Abstract
The history of studies on bubbles and crashes is mired in misleading and flawed theory elegantly packaged in mathematical economics approaches that plainly don’t work: That’s because trading is always conducted in an open nonlinear system that noticeably destabilize as extreme market events unfold. Moreover, the underlying and still predominant efficient-market and capital asset pricing models were never designed nor intended to be used as platforms for the study of such events. Flaws in the rationality approach thus run so deep as to render the entire framework—as here surveyed for reference and historical perspective purposes—bereft of any ongoing analytical benefits.
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Notes
- 1.
Assumptions are that agents are risk -neutral, have rational expectations, and require a constant (real) rate of return on the asset, EtRt+i = k. The Euler equation is then Pt = δ(EtPt + 1 + EtDt + 1), where P is the price, D is the dividends, E is the mathematical expectations operator, and δ = 1/(1 + k) is the discounting factor. Cuthbertson and Nitzsche (2004, p. 398) explain: “The price you are prepared to pay today for a stock depends on the price you think you can obtain at some point in the future. But the latter depends on the expected price even further in the future…[T]he Euler equation does not rule out the possibility that the price may contain an explosive bubble.” An Euler equation is a general theorem of the calculus that appears as x1f1 +x2f2 = kf(x1,x2).
After repeated forward substitutions and the assumption that what is known as a transversality condition holds (i.e., lim(δnEtDt + n = 0 as n → ∞), the formulation of a “rational bubble” that satisfies the Euler equation is expressed as:
\( {P}_t=\sum \limits_{i=1}^{\infty }{\delta}^i{E}_t{D}_{t+1}+{B}_t={P}_t^f+{B}_t \), where Bt is the “rational bubble,” and “the actual market price Pt deviates from its fundamental value, \( {P}_t^f \) by the amount of the rational bubble Bt” (p. 399). As Bailey (2005, p. 241) writes, this suggests that “asset prices need not equal the NPV of future payoffs but can become any one of an infinite number of values according to the size of the bubble. The ‘bubble’ term captures all the speculative and self-fulfilling aspects of potentially wild asset price changes.”
According to Cuthbertson and Nitzsche (2004, p. 401), individuals will pay prices above the fundamental price “as long as the bubble element yields them the required rate of return next period and is expected to persist…[I]n the real world, rational bubbles can really only exist if each individual’s horizon is shorter than the time period when the bubble is expected to burst…[O]ne would pay a price above the fundamental value because one believes that someone else will pay an even greater price in the future…investors are myopic.”
From this point onward, however, assumptions and complications with the various rational bubble models begin to compound in terms of both model-specification complexities and problematic econometric testing assumptions. Cuthbertson and Nitzsche (2004, p. 408), for example, include in their list of econometric challenges:
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Potentially nonstationary series using finite data sets
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Behavior of asymptotic test statistics in the presence of explosive regressors
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Obtaining precise estimates of nonlinear parameters
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Corrections for heteroscedasticity and moving average errors
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Difficulties in specification of correct equilibrium modes of expected returns
Poundstone (2010, p. 9) questions entirely the notion that prices of anything can be firmly established over time. “Relative valuations are stable and coherent, while actual dollar amounts can be wildly arbitrary.” And Prechter writes in the April 2011 Elliott Wave Theorist that “the stock market never attaches to any benchmark of value, be it dividends, earnings, book values or the bond/stock yield spread. It is ceaselessly dynamic.” By illustration, over the past century, the price of $1 worth of S&P annual earnings based on trailing 12-month P/E ratios has ranged from around six times to more than 50 times (excluding rare years of losses when the ratio is infinite). Similarly, the price of $1 worth of annual DJIA dividends has had a range of 14 times and for book value around 19 times. In the 1940s and the early 1980s, $1 worth of S&P 400 book value was priced around $1.50 and, in the early 2000s, around $9.00. Poundstone (2010, p. 263) makes the same point.
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- 2.
To take account of the cost of money and uncertainty of payoff, the general mathematical representation of the LOOP is simply stated as pt+1 = f (mt, pt), where mt are discount factors and parameters and pt are prices (with pt+1 being the price one period ahead).
- 3.
Interest in the rational expectations school of thought continued to grow and peaked in the 1980s, although it was not until 1995 that Robert Lucas, Jr., a leader in the field, was awarded the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel. But another Nobel Laureate, Gary Becker, says in O’Grady (2009), “There is a lot of debate in economics about whether we can understand bubbles within a rational framework.” And Garrison (2001) provides a detailed review of why RE doesn’t work from the Austrian school’s perspective. As Stiglitz (1990) explains with regard to the large body of work on rational expectations, the underlying issue is the movement of asset prices over time :
If the asset price increased more slowly than the discount factor, eventually the terminal price became of negligible importance as viewed from today. Under such circumstances, the value of the asset had to be just equal to the discounted value of the stream of returns it generated, and no bubbles could exist. But as long as no one in the economy has an infinite planning horizon, there was nothing to ensure that this condition on prices (called the transversality condition) would be satisfied…If, for some asset, the rate of price increase equals (or exceeds) the rate of interest, then the share of the value of all assets accounted for by this asset would grow without bound, a condition inconsistent with long run equilibrium.
