Employing conditional variance processes to examine the market efficiency of the gold rates of return

J ECO BUSN

355

1994; 46:355-365

Employing Conditional Variance Processes to Examine the Market Efficiency of the Gold Rates of Return Michael Marshall and Thanasis Stengos

The purpose of this paper is to examine the efficiency of spot gold rates of return for daily, weekly, biweekly, and monthly data. Market efficiency is tested in the context of Sims' instantaneous unpredictability property. We apply generalized conditional heteroskedasticity models to capture the conditional means and variances of the gold rates of return in the different frequency series that we examined and we also tested for out-of-sample performance. We found no evidence of out-of-sample forecastability in the series.

Introduction The purpose of this paper is to employ the autoregressive conditional heteroskedasticity (ARCH) model of Engle (1982) and its generalized version (GARCH) of Bollerslev (1986) to estimate time series models for the daily, weekly, biweekly, and monthly spot rates of return on gold for the period between the beginning of 1975 to the middle of 1993. The estimated models are then used to test the martingale version of the market efficiency hypothesis (EMH) as put forth by Sims (1984). The hypothesis of market efficiency suggests that asset prices are determined by the interaction of rational agents. In addition, one usually also requires that publicly available information cannot he used to construct profitable trading rules. In other words, market efficiency has come to he associated with the notion that information acquisition by individuals is a futile activity. Evidence from studies on variance bounds [see LeRoy (1989) for references] and from such studies as Fama and French (1988b) have suggested that rates of return have zero autocorrelations over short and long time intervals, hut negative ones over intermediate ones. This finding agrees with Sims (1984), who argues that the martingale model is consistent with "instantaneously unpredictable" processes, a property satisfied by many stochastic processes. Department of Economics, University of Guelph, Guelph, Ontario, Canada. Address reprint requests to Thanasis Stengos, Department of Economics, University of Guelph, Guelph, Ontario, Canada, NIG 2W1. Journal of Economicsand Business © 1994 Temple University

0148-6195/94/$07.00

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M M,~rshall and T. S~vm.',, Contrary evidence to the foregoing findings arc to be found in t~o ~-cc~m~ sources of the literature. Lo and MacKinlay (1988) found that weekly and montht~ stock returns had positive autocorrelations contradicting the assertion of zer~ autocorrelations suggested previously. Additional evidence on this has been pr~> vided by studies that use tools from nonlinear dynamics to test for nonlinear dependence in the series, such as Scheinkman and LeBaron (1989), Frank and Stengos (1989), and Hsieh (1989). These findings raise the possibility that systematic and usable nonlinear structure in the rates of return has yet to be accounted for. Brock (1988) discusses from a theoretical point of view some of the issues associated with the presence of nonlinearities in finance. It is only dependence in the first moment that violates the martingale hypothesis. Dependence in the higher conditional moments would only violate the more restrictive random walk hypothesis. Empirical studies that employ nonlinear tests to look for dependence in the entire conditional distribution of the process [see Frank and Stengos (1989)] may have uncovered dependence of higher conditional moments and as such they may not provide contradictory evidence to the martingale hypothesis. It is therefore important to try to identify the source for the possible violation of the independence assumption of the time series process under investigation using these nonlinear diagnostics. A rejection of the independently and identically distributed (i.i.d.) hypothesis could conceivably stem from time dependence of the conditional variance of the series, and in that case one could not reject the martingale hypothesis. Thus, a correct modeling process should not only incorporate volatile price changes but also should allow for this volatility to be time dependent. The ARCH process is one such process capable of incorporating these two requirements. In fact, there is a tremendous amount of evidence that suggests that higher conditional moments are correlated. For example, timevarying volatility has been captured by the ARCH and GARCH models of Engle (1982) and Bollerslev (1986). McCurdy and Morgan (1987) used the GARCH model to investigate the time series of several exchange rates and found support for the martingale hypothesis. Following Frank and Stengos (1989), we will conduct a thorough investigation of market efficiency in the gold market. Because of the diversity of gold uses (store of value, medium of exchange, value in production, and aesthetic value) and its importance in society both historically and at present, the question of efficiency in the gold market is an important one; see Jastram (1977) for a historical perspective. With new supply and demand for gold accounting for only approximately 1% of its existing stock annually, its current price is therefore heavily influenced by future spot price expectations. Thus, possible sales of large holdings of government-owned gold only add to the economic uncertainty and hence will cause expectations to be more unstable than is usually the case [see Fama and French (1988a)]. The implied price instability of gold prices puts the study of market efficiency in the gold market on a different footing than the study of market efficiency for other assets. To test for the presence of nonlinearities in gold rates of return, we employ a number of standard diagnostics including a recently developed nonparametric test by Brock, Dechert, and Scheinkman (1987), hereafter referred to as the BDS test. We proceed to estimate the most parsimonious t3ARCH specification for each of the series at our disposal by subjecting each specification to a battery of diagnos-

