measuring the strangeness of gold and silver rates of return

Reviewof Economic Studies (1989) 56, 553-567© 1989 The Review of Economic Studies Limited

0034-6527/89/00360553$02.00

Measuring the Strangeness of Goldand Silver Rates of Return

MURRAY FRANKUniversity of British Columbia

and

THANASIS STENGOSUniversity 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 donot 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 correlationdimension is between 6 and 7 while the Kolmogorov entropy is about 0·2 for both assets. Theevidence is consistent with a nonlinear deterministic data generating process underlying the ratesof return. The evidence is ce}1ainly not sufficient to rule out the possibility of some degree ofrandomness being present.

1. INTRODUCTION

The purpose of this paper is to examine the predictability of asset price changes. Inwell-functioning markets it is often held that asset price changes should be unpredictable.Otherwise speculators will take advantage of the predictable price change thereby forcingthe change to happen immediately. Any subsequent changes are then left unpredictable.This intuition was formalized by Samuelson (1965) and has been incorporated in standardtextbooks such as Brealey and Myers (1984). This intuition is often taken to be synonymous with the efficient market hypothesis. Analysis of intertemporal general equilibriummodels indicates that this intuition is only rigorously justified under fairly stringentconditions, see Lucas (1978) and Brock (1982). Despite being a rather special casetheoretically, considerable empirical support has been reported for the "martingalehypothesis", see Fama (1970).

Sims (1984) has provided an interesting rationalization of the empirical success ofthe martingale hypothesis. OUf 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 ofthe martingale hypothesis. These two approaches are quite different. Scheinkman andLeBaron (1986) tested for chaos using weekly returns on stocks taken from the Centerfor Research in Security Prices at the University of Chicago (CRSP). They devoted mostattention to an index of stocks but also considered a number of individual stocks. Theirresults are seemingly consistent with chaos.)

1. "The behavior ... seems to leave no doubt that past weekly returns help predict future ones ... Furtherit 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 fromnonlinearities as opposed to randomness and is not compatible with a theory that predicts that the returns aregenerated by independent random variables." Scheinkman and LeBaron (1986).

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554 REVIEW OF ECONOMIC STUDIES

The paper is organized as follows. Section 2 is concerned with some theoreticalbackground. First we give an indication of the restrictions needed to generate themartingale hypothesis in a standard economic setup. Then we discuss the approach ofSims (1984). That approach was developed in reaction to the restrictiveness of theassumptions required to generate the martingale hypothesis in a standard setup. Next adefinition of chaos is offered in order to iHustrate the sense in which chaos provides analternative to Sims (1984).

Section 3 is concerned with the testing methods. Our work is based on the correlationdimension and the Kolmogorov entropy. Since these are not yet standard tools in empiricaleconomic research we attempt to motivate their use. As the correlation dimension hasalready been analyzed by Brock (1986) and Brock and Dechert (1988) we focus moreattention on the Kolmogorov entropy. The empirical results are set out in Section 4, anda 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 modelsof Lucas (1978) and Brock (1982). In the Lucas model one derives a Euler equation forthe representative agent relating the marginal utility of consumption at date t to expectedmarginal utility of consumption at date t + 1. In order to obtain the martingale hypothesisin asset prices added structure is required. One needs to correct for dividends anddiscounting and then either assume risk-neutrality or else assume that there is no aggregaterisk.

Let Pt be the price of an asset at date t and let Et be the expectations operatorconditioned on the information available at date t. A process {Pt} is said to be a martingaleif Etpt+s = Pt for all s> O. The claim that asset prices constitute a martingale is frequentlyidentified in textbooks as the essence of the "efficient markets hypothesis". For theoreticalanalysis 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-Brockapproach 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 restrictiveassumptions. This led Sims to question whether the apparent empirical success of themartingale hypothesis is little more than a robust fluke. In place of the Lucas-Brockframework Sims (1984) provides an analysis in which the martingale hypothesis is obtainedfor very short time intervals. Lengthy time intervals need not satisfy the martingaleproperty.

Definition 2.1. A process {Pt} is said to be instantaneously unpredictable (I.U.) if

1· Et[pt+v - Et(pt+v)2] 1lmv-'oo 2 ~ a.s.

Et[pt+v - (Pt) ]

2. The consumer has constant relative risk-aversion and the joint distribution of consumption and returnsis lognormal.

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FRANK & STENGOS GOLD AND SILVER 555

If asset prices satisfy Definition 2.1 and there are stationary increments, then aregression of (PI+S - PI) on any variable known at date t will have an R 2 that approacheszero as s approaches zero. Changes over long time periods may be forecastable, butshort-term price changes should be unpredictable. In modern financial theory diffusionprocesses are often assumed. Diffusion processes satisfy the LV. property. Any processwith well-behaved derivatives will not satisfy the LV. property.

