measuring the strangeness of gold and silver rates of return.pdf

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The Review of Economic Studies, Ltd.

Measuring the Strangeness of Gold and Silver Rates of ReturnAuthor(s): Murray Frank and Thanasis StengosReviewed work(s):Source: The Review of Economic Studies, Vol. 56, No. 4 (Oct., 1989), pp. 553-567

Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297500.

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Review of Economic Studies (1989) 56,

553-567

0034-6527/89/00360553$02.00

?

1989 The

Review of

Economic

Studies

Limited

easuring

t h

Strangeness

o

o l d

n d

S i l v e r R a t e s

o e t u r n

MURRAY FRANK

University of British Columbia

and

THANASIS

STENGOS

University of Guelph

First

version received January

1988;

final

version

accepted

March

1989 (Eds.)

The

predictability

of rates of return on

gold

and

silver

are

examined.

Econometric tests do

not reject the

martingale hypothesis for either asset. This failure to

reject

is

shown to

be

misleading.

Correlation dimension estimates

indicate

a

structure not

captured

by

ARCH.

The

correlation

dimension is

between 6 and

7

while the

Kolmogorov entropy is about

0-2

for both assets.

The

evidence is

consistent with

a

nonlinear deterministic data

generating process underlying the rates

of

return. The evidence is

certainly not sufficient to rule out

the possibility of some

degree

of

randomness

being present.

1. INTRODUCTION

The purpose of

this paper is to examine the

predictability of asset price changes. In

well-functioning

markets it

is often

held

that asset

price changes should

be

unpredictable.

Otherwise

speculators

will take

advantage

of the

predictable price change

thereby forcing

the

change

to

happen

immediately. Any subsequent

changes

are then left

unpredictable.

This

intuition was formalized

by

Samuelson

(1965)

and

has been

incorporated

in

standard

textbooks

such as

Brealey

and

Myers

(1984).

This intuition

is often

taken

to be

synony-

mous

with

the efficient markethypothesis.

Analysis

of

intertemporal general

equilibrium

models

indicates

that

this intuition is

only rigorously

justified

under

fairly stringent

conditions, see

Lucas

(1978)

and Brock

(1982). Despite being

a rather

special

case

theoretically,

considerable

empirical support

has, been

reported

for the

martingale

hypothesis ,

see

Fama

(1970).

Sims

(1984)

has

provided

an

interesting

rationalization of the

empirical

success

of

the

martingale hypothesis.

Our focus is

on

the extent to which the views of

Sims,

or

chaos

in

the

sense

of Brock

(1986)

might

be

responsible

for the

empirical

success

of

the

martingale hypothesis.

These

two

approaches

are

quite

different.

Scheinkman and

LeBaron

(1986) tested

for

chaos

using weekly

returns on

stocks

taken from the Center

for Research

in

Security

Prices at the

University

of

Chicago (CRSP). They

devoted

most

attention to

an index of

stocks but also considered

a

number of

individual stocks. Their

results are seemingly consistent with chaos.'

1.

The

behavior ...

seems to

leave

no

doubt that

past

weekly returns

help

predict

future

ones ...

Further

it

seems

that most

of

the

variation on

weekly

returns is

coming from

nonlinearities as

opposed

to

randomness.

Or more

moderately,

the data

is not

incompatible

with

a

theory

where

most of

the

variation

would come

from

nonlinearities

as

opposed

to

randomness

and

is not

compatible

with a

theory that

predicts

that the

returns are

generated

by

independent

random

variables.

Scheinkman and

LeBaron

(1986).

553


3/16

554

REVIEW

OF

ECONOMIC

STUDIES

The paper

is

organized

as follows.

Section 2

is

concerned

with some

theoretical

background.

First

we give an

indication of

the

restrictions

needed

to

generate the

martingale

hypothesis

in

a standard

economic

setup.

Then we

discuss

the approach

of

Sims

(1984). That

approach

was

developed

in

reaction to

the

restrictiveness of the

assumptions required to generate the martingale hypothesis in a standard setup. Next a

definition of

chaos is offered

in

order

to

illustrate the

sense

in

which

chaos provides an

alternative

to Sims

(19984).

Section

3 is concerned with the

testing

methods. Our work

is

based

on the

correlation

dimension and the

Kolmogorov

entropy.

Since

these are

not

yet

standard

tools

in

empirical

economic

research we

attempt

to

motivate their use.

As

the correlation

dimension

has

already

been

analyzed by

Brock

(1986)

and Brock

and

Dechert

(1988)

we

focus

more

attention

on the

Kolmogorov

entropy.

