Complexity and the Economy
Page 13
units to gain a footing (see Eigen and Schuster, 1979; Kauffman, 1993). Thus,
Table 1. RETURNS AND VOLUME STATISTICS (MEDIANS) IN THE TWO
REGIMES COLLECTED FOR 25 EXPERIMENTS AFTER 250,000 PERIODS
Mean
Std. Dev.
Skewness
Kurtosis13
Vol. traded
R.e.e. Regime
0.000
2.1002
0.0131
0.0497
2,460.9
Complex Regime 0.000
2.1007
0.0204
0.3429
7,783.8
13. Kurtosis numbers are excess kurtosis (i.e., kurtosis-3).
[ 56 ] Complexity and the Economy
technical analysis can emerge if trend-following (or mean-reversion) beliefs are, by chance, generated in the population, and if random perturbations in
the dividend sequence activate them and subsequently validate them. From
then on, they may take their place in the population of patterns recognized
by the agents and become mutually sustainable. This emergence of structure
from the mutual interaction of system subcomponents justifies our use of the
label “complex” for this regime.
What is critical to the appearance of subpopulations of mutually reinforc-
ing forecasts, in fact, is the presence of market information to condition upon.
Market states act as “sunspot-like” signals that allow predictors to coordi-
nate upon a direction they associate with that signal. (Of course, these are
not classic sunspots that convey no real information.) Such coordination or
mutuality can remain in the market once it establishes itself by chance. We
can say the ability of market states to act as signals primes the mutuality
that causes complex behavior. There is no need to assume a separate class of
noise traders for this purpose. We can test this signaling conjecture in further experiments where we “turn off” the condition part of all predictors (by filling them with nonreplaceable #’s). Now forecasts cannot differentiate among
states of the market, and market states cannot act as signals. We find, con-
sistent with our conjecture that signaling drives the observed patterns, that
the complex regime does not emerge. As a further test of the significance of
technical-trading signals, we regress the current price on the previous peri-
ods plus the technical indicator (price > 500-period moving average). In the rational-expectations regime, the technical indicator is of course not significant. In the complex regime, the trend indicator is significant (with t-value of 5.1 for the mean of the sample of 25 experiments), showing that the indicator does indeed carry useful market information. The corresponding test on
actual financial data shows a similar result (see Brock et al., 1991).
One of the striking characteristics of actual financial markets is that both
their price volatility and trading volume show persistence or autocorrelation.
And volatility and volume show significant cross-correlation. In other words,
both volume and volatility remain high or low for periods of random length,
and they are interrelated. Our inductive market also shows persistence in
volatility or GARCH behavior in the complex regime (see Figure 4), with the
Chi-square statistic in the Engle GARCH Test significant at the 95% level.14 It also shows persistence in trading volume (see Figure 5), as well as significant cross-correlation between trading volume and volatility (see Figure 6). The
figures include corresponding correlations for the often-used market stan-
dard, IBM stock. Note that because our time period and actual market days do
not necessarily match, we should expect no exact overlap. But qualitatively,
14. Autocorrelated volatility is often fitted with a Generalized Autoregressive Conditional Heteroscedastic time series. Hence, the GARCH label. See Bollerslev et al.
(1990); and Goodhart and O’Hara (1995).
a sse t Pr icing under endogenous exPectat ion s [ 57 ]
0.225
0.175
0.125
Ree. Case
Complex Case
0.075
IBM Daily
0.025
–0.025 0
1
2
3
4
5
6
Lag
Figure 4:
Autocorrelation of volatility in rational-expectations and complex regimes, and in IBM daily returns.
persistence in our market and IBM’s is similar.) These correlations are not
explained by the standard model, where theoretically they are zero.
Why financial markets—and our inductive market—show these empirical
“signatures” remains an open question. We conjecture a simple evolutionary
explanation. Both in real markets and in our artificial market, agents are constantly exploring and testing new expectations. Once in a while, randomly,
more successful expectations will be discovered. Such expectations will
change the market, and trigger further changes in expectations, so that small
and large “avalanches” of change will cascade through the system. (Of course,
on this very short time-lag scale, these avalanches occur not through the
0.6
0.5
0.4
VolACF Ree
0.3
VolACF Complex
VolACF IBM
0.2
0.0
0 0
1
2
3
4
5
6
Lag
Figure 5:
Autocorrelation of trading volume in the rational-expectations and complex regimes, and in IBM daily returns.