Sims (1984) provides a detailed discussion of the LV. property and a very strongargument for it as the basis of the empirical success of the martingale hypothesis. Heconcludes that "Except for the easily identified exceptional cases, neither the real worldnor an analytically manageable economic model is likely to generate security prices whichfail to be instantaneously unpredictable".

C. Chaos

In some respects "chaos" or "strangeness" is the polar opposite to a process that isinstantaneously unpredictable. There are a number of different definitions of chaos incurrent use. Not all of the definitions are equally practical for empirical research.

Definition 2.2. Let 0 be a space with metric d and let f: 0 ~ 0 be a continuousmapping defined on O. A discrete dynamical system (O,f) is said to be chaotic (orstrange) if there exists a (5 > 0 such that for all wE 0 and all 8> 0 there is Wi En and ksuch that d(w, w') < 8 but d(fkW,fkW') ~ O.

In this definition fkw denotes the k-fold iteration of point w by the map f Definition2.2 contains less than is frequently included in definitions of chaos. Devaney (1986) forexample includes the requirement that there exist dense orbits and that periodic pointsare dense. We do not take such a definition since the additional conditions do not seemto be verifiable nor refutable for empirical systems. There seems little point to includingconditions that one cannot check. In practice the existence of dense orbits must beassumed in any case in order to characterize the system. Our definition follows the usageof Eckmann and Ruelle (1985) who take "chaos" to be synonymous with "sensitivedependence on initial conditions". Brock (1986) defines chaos in terms of the largestLyapunov exponent being positive. The Lyapunov exponent definition is related to theKolmogorov entropy. The Kolmogorov entropy is a lower bound on the sum of thepositive Lyapunov exponents, see Eckmann and Ruelle (1985). The Kolmogorov entropyis discussed in Section 3B.

A chaotic system will be quite predictable over very short time horizons. If howeverthe initial conditions are only known with finite precision, then over long intervals theability to predict the time path will be lost. This is despite the process being deterministically generated. Typically for chaotic systems nearby trajectories locally separate exponentially fast.

The mathematical theory of chaos is currently an active research area, see Lasotaand Mackey (1985), Devaney (1986) and Guckenheimer and Holmes (1986). There area great many ways in which chaos might enter an economic system. Our work is not tiedto any particular entry mechanism. If evidence of chaos is found then it becomes anatural 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 I + I =4XI(1- XI) and let PHI = PI + (X, -0'5).Simulate this two equation system starting with XI E (0,1) and PI = toO. On the simulated data test (P'+I - P,) =uo+ ut(PI - PI-I) + £" One will be unable to reject a o= at = 0 and the R 2 will be very close to zero. Howeverthis example is anything but unpredictable. This example is discussed more fully in Frank and Stengos (1988).Also consider footnote 12.

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We have three alternative possible interpretations of the observed irregularity of assetprices. The Lucas (1978) framework considered in section A does not tie the martingalehypothesis to the size of the time intervals being employed empirically. Sims (1984)generates unpredictability explicitly for short time intervals. For lengthy time intervalsSims' theory permits the asset price changes to be predictable. The chaos interpretationleaves unspecified the economic mechanism which generated the data. It is consistentwith many possible theories including versions of the Lucas-Brock setup. Chaos is notconsistent with Sims' approach. If chaos is present then asset price changes will (at leastin 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 assetprices will not be predictable.

It is worth emphasising that these three approaches by no means exhaust the set ofconceivable theories of the observed irregularity of asset prices. These three approachesare considered together since each has been suggested previously as a possible interpretation. In rejecting one or more of these possible views we do not establish that a particularalternative is true. This familiar methodological point is particularly pertinent with respectto any suggestion of deterministic chaos.

3. TESTING METHODOLOGY

In this section we describe the two measures on which our empirical work is based. Thetheoretical underpinning is an assumption of ergodicity. Such an assumption is requiredif we are to use time averages as representative of the system's behaviour. The firstinvariant of the system is the correlation dimension. The second invariant of interest isthe Kolmogorov entropy. We take these in turn.

A. Correlation dimension

The correlation dimension is originally due to Grassberger and Procaccia (1983) andTakens (1983). For more detail than we provide see Brock and Dechert (1988) andEckmann and Ruelle (1985).

Start by assuming that the system is on an "attractor". An attractor is a closedcompact set S with a neighbourhood such that almost all initial conditions in theneighbourhood have S itself as their forward-limit set. In other words, these initialconditions 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 calleda strange attractor or else a chaotic attractor.