Efficiency of Gold Returns

357

tics. Then we generate out-of-sample one period ahead forecasts in order to assess the forecasting ability of the chosen models. The results suggest that there is no forecasting ability present and they seem to confirm the evidence of Prescott and Stengos (1990), who used nonparametric kernel regression estimates to generate out-of-sample forecasts for the weekly spot rates of return gold series. Using nearest neighbor techniques, Frank and Stengos (1993) find a similar lack of forecastability for other precious metals daily series for the period 1975 to 1985. The paper is organized in the following way. The next section briefly discusses Sims' (1984) exposition of the martingale hypothesis, referred to as the property of instantaneous unpredictability as it applies to the gold market. Then we proceed to discuss the BDS test, which we use to test for the presence of nonlinear structure in the residuals. In the following section we report the empirical findings of in-sample estimation and out-of-sample forecasts. Finally, we present concluding remarks.

Market Efficiency: Martingale and Instantaneous Unpredictability We define a martingale in the following way. Let p, be a suitably transformed gold price to account for cumulated dividends discounted back to the present. For the case of gold, dividends are negligible. Let f~,_ 1 be the information set available at time t - 1. If rates of return r, are defined as the sum of dividend yield plus capital gains minus one, r, follows a martingale difference equation if E(r, lfl,_~) = O.

(1)

The formal relationship between the martingale and the efficient market hypothesis has been explored by Lucas (1978) and Brock (1982). Sims (1984) has also provided an explanation of the empirical success of the martingale hypothesis, such as documented by Fama (1970). Sims (1984) suggests that the martingale hypothesis is valid for short time intervals; however, lengthy time intervals need not satisfy this hypothesis. This is coined as instantaneously unpredictable (IU) and it is defined in the following manner:

Definition. A process Y~ is IU if and only if 1 almost surely.

Sims (1984) purports that if a process satisfies this definition, then regressing Y,+~. - Yt on any variable known in fl, will yield goodness of fit measures that approach zero as t' approaches zero. If Y, is taken to be the logarithm of the gold spot price, then Y,+~. - Y, denotes the rate of return. In this case, regressions of Yt+~.- Yt on variables in 1] t should produce goodness of fit measures that are closer to zero with higher frequency data than they are with lower frequency data. In the paper we will examine Sims' IU proposal as the main testable hypothesis in question for the case of gold returns.

M. Marshall and T. Stengo:~

358

A feature of gold is that the return distribution departs significantly from normality. Solt and Swanson (1981) showed that when compared to the normal distribution, weekly gold returns (January 1971-December 1979) were peaked and leptokurtic (fat-tailed), because of the odd number of extraordinarily large and small returns. Also, Aggarwal and Soenen (1988) showed that the distribution of daily gold returns (January 1975-December 1982) is positively skewed and leptokurtic when compared with the normal. Prescott and Stengos (1990) and Frank and Stengos (1989) found daily, weekly, and biweekly gold returns as well as the residuals from linear models and the standardized residuals from some simple ARCH conditional variance processes to reject the hypothesis of i.i.d. The high degree of heteroskedasticity found in gold returns by Solt and Swanson (1981), along with the nonnormai distribution, render linear estimation techniques far from satisfactory. The (G)ARCH models have mixed reviews as to their ability to account for all of the leptokurtosis present in asset prices; see Baillie and Bollerslev (1989) and McCurdy and Morgan (1988). However, it is clear that the standardized (G)ARCH residuals are closer to normal than the residuals from a linear process. The ARCH(q) model specifies the conditional variance of e t as a linear function of q lags of the squared innovations. The generalized ARCH (GARCH) model of Bollerslev (1986) extends the ARCH model by specifying the conditional variance h t as a linear function of its past values and lagged squared innovations. Thus, the conditional variance of e.t of a GARCH(p, q) model is parametrized as q

P

h, = oto + E ~ie,~i + E fljh, ;, i=l

j

(2)

I

where a 0 > 0, ai > 0 Vi, fli > 0 Vj, and the characteristic roots of (1 - Eai E / 3 j ) lie outside the unit circle. .Through substitution of ht_ j, it can be seen that a GARCH process is in e s s e n c e an infinite order ARCH process. Thus, a low order GARCH model will have properties similar to that of a high order ARCH process without problems of estimal~ing many parameters subject to nonnegativity constraints. Typically when applying GARCH models to financial data, the order is very small, usually p _< 2, q<2.

