Complexity and the Economy
Page 12
creates new predictors by “mutating” the values in the predictor array, or by
“recombination”—combining part of one predictor array with the comple-
mentary part of another.
The expectational system then works at each time with each agent observ-
ing the current state of the market, and noticing which of his predictors match this state. He forecasts next period’s price and dividend by combining statistically the linear forecast of the H most accurate of these active predictors, and given this expectation and its variance, uses Eq. (5) to calculate desired stock holdings and to generate an appropriate bid or offer. Once the market clears,
the next period’s price and dividend are revealed and the accuracies of the
active predictors are updated.
As noted above, learning in this expectational system takes place in two
ways. It happens rapidly as agents learn which of their predictors are accurate and worth acting upon, and which should be ignored. And it happens on a
slower time scale as the genetic algorithm from time to time discards non-
performing predictors and creates new ones. Of course these new, untested
predictors do not create disruptions—they will be acted upon only if they
prove accurate. This avoids brittleness and provides what machine-learning
theorists call “gracefulness” in the learning process.
We can now discern several advantages of this multibit, multipredictor
architecture. One is that this expectational architecture allows the market to have potentially different dynamics—a different character—under different
states or circumstances. Because predictors are pattern-recognizing expec-
tational models, and so can “recognize” these different states, agents can
“remember” what happened before in given states and activate appropriate
forecasts. This enables agents to make swift gestalt-like transitions in forecasting behavior should the market change.
Second, the design avoids bias from the choice of a particular functional
form for expectations. Although the forecasting part of our predictors is
linear, the multiplicity of predictors conditioned upon the many combina-
tions of market conditions yield collectively at any time and for any agent a
nonlinear forecasting expression in the form of a piecewise linear, noncon-
tinuous forecasting function whose domain is the market state space, and
whose accuracy is tuned to different regions of this space. (Forecasting is, of course, limited by the choice of the binary descriptors that represent market
conditions.)
Third, learning is concentrated where it is needed. For example, J = 12
descriptors produces predictors that can distinguish more than four thousand
different states of the market. Yet, only a handful of these states might occur often. Predictor conditions that recognize states that do not occur often will be used less often, their accuracy will be updated less often and, other things being equal, their precision will be lower. They are, therefore, less likely to survive in the competition among predictors. Predictors will, therefore, cluster
[ 50 ] Complexity and the Economy
in the more visited parts of the market state space, which is exactly what we want.
Finally, the descriptor bits can be organized into classes or informa-
tion sets which summarize fundamentals, such as price-dividend ratios or
technical-trading indicators, such as price trend movements. The design allows us to track exactly which information—which descriptor bits—the agents are
using or ignoring, something of crucial importance if we want to test for the
“emergence” of technical trading. This organization of the information also
allows the possibility of setting up different agent “types” who have access to different information sets. (In this chapter, all agents see all market information equally.)
A neural net could also supply several of these desirable qualities. However,
it would be less transparent than our predictor system, which we can easily
monitor to observe which information agents are individually and collectively
using at each time.
4. COMPUTER EXPERIMENTS: THE EMERGENCE
OF TWO MARKET REGIMES
Experimental Design
We now explore computationally the behavior of our endogenous-expectations
market in a series of experiments. We retain the same model parameters
throughout these experiments, so that we can make comparisons of the
market outcomes using the model under identical conditions with only con-
trolled changes. Each experiment is run for 250,000 periods to allow asymp-
totic behavior to emerge if it is present; and it is run 25 times under different random seeds to collect cross-sectional statistics.
We specialize the model described in the previous section by choosing
parameter values, and, where necessary, functional forms. We use N = 25
agents, who each have M = 100 predictors, which are conditioned on J = 12
market descriptors. The dividend follows the AR (1) process in Eq. (4), with
autoregressive parameter ρ set to 0.95, yielding a process close to a random walk, yet persistent.
The 12 binary descriptors that summarize the state of the market are the
following:
1–6 Current price × interest rate/dividend > 0.25, 0.5, 0.75, 0.875, 1.0, 1.125
7–10 Current price > 5-period moving average of past prices (MA), 10-period MA, 100-period MA, 500-period MA
11
Always on (1)
12
Always off (0)
a sse t Pr icing under endogenous exPectat ion s [ 51 ]
The first six binary descriptors—the first six bits—reflect the current price in relation to current dividend, and thus, indicate whether the stock is
above or below fundamental value at the current price. We will call these
“fundamental” bits. Bits 7–10 are “technical-trading” bits that indicate
whether a trend in the price is under way. They will be ignored if useless,
and acted upon if technical-analysis trend following emerges. The final two
bits, constrained to be 0 or 1 at all times, serve as experimental controls.