- 4.
Evans (2003, p. 15) mentions this, and in Chap. 2, drawing on the comprehensive Raines and Leathers (2000), he surveys and compares competing theories. Keynes , for instance, is shown to be unsurprised by volatile departures from “intrinsic” values, and Toporowski (1999) views the market in nonequilibrium terms antithetical to the EMH approaches.
- 5.
The Turnovsky (1996, pp. 59-60) representation of the rational expectations hypothesis is not much different from those used by others. Predictions generated by this model are conditional on information available at the time of prediction, with purely random price fluctuations around the forecast shown by the error term, et, which has zero mean. Turnovsky 1996, p. 137) writes that “most rational expectations models have been associated with saddlepoint behavior. That is, the dynamic system in which they are embedded involves both unstable and stable roots.”
- 6.
The quotation is from Azariadis (1981). Theoretical embellishments include overlapping generation (OLG) models that take account of presumed intertemporal optimizing changes of consumer utility functions, but that also include what are known as “sunspot ” equilibria, the meaning of which, according to Azariadis and Guesnerie (1986),
is still open to interpretation. One may view “ sunspots ” as a convenient label for a host of psychological factors (“animal spirits,” fears, Bayesian learning theories, etc.) that are unrelated to the preferences, endowment or production set of any individual, and yet come to influence the forecasts and actions of economic decision-makers…these factors have some pertinence for the explanation of…the Dutch tulip mania…and the Great Depression….Whatever it may denote, the concept of sunspot equilibrium seems to be of central importance for a thorough understanding of rational expectations as an equilibrium construct.
This says that in some circumstances economic fundamentals alone will not be sufficient to determine equilibrium allocations and that psychological factors (i.e., extraneous variables or “sunspots”) eventually do matter. How investors interpret, feel about, and react to fundamentals may thus be important. Thus, following Tirole (1982), Blanchard and Fischer (1989, pp. 218–223) explain that “bubbles cannot arise when there is a finite number of individuals who have infinite horizons…if bubbles can exist in a general equilibrium, it must be because new players come into the game over time.” Duffy and Fisher (2005) advise that the “difficulty lies in identifying sunspot variables and isolating their effects from those of the fundamentals.” Their lab experiment found direct evidence of sunspot equilibria in a closed-book call market in which price determination is centralized. Such equilibria are sensitive to the flow of information. As an aside, the phrase “self -fulfilling prophecy” was coined by sociologist Robert K. Merton (and father of financial economist Robert C. Merton) in the Antioch Review, 1948. See Dunbar (2000, p. 19). In the article by Cass and Shell (1983), a sunspot is something that is intrinsically irrelevant but nonetheless influences prices. Benhabib et al. (2015) formalize a Keynesian rational expectations model showing that aggregate demand driven by sentiments (i.e., “animal spirits”) can generate output fluctuations and boom-bust cycles.
- 7.
Kurz (1994) proposes that “the theory of expectations be reformulated under the assumption that agents do not know the structural relations (such as equilibrium prices) of the economy.”
- 8.
Different implications of rationality assumptions appear in Kindleberger and Aliber (2011, p. 40). Maddala (2001, p. 420) shows that Muth required the prediction error, \( {(e)}_t={y}_t-{y}_t^{\ast } \), to be unbiased, with E(ε)t = 0. He also required that the prediction error be uncorrelated with the entire information set that is available to the forecaster at the time the prediction is made. It would seem at the outset that the use of such rational expectations models for the purpose of studying bubbles is already doomed to failure by this very assumption, which requires that at the inception of every new trade, automaton-like investors immediately erase from their memories any and all past information and emotions relating to previous trades. Evans (1991) explains the econometric weaknesses of REH studies.
Several additional objections to the REH approach are raised because, for this approach to work, rather severe restrictive assumptions are required. For example, commonly shared information ought to be held unanimously and all relationships are assumed to be linear. But in a world in which information appears to be far from being perfectly distributed and relationships are probably far from linear, these model restrictions alone appear ab initio to limit practical applications. Lovell (1986) is also a skeptic, finding in a study of business firms that “the variance of anticipations is larger than the variance of the realizations, which is inconsistent with the rational expectations hypothesis…if the cumulative evidence is to be believed, we are compelled to conclude that expectations are a rich and varied phenomenon that is not adequately captured by the concept of rational expectations.”
