much ado about nothing

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International Review of Financial Analysis8:2 (1999) 139–151

Much ado about nothingLong-term memory in Pacific Rim equity markets

John S. Howea,*, Deryl W. Martinb, Bob G. Wood, Jr.b

aDepartment of Finance, College of Business and Public Administration, University of Missouri-Columbia,Columbia, MO 65211, USA

bDepartment of Economics, Finance, and Marketing, College of Business Administration, TennesseeTechnological University, Cookeville, TN 38505, USA

Abstract

Using classical and modified rescaled range analyses (R/S analysis), this study examinedthe equity markets of Japan, Australia, Hong Kong, Singapore, Korea, and Taiwan. Using theclassical rescaled-range method of analysis, we documented the presence of a long-rangenonlinear deterministic structure in the returns of the Japanese, Singaporean, Korean, andTaiwanese indices, ranging from 3 to 4 years in duration. However, after correcting for short-range dependence using Lo’s (1991) modified R/S analysis technique, all evidence of long-term memory disappeared. The absence of long-range dependence is consistent with marketefficiency, and these findings call into question patterns in other asset streams documentedusing the classical method of rescaled range analysis. These findings also raise the generalspecter of significant sensitivity of empirical findings to the choice of method of analysis. 1999 Elsevier Science Inc. All rights reserved.

JEL classification: G12, G15

Keywords: Chaos; Nonlinear dynamics; Fractal structure; Rescaled range

1. Introduction

Asset pricing theories and econometric tests of those theories frequently assumethat security returns are independently and identically distributed (IID) with a normaldistribution.1 This assumption allows for tractable theories and empirical testing. In-deed, research on asset pricing based on the normality assumption has significantlyadvanced the understanding of the returns-generating processes in financial markets.However, a linear paradigm is built into the normality assumption. Even if existing

* Corresponding author. Tel.: 573-882-5357.

1057-5219/99 $ – see front matter 1999 Elsevier Science Inc. All rights reserved.PII: S1057-5219(99)00015-0

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140 J.S. Howe et al. / International Review of Financial Analysis 8 (1999) 139–151

(linear) models are reasonable approximations of market dynamics, nonlinear methodsmay provide better representations of reality, thereby generating further insight intothe nuances of market processes.

Unlike linear methods of analysis, nonlinear modeling makes no a priori assumptionsabout data distributions. Recent research in this area suggests that the residuals ofconventional linear models may actually be the result of nonlinear processes ratherthan random noise. If part (or all) of the process is nonlinear, chaos analysis can beused to determine if the process is deterministic in nature.2 If the process has anonlinear cyclic time path that can be identified, financial economists can better modelasset pricing in those markets that exhibit chaotic tendencies, as well as relatedderivatives markets.

The extant literature contains many studies that examine the high frequency (orshort-term) dynamics of security returns, such as daily and intradaily return patterns.Research focused on low-frequency dynamics—return regularities occurring overlonger horizons—is less common. Long-term persistence (or dependence) is presentwhen apparently random short-term return variations contain long-term structure.The paucity of literature in this area arises from a lack of data: the analysis is extremelydata intensive and therefore requires examination of rather long time intervals.

Long-term dependencies (also referred to as fractal dynamics; see Mandelbrot,1977a,b) have been found in the returns of a variety of asset classes. Booth, Kaen,and Koveos (1982) found evidence of a deterministic process in exchange rate changes.Helms, Kaen, and Rosenman (1984), Milonas, Koveos, and Booth (1985), Kao andMa (1992), Eldridge, Bernhardt, and Mulvey (1993), Fang, Lai, and Lai (1994), andCorazza, Malliaris, and Nardelli (1997) found long-term dependence in index andcommodity futures returns. Greene and Fielitz (1977), Aydogan and Booth (1988),and Nawrocki (1995) examined nonlinear regularities in U. S. equity market returns.Cheung, Lai, and Lai (1993) examined long-term dependence in developed Europeanand Asian markets. Lo (1991) found that, after correcting for short-range dependence,there is no evidence of long-range dependence in U. S. equity markets. Jacobsen(1996) also found that findings of long-range patterns of dependence in the indicesof five European countries, the United States, and Japan are biased by short-rangedependence. The lack of demonstrable long-term dependence in return streams aftercorrection for short-term dependence questions the findings generated by classicalR/S analysis.3

Analyses of market return patterns in non-U.S. markets have been, to date, limited.The first purpose of this study was to determine if there exists some type of a cyclictime path in the equity index returns of Japan, Australia, Hong Kong, Singapore,Korea, and Taiwan. The second purpose was to examine the sensitivity of the findingsto the choice of method of analysis.