Consider a time-series of rates of return rl , t = 1,2,3, ... , T. We suppose that thesewere generated by an orbit or trajectory that is dense on the attractor. Use the time-seriesto create an embedding. In other words create "M-histories" as r~ = (rt , rl +1>"" rt+M-t).

This converts the series of scalars into a series of vectors with overlapping entries. If thetrue system which generated the time-series is n-dimensional,4 then provided M ~ 2n + 1generically the M-histories recreate the dynamics of the underlying system (there is adiffeomorphism 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 R 5

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established by Takens (1980). It is this result which permits the empirical work. Broomhead 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 theattractor by calculating the correlation integral, eM (e). For a particular embeddingdimension M, the correlation integral is defined to be

eM (e) ={the number of pairs (i,j) whose distance Ilr~ - rfl12 ell T 2• (3.1)

Here II· II denotes the distance induced by the selected norm. We use the Euclideandistance. The other distance function that is sometimes employed is the sup-norm. ByTheorem 2.4 of Brock (1986) the correlation dimension is independent of the choice ofnorm. In principle T should go to infinity, but in practice T is limited by the length ofthe available time series. This will in turn place limitations on the choice of B.

To obtain the correlation dimension, D M take

(3.2)

As a practical matter one searches to see if the values of D M stabilize at some valueDas M increases. If so, then D is the correlation dimension estimate. If however, as Mincreases the D M 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 D M is found thenthe system is substantially deterministic even if complicated. In principle an independentlyand identically distributed stochastic system is infinite dimensional. Each time oneincreases the available degrees of freedom, the system utilizes that extra freedom. Withfinite data sets, high dimensionality will be indistinguishable from infinite dimensionalityempirically, see Ramsey and Yuan (1987) concerning small data sets.

Two practical problems concerning € should be noted. If e is too large, theneM (e) = 1 and no information about the system is obtained. It is also possible for e tobe too small. With finite data sets there is a limit to the degree of detail that one maydiscern. This limitation on the ability to get a detailed focus means that even in principleone can never exclude the possibility of the system containing some degree of additivenoise. However, the test still can find out if there are substantial nonlinearities movingthe system. Empirically finding an appropriate range of values for e is not difficult forthese series.

There are several papers in economics that use the correlation integral. Barnett andChen (1988) used it to examine monetary aggregates, a low correlation dimension estimatewas obtained. Brock and Sayers (1988) investigated American macroeconomic time-series.They reject chaos but find some evidence of nonlinear structures. As previously indicatedScheinkman and LeBaron (1986) examined American stock market data and obtainedresults strikingly similar to those that we obtain. Since the number of related papers islarge and rapidly growing we do not carry out a full survey here. For an overview of theliterature see Frank and Stengos (1988).

B. Kolmogorov entropy

Dimension measures the degree of complexity of a system. Entropy is a measure of timedependence. The Kolmogorov5 entropy, K, quantifies the concept of "sensitive dependence 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".

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information is created. This is due to the following argument. Consider two trajectoriesthat are so close initially as to be indistinguishable to an observer. As time passes, thetrajectories may separate and become distinguishable. The entropy measures how rapidlythis happens.

For an ordered system, that is to say quasi-periodic or less erratic still, K = 0, whilefor an independent and identically distributed stochastic system K = +00. For a deterministic chaotic system 0 < K < 00 and K may be thought of as a measure of predictabilityof the system. Let T be the average length of time for which knowledge of the currentstate of the system (at specified accuracy) can be used to predict the future evolution,then K - T~l.

Since the Kolmogorov entropy is not yet a standard tool in the economics literature,we provide a formal definition to help clarify matters. Let (0, F, J.L) be a probabilityspace, with 0 the space of states of nature, F a sigma-algebra and p., a measure. We areinterested in partitions of (0, F, p.,) into disjoint collections of the elements of F whoseunion is n. Let A = {A j , A 2 , ••• , An} represent such a finite partition. There is a one-toone correspondence between finite partitions and finite sub-sigma-algebras. We can definethe entropy of the partition A by

(3.3)

and a In a = 0 when a = O. Hess (1983) provides an economic interpretation of K (A)where the particular partition is directly into the states of nature.

Next let f:O~O and f-kA j is the set of points mapped by fk into A j • If B={B}, B2 , ••• , Bk } is another finite partition of (0, F, p.,) then the "join" of A and B is

A vB = {A j n Bj Ii = 1,2,3, ... , n;j = 1,2,3, ... , k}. (3.4)

Now take Am = A v f-1A V f- 2A v· .. V f-m+1A which is a partition composed of piecesAi nf-1A j n' .. nf-m+1A j " with it E {I, 2, 3, ... , n}. Am is the "least common refinement"of the partitions A through f~m+lA. A partition f-kA is derived from A by iteration ofthe map f backwards in time k steps. The partition Am is the partition generated by Aover a time interval of length m.

