price remains unchanged. Therefore, public signals, in expectation, lead to price movements in the same direction as the initial price response to the private signal. These subsequent price moves are not justified by fundamentals and represent a bubble. The bubble starts to deflate after the accumulated public signals force investors to eventually grow less confident in their private signal.

Scherbina describes the representativeness heuristic as follows.

 The fourth model combines two behavioral phenomena, the representativeness heuristic and the conservatism bias. Both phenomena were previously documented in psychology and represent deviations from optimal Bayesian information processing. The representativeness heuristic leads investors to put too much weight on attention-grabbing (“strong”) news, which causes overreaction. In contrast, conservatism bias captures investors’ tendency to be too slow to revise their models, such that they underweight relevant but non-attention-grabbing (routine) evidence, which causes underreaction… In this setting, a positive bubble will arise purely by chance, for example, if a series of unexpected good outcomes have occurred, causing investors to over-extrapolate from the past trend. Investors make a mistake by ignoring the low unconditional probability that any company can grow or shrink for long periods of time. The mispricing will persist until an accumulation of signals forces investors to switch from the trending to the mean-reverting model of earnings.

Interesting, several of these “irrationalities” can generate negative, as well as positive bubbles.

Finally, Scherbina makes an important admission, namely that

 The behavioral view of bubbles finds support in experimental studies. These studies set up artificial markets with finitely-lived assets and observe that price bubbles arise frequently. The presence of bubbles is often attributed to the lack of common knowledge of rationality among traders. Traders expect bubbles to arise because they believe that other traders may be irrational. Consequently, optimistic media stories and analyst reports may help create bubbles not because investors believe these views but because the optimistic stories may indicate the existence of other investors who do, destroying the common knowledge of rationality.

I dwell on these characterizations because I think it is important to put to rest the nonsensical “perfect information, perfect foresight, infinite time horizon discounting” models which litter this literature. Behavioral economics is a fresh breeze, for sure, in this context. And behavioral economics  seems to me linked with the more muscular systems dynamics and complexity theory approaches to bubbles, epitomized by the work of Sornette and his coauthors.

Let me then leave you with the fundamental equation for the price dynamics of an asset bubble.