The Cagan (1956) model of hyperinflation is an early example suggesting that adaptive inflation expectations can be “rational,” but only under special assumptions. The Cagan REH approach allows, for example, according to Blanchard and Fischer (1989, pp. 218–23), “that the announcement of a future increase in the money stock itself increases the price level today. Real money balances decrease, and the price level slowly increases to its new higher level over time. Inflation therefore takes place in advance of the increase in the money stock.”
- 9.
- 10.
The question and the arguments harken back to Friedman’s Essays in Positive Economics (1953) in which Keynes is quoted as saying that positive economics deals with what is not what ought to be. See also Brock (1991).
Again, Shiller (2002b) places rational expectations in historical perspective:
The efficient markets theory reached its height of dominance in academic circles around the 1970s. At that time, the rational expectations revolution in economic theory was in its first blush of enthusiasm…Prominent finance models of the 1970s related speculative asset prices to economic fundamentals , using rational expectations to tie together finance and the entire economy in one elegant theory…
…efficient markets theory may lead to drastically incorrect interpretations of events such as major stock market bubbles. (p. 31)
- 11.
- 12.
According to Greenwald et al. (2001, pp. 148–9), a further implication with respect to the EMH/CAPM is that “value investors reject both parts of the theory. They think stock selection does matter, and they do not accept the definition of risk as simply relative volatility…volatility is not the only and perhaps not even the best measure of risk.” Ederington and Guan (2005), in fact, indicate that adjusted mean absolute deviation might be a better measure. Also, in a BusinessWeek interview (April 9, 2007) with Peter Bernstein , Professor Elroy Dimson is quoted as saying “Risk means more things can happen than will happen.” This quotation also appears in Bernstein (2005b, p. 47) and is from Dimson’s earlier work. See also Zhang et al. (2016). The DCF methodology is also deceptive because both the cash flow estimates and discount-rate assumptions are highly uncertain (i.e., stochastic).
Taleb (2007, p. 161) emphasizes that when making forecasts, “variability matters.” Bogle (2008) highlights the work of economist Frank H. Knight, who made the first distinction between the terms “risk”—a measureable quantity in which probabilities and distributions are known—and “uncertainty ,” which is immeasurable and thus not subject to probabilities. And commodity trader, Richard Dennis , is quoted in Covel (2007, p. 23) as saying, “One man’s volatility is another man’s profit.”
- 13.
Hansen and Renault (2009) describe pricing kernels or stochastic discount factors as being used to represent valuation operators. Such operators are built as “mappings that assign prices that trade in competitive markets to payoffs on portfolios of assets.” This is all related to the law of one price.
- 14.
See also Miao (2014).
- 15.
See Henriques (2008).
- 16.
Among the many studies of short-selling bans and regulations, see Saffi and Sigurdsson (2011), Beber and Pagano (2013), and Jain et al. (2013). Napoleon labeled short-sellers “enemies of the state,” England banned the practice from 1733 until the middle of the nineteenth century, and short-selling was illegal in New York State in the early 1800s. See Surowiecki (2004, p. 225).
- 17.
Tirole (1982) adds, “the price has to grow faster during the duration of the bubble…for the asset holders to be compensated for the probability of a crash.” McQueen and Thorley (1994) similarly say, “The probability of a high return exactly compensates investors for the probability of a crash; the model shows the rationality of staying in the market despite the overvaluation.”
- 18.
Using traditional methods—in which the number of positive abnormal returns within a period suspected of containing bubbles is counted—Hardouvelis (1988) further notes that “[I]nstead of testing directly the null hypothesis of speculative bubbles, they test the null hypothesis that stock prices are priced rationally, and then interpret a possible rejection of this null hypothesis as evidence that speculative bubbles may be present. Since these methods are indirect, they have very little power to detect speculative bubbles.” See also Scheinkman and Xiong (2003).
- 19.
This appears in Gay et al. (1994).
- 20.
Hillier (1997, pp. 3–5) explains that asymmetric information problems fall into three categories: selection of projects, hidden actions, and costly state verification.
- 21.
They go on to show that “[T]he simplest is that of a deterministic bubble, ct = c0θ−t.” In this case the higher price is justified by the higher capital gain and the deviations grow exponentially.
- 22.
In comparing the Blanchard and Watson (1982) to the Abreu and Brunnermeier (2003) approach, Chamley (2004, p. 383) explains that in “Blanchard , the growth rate of the bubble price must be higher than the market rate to compensate for the probability of the crash, as all agents are informed.” But in Abreu and Brunnermeier , “the growth rate of the price bubble does not need to be as high, because only a fraction of the agents are informed.”
- 23.