The

empirical results are set out

in

Section

4,

and

a brief

conclusion is offered

in

Section 5.

2.

THEORETICAL

UNDERPINNINGS

A.

Traditional

efficient

markets

hypothesis

A

standard

context

in

which to

consider asset

pricing

issues are the

intertemporal

models

of

Lucas

(1978)

and

Brock

(1982).

In

the Lucas

model one

derives a Euler

equation

for

the

representative

agent

relating

the

marginal

utility

of

consumption

at

date

t

to

expected

marginal

utility

of

consumption

at date

t

+

1.

In

order to

obtain the

martingalehypothesis

in

asset prices added

structure

is

required. One

needs to correct

for

dividends

and

discounting

and then either

assume

risk-neutrality

or

else

assume that there is no

aggregate

risk.

Let

Pt

be

the

price

of an

asset

at date

t

and

let

Et be the

expectations operator

conditioned on

the

information

available at

date t. A

process

{Pt}

is said

to

be a

martingale

if

E,p,+?

=

Pt

for

all s

>

0.

The claim

that asset

prices constitute

a

martingale

is

frequently

identified

in

textbooks as the essence of

the efficient

markets

hypothesis .

For

theoretical

analysis of such

economies see the

general

equilibrium models

of

Lucas

(1978)

and

Brock

(1982).

Hansen and

Singleton

(1983)

have

an

important

special

case of the

Lucas-Brock

approach

for which

there is a

closed-form

solution.2

Hansen

and

Singleton

(1983)

empirically

test their solution.

B.

Sims'

approach

To

obtain the

martingale

hypothesis

in

a

Lucas-Brock

setup requires

very

restrictive

assumptions. This led

Sims to

question whether

the

apparent

empirical

success of the

martingale

hypothesis

is little more

than a

robust

fluke. In

place of

the

Lucas-Brock

framework

Sims

(1984) provides

an

analysis

in

which

the

martingale

hypothesis

is

obtained

for

very

short

time

intervals.

Lengthy

time

intervals

need not

satisfy

the

martingale

property.

Definition 2.1. A process

{Pt}

is said to be instantaneously unpredictable (I.U.) if

li,

Et[pt+v

-

Et

(pt+v)2]

as

V-0Et[pt+v

-

(Pt)2]

2.

The

consumer

has constant relative

risk-aversion

and the

joint

distribution of

consumption

and

returns

is

lognormal.


4/16

FRANK & STENGOS GOLD AND SILVER 555

If asset prices satisfy Definition 2.1

and there are stationary increments, then a

regression of

(p,?

-p,)

on any variable known

at date t will have an R2 that approaches

zero as

s

approaches zero. Changes over

long time periods may be forecastable, but

short-term price changes should be

unpredictable. In modern financial theory diffusion

processes are often assumed. Diffusion processes satisfy the I.U. property. Any process

with

well-behaved derivatives will not satisfy

the I.U. property.

Sims

(1984) provides

a

detailed

discussion of the I.U. property and a very strong

argument

for

it

as

the basis

of

the empirical success of the martingale hypothesis. He

concludes that Except for the easily identified exceptional cases, neither the real world

nor an analytically manageable economic

model is likely to generate security prices which

fail to

be

instantaneously unpredictable .

C.

Chaos

In some respects chaos or strangeness is the polar opposite to a process that is

instantaneously unpredictable. There are a

number of different

definitionsg-of

chaos in

current use. Not all of the definitions are equally practical for empirical research.

Definition

2.2. Let fl be. a

space

with

metric d and

let

f

:

fl

-

fl be a continuous

mapping

defined

on

ft.

A

discrete

dynamical

system

(fQ,f) is

said to be chaotic

(or

strange)

if there exists a

8 >

0 such

that for

all c E: l and

all

?

>

0

there is

co

fE

and

k

such

that

d

(o, co')

< E

but

d

(fkc,fkw')

_8

In

this definition fkco denotes the k-fold iteration

of point Ctby the map

f

Definition

2.2

contains less

than

is frequently included

in

definitions of chaos. Devaney (1986) for

example includes the requirement that there exist dense orbits and that periodic points

are dense. We do

not take such

a

definition since the

additional conditions do not seem

to

be

verifiable nor refutable for empirical

systems. There seems little point to including

conditions that one cannot check. In practice

the existence of dense orbits must be

assumed

in

any case

in

order

to characterize

the system. Our definition follows the usage

of

Eckmann and

Ruelle

(1985)

who

take

chaos to be

synonymous

with sensitive

dependence

on

initial conditions .