We can now distinguish the entropy of a particular partition from the entropy of thesource K(A,f):

K(A,f)=limm->oo K(Am)jm. (3.5)

Petersen (1983) gives condition under which this limit exists. Now K(A,f) measuresthe average uncertainty per unit time about which part of the partition A, a trajectorywill enter next given the past history of that trajectory. K(A,f) is the entropy of themapping f with respect to the partition A.

So far not much has been asserted about the nature of the partition A. In a generalsetup the choice of partitions could themselves be economic issues. A poorly selectedpartition could induce poor decisions. For present purposes we simply take the entropyof the mapping f to be

K(f) = supremum A K(A,f), (3.6)

where A is from some feasible set of possible partitions. In K (f) we 4ave a measurementof the average uncertainty concerning where f sends the various points of O. Themagnitude of K (f) is a measure of chaos since it measures the extent to which f createsdisorder. For a derivation of some properties of K(f) see Petersen (1983).

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Between 1979 and 1981 there were several attempts to directly implement K(f). Asdiscussed by Eckmann and Ruelle (1985) these efforts proved problematic in part dueto the difficulty in implementation of the supremum. An empirical approximation ofK(f) has been suggested by Grassberger and Procaccia (1983b) which is much morereadily implemented. This approximation, denoted K 2 , is based on their correlationintegral eM (e). First define:

K 2(A) = -In 2:7=1 jL(A;)2.

Letting e represent the size of the partitions in A we then have

K 2(JL) = lim HO limm~oo KiA m)/ m.

How does K 2 relate to K (f)?

O~KiJL)~K(f)·

(3.7)

(3.8)

(3.9)

(3.10)

For an i.i.d. system K? will be infinite while for a chaotic system it will be positive but finite.Grassberger and Procaccia (1983b) suggest that K 2 is preferable to K. Typically K 2

and K are numerically close. The implementation of K 2 is simply to find

. . . ( eM(e) )K2=hmE~ohmM_oohmN_ooln CM+1(e) .

The practical limitations on the implementation of K 2 are much the same as forimplementation of the correlation dimension. The fact that the length of the time-seriesis finite is perhaps the most serious difficulty. Due to the finiteness of the time-series, asM increases, the correlation integral will be reduced to counting only the points themselves. Accordingly eM (e) and eM+1(e) will each converge to the same value. and thenIn 1 = O. Thus if one examines too large an embedding dimension then the estimatedvalue of K 2 will be biased towards zero. For further background concerning entropy,see Cohen and Procaccia (1985) as well as Eckmann and Ruelle (1985).

How is one to determine whether one is using "too large" an embedding dimension?Since there is no strong a priori theory to appeal to we took the following approach. Weused iterations of the "tent map" to create series with various degrees of complexity andwith the same length of time series as our data sets. Using this generated data we thencheck to see at what level of M the bias toward zero starts to take effect. We then limitattention to values of M below those at which the bias appears to become important.

The "tent map" is given by Xr +1= f(xr) where

f(x)=2x. XE[O,!], f(x)=2(1-x), xE[!,1]. (3.11)

The tent map is a one-dimensional map. If one uses a series of tent maps with the outputof one map fed into the next, then one can create systematically more complex maps.Such iterations are sometimes termed as the "extrapolation time". From Figure 1 we seethat as expected the bias towards zero takes effect earlier for greater extrapolation times.Also observe that K 2 does distinguish simpler from more complex objects as theorysuggests.

4. EMPIRICAL RESULTS

The data that we used are the daily prices for gold and ·silver. For gold we used theclosing price in London, England measured in U.S. dollars per fine ounce. The seriesstarts at the beginning of 1975 and runs through June 1986. The data was obtained from

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I. P. Sharp commodities data base identified as comdaily series EAUD. For silver wealso used the closing price in London, but starting at the beginning of 1974 through June1986. Silver is measured in British pence per troy ounce. The silver data also comesfrom I. P. Sharp commodities database and is identified as comdaily series EAGDSPB.

Following Sims (1984) the null hypothesis being tested has short time interval dataseries possessing the martingale property (LU.). However longer time interval series areallowed to be more predictable and consequently less likely to follow a martingale process.

