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The evolution of the system is determined by individuals who meet
and there is a probability (1 - 6) that a person is converted (8 = prob
converted) from black to white or vice versa. There is also a small pr
that an agent changes his ˜colour™ independently before meeting anyon
to exogenous ˜news™ or the replacement of an existing trader by a new
a different view).
The above probabilities evolve according to a statistical process known a
+
chain and the probabilities of a conversion from k to k 1, k - 1 or


-
given by:
+ +
k 1 with probability p1 = p ( k , k 1)
Y
no change with probability = 1 - p1 - p2
k
L - 1 with probability p2 = p ( k , k - 1)
k
where
(1 - 6 ) k


(1 - S)(N - k )
1
k
p2 = N [E+
N-1
In the special case E = 6 = 0 the first person always gets recruited to the secon
viewpoint and the dynamic process is a martingale with a final position at k = 0
Also when the probability of being converted (1 - 8) is relatively low and the
of self-conversion E is high then a 50-50 split between the two ensues (see F
Kirman works out what proportion of time the system will spend in each
the equilibrium distribution). The result is that the smaller the probability of sp
conversion E relative to the probability of not being converted 6, the more time
spends at the extremes, that is 100 percent of people believing the system is in o
of the two states. (The required condition is that E < (1 - 6 ) / ( N - l), see F
0 0
1200 1600 2000
800
400

Figure 8.5 100000 Meetings Every Fiftieth Plotted: E = 0.15,6= 0.8. Source: A. Kirman (199
terly Journal of Economics. 0 1993 by the President and Fellows of Harvard College and the M
Institute of Technology. Reproduced by permission

100


80


60
K
40


20



1200
0 800 1600
400 2000

Figure 8.6 1OOOOO Meetings Every Fiftieth Plotted: E = 0.002, 6 = 0.8. Source: A. Kirman
Quarterly Journal ofEconomics. 0 1993 by the President and Fellows of Harvard College and the
Institute of Technology. Reproduced by permission


The absolute level of 6 , that is how ˜persuasive™ individuals are, is not impo
only that E is small relative to 1 - 6. Although persuasiveness is independ
number in each group, a majority once established will tend to persist. Hence
are more likely to be converted to the majority opinion of their colleagues in
and the latter is the major force in the evolution of the system (i.e. the prob
any single meeting will result in an increase in the majority view is higher th
the minority view).
Kirman (1991) uses this type of model to examine the possible behaviour
price such as the exchange rate which is determined by a weighted average of
talists and noise traders™ views. The proportion of each type of trader wi depe
above evolutionary process of conversion via the Markov chain process. He si
115
t
8 111
$3
{ 107
w
103

99

1 I l
I I l l l I I I I
I I I I I I I
50 70
20 80
11 30 901
40 60
Period

Figure 8.7 Simulated Exchange Rate for 100 Periods with S = 100. Source: Kirm
Taylor, M.P. (ed) Money and Financial Markets Figure 17.3, p. 364. Reproduced by p
Blackwell Publishers

model and finds that the asset price (exchange rate) may exhibit periods of
followed by bubbles and crashes as in Figure 8.7.
In a later paper Kirman (1993) assumes the fundamental™s price, Pf, is det
some fundamentals, P,, while the chartists™ forecast by simple extrapolation,
change in the market view, is