Brock

(1986)

defines

chaos

in

terms

of

the

largest

Lyapunov exponent being positive.

The

Lyapunov exponent definition is

related

to the

Kolmogorov entropy.

The

Kolmogorov

entropy

is a lower

bound

on the sum of the

positive Lyapunov exponents,

see Eckmann and Ruelle

(1985).

The

Kolmogorov entropy

is discussed in Section 3B.

A chaotic

system

will be

quite predictable

over

very

short time horizons.

If

however

the

initial

conditions

are

only

known

with finite

precision,

then over

long

intervals

the

ability to predict

the

time path

will be lost. This is

despite

the

process being

deterministi-

cally generated. Typically

for chaotic

systems nearby trajectories locally separate exponen-

tially

fast.

The

mathematical

theory

of chaos is

currently

an active research

area,

see

Lasota

and

Mackey (1985), Devaney (1986)

and

Guckenheimer and Holmes

(1986).

There are

a

great many ways

in

which

chaos

might

enter an

economic

system.

Our work is not

tied

to any particular entry mechanism. If evidence of chaos is found then it becomes a

natural

topic

for

further

research

to

attempt

to

identify

its

source

or

sources.3

3. A

particularly simple

ad

hoc

example is as follows. Let

X,+,

4X,(1

-

X,) and

let

P,,1

3 P,

+ (X, -0.5).

Simulate this two

equation system

starting

with

Xl

E

(0, 1)

and P1 =

100. On the

simulated data

test

(P,,,

-

P,) =

ao+

a1(P,

-

P,1)

+

E,. One will

be unable to

reject

a0

=

a,

=

0

and the R2

will be very

close to zero.

However

this

example

is

anything

but

unpredictable.

This

example

is

discussed more

fully

in

Frank

and

Stengos (1988).

Also

consider footnote

12.


5/16

556

REVIEW OF ECONOMIC STUDIES

We have

three alternative

possible

interpretations

of the observed

irregularity

of asset

prices. The Lucas

(1978) framework considered

in

section

A does not tie the martingale

hypothesis to the

size

of the time intervals being employed empirically. Sims (1984)

generates unpredictability explicitly

for short time intervals. For lengthy time intervals

Sims' theory permits the asset price changes to be predictable. The chaos interpretation

leaves unspecified

the economic mechanism which generated

the data. It is consistent

with many possible theories including

versions

of

the

Lucas-Brock

setup. Chaos is not

consistent with Sims'

approach. If chaos is present then asset

price changes will (at least

in

principle) be predictable

over short

time intervals, but not over long time intervals.

Over long

time intervals,

due

to

the sensitive

dependence

on initial conditions,

the asset

prices will not be predictable.

It is worth emphasising that these three

approaches by no means exhaust the

set of

conceivable theories of

the observed irregularity

of

asset prices. These three approaches

are considered together

since each

has been suggested previously as

a

possible

interpreta-

tion. In rejecting one or more of these possible views we do not establish that a particular

alternative is true.

This familiar methodological

point is particularlypertinent with respect

to

any suggestion

of deterministic chaos.

3. TESTING METHODOLOGY

In

this section

we

describe

the two measures

on which

our empirical

work

is based.

The

theoretical

underpinning

is an

assumption

of

ergodicity.

Such an

assumption

is

required

if

we are to use time averages as representative

of

the

system's

behaviour. The first

invariant

of

the

system

is the correlation

dimension.

The

second invariant

of interest

is

the Kolmogorov entropy.

We take these

in

turn.

A. Correlation

dimension

The correlation

dimension is

originally

due to

Grassberger

and Procaccia

(1983)

and

Takens (1983).

For

more

detail than

we

provide

see

Brock and

Dechert

(1988)

and

Eckmann

and Ruelle

(1985).

Start

by

assuming

that the

system

is on an attractor .

An

attractor is

a

closed

compact

set

S with

a

neighbourhood

such that almost

all initial conditions

in the

neighbourhood have S itself as their forward-limit set. In other words, these initial

conditions

are

attracted

to S as time progresses. The neighbourhood

is termed the

basis of attraction

for

the attractor.

An attractor

satisfying

Definition 2.2 is then called

a

strange attractor

or

else

a chaotic attractor.

Consider

a

time-series

of rates of return

r,,

t

=

1, 2, 3,

.

.

.,

T.

We

suppose

that

these

were

generated by

an orbit or

trajectory

that is dense on the

attractor.

Use the time-series

to create

an

embedding.

In

other words create M-histories

as

rm

=

(r,,

rt+?,

*,

rt+M-1).

This converts

the series of scalars into a series of vectors with overlapping

entries.