Testing for Nonlinearities The BDS statistic is designed to test the null hypothesis that a time series is i.i.d. against a variety of alternative nonlinear hypotheses. We will discuss briefly its structure. Let {xt: t = 1, 2 , . . . , T} be a sequence of observations that are i.i.d. From this series, construct the m-dimensional vector, or "m-history"

x m = (x,,

x,÷ ~.....

x,÷m_

1 ).

Efficiency of Gold Returns

359

Using these m-histories, we can compute th6 following quality, known as the sample correlation integral: 2 C,,,(e) = T , . ( T m _ 1) ]~I,~(x'~,x'~),

(3)

t
where Tm = ( T - m + 1) and I,(x, z) is an indicator function that equals unity if Ix - zl < e. Here I" l is the supnorm. The sample correlation integral measures the proportion of the m-dimensional points that are "close" to each other, where close is defined in terms of the supnorm criterion. It can be shown that if {x t} is i.i.d, with a nondegenerate density function, then for fixed m and e, C,,(e, T ) ---, [Cl(e)] m with probability 1, as T ~ oo. Furthermore,

¢T(Cm(e, T) - Cl(oe , T) m) ~ N(O, Vm(~,)). Brock, Dechert, and Scheinkman (1987) derived the expression for the variance term Vm(E), which allows the following asymptotically normally distributed test statistic (the BDS statistic) to be calculated as Win(e, T ) = f I ' ( C m ( Z , T ) - Cl(,~ , T)m)/~/Vm(e).

(4)

It is clear from the form of the BDS statistic in equation (4) that, in practice, numerical values must be assigned to the two parameters ~ and m. For a given value m, e should not be too small; otherwise, that sample correlation integral will capture too few points. Similarly, e should not be chosen too large. Because there is no unique choice for these two parameters, users report a number of statistics. Although these statistics are not independent, a battery of significant BDS statistics does provide strong evidence against the null hypothesis. 1 Monte Carlo simulations by Brock, Dechert, and Scheinkman (1987) provide evidence that the BDS statistic has good power against a variety of nonlinear alternatives. More recently, extensive simulations by Brock, Hsieh, and LeBaron (1991) indicated that the BDS statistic has good size and power characteristics even in moderately sized samples. Moreover, the statistic has good power against a wide variety of nonlinear alternatives, including test map chaotic processes and stochastic processes, such as autoregressive, threshold autoregressive, nonlinear moving average, and ARCH. However, the test statistic seems to lack power when applied to standardized residuals from (G)ARCH processes; see Brock, Hsieh, and LeBaron (1991). We conducted a simple Monte Carlo simulation where the BDS is applied on the standardized residuals of a GARCH(1, 1) process for 500 and 1000 observations over 2000 replications. The critical values we obtained were below the asymptotic ones. Table 1 reports the critical values from the experiments that were performed.

1The null hypothesis is accepted at the 1% and 2.5% significance levels for the monthly spot price and spot return, respectively.

360

M. Marshall

Table 1. B D S Critical V a l u e s from S t a n d a r d i z e d G A R C H ( 1 ,

1)

and

T. Stengos

Residuals

Quantiles ( % ) m, e a

1

2.5

97.5

99.0

3, 1

-~ 1.789

- 1.460

1.423

5, 1

-- 1.511

- 1.322

1.411)

1.801

10, 1

- 1.920

- 1.611

2.12(1

2.752

3, 1.5

- 1.610

- 1.333

1.332

1.650

5, 1.5

- 1.440

- 1.234

1.383

1.771

10, 1.5

- 1.812

- 1.515

1.970

2.451

3, 1

- 1.630

- 1.451

1.400

1.781

5, 1

- 1.461

- 1.201

1.210

1.491

10, 1

- 1.650

- 1.431

1.731

2.202

3, 1.5

- 1.570

- 1.410

1.380

1.710

5, 1.5

- 1.440

- 1.181

1.170

1.771

10, 1.5

- 1.602

- 1.371

1.661

2.151

F = 500

T=

1.790

1000

am ------embedding dimension; e ~- neighboring parameter, which is measured in standard deviations of the respective series.