To test this hypothesis we considered daily, weekly and biweekly series of gold andsilver rates of return. Consideration of weekly data series is useful as a check on whetherresults in the daily data might be due to a weekend effect. The null hypothesis is thatthe daily series should be more random than the weekly series while the biweekly datashould possess the. most (if any) structure. This theory also allows for the possibility thatthere will be no structure in any of these series. The traditional martingale hypothesisas in Section 2 implies that none of these series should have low dimensionality.

We first estimated the following equation for both the gold and silver series rates ofreturn and for each of the daily, weekly and biweekly time intervals

(4.1)

In (4.1) we have 't = (Pt - Pt-I)/Pr-l where Pt is the price of the asset at date t.According to the martingale hypothesis 130 = 131 = O. According to Sims (1984) for shorttime intervals these regressions ought to have R 2 values near zero.

Table I reports the results of the estimations of (4.1) for the various series.6 We alsoexamined the error structure of (4.1) and tested for the possible presence of ARCH(Autoregressive Conditional Heteroskedastic) effects. The ARCH model is a nonlinearstochastic specification due to Engle (1982). It is often fitted to economic time-series

TABLE I

AR (I) results for gold and silver

r, 130 131 R 2 A(6) A(12)

Gold (0) 0·0004 -0-0606 0·0003 88'43(S.E.) (0'3199) (0'0866)

Silver (0) 0·0284 -0,0609 0·0008 51·43(S.E.) (0'1419) (0,0793)

Gold (W) 0·0026 0·0668 0·0035 50-24(S.E.) (0·0253) (0'0045)

Silver (W) 0·0046 -0,0284 0·0057 46·89(S.E.) (0,0264) (0'0398)

Gold (BW) 0·0029 0·1093 0·0053 39·06(S.E.) (0'0208) (0,5843)

Silver (BW) 0·0051 -0,0192 0·0084 51.34(S.E.) (0'0370) (0'0567)

Notes. In this table 0 = daily series, W = weekly series, BW = biweekly series. Thecolumns beneath A(6) and A(12) give the results of ARCH tests for an ARCH (6) and anARCH (12) process. These were computed as nR 2 from regressing the squared residualsof the AR (l) regressions given above on their own 6 lags or 12 lags respectively_ And nis the number of observations. The critical values of X2 with 6 and 12 degrees of freedomat the 5% level are 12·59 and 21·03 respectively.

6. We also looked at longer AR specifications. All had R 2 values very close to zero.

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when these series are observed to be quite volatile, see Engle and Bollerslev (1986). Wetested for the presence of a general ARCH (p) process by obtaining the residuals fromthe AR (1) specification reported in Table I, and running a regression of the squaredresiduals on a constant and the squared residuals lagged p times. Then nR 2 is the teststatistic distributed as chi-square with p degrees of freedom, where n is the number ofobservations.

For the daily series p was found to be 12 for both gold and silver since any additionallags did not noticeably improve the test statistic, they only affected the degrees of freedom.Similarly, p was found to be 6 for both weekly and biweekly series. Table I reports theresults of these tests. In the appendix, Tables Al and A2present the maximum likelihoodestimates of the ARCH models for the relevant series.

As expected the results in Table I appear supportive of the martingale hypothesis.The finding of ARCH effects is also fairly common in financial time-series, see Engleand Bollerslev (1986). These results suggest that the returns on both gold and silver arefairly typical of financial data.

A. Correlation dimension estimates-Daily data

We have already indicated that a certain amount of judgement is required to measurethe correlation dimension. One must choose reasonable values for e and M. For ourdata the relevant range7 for values of e was between (o'9fo and (0'9)33.

For each embedding dimension one plots In C M (E) against In c. Over the relevantrange of values of e one calculates the slopes by ordinary least squares. In Table II theresults of these calculations are reported for embedding dimensions M = 5, 10, 15,20,25.We stop at an embedding dimension of 25 since the system did not seem to use the extrafreedom available to it when higher embedding dimensions were tried. It should also benoted that an embedding dimension of 25 is also the highest embedding dimension beforethe Kolmogorov entropy estimates started to be biased towards zero for the tent mapwith an extrapolation time of ten. At an embedding dimension of 25 both the gold andthe silver series have a correlation dimension between 6 and 7. Column R of Table IIreports the results for estimates from a computer-generated series of length 3000 observations with the same mean and variance as the gold series. The computer-generated randomnumbers are higher dimensional than the actual asset rates of return.