They convey no useful market information, but can tell us the degree to
which agents act upon useless information at any time. We say a bit is “set”
if it is 0 or 1, and predictors are selected randomly for recombination, other things equal, with slightly lower probabilities the higher their specificity—
that is, the more set bits they contain (see Appendix A). This introduces a
weak drift toward the all-# configuration, and ensures that the informa-
tion represented by a particular bit is used only if agents find it genuinely
useful in prediction. This market information design allows us to speak of
“emergence.” For example, it can be said that technical trading has emerged
if bits 7–10 become set significantly more often, statistically, than the con-
trol bits.
We assume that forecasts are formed by each predictor j storing values for the parameters a , b , in the linear combination of price and dividend, E [ p +
j
j
j
t + 1
d | I ] = a ( p + d ) + b . Each predictor also stores a current estimate of its t + 1
t
j
t
t
j
forecast variance. (See Appendix A).
Before we conduct experiments, we run two diagnostic tests on our
computer-based version of the mo
del. In the first, we test to see whether
the model can replicate the rational-expectations equilibrium (r.e.e.) of
standard theory. We do this by calculating analytically the homogeneous
rational-expectations equilibrium (h.r.e.e.) values for the forecasting param-
eters a and b (see Appendix A), then running the computation with all predictors “clamped” to these calculated h.r.e.e. parameters. We find indeed that such predictions are upheld—that the model indeed reproduces the h.r.e.e.—
which assures us that the computerized model, with its expectations, demand
functions, aggregation, market clearing, and timing sequence, is working
correctly. In the second test, we show the agents a given dividend sequence
and a calculated h.r.e.e. price series that corresponds to it, and test whether they individually learn the correct forecasting parameters. They do, though
with some variation due to the agents’ continual exploration of expectational
space, which assures us that our agents are learning properly.
The Experiments
We now run two sets of fundamental experiments with the computerized
model, corresponding respectively to slow and medium rates of exploration
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by agents of alternative expectations. The two sets give rise to two different regimes—two different sets of characteristic behaviors of the market. In the slow-learning-rate experiments, the genetic algorithm is invoked every
1,000 periods on average, predictors are crossed over with probability 0.3,
and the predictors’ accuracy-updating parameter θ is set to 1/150. In the medium-exploration-rate experiments, the genetic algorithm is invoked every
250 periods on average, crossover occurs with probability 0.1, and the pre-
dictors’ accuracy-updating parameter θ is set to 1/75.10 Otherwise, we keep the model parameters the same in both sets of experiments, and in both
we start the agents with expectational parameters selected randomly from
a uniform distribution of values centered on the calculated homogeneous
rational-expectations ones. (See Appendix A.) In the slow-exploration-rate
experiments, no non-r.e.e. expectations can get a footing: the mar-
ket enters an evolutionarily stable, rational-expectations regime. In the
medium-exploration-rate experiments, we find that the market enters a com-
plex regime in which psychological behavior emerges, there are significant
deviations from the r.e.e. benchmark, and statistical “signatures” of real financial markets are observed.
We now describe these two sets of experiments and the two regimes or
phases of the market they induce.
The Rational-Expectations Regime
As stated, in this set of experiments, agents continually explore in prediction space, but under low rates. The market price, in these experiments, converges
rapidly to the homogeneous rational-expectations value adjusted for risk,
even though the agents start with nonrational expectations. In other words,
homogeneous rational expectations are an attractor for a market with endog-
enous, inductive expectations.11 This is not surprising. If some agents forecast differently than the h.r.e.e. value, then the fact that most other agents are
using something close to the h.r.e.e. value will return a market-clearing price that corrects these deviant expectations: There is a natural, if weak, attraction to h.r.e.e. The equilibrium within this regime differs in two ways from
the standard, theoretical, rational-expectations equilibrium. First, the equilibrium is neither assumed nor arrived at by deductive means. Our agents instead
arrive inductively at a homogeneity that overlaps that of the homogeneous,
10. At the time of writing, we have discovered that the two regimes emerge, and the results are materially the same, if we vary only the rate of invocation of the genetic algorithm.