Empirical Results The data that we used came from the I. P. Sharp daily commodities data base called Comdaily, series EAUD. It is the closing price (p,) of gold in London, England, denominated in U.S. dollars per fine ounce and runs from January 1, 1975 to April 30, 1993. Let us define Yt to be the natural logarithm of Pr From the daily prices, daily (d) rates of return (rd, t) were constructed by taking the day to day changes in the logarithm of the price (Yt). The weekly (w), biweekly (b), and monthly (m) rates of return (rw,t, rb, t, rm, t) were constructed by taking every Wednesday, every other Wednesday, and every fourth Wednesday's change in the logarithm of the price, respectively. In the data set there were 11 Wednesdays in which the market was closed. When this occurred, two Tuesday, seven Thursday, and two Friday observations were used instead. We tested for the stationarity of the rates of return using the augmented Dickey-Fuller (ADF) test statistic. The results show that the natural logarithm of gold prices in all series were integrated of order 1 [I(1)]. However, the first difference in the natural logarithm of the price (i.e., the rates of return) for all series were found to be I(0). Lagrange multiplier (LM) test statistics for the null hypothesis of no autocorrelation for the residuals of the ADF regressions suggest that there is no residual autocorrelation in the preceding regressions. 2 The next thing that was examined was the BDS statistic for the raw data of all series. These results are presented in Table 2, where it can be seen that none of the series is found to be i.i.d.

2 T o save space we do not report the detailed regression estimates for the daily series. The daily B D S statistics are reported in Table 2 and the rest of the diagnostics in Table 4. The detailed daily series regression results are available from the authors on request.

361

Efficiency of Gold Returns Table 2. BDS Statistics for all Series, 1975-1993 Raw series

OLS residuals

Standardized G A R C H residuals

10.853 13.194 7.406 3.924

10.336 13.337 7.822 4.165

0.579 2.846 2.060 - 0.921

19.605 26.032 11.992 6.743

18.445 26.396 12.581 6.894

1.03 2.275 0.972 - 0.574

10.2.87 10.801 7.215 4.879

10.152 10.822 7.625 5.085

0.694 2.369 2.445 - 0.805

14.159 15.321 9.758 6.155

13.906 15.485 10.352 6.470

0.733 1.877 1.454 - 0.274

m=5, e=l a Daily Weekly Biweekly Monthly m = 10, ~ = 1a Daily Weekly Biweekly Monthly m = 5, e = 1.5 a Daily Weekly Biweekly Monthly m = 10, e = 1.5 a Daily Weekly Biweekly Monthly

am -= embedding dimension; e -~ neighboring parameter is measured in standard deviations of the respective series. The critical values for the BDS statistics for the 200 series and the OLS residuals is 1.96 at the 5% level. For the GARCH standardized residual series, the critical values are given in Table 1.

All the series are modeled as autoregressive (AR) type processes with differing (G)ARCH specifications. The lag length for each series was determined using Akaike's information criterion (AIC), Schwarz-Bayesian information criterion (SBIC), and likelihood ratio (LR) tests. The daily series was modeled as 3

5

rd, t = 190 t- E pira,t-i + E ~jOj, t + Ca,t-

i=1

(5)

j=l

In the case of the OLS estimation of equation (5), we assume that ea, t is zero mean i.i.d., whereas in the case of the GARCH specification, we model ea, t as a GARCH(1, 1) process. The specifications of the conditional means of the gold rates of return for the reported series are characterized by an AR(1) process with a constant and can be represented by rj, t = Po + plrj, t-1 + 5 , t ,

Vj = w,b,m.

(6)

In the case of OLS estimation of (6), it is assumed that ej, t is i.i.d, with mean zero. The (G)ARCH specifications of the weekly, biweekly, and monthly series were found to be GARCH(1, 1), GARCH(1, 1), and ARCH(4), respectively. The results of these series and diagnostics checks are reported in Tables 3 and 4, respectively. A GARCH specification was found to be the best choice for the weekly and biweekly series. When the Engle test was performed on the OLS residuals as far back as the twelfth and the sixth lag in the biweekly and monthly series, respectively, it was found to be significant. Furthermore, as ARCH(4) specification for the

362

M. Marshall and 1. Slcng~> Table 3. L i n e a r and N o n l i n e a r E s t i m a t e s for Weekly(w), Biweekly (/~ ), and M o n t h l y I m t G o l d R a t e s o f R e t u r n for the P e r i o d ,lanuary 1975- April 1993 Cocff. t3. (std. error) [~