TABLE II

Correlation dimensions for daily data

M G S R ARCHG ARCHS SHG SHS

5 1·11 2·01 3·15 1·25 1·13 2·85 3·0110 2·45 3·91 5·42 2·23 3·58 5·12 5·25IS 3·83 5·32 8·91 4·51 4·91 7·01 6·7720 5·41 6·31 11·52 5·82 6·22 9·51 9·8125 6·31 6·34 13·48 6·65 6·71 11·27 10·59

Notes. M = embedding dimension, G = daily gold series, S = daily silver series, R = computer-generated(pseudo-) random numbers, ARCHG = residuals from ARCH (12) of the gold series, ARCHS = residuals fromARCH (12) of the silver series, SHG = reshuffled ARCHG series, SHS = reshuffled ARCHS series.

7. A more detailed presentation of our calculations can be found in an earlier version of this paper whichcirculated as Discussion Paper # 1986-13, Economics Department, University of Guelph.

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We next carried out Brock's (1986) Residual Test. If a series is generated bydeterministic chaos then the residuals from a linear, or smooth nonlinear transformationof the data, should yield the same correlation dimension as the original series. Toimplement this test we took the residuals from the ARCH (12) specification for both goldand silver series and calculated their correlation dimensions. As reported in Table II atan embedding dimension of 25 the correlation dimension estimates again turn out to bebetween 6 and 7. The daily series appears to pass Brock's Residual Test.

Scheinkman and LeBaron (1986) have proposed a "shuffle diagnostic" for chaos.One recreates the data series by sampling randomly with replacement from the data untilyou have a shuffled series of the same length as the original. The shuffled series shouldbe considerably more random than the original data if the original data is strange. If theoriginal data is not strange but rather is independently and identically distributed thenthe shuffling will leave the correlation dimension unchanged since there was no structurepresent to be destroyed by shuffling.

To carry out the shuffle diagnostic we resampled from the ARCH residuals using auniform pseudo-random number generator. If the ARCH procedure picked up the relevantstructure then the reshuffled data will have the same correlation dimension. In Table IIestimates are reported for the typical reshuffled series. The dimension estimates are raisedsubstantially.

Motivated by the methodology of "bootstrapping"g we attempted to generate anempirical distribution for the shuffle diagnostic. We performed repeated shuffling of theoriginal series as well as for the ARCH residuals. At an embedding dimension of 25 weperformed 30 replications of each shuffle. For each of these 30 replications for both theoriginal and the ARCH residuals we calculated the correlation dimensions. The rangeof estimates thereby obtained is reported in Table III. Evidently, at M = 25 all of the

TABLE III

Correlation dimension (D) estimates at M = 25

Series name Unshuffled 0 Maximum shuffled 0 Minimum shuffled 0

Daily DataG 6·31 11·44 9·23S 6·34 11·71 9·12

ARCHG 6·65 12·01 9·45ARCHS 6·71 11·35 9·43

Weekly DataG 6·47 11·62 8·98S 6·49 11·50 8·38

ARCHG 6·59 11·18 8·15ARCHS 6·79 11·38 8·44

Biweekly dataG 6·62 10·27 8·32S 6,38 10'55 8'11

ARCHG 6·66 10·71 8·44ARCHS 6·73 10·44 8·30

Notes. For each series we report the unshuffled dimension estimate as well as the maximumand minimum values obtained from 30 reshufflings of the data. G = gold series. S = Silver series.ARCHG = residuals from the ARCH (12) gold series for the daily data and from ARCH (6) goldseries for the weekly and biweekly data. ARCHS = residuals from the ARCH (12) silver seriesfor the daily data and from ARCH (6) silver series for the weekly and biweekly data.

8. See for example Bickel and Friedman (1981).

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original estimates are below the estimates from the shuffled series. Since the intent ofthe bootstrap method is to permit confidence interval type statements these results arestrong. It seems that there is considerable structure remaining in the data even after useof the ARCH procedure.

B. Correlation dimension estimates- Weekly and biweekly data

To create the weekly and the biweekly time-series we took data from Wednesdays andfrom alternate Wednesdays, respectively. The results of the AR (l) regressions are reportedin Table 1. As with the daily data these results appear to be consistent with the martingalehypothesis and with Sims (1984) approach. It would appear from the results in Table Ithat even biweekly data might represent a "short" time interval in the sense of Sims (1984).