The focus on Pacific Rim equity indices is appropriate for a number of reasons.First, the Pacific Rim economies are becoming an increasingly important componentof the global economy. The equity markets of Japan, Australia, Hong Kong, Singapore,Korea, and Taiwan are integral segments of an increasingly integrated world financialmarket, as evidenced by the spillover effects from Hong Kong to other markets in

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J.S. Howe et al. / International Review of Financial Analysis 8 (1999) 139–151 141

October 1997. Understanding the behavior of these markets is thus an importantundertaking. Second, this set of markets allows comparison of developed markets(Japan and Australia) with maturing markets (Hong Kong, Singapore, Korea, andTaiwan) in the same region to determine if the returns-generating processes andpresence or absence of chaos depends on the degree of market development. Third,the presence of a deterministic process in stock returns (if confirmed) suggests adegree of predictability that may be of value to investors in deciding the portfolioweights to be assigned to each market. Specifically, the presence of a nonlineardeterministic component of returns may allow investors to improve portfolio perfor-mance via tactical asset allocation. Fourth, the presence of fractal structure in financialprices may reflect fractal dynamics in the underlying economy which, in turn, wouldbe of value in modeling business cycles. Fifth, the nature of stock returns is directlyrelevant to the pricing of derivative instruments, such as index options and futures.Finally, the presence of fractal behavior has implications for time-series modeling.

Our findings can be summarized as follows. Using the classical rescaled-rangemethod of analysis, we documented the presence of a long-range nonlinear determinis-tic structure in the returns of the Japanese, Singaporean, Korean, and Taiwaneseindices, ranging from 3 to 4 years in duration. However, after correcting for short-range dependence using Lo’s (1991) modified R/S analysis technique, all evidence oflong-term memory disappeared. The absence of long-range dependence is consistentwith market efficiency, and these findings call into question patterns in other assetstreams documented using the classical method of rescaled range analysis. Thesefindings also raise the general specter of significant sensitivity of empirical findingsto the choice of method of analysis.

These results are similar in spirit to those of Aydogan and Booth (1988) and Cheunget al. (1993). Aydogan and Booth examined U.S. equity returns, both for individualstocks as well as the market, from July 1962 through December 1980. They reportedthat a “. . . large number of the Hurst coefficients that appear to signify some typeof dependence are likely to occur by chance only” (p. 148). Similarly, Cheung, Lai,and Lai tested the national equity indices of Germany, Italy, Japan, and the UnitedKingdom. They report that: “While the conventional R/S analysis seems to indicatethe presence of long cycles in stock return, no significant evidence of long cycles canbe found using the modified R/S analysis once short-term dependence and conditionalheteroskedasticity in the data are adjusted for” (p. 45). One important contributionof our paper is to document a similar effect in six Pacific Rim indices over a differentperiod. The problem of choosing the appropriate method of analysis appears to bepervasive, rather than isolated.

The paper is organized as follows. The next section reviews the data and methodof analysis. Section 3 reports the empirical results, and section 4 concludes.

2. Data and methods

We analyzed daily returns of the Nikkei Stock Average (Japan), the All OrdinariesIndex (Australia), the Hang Seng Index (Hong Kong), the Strait Times Index (Singa-

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pore), the Korea Composite Stock Price Index (KCSPI), and the Weighted StockIndex (Taiwan) from January 1981 through May 1994. The sample data are from TheWall Street Journal, The Times of London, the Australian Stock Exchange, HSIServices, Inc. of Hong Kong, the Stock Exchange of Singapore Ltd., the Korea StockExchange, and the Taiwan Stock Exchange.