(;ARC11 tl.l)(w)

OI,S (w) (k000723 0.000943 0.02848 0.032394

- (k000107 (k000099 (k021558 0,(134792 0.t100007 (k~}00003 11.07411 0.007829

5o 51

( ;A RCft (I,l)(b)

OLS (hi (I.1i01309 0.001924 0.074929 0.045771

(k0(111252 0,001455 11.(}774~ 0.04806 0.(l(100(15 0.( 1001)07 i).063125 0.012(188

(JAR( '1 t II,I){m}

()LS (m) !1.002613 0.003986 0.078621 11.1164917

0.0020l)2 (L0035t)9 ().(179211~ (I.1176il2t 0.0013g I 0.0(103114 0.22508 0.095826 0.081644 0.071}957 0.(163193 0.1166406 0.2705(17 (I. 1(i255

&2 &3 &a (~l R2 L" Tb

0.00081 2022.36 955

0.91973 0.0090484 0.00081 2159.04 954

0.934332 0.01131 0.00561 898.066 476

0.00561 836.231 477

0.006177 327.271 238

0.006177 346.747 234

aL is the log of the likelihood function. bT is the sample size for w = weekly series, b = biweekly series, m = monthly series for the period January 1, 1975 to April 30, 1993.

T a b l e 4. D i a g n o s t i c C h e c k s for Daily, W e e k l y , B i w e e k l y , a n d M o n t h l y R e t u r n s for 1975-1993 a

OLS (d) d.f. GARCH(1, 1) (d) d.f. OLS (w) d.f. GARCH(1, 1) (w) d.f. OLS (bw) d.f. GARCH(1, 1) (bw) d.f. OLS (m) d.f. ARCH(4) (m) d,f.

BG(5, 10)

Q(5,10)

Q2(5,10)

SK

KU

BJ

H

E

23.47 10 14.818 10 19.731 10 11.703 10 18.493 10 6.044 10 12.631 5 7.454 5

23.435 7 14.781 7 20.199 9 111.91 9 19.194 9 6.173 9 13.741 4 6.919 4

2720.7 7 13.026 7 242.294 9 9.546 9 119.156 9 11.168 9 86.233 4 4.177 4

2.725 1 24.29 1 40.014 1 1.091 1 0.649 1 2.574 1 9.23 1 1.575 1

22252.5 1 3027.9 1 1406.7 1 243.975 1 268.243 1 39.738 1 80.459 1 11.247 1

2~255.3 2 3052.2 2 1446.7 2 245.066 0 268.892 2 42.312 2 89.689 2 12,822 2

1026.5 43

1076.3 211 2.408 1(1 137.81 20 4.516 10 58.989 10 8.854 10 55.036 5 3.449 5

aTen lags are used for BG, Q, and Q2 for the weekly and biweekly series and 5 monthly series. Q(5,10) is the Ljung-Box (19781 portmanteau test on the first (5, 10) lags function; Q2(5,10) is the Ljung-Box test on the squared (standardized) residuals; SK skewness and excess kurtosis; BJ is the Bera-Jarque test for nonnormality; H is White's ticity of unknown form; and E is Engle's (1982) 'test for A R C H disturbances with P w = weekly series, b = biweekly series, and m = monthly series for the period.

45.136 2

19.228 2

49.309 2

lags are used for the of the autocorrelation and KU are tests for test for heteroskedaslags equal to the d.f.