As Jeremy Grantham said in GMO’s January 2005 Quarterly Letter (p. 5), “one of the paradoxes…is that reducing or avoiding real risk in portfolios can seriously increase career and business risk, which rises with any deviation from standard behavior.” Hong et al. (2000) cited in Cassidy (2009a, p. 178) found that “being bold and good does not significantly improve an analyst’s future career prospects.” A similar finding for fund managers appears in Chevalier and Ellison (1999). See also Guerrieri and Kondor (2012) and Sato (2016).
Grantham , in the GMO October 2006 Quarterly Letter (p. 4) and in Barron’s of November 6, 2006, later observes that “Great Lakes Dock & Dredge, Hartford Steam Boiler and Twin Disc Clutch made clients feel much worse, apparently, than losing the same money in Avon, IBM and Johnson & Johnson.” The implication is that investment in the most popular stocks carries a lower career risk even though the losses may be the same with less popular names. In the GMO April 2007 Quarterly Letter, he observes that the necessary conditions for bubbles to form are that “the fundamental economic conditions must appear to be excellent…and liquidity must be generous in quantity and price: it must be easy and cheap to leverage.” He calls the early 2007 experience the “first truly global bubble” in which the risk premiums for the three major asset classes—stocks, bonds, and real estate—reached historic lows everywhere.
- 24.
In using equal-weighted and value-weighted monthly data of major exchange indices, Harman (2000, p. 266) questions whether duration dependence tests have the power adequate to detect speculative bubbles in stock prices. “[T]he type of hazard function used and the frequency at which returns were measured had an effect on the results.” Serial dependence is covered by Kritzman (1994). Another study relevant to duration dependence by Harman and Zuehlke (2004) points to “some troublesome issues” and “calls into question the efficacy of using hazard models to test for speculative bubbles.”
- 25.
Pierdzioch (2010) found that periodically collapsing bubbles in the pre-WWI German stock market cannot be ruled out.
- 26.
- 27.
- 28.
Nobel Laureate Vernon Smith began experimental economics in Smith (1962). An example of its application is in Henker and Owen (2008). Laboratory asset market experiments by Lei et al. (2001) suggest (counterintuitively and unrealistically, in my opinion) that even when speculation is not possible, bubbles and crashes still emerge. Gjerstad and Smith (2009) write, “Bubbles can arise when some agents buy not on fundamental value, but on price trend or momentum. If momentum traders have more liquidity, they can sustain a bubble longer.” Smith’s laboratory approach also suggests that people who experience bubbles more than once tend to learn from their first-round experience, with later bubbles not as pronounced. This is discussed in Hussam et al. (2008) and in Caginalp et al. (2000). Greenwood and Nagel (2008), also showed that more experienced fund managers in the TMT bubble were better at recognizing the bubble and avoiding it than were the younger ones. Experiments in Haruvy et al. (2007) indicate that individuals hold beliefs conditioned on past trends and that traders don’t initially anticipate market downturns. Palan (2009) surveys experimental economics literature and suggests that the efficiency of markets might be improved through the use of derivative forward-looking options. Poundstone (2010, p. 263) cites similar experiments by Colin Camerer. See also Ackert et al. (2009) and Andrade et al. (2016) on types of recently seen films affecting subsequent trading activity.
- 29.
Several Ph.D. dissertations have focused on bubbles include Harman (2000), who analyzes “characteristics of securities to determine whether bubbles form more readily in the prices of some securities than in the prices of others.” Harman examines bubbles from the standpoint of institutional investors and finds “evidence of skewness , kurtosis, and serial dependence in returns that is consistent with asset price bubbles.” She also finds that “investor herding is present during periods identified as having characteristics consistent with speculative bubbles,” that “investors herd on both positive and negative returns,” and that there is evidence “of a positive relationship between changes in institutional ownership and measures of abnormal volume.” Other dissertations of interest include those of Golden (1995), Aksoy (1997), Porras-Gonzalez (2000), Ali (2003), and Smith (2003).
- 30.
The definitional approach that comes mathematically closest to the current work was provided by Watanabe et al. (2007), in which divergent and convergent expressions of recent prices are exponentially fitted for the purpose of defining bubbles and crashes emanating from the TMT mania of the late 1990s. The Watanabe approach, however, is entirely mechanistic and, like vector autoregression models, atheoretical.
- 31.
West (1987) tests for bubbles using the specification test of Hausman by comparing two sets of estimated parameters that calculate the expected present discounted value (PDV ) of a stock’s dividend stream. The null hypothesis is that prices are in accord with a standard EMH model. “Speculative bubbles are tested for, then, by seeing whether the two sets of estimates are the same…The data reject the null hypothesis of no bubbles.” Another approach is shown in Engsted and Tanggaard (2004), who constructed a test for speculative bubbles based on the price/dividend ratio, with the idea that if there are no bubbles, the P/D ratio can be decomposed into different covariances. The results suggest that “up to the late 1980s, there are not strong indications of bubbles in U.S. stock prices. However, by including data from the 1990s, there is some evidence of the presence of speculative bubbles.”