The Nikkei Stock Average is a price-weighted average containing 225 stocks repre-senting approximately 60% of the capitalization of the First Section of the TokyoStock Exchange. Over 320 companies’ common shares make up the All OrdinariesIndex of Australia. Hong Kong’s Hang Seng Index represents approximately 70% ofthe Stock Exchange of Hong Kong’s total market capitalization. The Strait TimesIndex of the Stock Exchange of Singapore is a relatively narrow capitalization-weightedindex of 30 companies. The Korea Composite Stock Price Index is a broad-basedindex representing all listed companies of the Korea Stock Exchange. The TaiwanWeighted Stock Index is a value-weighted index of essentially all stocks traded onthe Taiwan Stock Exchange. See Berlin (1990) for a more detailed discussion of theseindices.

This paper uses both (1) classical rescaled range (R/S) analysis first advanced byHurst (1951) and subsequently refined by Mandelbrot and Wallis (1969) and Wallisand Matalas (1970); and (2) modified R/S analysis (Lo, 1991). R/S analysis detectslong-term patterns by eliminating short-term noise from the return stream. The calcula-tions used in the analysis are not rigorous, but, as noted earlier, the process is highlydata intensive.

Although a thorough statistical delineation of the R/S technique is beyond thescope of this paper, a brief description of the estimation process is necessary. Assummarized by Peters (1991, 1994), a price time series of length M is transformedinto a time series of logarithmic ratios of length N 5 M 2 1:

Ni 5 log1M(i11)

Mi2, i 5 1,2,3, . . . (M 2 1). (1)

The time series shown in Eq. (1) is divided into A contiguous subperiods of lengthn where N 5 A*n. Each subperiod (Ia) is labeled with a 5 1, 2, 3, . . . A, and eachelement is labeled (Nk,a), where k 5 1, 2, 3, . . . n. The mean of each Ia is defined inEq. (2):

ea 5 11n2ok

i51

Nk,a. (2)

The accumulated departure from the mean of subperiod Ia is defined in Eq. (3):

Xk,a 5 ok

i51

(Ni,a 2 ea). (3)

The range within each subperiod Ia is defined in Eq. (4):

RIa 5 max(Xk,a) 2 min(Xk,a). (4)

The sample standard deviation is calculated for each Ia as

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SIa 5 111n2on

k51

(Nk,a 2 ea)220.50

. (5)

Each range is then normalized by dividing by its corresponding SIa. The rescaled rangefor each Ia is equal to RIa/SIa. For A contiguous subperiods of length n for any series,the rescaled range is shown in Eq. (6):

1RS2n

5 11A2o

A

a511Ria

SIa2 . (6)

The process is repeated by increasing n to the next higher value so that (M 2 1)/nis an integer value until n 5 (M 2 1)/2. The log(R/S)n is then regressed, with log(n)as the independent variable. The regression is run for all n > 10 integer values; smallervalues of n produce unstable estimates. The statistic of interest, the Hurst exponent,is estimated by the slope of the equation.

A Hurst exponent of 0.50 implies an IID process. An exponent of 0.50 , H < 1.00suggests persistence or a long-term memory in the returns-generating process, wherecurrent price movements are positively correlated with future price movements. Theprocess is independent of time scale—a key characteristic of a fractal time series. AHurst exponent of less than 0.50 (0 < H , 0.50) indicates antipersistence, a returnpattern resembling a mean-reverting process.

A problem with early applications of R/S analysis was the lack of a tractable testof significance. More advanced statistical techniques efficiently test the significanceof the Hurst exponent. We begin by estimating an expected R/S (E(R/Sn)) value andexpected H (E(H)) to determine if the calculated R/Sn and H deviate significantlyfrom expected values. The E(R/Sn) is from Hurst (1951) as modified by Anis andLloyd (1976), and is defined in Eq. (7):

E1RSn2 5 1(n)1p222

20.50

* on21

r511(n 2 r)

r 20.50

. (7)

The Anis and Lloyd (1976) model of the expected Hurst exponent is used rather thanthe Peters (1994) version because of the latter’s failure to characterize properly theasymptotic behavior of the adjusted rescaled range (see Ellis (1996) for a thoroughexplanation of the Peters’ bias). If a system is IID, the calculated H should not deviatesignificantly from the expected value of H, E(H). Significance is determined by Eq. (8):

(H 2 E(H))(Var(E(H))0.50

, (8)

where the variance of E(H) is as shown in Eq. (9):

Var(H)n 51T

. (9)

Because the E(H) is IID, the variance is dependent on total sample size (T), not nor H.