where A Pf, + ˜ v(p, - p r ) ,Ap;+l = p , - pt-l and the weights of depend on
=
eters governing the rate of conversion of market participants. The weights are e
and incorporate Keynes™ beauty queen idea. Individuals meet each other and
converted or not. They then try and assess which opinion is in the majority
their forecasts on who they think is in the majority, fundamentalists or char
the agent does not base his forecast on his own beliefs but on what he perce
majority view. This is rational since it is the latter that determines the ma
not the individual™s minority view. The model is then simulated and exhibit
that resembles a periodically collapsing bubble. When the chartists totally dom
constant and when the fundamentalists totally dominate P, follows a random w
dard tests for unit roots are then applied (e.g. Dickey and Fuller (1979) and P
Perron (1988)) and cointegration tests between P , and pt tend (erroneously)
there are no bubbles present. A modification of the test by Hamilton (1989
designed to detect points at which the system switches from one process to a
only moderately successful. Thus as in the cases studied by Evans (1991, see
when a periodically collapsing bubble is present, it is very difficult to detect.
There is very little hard evidence on the behaviour of noise traders and the d
opinions in financial markets. Allen and Taylor (1989) use survey techniques to
the behaviour of chartists in the FOREX market. These ˜players™ base their v
important. Also there is a tendency for chartists to underpredict the spot rate
market and vice versa. Hence the elasticity of expectations is less than one (i
the actual rate does not lead to expectations of a bigger rise next period). They
the heterogeneity in chartists™ forecasts (i.e. some forecast ˜up™ when others are
˜down™) means that they probably do not as a group influence the market ov
and hence are not destabilising. The evidence from this study is discussed in
in Chapter 12.


8.3 CHAOS
Before commencing our analysis of chaotic systems it is usefal briefly to review
of the solutions we have obtained so far, to explain returns on stocks and the s
In earlier chapters we noted that stochastic linear systems, even as simple as
walk with drift, can generate quite complex time series patterns. In contrast,
dynamic linear deterministic system such as


Equation (8.22) is a second-order difference equation. Given starting values yo,
parameters (a,b, c) we can determine all future values of yt to any degree of a
repeated substitution in (8.22). The time path of yt can converge on a stable e
value 7 = a/(l - b - c ) or for certain parameter values may either have an
path or a monotonic path. For some parameter values the path may either be
and explosive (i.e. cycles of ever increasing amplitude) or monotonic and exp
problem in basing models on deterministic differential equations like (8.22
the ˜real world™ we do not appear to observe deterministic paths for economi
Hence equations like (8.22) require an additional additive (linear) stochastic e

+ +˜ +˜
Yt = a y - ˜
Et

or
- cL2)-™(a+ e t ) = f(e,, et-l, ˜
yt = (1 - bL ˜-2)

If we assume E˜ is white noise then we can see that yt is generated by a infin
average of the random disturbances The latter can produce a time series tha
have cyclical elements (see Part 6) on which are superimposed random shocks
these cycles are not of fixed periodicity (unlike the deterministic case). The ra
with drift is a special case of (8.24) with a # 0, b = 1, c = 0.
In our ˜first look™ at empirical results on stock prices and stock returns we
the apparent randomness in the behaviour of these series. Indeed the EMH su
stock prices and (excess) returns should only change on the arrival of new info
news (about future fundamentals such as dividends). Hence the randomness
in the data is given an explicit theoretical basis and is represented by linear
models such as the random walk.
The course of the bubble is unpredictable, so unpredictability of stock returns
Intrinsic bubbles again yield a solution for stock prices which consists of two