Table IV presents the various correlation dimension estimates for both weekly andbiweekly data. The results again indicate dimensionality between 6 and 7 and that suchdimensionality is not accounted for by an ARCH structure. These series again seem topass Brock's Residual Test. In Table III the outcomes of 30 replications for each seriesof the shuffle diagnostic are reported. For all cases the shuffled series were of higherdimensionality than the unshuffled series.9

TABLE IV

Correlation dimensions for weekly and biweekly data

M G S R ARCHG ARCHS SHG SHS

Weekly5 1·43 1·56 3·12 1·14 1·07 2·71 2·63

10 3·49 3·83 5·70 2·28 2·30 5·03 4·9515 3·99 4·55 9·50 4·27 3·81 7·12 6·9120 5·41 5·95 12·11 5·78 5·51 9·11 9·2325 6·47 6'49 14'01 6·59 6·79 10·80 11·01

Biweekly5 1·01 1·64 3·01 1·21 1·71 2·65 3·01

10 2·72 3·08 6·25 2·81 3·01 4·93 5·0115 4·31 4·92 10·01 4·71 5·31 6·91 6·8320 5·82 5·37 12·45 5·61 5·85 8·12 8·32 -25 6·62 6·38 13·75 6·66 6·73 9·48 10·31

Notes. M = embedding dimension, G = gold series, S = silver series, R = computer-generated (pseudo-) randomnumbers, ARCHG = residuals from ARCH (6) of gold series, ARCHS = residuals from ARCH (6) of silverseries, SHG = reshuffled ARCHG series, SHS = reshuffled ARCHS series.

The results of this analysis offers quite a clear pattern. Weekend effects are notresponsible for the structures detected in the daily data. The data possess a structurethat is somewhere between 6 and 7 dimensional. This structure is not accounted for bythe ARCH procedure. The structure is as evident in the weekly and biweekly data as itwas in the daily.

9. Sims (1984) notes that in moving to longer time intervals one is also typically moving to fewerobservations. One does not want the number of observations to be responsible for one's empirical results. Wetherefore also conducted the analysis for shorter daily series consisting of the last 600 observations for bothgold and silver series. This length of series agrees with the actual length of the weekly data series. The correlationdimension estimates obtained from these shorter daily series were not noticeably different from the originaldaily series. The number of observations is not responsible for our findings.

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C. Kolmogorov entropy estimates

As previously indicated we calculate the Grassberger and Procaccia (l983h) approximation to the Kolmogorov entropy, denoted K 2 • Recall the interpretation of K 2• It measureshow strange a time-series is. If a time-series is completely random then K 2 = 00. If atime-series is completely smooth then K 2 = O. Entropy measures the rate at which indistinguishable paths become distinguishable when the system is observed with only some finitelevel of accuracy. The lower the value of K 2 the more predictable the system is, at leastin principle.

Empirically, for daily data, for gold our estimate of K 2 is 0·15±0·07. For daily silverdata the value of Kz is 0·19 ± 0·09. These values are not too different from each otherand their ranges overlap. For weekly data we get that Kz is 0·20± 0·04 for gold and0'22±0'05 for ~ilver. The corresponding estimates for the biweekly data were 0'24±0'03for gold and O'24±0'05 for silver.

A variety of comparison series were examined for K 2 • Comparison with Figure 1shows that at an embedding dimension of 25 the empirical findings are similar to a tentmap at an extrapolation time ranging from 3 to 6. They are lower than the tent mapiterated ten-fold. We also examined iterated versions of the Henon map and the logisticmap. Their behaviour was similar to the iterated tent map in two respects. Higher numbersof iterations led to higher estimates for K 2 • With 3000 observations the bias towards zerostarts to take effect at an embedding dimension of about 25.

---/=3

--/=6

-/=100.5

0.4

0.3

0.2

0.1

I = Iteration of the Tent Map

_::::=::=::.;;::;.:::=::.;;:.:::.=::::,:._c:=:_==_,- ......._---- ..... ------

0.0

16 20 24

Embedding Dimension

28 32

FIGURE 1

Simulation results for the Kolmogorov entropy approximation (K 2 )

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In consideration of the dimension, shuffling provided a useful diagnostic. Shufflingdid not work so well for the Kolmogorov entropy estimates. It led to numerically sensitiveresults. Small changes in e and the embedding dimension produced fairly large changesin the estimates for K 2•

The K 2 estimates bolster the results of the correlation dimension estimates. Onceagain the patterns are consistent with the chaos interpretation of these time-series. Thesefindings may also be consistent with some nonlinear stochastic processes.

5. CONCLUSION

In this study we examined certain properties of the rates of return on gcJd and silverfrom the mid 1970's to the mid 1980's. Using familiar econometric techniques it wasshown that, as expected, neither series caused one to reject the martingale hypothesis.We examined daily, weekly, and biweekly series.

While the martingale hypothesis is not rejected by the standard techniques, this resultwas shown to be misleading. By estimation of the correlation dimension and the Kolmogorov entropy, evidence of structures not captured in the usual approach was found.This evidence calls into question Sims (1984) rationalization of the empirical success ofthe martingale hypothesis. lo Our evidence is consistent with certain of the earliest testsof the efficient markets hypothesis. II Over all there appears to be some sort of nonlinearprocess of between 6 and 7 dimensions which generated the observed gold and silverrates of return. Such a structure might be rationalizable within the Lucas- Brockframework, but not in the currently available versions. 12 The findings seem to be robust.They require a satisfactory theoretical explanation.