Efficiency of Gold Returns

363

monthly series accounted for the ARCH effects as can be seen from the diagnostics in Table 4. From Table 3, it can be seen that (G)ARCH models are better representations of the respective series from the improvement in all the logs of the likelihood functions and the significance of the estimated coefficients in the conditional variance. The (G)ARCH models in all series substantially improve upon the excess kurtosis in the standardized residuals over the linear model. There was no skewness detected in the residuals of the linear model for the biweekly series. However, the test for skewness was significant for the OLS residuals in the weekly and monthly series. The (G)ARCH specification produced insignificant skewness statistics for all the series except the daily. Also, standardized (G)ARCH residuals exhibit no sign of correlation for any of the series. The BDS statistics for the OLS residuals and (G)ARCH standardized residuals are reported in Table 2. As can be seen from the BDS statistics, standardized residuals from the (G)ARCH specification clearly accept the null of i.i.d, in all the series. Recall that we are interested in testing Sims' (1984) proposition that market efficiency can be described by IU rates of return processes. In order to test the foregoing hypothesis, the Conditional mean of the daily returns was respecified as a AR(1) process with a constant thus being identical to the other series. This was then estimated with the conditional variance specified as GARCH(1, 1)~ From this nonlinear regression, a goodness of fit measure of 0.003931 was obtained. Therefore, from the R2s in the other series, it can be concluded that Sims, IU hypothesis of a martingale is valid when moving from the monthly to the weekly series. However, the R 2 from the daily series is higher than that of the weekly. This finding by itself is not sufficient evidence against the IU hypothesis, but is an anomaly that must be investigated further. To further examine Sims' (1984) IU proposition, we will analyze the predictive power of the out-of-sample one-periodahead forecasts for all the series under consideration. We constructed 1000, 200, 100, and 50 out-of-sample one-period-ahead forecasts of the returns of the daily, weekly, biweekly, and monthly series, respectively. The actual returns were then regressed against the forecasted returns. If the conditional means of the respective series have any predictive power whatsoever, then the coefficient on the forecasted return will not be significantly different from 1. Another good indicator of the predictive power of these models are the R2s obtained from the actual-forecast regressions. The results of these regressions are reported in Table 5. From this table, notice that none of the series exhibit any sort of predictive power. The daily series is the only one that has an estimated

Table 5. Out-of-Sample One-Period-Ahead Forecasts. M°del: rLt = aj.o + °tj, lrft + ~j,t V j = d , w , b , m Coeff.

Daily

Weekly

Biweekly

Monthly

ao (std. error) oq

0.000017 0.000259 - 0.109405 0.349844 0.000098 1000

0.000393 0.00123 - 2.90012 2.86155 0.005161 200

- 0.000838 0.002362 0.353987 1.13514 0.000991 100

- 0.00408 0.005798 - 1.31373 1.27211 0.021736 50

R 2

T

364

M. Marshall and T. Steng~>:. coefficient on the forecasted return that is not significantly different from i However, the goodness of fit is extremely poor at 0.01%, thus suggesting that there exists no forecasting ability in this series. The results of the out-of-sample predictions are taken to be evidence for Sims' (1984) instantaneously unpredictable proposal to show that the gold bullion spot market is efficient.

Conclusion Nonlinear conditional variance processes have been applied to daily, weekly, biweekly, and monthly gold rates of return. These processes have been shown to be good representations of the conditional variances of the returns in all series. Several diagnostic checks were performed on the standardized residuals as well as the BDS statistic of Brock, Dechert, and Scheinkman (1987). The standardized residuals from the (G)ARCH models in all series were found to be i.i.d. Also, the (G)ARCH models seem to capture much of the excess kurtosis, although not all. This would suggest that there must exist better representations that would account for all the excess kurtosis. All but one diagnostic test statistic is improved upon when using the standardized residuals from the conditional variance models. It also was found that the A R C H effects tended to decrease with the length of the return examined. The conditional variance models had no forecasting ability in the out-of-sample one-period-ahead forecasts. From the BDS statistic, it is evident that the random walk version of the E M H is not valid; however, this result was not surprising. The instantaneously unpredictable proposal of Sims (1984) is used to conclude that the market is efficient in the case of gold returns.

We would like to thank two referees and the editor for their helpful comments.The second author also acknowledges financial support from SSHRC of Canada.

References Aggarwal, R., and Soenen, L. A. (1988). The nature and efficiency of the gold market. The Journal of Portfolio Management 14(3):18-21. Baillie, R. T., and Bollerslev, T. (1989). The message in daily exchange rates: A conditionalvariance talc. Journal of Business and Economic Statistics 7:297-305. Bollcrslev, T. (1986). Generalized autoregrcssive conditional heteroskcdasticity. Journal of Econometrics 31:307-327. Brock, W. A. (1982). Asset prices in a production economy. In The Economics of Information and Uncertainty (J. J. McCall, ed.). Chicago: The University of Chicago Press. Brock, W. A. (1988). Nonlinearity and complex dynamics in economics and finance. In The Economy as an Evolving Complex System, Santa Fe Institute Studies in the Science of Complexity (P. W. Anderson, ed.). Reading, MA: Addison-Wesley. Brock, W. A., Dechert, W. D., and Scheinkman, J. (1987). A test for independence based on the correlation dimension. Discussion paper 8702, Department of Economics, University of Wisconsin.

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