- 32.
Dezhbakhsh and Demirguc-Kunt (1990) applied what is known as a regression equation specification error test (RESET) even though the test, because of its behavioral symmetry, admittedly does not discriminate between bubbles and fads. Additional tests for self-fulfilling speculative price bubbles, discussed in Hamilton (1986), may also be applicable here. Fukuta (2002) examines via cointegration techniques two necessary conditions for the absence of rational bubbles while assuming that the discount rate is stationary.
- 33.
With the Sola and Driffill approach, the explanatory importance of a bubble in accounting for the divergence of stock price trends from expected dividend trends is relatively low. A Markov-switching unit root test approach for detecting periodically collapsing bubbles is further explored by Hall et al. (1999), who propose use of a generalized Dickey -Fuller procedure to identify collapsing periods from expanding ones. On regime switching see also Driffill and Sola (2001). Evans (1991) examines pitfalls in testing for explosive bubbles. Charemza and Deadman (1995) highlight the failure of unit root testing, and Psaradakis et al. (2001) note that “tests for unit roots and cointegration may fail to detect the presence of explosive rational bubbles that collapse periodically.” On using unit root tests (augmented Dickey -Fuller), see also Hamilton and Whiteman (1985), Campbell and Perron (1991), Craine (1993), and Taipalus (2012).
- 34.
Cointegration, for example, is a method for analyzing long-run relationships between nonstationary variables and has often been used to test the rational expectations and market efficiency hypotheses. The methodology is covered by Dickey et al. (1991) and by Maddala and Kim (1999). Cointegration aspects of bubbles are noted in Campbell and Shiller (1987), Froot and Obstfeld (1991), Barsky and DeLong (1993), and Gujarati (1995, p. 726). From Sarno and Taylor (1999):
If stock prices and dividends are realizations of I(1)(difference-stationary) processes, then in the absence of bubbles the standard present value model of stock prices implies cointegration between the stock price and dividend series…If the stock price series contains an explosive bubble term, however, which is not by definition in the dividend price series, then this will drive a wedge between prices and dividends so that they will not be cointegrated.
The vector autoregression (VAR) approach, as described by Maddala (2001, p. 544), “is a multiple time-series generalization…easy to estimate because we can use the OLS [ordinary least squares] method.”
- 35.
- 36.
A test that does not require a detailed specification of the underlying equilibrium model was designed by Diba and Grossman (1988). In this study, the explosive price action of a rational bubble is such that prices and fundamentals follow a different path, that is, prices and fundamentals will not be cointegrated. The advantage of this approach is that the researcher does not have to specify all the details of the equilibrium model. Still, the main drawback of the Diba and Grossman method is that it will generally have difficulty in discovering bubbles that successively grow over time and partially burst.
West (1987) provides a specification test for rational speculative bubbles based on estimates of the underlying equilibrium model using two different techniques. From Engsted and Tanggaard (2002):
…one technique gives consistent estimates of the model parameters both with and without bubbles. The other technique gives consistent estimates if there are no bubbles, but inconsistent estimates if there is a bubble (provided the bubble is correlated with fundamentals ). Thus, under the null hypothesis of no bubbles, the two sets of estimates should be equal. If the null is rejected, it indicates the presence of bubbles. A drawback of this test is that it requires detailed specification of the underlying equilibrium model, so rejection of the no bubble hypothesis may not be due to bubbles but may instead be due to imposition of the wrong model.
- 37.
- 38.
Econometricians have thus devised tests for stationarity, among which the unit root test is perhaps the most popular. In West’s tests, for instance, a model using discounted stock dividends requires a specific assumption about whether the discount rate is stationary or time-varying (and also a model of what generates such variation), whereas in the Diba and Grossman (1988) approach it doesn’t matter whether the discount factor applied to future dividends varies as long as the series itself is stationary.
- 39.
In media-related stocks, Time Warner had responded to the 2006 breakup threat by corporate activist Carl Icahn by repurchasing $20 billion worth of shares at prices around $18. By 2008, the remaining shares had traded down to $9. Percentage declines of Viacom and CBS shares in the face of sizable repurchase programs were even steeper. And from 2006 onward GE had been repurchasing nearly $25 billion of shares at as high as $40 in 2007/2008, but this didn’t prevent shares from falling to $13 in late 2008. At the urging of activist Trian Fund, GE then from 2014-17 wasted $29 billion on share repurchases at prices averaging $30, more than double the early 2018 price. GE was then unable to comfortably support pension liabilities and an already reduced dividend. See also Grullon and Michaely (2002).