If persistence is evident in market returns, the next step is estimation of the length

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of the cycle underlying the returns-generating process. The approximate cycle lengthis estimated by the V statistic, defined as shown in Eq. (10):

Vn 51RS2

n

n0.50. (10)

This ratio is simply the rescaled range scaled by the square root of time. If the processis IID, the ratio will be constant. As long as V is scaling faster than time, a deterministicprocess continues. If V declines significantly, the long-term memory process fadesand the cycle ends.

The distribution of the classical R/S test statistic is not well defined, and the resultsof the analysis are sensitive to the presence of heteroskedasticity or short-term depen-dence in the data. To ensure that deviation from the expected R/S statistic is theresult of only long-term dependence, we incorporated Lo’s (1991) modified rescaledrange technique. Lo’s technique involves modifying the classical R/S statistic to makeits behavior invariant to a general set of short-term memory processes, yet still be ofa form that deviates for the existence of longer-term memory. To accomplish this,Lo altered the classic R/S devisor (our Eq. (5)) to include not only the usual (maximumlikelihood) standard deviation but also added to that a weighted average of autocovari-ance estimators. Thus for the modified R/S statistic the divisor becomes that shownin Eq. (11):

SIa 5 ((1/n)* on

k51

(Nk 2 e)2 1 2 oa

k51

vk(q)gk)0.50, (11)

where the weights vk(q) are as specified by Andrews (1991) for AR(1) processes atthe data’s optimal truncation lag, q. Critical values for the modified statistic are foundin Lo (1991). Long-term mean reversion (negative dependence) is indicated by asignificant statistic on the left tail; long-term mean aversion is indicated by a significantstatistic on the right tail.

3. Empirical results

We first examined the six Pacific Rim equity market return series for deviationsfrom normality. The skewness and kurtosis measures will be indistinguishable fromzero if the data are normally distributed. Table 1 shows significant departures fromnormality in Pacific Rim equity markets. Five of the six markets (Australia, HongKong, Singapore, Korea, and Taiwan) exhibit significant skewness. Of these, one ispositively skewed (Korea), and the other four are negatively skewed. All markets aresignificantly leptokurtotic, that is, they display significant kurtosis. These findingssuggest the inappropriateness of the normality assumption for modeling Pacific Basinmarkets. Interestingly, the mean returns for Japan and Australia over the sampleinterval are not significantly different from zero at conventional confidence levels(the p-values are 0.062 and 0.055, respectively); the average returns for the emergingmarkets are significantly positive at the 95% confidence level.

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Table 1Summary statistics for Pacific Rim equity market returns

Developed markets Emerging markets

Statistic Japan Australia Hong Kong Singapore Korea Taiwan

Observations (N) 3564 3400 3330 3360 3948 3876Mean return (%) 0.0365 0.0359 0.0694 0.0444 0.0654 0.0781Maximum 13.2359 16.9442 9.3025 12.8819 12.0797 8.2189Minimum 214.9001 224.9092 233.3305 220.8726 210.4073 29.6993Median 0.0617 0.0426 0.0932 0.0495 0.0048 0.0650Standard deviation 1.1681 1.0922 1.8042 1.2708 1.1893 1.8334t-test (mean 5 0) 1.8686 1.9221 2.2214 2.0255 3.4587* 2.6551*(p-value) 0.0618 0.0547 0.0264 0.0429 0.0005 0.0080Skewness 20.0037 23.1788* 22.7952* 21.9735* 0.3671* 20.2293*Kurtosis 17.0526* 104.1568* 44.7774* 35.4144* 5.8044* 2.8192*

* Significant at the 1% level.

Both classical and modified rescaled range estimates are shown to be upwardlybiased if the mean is nonstationary (Klemes, 1974; and Kao & Ma, 1992). We thereforetested the return streams to determine if the data is indeed stationary by using a Box-Jenkins-type stationarity test (Bowerman & O’Connell, 1979). The data is stationaryif the test t-statistic loses significance or drops off after 3 or 4 lags with no subsequentspiking. As seen in Table 2, all of the return streams display stationarity with theexception of the Australian market. Because R/S estimates are biased upward if themean is nonstationary, any pattern in the Australian returns must be suspect.