If

the

true

system

which

generated

the

time-series is

n-dimensional,4

then

provided

M

?

2n

+

1

generically

the M-histories recreate

the

dynamics

of the

underlying system (there

is a

diffeomorphism between the M-histories and the underlying data generating system).

This extremely useful

mapping

between the underlying system

and

the M-histories

was

4. For an intuitive discussion of the meaning of dimension see Frank and Stengos (1988). Familiar,

smooth examples include:

a

point is zero-dimensional,

a

line is one-dimensional,

a

plane is two-dimensional.

These objects retain their dimensionality even when embedded in less restricted spaces, say R5.


6/16

FRANK & STENGOS GOLD

AND SILVER 557

established by

Takens (1980). It

is this result which permits the empirical

work. Broom-

head and King (1986) discuss certain

practical limitations on the use of

Takens' theorem.

Next one measures the spatial correlations

amongst the points (M-histories) on the

attractor by calculating the correlation

integral, CM(8). For a particular

embedding

dimension M, the correlation integral is defined to be

CM(E)={the

number of

pairs (i,j) whose distance

IIrM_rjMl

C??}/ T2. (3.1)

Here 1. denotes

the distance induced by the selected

norm. We

use the Euclidean

distance. The

other distance function

that is sometimes employed is

the sup-norm. By

Theorem

2.4

of

Brock (1986) the

correlation dimension is independent

of the choice of

norm.

In

principle

T

should

go to infinity,

but

in

practice

T

is

limited

by the length of

the available time

series.

This will

in

turn

place

limitations on the choice

of 8.

To obtain

the correlation dimension,

DM

take

D

=

lim6?O

{ln

Cm

()/ln

E}.

(3.2)

As

a

practical

matter

one

searches to see if the values of

DM

stabilize at some value D

as

M

increases.

If

so,

then D is the correlation dimension estimate. If

however,

as M

increases the

DM

continues to

increase at the same rate then the system

is taken to be

high

dimensional or

in

other words stochastic.

If

a low value for

DM

is found then

the

system

is

substantially

deterministic even

if

complicated.

In principle

an independently

and

identically

distributed stochastic

system is infinite dimensional.

Each time one

increases the available degrees of freedom, the system

utilizes that extra

freedom. With

finite data sets, high dimensionality

will be indistinguishable from infinite

dimensionality

empirically,

see

Ramsey

and Yuan

(1987) concerning

small data sets.

Two practical problems concerning

8

should be noted. If

8

is too large, then

CM(,)

=

1

and no information about the system is

obtained. It- s also possible for

8

to

be

too small.

With

finite

data sets there is a limit to the degree

of

detail that one may

discern. This limitation on the

ability

to

get

a detailed focus means that even

in

principle

one

can never exclude the

possibility

of the

system

containing

some

degree

of additive

noise. However,

the test still can

find

out

if

there are substantial nonlinearities moving

the

system.

Empirically finding

an

appropriate range

of

values

for

?

is

not

difficult for

these series.

There are

several papers

in

economics that use

the correlation integral. Barnett and

Chen (1988) used it to examine monetary aggregates, a low correlation dimension estimate

was

obtained. Brock

and

Sayers (1988) investigated

American

macroeconomic

time-series.

They reject

chaos

but

find

some evidence

of nonlinear

structures.

As

previously

indicated

Scheinkman and

LeBaron

(1986)

examined

American

stock

market data

and

obtained

results

strikingly

similar

to

those

that

we

obtain. Since the number

of related

papers

is

large

and

rapidly

growing

we

do not

carry

out a full

survey

here. For an overview of the

literature

see

Frank and

Stengos

(1988).

B.

Kolmogorov

entropy

Dimension measures the degree of complexity of a system. Entropy is a measure of time

dependence.

The

Kolmogorov5

entropy,

K,

quantifies

the

concept

of

sensitive depen-

dence on initial

conditions .

It

is

frequently

described

as

measuring

the

rate at which

5. It

is

also

termed

Kolmogorov-Sinai invariant ,

measure theoretic

entropy

or sometimes

simply

entropy .


7/16

558

REVIEW OF

ECONOMIC

STUDIES

information is

created. This is due to the

following argument. Consider two

trajectories

that are so close

initially

as

to

be

indistinguishable

to an

observer. As time passes, the

trajectories

may separate

and become

distinguishable.

The

entropy measures how

rapidly

this happens.

For an ordered system, that is to say quasi-periodic or less erratic still, K = 0, while

for

an independent and

identically distributed stochastic system K

=

+00. For a

deter-

ministic

chaotic system 0

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