- 40.
Higher regulatory and other related costs, meanwhile diminished the number of IPOs even as mergers and private equity takeovers reduced the number of domestic companies listed on U. S. exchanges from a peak of 8,025 in 1996 to 4,333 in 2016.
- 41.
- 42.
Malliaris (2012) breaks this literature into four categories:a) Papers that assume all investors to be rational and have identical information. In these, finite bubbles cannot exist;b) Investors have rational expectations but asymmetric information. Bubbles might develop more easily under such conditions;c) Investors might be either rational or behavioral in reactions and bubbles can last a long time;d) Traders have heterogeneous beliefs and bubbles can emerge.
- 43.
Sornette (2003a, p. 153) suggests, “The market return from today to tomorrow is proportional to the crash hazard rate…the higher the risk of a crash, the larger is the price return. In essence, investors must be compensated by a higher return in order to be induced to hold an asset that might crash.”
- 44.
About Elliott’s work, see the February and June 1999 Scientific American and also Elliott (1938), who apparently was first to describe stock market price patterns as being “fractal .” Prechter (2004) says that such self-similarity at different scales was foreshadowed by Goethe in 1790 and then by the mathematician Georg Cantor a hundred years later. The basic Elliott idea can be described as follows: A line of length c is partitioned into segment lengths a and b. The ratio of the first segment, a, to the second, b, is made equal to the ratio of the second segment, b, to the total, c. The equation is then a/b = b/(a+b) = b/c. Setting b = 1 as the scale of measurement, the equation becomes a = 1/(a + 1), or a2 + a – 1 = 0. The two solutions for a are then 0.618 (the “golden mean”) and −1.618. See also Chap. 10, note 15 and Sornette (2003b) and Gresnigt et al. (2015) in which crashes are modeled as being akin to earthquakes.
- 45.
Bourke (1998) describes 1/f noise.
- 46.
Pareto (Zipf’s) law has also been used to compare the number of cities in the world and the number of inhabitants. The income or revenue of a company as a function of the rank is another example of the law, which as Li (1999) has noted should be called Pareto’s law because Pareto observed this at the end of the 1800s.
- 47.
- 48.
There may be instances in which a function seems to follow a power law (i.e., be exponential) but in actuality does not. But a study by Ijiri and Simon (1974) showed that rejecting the Pareto law (i.e., a power law) in favor of a downwardly concave relationship between log-size and log-rank is consistent with positively autocorrelated growth rates. This would then suggest that a power law might still be used as an approximation and that autocorrelation might be more broadly applied. Among the papers in this area are those by Laherrère and Sornette (1998), Li (1999), Rousseau (1999), Urzua (2000), and Limpert et al. (2001).
Skepticism as to the full applicability of power laws to a wide variety of physical and economic data is also the theme of Laherrère and Sornette (1998), who instead propose use of a complementary alternative known as a stretched exponential family of probability density functions (pdfs). The problem is that such log-log plots “often display linearity over a limited range of scales and/or exhibit noticeable curvature.” In their comparisons of fit, they found that stretched exponential pdfs “account remarkably well for the center of most analyzed distributions…and have a tail that is ‘fatter’ than the exponential but much less so than a pure power law distribution…being economical in their number of adjustable parameters.” See also Geraskin and Fantazzini (2013).
- 49.
From West (2017, p. 142): “…it turns out that the patterns of fluctuations in financial markets…are simply nonlinearly scaled versions of one another...the behavior of the stock market is a self-similar fractal pattern that repeats itself across all timescales following a power law that can be quantified by its exponent.
- 50.
- 51.
An attractor, as Hilborn (2000, p. 22) explains, “is that set of points to which trajectories approach as the number of iterations goes to infinity.” The trajectory or orbit is the sequence of position values as measured over time and is analogous to, say, the path of a planet orbiting the sun . Williams (1997, p. 14) has additionally advised that “chaos analysis can reveal the time-limits of reliable predictions and can identify conditions where long-term forecasting is largely meaningless.” There is indeed a value to knowing ahead of time when something cannot be predicted, but chaos theory is weak in revealing details of a particular underlying physical law or governing equation.
- 52.
Serletis (1996) adds that “chaotic processes have first and second moment properties that are the same as for white noise processes.”
- 53.
Chaos theory methods, according to Hsieh (1991), make possible the description of a rich variety of nonlinear situations that are commonly found in financial market economics. Says Serletis (1996, p. 211), “[T]he existence of chaos creates the possibility that profitable nonlinearity-based trading rules may exist, raising questions about the efficient markets hypothesis.”
- 54.
Serletis: “The most important tool for diagnosing the presence of sensitive dependence on initial conditions (and thereby chaoticity) is provided by the dominant Lyapunov exponent. The exponent measures the average exponential divergence or convergence between (state-space) trajectories that have ‘infinitesimally’ small differences in their initial conditions. A positive Lyapunov exponent is an operational definition of chaotic behavior.”