Table 3 summarizes the regression analyses of the calculated and expected R/Svalues for the Nikkei Stock Average. The rescaled range for the Nikkei Stock Averageshows a systematic deviation from its expected values. The regression coefficients forthe sample periods 11 < n < 297 are H 5 0.567076 and E(H) 5 0.582083. Thedifference in the calculated and expected values is not significant. However, thedifference in the coefficients from the second regression (297 < n < 891) is highly

Table 2Stationarity test results for Pacific Rim equity market returns

t-values

Developed markets Emerging markets

Lag Japan Australia Hong Kong Singapore Korea Taiwan

1 1.74 4.26* 2.66* 7.35* 4.81* 7.66*2 25.49* 21.68 20.01 4.95* 1.10 21.403 0.51 4.14* 5.24* 2.85* 1.61 6.01*4 1.88 4.87* 0.48 2.07 1.43 1.435 0.83 3.53* 0.36 1.12 0.45 0.21

* Significant at the 1% level.

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Table 3Classical R/S analysis results: Nikkei Stock Average (Japan)

R/Sa E(R/S)

11 < n < 297Observations 17 17Degrees of freedom 16 16Hurst exponent 0.567076 0.582083Standard error 0.004839 0.005834Significance 20.8959


t 5 3564 total observations.a Complete R/S analysis results are available from the authors.

significant, with the calculated H more than three standard deviations larger thanE(H). These results suggest that the Nikkei Stock Average has a cycle of 891 tradingdays, or approximately 3 years, over the period examined (the Tokyo Stock Exchangewas open for trading on selected Saturdays from 1981 until 1989; hence, 891 daysrepresents only approximately 3 years).

An interruption in the deviation pattern of the R/S and E(R/S) for the AustralianAll Ordinaries Index appears to occur at 425 days. Regression analysis of the tradingperiod (10 < n < 425) shows a difference between H and E(H) (0.597059 and 0.573743,respectively) but the difference is not statistically significant. Complete regressionresults are contained in Table 4. No fractal pattern is found in the Australian market.

The V statistic ratio of the R/S and the E(R/S) of the Hang Seng Index changesat 555 trading days. As shown in Table 5, regression coefficients for this period (10 <n < 555) are H 5 0.562486 and E(H) 5 0.571105; the difference is not statisticallysignificant. Hence, no deterministic cycle is evident in the Hong Kong market.

Table 4Classical R/S analysis results: All Ordinaries Index (Australia)

10 < n < 425

R/Sa E(R/S)

Observations 15 15Degrees of freedom 14 14Hurst exponent 0.597059 0.573743Standard error 0.006850 0.006500Significance 1.3595

t 5 3400 observations.a Complete R/S analysis results are available from the authors.

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Table 5Classical R/S analysis: Hang Seng Index (Hong Kong)

10 < n < 555

R/Sa E(R/S)



The V statistic ratios for the Strait Times Index change at 420 and 1120 tradingdays. Table 6 shows the regression analysis results. The difference between the actualand expected Hurst exponents (0.597644 and 0.575970, respectively) is not significantfor the first regression period (10 < n < 420), that is, there is no discernible cyclepresent for this period. In contrast, a significant difference is demonstrated betweenH and E(H) in the second regression. For 420 < n < 1120, the actual value of H isalmost four standard deviations larger than the expected value. This trading periodof 1120 days (roughly 4 years) in the Singapore Stock Market is similar to the cyclefound in the Dow Jones Industrial Average (Peters, 1994).

The V statistic for the Korean market index shows that the ratio ceases to rise at1316 days. The regression analysis for Korea (Table 7) reveals a significant difference:R/S is over two standard deviations above E(R/S). The Korean cycle is thus also ofapproximately 4 years duration.