[ 58 ] Complexity and the Economy
0.4
0.35
0.3
0.25
0.2
Complex Case
Ree. Case
0.15
IBM Daily
0.1
0.05
0
–0·05 –4 –3
–2
–1
0
1
2
3
4
Lag
Figure 6:
Cross-correlation of trading volume with volatility, in the rational-expectations and complex regimes, and in IBM daily returns.
genetic algorithm, but by agents changing their active predictors.) Changes
then manifest in the form of increased volatility and increased volume. One
way to test this conjecture is to see whether autocorrelations increase as
the predictor accuracy-updating parameter θ in Eq. (A.1) in Appendix A is increased. The larger θ is, the faster individual agents “switch” among their predictors. Thus, the more such switches should cascade. Experiments confirm that autocorrelations indeed increase with θ. Such cascades of switching in time are absorbed by the market, and die away. Hence, our evolutionary
market exhibits periods of turbulence followed by periods of quiescence, as
do actual markets.15
5. DISCUSSION
To what extent is the existence of the complex regime an artifact of design
assumptions in our model? We find experimentally, by varying both the mod-
el’s parameters and the expectational-learning mechanism, that the complex
regime and the qualitative phenomena associated with it are robust. These are
not an artifact of some deficiency in the model.16
15. For a discussion of volatility clustering in a different model, see Youssefmir and Huberman (1995); and a
lso Grannan and Swindle (1994).
16. One design choice might make a difference. We have evaluated the usefulness of expectational beliefs by their accuracy rather than by the profit they produce. In practice, these alternatives may produce different outcomes. For example, buying into a price rise on the basis of expectations may yield a different result if validated by profit instead of by accuracy of forecast when “slippage” is present, that is, when traders on the other side of the market are hard to find. We believe, but have not proved, that the two criteria lead to the same qualitative results.
a sse t Pr icing under endogenous exPectat ion s [ 59 ]
It might be objected that if some agents could discover a superior means of forecasting to exploit the market, this might arbitrage complex patterns
away, causing the market again to converge to rational expectations. We
believe not. If a clever metaexpectational model was “out there” that might
exploit others’ expectations, such a model would, by aggregation of others’
expectations, be a complicated nonlinear function of current market informa-
tion. To the degree that the piecewise linear form we have assumed covers
the space of nonlinear expectational models conditioned on current market
information, agents would indeed, via the genetic algorithm, pick up on an
approximate form of this superior metamodel. The complex regime owes its
existence then not to limitations of forecasting, but rather to the fact that in our endogenous-expectations model market information can be used as signals, so that a much wider space of possibilities is open—in particular, the
market can self-organize into mutually supporting subpopulations of predic-
tors. (In fact, in a simpler, analytical model, with a small number of classes of trader whose beliefs adapt endogenously, Brock and Hommes (1996) find
similar, rich, asset-price dynamics.) There is no reason these emergent sub-
populations should be in stochastic equilibrium. Indeed, agents may mutually
adapt their expectations forever, so that the market explores its way through
this large space, and is nonstationary. In some early exploratory experiments, we “froze” successful agents’ expectations, then reinjected these agents with
their previously successful expectations much later. The reintroduced agents
proved less successful than average, indicating that the market had evolved
and was nonstationary.
It might be also objected that by our use of condition bits in the predictors, we have built technical trading into our model. And so it is no surprise that it appears in the complex regime. But actually, only the possibility of technical trading is built in, not its use. The use of market descriptors is selected against in the model. Thus, market signals must be of value to be used, and technical
trading emerges only because such market signals induce mutually supporting
expectations that condition themselves on these market signals.
If the market has a well-defined psychology in our model, does it also expe-
rience “moods?” Obviously not. But, notice we assume that agents entertain
more than one market hypothesis. Thus, we can imagine circumstances of a
prolonged “bull-market” uptrend to a level well above fundamental value in
which the market state activates predictors that indicate the uptrend will continue, and simultaneously other predictors that predict a rapid downward cor-
rection. Such combinations, which occur easily in both our market and actual
markets, could well be described as “nervous.”
What about trade, and the motivation to trade in our market? In the
rational-expectations literature, the deductively rational agents have no motivation to trade, even where they differ in beliefs. Assuming other agents have access to different information sets, each agent in a prebidding arrangement
[ 60 ] Complexity and the Economy
arrives at identical beliefs. Our inductively rational agents (who do not communicate directly), by contrast, do not necessarily converge in beliefs. They
thus retain a motivation to trade, betting ultimately on their powers as mar-
ket statisticians. It might appear that, because our agents have equal abilities as statisticians, they are irrational to trade at all. But although their abilities are the same, their luck in finding good predictors diverges over time. And
at each period, the accuracy of their predictors is fully accounted for in their allocations between the risk-free and risky asset. Given that agents can only
act as market statisticians, their trading behavior is rational.