Lyapunov-exponent methods are widely used in searching for such attractors. These exponents are based on the idea that in time-series regions governed by chaos, the most significant feature is sensitive dependence on initial conditions (SDIC). This means that even a small change at the beginning of the data series results in enormous changes down the line. But it also means that long-term forecasts cannot be made.
Bulow and Klemperer (1994) explain frenzies by focusing on the idea that “in the real world, buyers and sellers can choose when to trade…A further result is that the price path is highly sensitive to small changes in the underlying demand structure.” See also Williams (1997, pp. 355–70) and Theiler et al. (1992).
- 55.
Liu et al. (1992) found that the correlation exponent test of chaos theory could distinguish white noise from chaos (but could not distinguish white noise from chaos mixed with a small amount of white noise). Their paper advises instead that the well-known BDS test (Brock et al. 1996) is perhaps more effective in distinguishing between linear and nonlinear stochastic processes. The classic papers of Grassberger and Procaccia (1983a, b) showed how the correlation dimension could be used to detect attractors in both standard and lagged phase space. See also Takens (1981), Wolf et al. (1985), Rasband (1990, p. 75), and Hilborn (2000, p. 376). Gleick (1987) provides a popular overview.
As for the BDS test, Serletis and Dormaar (1996, p. 116) nevertheless say, “The BDS test does not currently provide a direct test for nonlinearity or for chaos, because the sampling distribution of the test statistic is not known (either in finite samples or asymptotically) under the null hypothesis of nonlinearity, linearity, or chaos. It is, however, possible to use the BDS test to produce indirect evidence about nonlinear dependence…which is necessary but not sufficient for chaos.” See also Brock (1986).
- 56.
- 57.
Falconer (2013, p. 87). Very small such random steps taken rapidly at small intervals t with length \( \sqrt{t} \) lead to a fractal Brownian motion form. See also Hassler (2016), Cox and Hobson (2005), Heston et al. (2007), Jarrow et al. (2010, 2011), Cieplinski et al. (2012), and Herzog (2015). Brownian noise is, of course, the random walk under a different name. It is an example of a martingale process, in which the best forecast of xt+1 that could be determined on the basis of current information, Ωt, equals xt.
- 58.
The EMH construct crumbles when the correlations among the returns of individual securities and different asset classes rise toward unity, as they alway do in extreme events (e.g., Fig. 2.8).
- 59.
Dardik (2017, p. 130) writes in relation to science in general that “[M]athematics does not work where there is high variability, or a high degree of complexity, within or across scales.” That is, for instance, bubbles and crashes.
- 60.
Wilmott and Orrell (2017, pp. 140–7) write: “…it is no longer correct to say that the volatility is a single parameter– it is a whole series of separate parameters, which apply to different prices and dates.” Bookstaber (2017, pp. 17–8) adds that markets are not ergodic, a term that describes a process that doesn’t vary with time or experience and continues to follow the same probabilities as in the past. “This works for physics. And for the game of roulette…Our world is not ergodic – yet economists treat it as though it is.”
- 61.
Brunnermeier (2001, p. 59) adds: “…almost all bubbles can be ruled out in a symmetric information setting.”
- 62.
Buchanan (2013, pp. 29–49) provides a concise history of equilibrium-thinking economics and of the different meanings ascribed to the word “efficiency.”
- 63.
- 64.
Spotton and Rowley (1998) write that the simple premise that “fundamental ” present value of discounted stream of dividends and earnings governs asset and prices is a corresponding
implicitly compromised by the severity of its main requirements for constituents and a corresponding reliance on supplemental assumptions (representative individuals, atomistic markets, very rapid and sensitive adjustments…and an effective and immediate…dynamic process of arbitrage )…Trading risk against anticipated return, the rational investors choose current portfolios, seemingly free from the activities of others…the commitment to EMH often stems from a prior conviction that efficiency is clearly desirable…rather than from a clear evidential basis.
The random -walk/EMH approach and the presumed rationality of investors in the REH are actually separate issues that are not necessarily linked, even though the two notions often confusingly appear together in the literature.
- 65.
Taylor (2005, p. 69) agrees, writing, “It is very clear that the returns -generating process is not even approximately Gaussian.” And in Baker-Said (2008), Taleb explains that quant models typically make at least four errors: (a) narrativity , in which people hedge for what makes sense even though things that can happen don’t make sense at the time; (b) low volatility, in which the mistake is that low volatility does not mean low risk; (c) blindfoldedness, in which most people take risks even when they don’t know what risks they are incurring; and (d) stress testing, in which the most dangerous unforeseen events are not tested.