Table 6Classical R/S analysis: Strait Times Index (Singapore)

R/Sa E(R/S)




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Table 7Classical R/S analysis: Composite Stock Price (Korea)

12 < n < 1316

R/Sa E(R/S)



The deviation of the R/S from the E(R/S) of the Weighted Stock Index of Taiwanis confirmed by the V statistic (Table 8); the ratio changes at 114 and 1292 tradingdays. Regression analysis of the two periods (12 < n < 114 and 114 < n < 1292)shows that the differences between the Hurst exponents and expected Hurst exponentsare significant in both periods. The calculated Hurst exponent is over three standarddeviations above the expected value for the first cycle; the difference is over sevenstandard deviations for the second cycle. Hence, the Taiwan stock exchange showstwo cycles: one of approximately 114 trading days (approximately 4 months) and theother of about 1292 trading days (approximately 4 years). This longer cycle correspondsto the cycle found in the U.S. and Singapore markets.

We next reexamined the data using Lo’s modified R/S procedure. Recall fromsection 2 that Lo’s procedure includes in the statistic divisor a weighted average ofautocovariance estimators. Although this modification may seem modest, the results

Table 8Classical R/S analysis: Weighted Stock Index (Taiwan)

R/Sa E(R/S)




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Table 9Modified R/S analysis: Pacific Rim equity markets

Market Classical R/S Modified R/S q V p-valuea

Nikkei Stock Index (Japan) 101.768 102.181 2 1.71159 0.93All Ordinaries Index (Australia) 112.117 103.939 4 1.78255 0.96Hang Seng Index (Hong Kong) 85.970 82.304 3 1.42020 0.79Strait Times Index (Singapore) 74.912 62.284 6 1.07450 0.29Composite Stock Price (Korea) 119.366 112.351 5 1.78808 0.96Weighted Stock Index (Taiwan) 118.896 100.754 7 1.61834 0.89

a From Lo (1991).

are strikingly different, as shown in Table 9. That table displays the classical R/Sstatistic, the modified R/S statistic, the optimal truncation lag for each market’s data(q), the asymptotic V statistic associated with each modified R/S statistic (equal tothe modified R/S statistic divided by the square root of the number of observations),and the probability of the existence of long-term memory after adjusting for AR(1)processes for each of the Pacific Rim markets.

As Table 9 shows, the classical and modified R/S statistics are of roughly the samemagnitude. However, the p-values indicate no significance at any conventional levelwhen the modified R/S method is used. Indeed, the lowest p-value is 0.29, for theWeighted Stock Index of Taiwan, and four of the six p-values are above 80%. Thatis, we cannot reject the null hypothesis of no long-term dependence for any of thesix equity markets. When AR(1) processes are removed from the data, Pacific Rimfinancial markets exhibit no long-term patterns. What is most striking about this resultis not the lack of significant long-term dependence, but rather the dramatic changeof conclusion induced by the use of Lo’s modification.

4. Conclusions

Using classical R/S analysis, this study examined the equity markets of Japan,Australia, Hong Kong, Singapore, Korea, and Taiwan. We found evidence of long-term dependence in the returns of the Japanese, Singaporean, Korean, and Taiwaneseindices using this standard method of analysis. These patterns are of similar durationto those reported in earlier analyses, and all seems well with the world. But as is socommon, not all is as it seems. After modifying the classical technique to account forshort-term dependence using an AR(1) model, all evidence of long-term memory inPacific Rim equity index returns disappears. This absence of dependence in equitymarkets is consistent with informational efficiency and the inability to capture abnor-mal returns from long-cycle patterns.

The evidence presented here does not ultimately resolve the question of the presence(or absence) of long-term dependence in Pacific Basin equity indices. Rather, webelieve that the major contribution of this paper is to provide evidence that conclusionsabout long-term memory in asset prices are sensitive to the method of analysis chosen

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by the researcher, a finding that supports and expands the earlier work of Aydoganand Booth (1988). As a consequence, the profession’s empirical knowledge of long-term processes is not as solid as previously believed. Earlier studies must be reworkedto assess the robustness of their conclusions, and the search for a more powerful anddefinitive method of analysis must be intensified.

Notes

1. Consider, for example, the Sharpe-Lintner model of market equilibrium (Sharpe,1964; Linter, 1965), the Black-Scholes theory of option pricing (Black & Scholes,1973), and the empirical tests of Fama and MacBeth (1973).

2. Chaos is a nonlinear deterministic process that can yield a rich variety of timeseries that often appear to be random. See Hsieh (1991) and Baumol and Ben-habib (1989).

3. An alternative approach is taken by Batten and Ellis (1999).

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