where f ( D , ) is a non-linear function of dividends. An arbitrary linear stochas
for dividends (e.g. random walk with drift) then completes the model and w
(8.25) yields a stochastic process for P,. Because the function f(D,) non-line
is
linear stochastic process for D, is ˜transformed™ by (8.25) to yield a non-linear
process for P,.
So far, therefore, our models to explain the random nature of stock price (re
have involved introducing explicit stochastic processes somewhere into the
example, the equilibrium model in which expected returns are assumed to be co
the assumption of RE yields the random walk model. The latter equilibrium ret
via the RVF implies that stock prices only move in response to news about
that is the random forecast errors in the stochastic dividend process. In con
above, in chaotic models, apparent random patterns observed in real world d
generated by a non-linear system that is purely deterministic.
There is no commonly agreed definition of chaos but loosely speaking chao
are deterministic yet they exhibit seemingly random and irregular time serie
The time series produced by chaotic systems are highly sensitive to the initial
(i.e. the starting point yo of the system) and to slight changes in the parame
However, this sensitivity to initial conditions and parameter values does not r
possibility of producing reasonably accurate forecasts over short horizons. This
the time series from a chaotic system will be broadly repetitive in the early
time series, even if the initial conditions differ slightly.
The ˜sensitivity™ of chaotic systems is such that if the same chaotic system i
on two ˜identical™ computers (which estimate each data point to a precision of
then after a certain time, the path of the two series will differ substantially bec
minute rounding errors reacting with the highly non-linear system. This kind
the source of the observation that if the weather can be represented as a chao
then a butterfly flapping its wings in China can substantially influence weath
and hence may result in a hurricane in Florida.
Although chaotic systems produce apparent random patterns in the time do
nevertheless have a discernible structure (e.g. a specific frequency distribution)
be used to provide statistical tests for the presence of chaos. Space constraint
shall not analyse these tests (but see De Grauwe et a1 1993). As one might ima
be very difficult to ascertain whether a particular ˜random looking™ time serie
generated from a deterministic chaotic system or from a genuinely stochastic s
latter becomes even more difficult if the chaotic system is occasionally hit
random shocks: this is known as ˜noisy chaos™. Tests for chaotic systems requ
amount of data, if inferences are to be reliable (e.g. in excess of 20000 data p
hence with the ˜length™ of most economic data this becomes an acute problem
in the variables one can always ˜add on™ stochastic elements to represent the r
in human behaviour. Hence as a first step we need to examine the dynamics p
non-linear systems. If asset prices appear random and returns are largely un
we must at least entertain the possibility that these results might be generated
systems.
We have so far spoken in rather general terms about chaos. We now briefly
explicit chaotic system and outline how an economic model based on noise
smart money is capable of generating a chaotic system.

The Logistic Equation
About the simplest representation of a system capable of chaotic behaviour i
linear) logistic equation:
Yt+l = m ( 1 - Y r )
The steady state Y* is given when Yt+l = Y , = Y,-1 . . .
Y * = AY*(1 - Y * )

and the two solutions are:
Y* = 1 - l / h
Y* = 0

Not all non-linear systems give rise to chaotic behaviour: it depends on the
values and initial conditions. For some values of h the system is globally stabl
any starting value Y o , the system will converge to one of the steady state soluti
other values of h the solution is a limit cycle whereby the time series eventuall
(for ever) between two values U; and Y ; (where Y r # Y * , i = 1,2): this is k
two-cycle (Figure 8.8). The ˜solution™ to the system i therefore a differenti
s
and is known as a bifurcation. Again for different values of A the series can a
two-cycle to a 4, 8, 16 cyclic pattern. Finally, for a range of values of h the tim
Y , appears random and chaotic behaviour occurs (Figure 8.9). In this case if
value is altered from Y O = 0.3 to 0.30001, the ˜random™ time path differs afte
time periods, demonstrating the sensitivity to very slight parameter changes.
The dynamics of the non-linear logistic system in the single variable Y
solved by simulation rather than analytically. The latter usually applies a fortio
complex single variable non-linear equations and to a system of non-linear
where variables X , Y , 2, say, interact with each other. A wide variety of v
patterns which are seemingly random or irregular can arise in such models.

Noise Traders and Smart Money
It is not difficult to set up ad-hoc models of the interaction of noise traders (NT
money (SM) that are non-linear in the variables and hence that may exhibit ch
we say ad hoc we imply that the NT and SM need not necessarily maximise
defined function (e.g. utility of end of period wealth). De Grauwe et a1 (1993)
0.6
"Q˜ooowooow
0.4




L
I
I
I
40.0
30.0
20.0
10.0

Generation Number

Figure 8.8 Period 2: Limit Cycle. Source: De Grauwe (1993). Reproduced by pe
Blackwell Publishers