Our endogenous-expectation theory fits with two other modern
approaches. Our model generalizes the learning models of Bray and others
(Bossaerts, 1995; Sargent, 1993), which also assume endogenous updating
of expectations. But while the Bray models assume homogeneous updating
from a shared nonrational forecast, our approach assumes heterogeneous
agents who can discover expectations that might exploit any patterns pres-
ent. Our evolutionary approach also has strong affinities with the evolution-
ary models of Blume and Easley (1982, 1990). These assume populations of
expectational (or more correctly, investment) rules that compete for survival
in the market in a given population of rules, and that sometimes adapt. But
the concern in this literature is the selective survival of different, compet-
ing, rule types, not the emergence of mutually supportive subpopulations
that give rise to complex phenomena, nor the role of market signals in this
emergence.
Our inductively rational market, of course, leaves out many details of real-
ism. In actual financial markets, investors do not perfectly optimize portfo-
lios, nor is full market clearing achieved each period. Indeed, except for the formation of expectations, our market is simple and neoclassical. Our object,
however, is not market realism. Rather it is to show that given the inevitable inductive nature of expectations when heterogeneity is present, rich psychological behavior emerges—even under neoclassical conditions. We need not,
as in other studies (see Kirman, 1991; Friedman and Aoki, 1992; Sargent,
1993), assume sharing of information nor sharing of expectations nor herd
effects to elicit these phenomena. Nor do we need to invoke “behaviorism”
or other forms of irrationality (see Thaler, 1993). Herding tendencies and
quasi-rational behavior may be present in actual markets, but they are not
necessary to our findings.
6. CONCLUDING REMARKS
In asset markets, agents’ forecasts create the world that agents are trying to forecast. Thus, asset markets have a reflexive nature in that prices are generated by traders’ expectations, but these expectations are formed on the basis
a sse t Pr icing under endogenous exPectat ion s [ 61 ]
of anticipations of others’ expectations.17 This reflexivity, or self-referential character of expectations, precludes expectations being formed by deductive
means, so that perfect rationality ceases to be well defined. Thus, agents can only treat their expectations as hypotheses: they act inductively, generating individual expectational models that they constantly introduce, test, act
upon, discard. The market becomes driven by expectations that adapt endog-
enously to the ecology these expectations cocreate.
Experiments with a computerized version of this endogenous-expectations
market explain one of the more striking puzzles in finance: Standard theory
tends to see markets as efficient, with no rationale for herd effects, and no
possi
bility of systematic speculative profit, whereas traders tend to view the market as exhibiting a “psychology,” bandwagon effects, and opportunities
for speculative profit. Recently the traders’ view has been justified by invoking behavioral assumptions, such as the existence of noise traders. We show,
without behavioral assumptions, that both views can be correct. A market of
inductively rational traders can exist in two different regimes: Under a low
enough rate of exploration of alternative forecasts, the market settles into
a simple regime which corresponds to the rational-expectations equilibrium
of the efficient-market literature. Under a more realistic rate of exploration of alternative forecasts, the market self-organizes into a complex regime in
which rich psychological behavior emerges. Technical trading appears, as
do temporary bubbles and crashes. And prices show statistical features—in
particular, GARCH behavior—characteristic of actual market data. These
phenomena arise when individual expectations that involve trend following
or mean reversion become mutually reinforcing in the population of expec-
tations, and when market indicators become used as signaling devices that
coordinate these sets of mutually reinforcing beliefs.
Our endogenous-expectations market shows that heterogeneity of beliefs,
deviations from fundamental trading, and persistence in time series can be
maintained indefinitely in actual markets with inductively rational traders.
We conjecture that actual financial markets lie within the complex regime.
17. This point was also made by Soros (1994) whose term reflexivity we adopt.
[ 62 ] Complexity and the Economy
APPENDIX A
Details of the Market’s Architecture
Model Parameters. Throughout the experiments we set the interest rate r to 0.1, and agents’ risk-aversion parameter λ to 0.5. The parameters of the dividend process in Eq. (4) are ρ = 0.95, d = 10 , r = 0.1, σ 2 ε = 0.0743. (This error variance value is selected to yield a combined price-plus-dividend variance of 4.0 in the h.r.e.e.)
Predictor Accuracy. The accuracy, or precision, of agent i’s j th predictor is updated each time the predictor is active, and is recorded as the inverse of the moving average of squared forecast error:
e 2 = (1 − θ) 2