11. Within a simpler model, Blume and Easley (1982) prove analytically the evolutionary stability of r.e.e.
a sse t Pr icing under endogenous exPectat ion s [ 53 ]
theoretical rational expectations. Second, the equilibrium is a stochastic one.
Agents continually explore alternatives, albeit at low rates. This testing of
alternative explorations, small as it is, induces some “thermal noise” into the system. As we would expect, in this regime, agents’ holdings remain highly
homogeneous, trading volume remains low (reflecting only variations in fore-
casts due to mutation and recombination), and bubbles, crashes, and technical
trading do not emerge. We can say that in this regime the efficient-market
theory and its implications are upheld.
The Complex or Rich Psychological Regime
We now allow a more realistic level of exploration in belief space. In these
experiments, as we see in Figure 1, the price series still appears to be nearly identical to the price in the rational—expectations regime. (It is lower because of risk attributable to the higher variance caused by increased exploration.)
On closer inspection of the results, however, we find that complex patterns
have formed in the collection of beliefs, and that the market displays charac-
teristics that differ materially from those in the rational-expectations regime.
For example, when we magnify the difference between the two price series,
we see systematic evidence of temporary price bubbles and crashes (Figure 2).
We call this new set of market behaviors the rich psychological, or complex,
regime.
This appearance of bubbles and crashes suggests that technical trading, in
the form of buying or selling into trends, has emerged in the market. We can
check this rigorously by examining the information the agents condition their
forecasts upon. Figure 3 shows the number of technical-trading bits that are
used (are 1’s or 0’s) in the population of predictors as it evolves over time. In both sets of experiments, technical-trading bits are initially seeded randomly in the predictor population. In the rational-expectations regime, however,
technical-trading bits provide no useful information and fall off as useless
predictors are discarded. But in the complex regime, they bootstrap in the
population, reaching a steady-state value by 150,000 periods. Technical trad-
ing, once it emerges, remains12.
Price statistics in the complex regime differ from those in the
rational-expectations regime, mainly in that kurtosis is evident in the com-
plex case (Table 1) and that volume of shares traded (per 10,000 periods) is
about 300% larger in the complex case, reflecting the degree to which the
agents remain heterogeneous in their expectations as the market evolves. We
12. When we run these experiments informally to 1,000,000 periods, we see no signs that technical-trading bits disappear.
[ 54 ] Complexity and the Economy
100
95
90
85
80
Price
75
70
65
60
253000
253050
253100
253150
253200
Time
Figure 1:
Rational-expectations price vs. price in the rich psychological regime. The two price series are generated on the same random dividend series. The upper is the homogeneous r.e.e.
price, the lower is the price in the complex regime. The higher variance in the latter case causes the lower price through risk aversion.
note that fat tails and high volume are
also characteristic of price data from actual financial markets.
How does technical trading emerge in psychologically rich or complex
regime? In this regime the “temperature” of exploration is high enough to
100
R.e.e Price
80
60
40
20
Price Difference
0
–20
253000
253050
253100
253150
253200
Time
Figure 2:
Deviations of the price series in the complex regime from fundamental value. The bottom graph shows the difference between the two price series in Figure 1 (with the complex series rescaled to match the r.e.e. one and the difference between the two doubled for ease of observation). The upper series is the h.r.e.e. price.
a sse t Pr icing under endogenous exPectat ion s [ 55 ]
600
500
Complex Case
400
300
Bits Used
200
100
R.E.E. Case
0
0
50000
100000
150000
200000
250000
Time
Figure 3:
Number of technical-trading bits that become set as the market evolves (median over 25
experiments in the two regimes).
offset, to some degree, expectations’ natural attraction to the r.e.e. And so, subsets of non-r.e.e. beliefs need not disappear rapidly. Instead they can
become mutually reinforcing. Suppose, for example, predictors appear early on
that, by chance, condition an upward price forecast upon the markets showing
a current rising trend. Then, agents who hold such predictors are more likely
to buy into the market on an uptrend, raising the price over what it might
otherwise be, causing a slight upward bias that might be sufficient to lend validation to such rules and retain them in the market. A similar story holds for
predictors that forecast reversion to fundamental value. Such predictors need
to appear in sufficient density to validate each other and remain in the popu-
lation of predictors. The situation here is analogous to that in theories of the origin of life, where there needs to be a certain density of mutually reinforcing RNA units in the “soup” of monomers and polymers for such replicating