Patterson (2010b, p. 57) writes that Lévy studied distributions in which a single sample made an important change to the curve. Fox (2009, p. 7) writes that Henri Poincaré, the great French mathematician and thesis supervisor of Bachelier, thought with regard to the Gaussian distribution that “caution needed to be exercised in applying it to human behavior.” Jovanovic and Schinckus (2017, p. 2) recall that a French banker’s assistant, Jules Regnault, in 1863 first began to represent market variations using a random-walk approach.
- 66.
The belief remains that there is an equilibrium “fundamental” value that might somehow be found if only the right models were to be specified . Gilles and LeRoy (1992) indicate that the difficulties already begin with coming to an understanding of what is meant by the terms “fundamental” and “speculative bubble.” A source of the confusion, they say, is that “fundamental” is used to describe “sunspot as well as bubble equilibria.” See also Allen et al. (1993) and Brunnermeier (2001, p. 50), who say that sunspots “serve as a coordination device for the agents in the economy to select a particular static price equilibrium.”
Bailey (2005, Chapter 1) describes markets that can be “efficient” in several different ways: allocative, operational, informational, and portfolio structure. And Bray (1985) says that “Employing the rational expectations hypothesis imposes two logical requirements, that objective probability distributions exist, and that rational expectations equilibrium exists.”
- 67.
Robert Lucas, Jr., a pioneer proponent of the REH, in fact later found the theory to be flawed because it is not reasonable to believe that humans are perfectly rational or perfectly informed. Much of this theory “worked” only after making such idealistic assumptions. In the strict Lucas REH approach, there are no bubbles and banking crises.
- 68.
Kindleberger (1996, [1989], p. 21).
- 69.
Kahneman (2013, pp. 49, 411). “ The only test of rationality is not whether a person’s beliefs and preferences are reasonable, but whether they are internally consistent.”
- 70.
Lim (2015, p. 139 and 155).
- 71.
In an overview of modern financial economics, Cochrane (2005, p. 4) states that if the marginal utility of consumption, c, in period t+1 versus that in period t is taken into account, it can be shown that “bubbles”—“in which the prices grow so fast that people will buy now just to resell at higher prices later, even if there are no dividends”—are ruled out (p. 25). By contrast, the portfolio approach to asset pricing as in the CAPM and ICAPM relies heavily on the assumption that the investor has no non-asset income (p. 36)…the CAPM and ICAPM are not alternatives to the consumption -based model: “they are special cases of that model” (p. 169). Even Fama and French (2004) conclude that the early versions of the CAPM have “never been an empirical success…[T]he problems are serious enough to invalidate most applications of the CAPM.”
- 72.
From Cochrane (2005, p. 23). Barberis et al. (2001) develop a model in which investors derive direct utility not only from consumption but also from fluctuations in wealth. Their model allows for changes in risk aversion, incorporates aspects of prospect theory from behavioral finance, and also helps to explain excess volatility, predictability of returns, and a low correlation with consumption growth.
- 73.
Bailey (2005, p. 65) adds: “Without a criterion for separating what is fundamental from what is not, the distinctiveness of the EMH evaporates.”
- 74.
As per Prechter (2016, pp. 506–10).
- 75.
The emotional tail is explored in a somewhat different context by Haidt (2001).
- 76.
- 77.
The quotation also appears in Mehrling (2005, p. 13) and in Triana (2009, p. 53). Bernstein (2007, p. 214) adds that when Black moved from MIT near Boston to Goldman Sachs in New York in 1984, he said, “The market appears a lot more efficient on the banks of the Charles River than it does on the banks of the Hudson .” The quote is also in Mehrling , p. 246. Sharpe would also obviously agree with the sentiment. In a discussion with Bernstein (2007, p. 94), Sharpe says, “…how in the world can you measure expectations, which are a look forward, not backward? You cannot just look at history and deduce much about what expectations have been – or would be. The whole matter revolves around the future. Therefore, the historical data on which we all depend so heavily may be useless for asset pricing.”
Lowenstein (2008a) similarly writes, “Modern finance is an antiseptic discipline; it eschews anecdotes and examples, which are messy and possibly misleading — but nonetheless real. It favors abstraction, which is perfect but theoretical. Rather than evaluate financial assets case by case, financial models rely on the notion of randomness, which has huge implications for diversification. It means two investments are safer than one, three safer than two.”
- 78.
In commenting on the EMH, Buchan (1997, p. 240) says, “The efficient -markets doctrine is merely another attempt to apply rational laws to an arena that is self-evidently irrational. A market cannot operate by laws, for the laws would be discovered, and it would cease to be a market.”
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Vogel, H.L. (2018). Rationality Rules. In: Financial Market Bubbles and Crashes, Second Edition. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-71528-5_6
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