10. In his discussion Sims (1984) takes one week as an illustrative example of a small time interval. Ourresults show that if Sims' theory works it must be on a scale in which a small time interval is less than one day.Over large time intervals Sims (1984) neither predicts nor precludes the presence of structure. Sims takes ayear as an illustrative example of a large time interval.

II. " ... it is possible to devise trading schemes based on very short-term (preferably intra-day but at mostdaily) price swings that will on average outperform buy-and-hold". Fama (1970). Fama's discussion is concernedwith linear dependence and he argues that trading costs make such schemes unprofitable due to frequent tradingcosts. Present analysis indicates that nonlinear dependence may be more to the point.

12. Take the model in Brock (1982) and replace the stochastic shock process that affects the productionfunctions by a chaotic deterministic process. Agents knowing this would be unable to obtain excess utility ofprofits despite the advance knowledge of the changes. Such an approach focuses attention on the question ofwhat economic forces induce the chaotic process in the first place.

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APPENDIX

TABLE Al

ARCH results from the Table I estimation for gold

Standard Standard StandardGold (D) error Gold(W) error Gold (BW) error

f30 0·0002 (0,0004) 0·0016 (0,0014) 0·0019 (0'0021)f31 -0·0473 (0'0513) 0·0568 (0'0413) 0·1093 (0,5736)a o 0·0447 (0'0320) 0·1344 (0'0420) 0·2732 (0'0586)a l 0·0788 (0'0331) 0·1677 (0'0430) 0·0a 2 0-0621 (0-0342) 0·0446 (0-0473) 0-0036 (0,0064)a3 0-0741 (0-0574) 0·0 0; 1180 (0,0606)a4 0-0198 (0-0318) 0·1250 (0-0356) 0-0828 (0,0668)as 0-0 0·0101 (0-0431) 0-2577 (0'0589)a 6 0-0214 (0'3281) 0·0320 (0-0444) 0-0132 (0'0523)a 7 0-0248 (0'4443)a 8 0·1632 (0-0317)a9 0·0004 (0'0321)a to 0·0581 (0,0255)all 0·0247 (0,0322)a\2 0·0345 (0'0333)

log L 2790·08 1265-64 563-81

Note_ The a terms represent the coefficients of the ARCH covariance structure_ Lower bounds of zero areused. A zero entry means that the model included the corresponding variable but the coefficient reached itslower bound of zero. The ARCH model following Engle (1982) is given below. Let X, be the vector ofexplanatory variables for the dependent variable and let the error term E, - N(O, h,). Then y, = X~ y + E, andh,=ao+I):1 ajE~_j'

TABLE A2

ARCH results from the Table I estimation for silver

Standard Standard StandardSilver (D) error Silver (W) error Silver (W) error

f30 0·0002 (0'0007) 0·0036 (0'0026) 0·0041 (0,0041)f31 -0,1109 (0' 3148) -0·0284 (0'0398) -0,0172 (0'0567)ao 0-0026 (0'0005) 0·2420 (0'0447) 0·2356 (0'0582)a l 0·1833 (0-0333) 0·0774 (0-0416) 0·1049 (0'0432)a 2 0-0145 (0-0345) 0·0169 (0'0423) 0·0034 (0'0578)a 3 0-0866 (0-0415) 0·0394 (0-0514) 0·0152 (0'0073)a4 0-0221 (0-0367) 0·0179 (0-0053) 0·2805 (0-0573)as 0-0317 (0-0327) 0·0097 (0-0035) 0·0a6 0·0 0·0073 (0-0041) 0·0053 (0,0054)a 7 0-0671 (0'0324)a 8 0·0413 (0,0334)a9 0·0729 (0,0354)

a lO 0·0368 (0'0388)all 0·0801 (0-0324)a\2 0·0

log L 5124·98 3253·41 1121·21

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Acknowledgement. We would like to thank Richard Arnott, William Brock, Roger Farmer, David Fowler,Clive Granger and Jose Scheinkman for valuable discussions. Lars Hansen's criticisms of an earlier draft wereextremely helpful. The comments by the referees are appreciated. This research was partly supported by agrant from the Research Excellence Program of the University of Guelph. We are grateful to all of theseindividuals, but none of them are responsible for any remaining deficiencies in our work.

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measuring the strangeness of gold and silver rates of return

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