1.o


0.8


0.6

Yfr)
0.4


0.2


0.0
20.0 30.0
10.0
0.0 40.0

Generation Number

Figure 8.9 Chaotic Regime. Start Value Yo = 0.3 (Solid Line), Start Value Y O= 0.30
Line). Source: De Grauwe (1993). Reproduced by permission of Blackwell Publishers

interesting model of this interaction where NT exhibit extrapolative behaviou
feedback) and the SM have negative feedback, since they sell when the pric
fundamentals. The model they develop is for the exchange rate St, although
can replace this by the price of any speculative asset, without loss of genera
aspect of the model is the heterogeneity of expectations of different traders. T
where E,S,+l is the market expectation at time t. The dating of the variable
needs some comment. The time period of the model is short. Agents are assum
information for time t - 1 and they take positions in the market at time t, bas
+
forecasts for period t 1. This is required because S, is the market solution of
and is not observable by the NT and SM when they take their positions in t
The expectations of NT are extrapolative so that

.&NI
E;[S,+1ISr-11 = f[S,-l, Sr-27 **



where f is a non-linear function. A simple form of extrapolative predicto
De Grauwe et a1 is based on chartists' behaviour, who predict that the pric
in the future if the short moving average SMA crosses the long moving ave
from below. (Hence at point A in Figure 8.10, chartists buy the asset since the
price rise.) A simplified representation of this NT extrapolative behaviour is:




In contrast, the SM have regressive expectations, relative to the long-run e
value S:-l given by fundamentals:


and adjust their expectations at the rate given by the parameter a. Note that
not have RE since they do not take into account the behaviour of the NT and




Moving Average




Time

Figure 8.10 Moving Average Chart. A, Chartists Buy Foreign Exchange; B, Chartists
Exchange. Source: De Grauwe (1993). Reproduced by permission of Blackwell Publi
De Long et a1 (1990) the relative weight given to NT depends on the profi
NT relative to those of the SM. The model of De Grauwe et a1 is rather aki
De Long et a1 in that the relative weight of NT and SM varies over time de
economic conditions and it provides the key non-linearity in the model. The
NT in the market m, is given by:


The SM are assumed to have different views about the equilibrium value S
views are normally distributed around St. Hence if the actual market price
the true equilibrium rate then 50 percent of the SM think the equilibrium rate
and 50 percent think it is too high. If we assume all the SM have the same
risk aversion and initial wealth then they will exert equal and opposite pres
market rate. Hence when Sr-1 = SF-l the SM as a whole do not influence
price and the latter is entirely determined by the NT, that is m, in (8.32) e
and the weight of the SM = (1 - m,)is zero (Figure 8.11). On the other ha
falls below the true market rate then more of the SM will believe that in the
equilibrium market rate is above the actual rate and as a group they begin t
the market rate, hence m in (8.32) falls and the weight of the SM = 1 - m,
t
Finally, note that /3 measures the degree of confidence about the true
market rate held by the SM as a whole. As # increases then mf+ 0 and (1 -
I
Hence the larger is B the greater the degree of homogeneity in the SMs™ view
the true equilibrium rate lies. In this case a small deviation of S,-1 from S:-l
to a strong influence of SM in the market. Conversely if /3 is small there i
dispersion of views of the SM about where the true equilibrium market rate l
weight of the SM in the market (1 - m,)increases relatively slowly (Figure 8
To close the model we need an equation to link the markets™ non-linear e
formation equation:




0


Figure 8.11 The Weighting Function of Chartists. Source: De Grauwe et a1 (1993).
by permission of Blackwell Publishers
If for simplicity we assume E,D,+1 = constant and substitute for E,P,+1 (i.e. eq
E,S,+1) from (8.33) we have a non-linear difference equation in P,. However
to our exchange rate example, we see in Chapter 13 that the Euler equation li
E,(S,+1 j is of the form:
s, = Xf[Jw,+llh
where X , = fundamentals, for example relative money supplies, that inf
exchange rate (and De Grauwe et a1 set b = 0.95 and X , = 1, initially). De G
simulate the model for particular parameter values and find that when the ex
parameter y is sufficiently high, chaotic behaviour ensues. This can be clea
Figure 8.12(a) where the equilibrium exchange S rate is normalised to unity.
T
initial condition is changed by 1 percent the time series is similar for about th
periods but then the two patterns deviate quite substantially (e.g. compare the
Figures 8.12(a) and 8.12(b) for periods 200-400).

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