stock returns and volatility on china's stock markets

The Journal of Financial Research. Vol. XXIV, No.4. Pages 523-543 • Winter 2001

STOCK RETURNS AND VOLATILITY ON CHINA'S STOCK MARKETS

Cheng F. LeeThe State University of New Jersey, Rutgers

Gong-meng Chen and Oliver M. RuiHong Kong Polytechnic University

Abstract

We examine time-series features of stock returns and volatility, as wellas the relation between return and volatility in four of China's stock exchanges.Variance ratio tests reject the hypothesis that stock returns follow a random walk.We find evidence of long memory of returns. Application of GARCH andEGARCH models provides strong evidence of time-varying volatility and showsvolatility is highly persistent and predictable. The results of GARCH-M do notshow any relation between expected returns and expected risk. Daily tradingvolume used as a proxy for information arrival time has no significant explanatorypower for the conditional volatility of daily returns.

JEL classification: G15

I. Introduction

Harvey (1995) reports that emerging markets have high average returns, lowoverall volatility, low exposure to world risk factors, and little integration. Heconcludes that emerging markets are less efficient than developed markets and thathigher return and lower risk can be obtained by incorporating emerging marketstocks in investors' portfolios. From a U.S.-based investor's point of view, it isimportant to understand the potential portfolio implications of investing in stocksin these countries. Additionally, it is desirable to understand the behavior of themajor equity performance indicators for these countries over time.

China's stock markets attract foreign investors' attention because of thatcountry's fast development and potential opportunities. Since the establishment ofthe Shanghai Stock Exchange on December 19, 1990, and the Shenzhen Stock

We would like to thank an anonymous reviewer and William T. Moore, the editor of the Journal ofFinancial Research, for their helpful comments and suggestions. Financial support provided by a grant fromthe Departmental Research GrantofHong Kong Polytechnic University (Account No. A-PA 87) isgratefullyacknowledged.

523

524 The Journal of Financial Research

Exchange on July 3, 1991, China's stock markets have expanded rapidly.' ByDecember 31, 1997, China had 720 A-share listed stocks, of which 372 traded onShanghai and 348 traded on Shenzhen, and 101 B-share listed stocks, ofwhich 50traded on Shanghai and 51 traded on Shenzhen. A-share stocks are traded amongChinese citizens and B-share stocks are traded among non-Chinese citizens oroverseas Chinese.' Total market capitalization exceeds $200 billion, or nearly onefifth of the country's gross domestic product. Daily trading volume typically hits750 million shares. China will, therefore, provide a major investment avenue forinternational and global investors after its accession to the World TradeOrganization (WTO). One concern international investors have, however, is the lackof knowledge of China's markets. Institutional characteristics of China's marketsdiffer from those in other countries, so that the research results from other countriescannot be automatically extended to China. One distinguishing feature of China'smarkets is that some shares are restricted to domestic investors and others arerestricted to foreign investors.

In this study we characterize the dynamics ofstock returns and conditionalvolatility in China's stock exchanges. We focus on whether stock returns follow therandom walk hypothesis in China. Also, we examine whether stock return volatilitychanges over time and whether it is predictable. We then study the relation betweenmarket risk and expected return. Finally, we examine whether daily trading volumeused as a proxy for information arrival time has significant explanatory power forthe conditional volatility of daily returns.

Many studies examine whether stock returns are predictable from the past.One hypothesis widely tested is that stock prices follow a random walk, whichimplies returns are independent. Lo and MacKinlay (1988) and Poterba andSummers (1988) provide empirical evidence against the random walk hypothesis forstock returns in the U.S. stock markets. Recently, the attention of research hasshifted to recognizing the possibility oflong-term dependence ofreturns. Lo (1991)conducts long-memory analysis for stock prices.

Financial economists study returns and conditional volatility of stockmarkets extensively. Baillie and DeGennaro (1990) study the dynamics ofexpectedstock returns and volatility in the U.S. stock markets; Poon and Taylor (1992)investigate the same relation in the U.K. stock market. Both studies find clustering,predictability, and persistence in conditional volatility in these markets. Conditionalsecond moments playa key role in various financial activities. Many asset pricingmodels predict that the expected return of any asset is directly related to itscovariance with one or more pricing factors. Most portfolio-diversification and risk-

'The Shanghai and Shenzhen stock exchanges are self-regulated and cross-listing is not allowed. In theShanghai Exchange, orders are automatically matched on a time-price priority basis, whereas in the ShenzhenExchange, orders are matched on a price-time-order priority basis.

2Forthe purpose ofB shares on the Shanghai and Shenzhen exchanges, overseas investors are describedas: foreign legal and natural persons; legal and natural persons from Hong Kong, Macau, and Taiwan; andother investors approved by the People's Bank of China.

China's Stock Markets 525

hedging strategies are based on the ability to predict variance and covariance. Theevidence on non-U.S. markets is limited. Therefore, investigating China's stockmarket, which has different economic, institutional, and microstructural features,becomes appealing to academies and practitioners. The investigation ofreturns andthe conditional volatility of China's stock markets provides an opportunity to addto the evidence.

We find that the variance ratio test rejects the random walk hypothesis. Thefractional differencing test for long memory devised by Geweke and Porter-Hudak(1983) is employed to detect long memory in stock returns in China's stock markets.This test provides support for long memory. The generalized autoregressiveconditionally heteroskedastic (GARCH) and the exponential generalizedautoregressive conditionally heteroskedastic (EGARCH) models are used to obtainappropriate series of conditional variances that can be used as expected volatilityestimates. We find strong evidence of time-varying volatility and clustering ofhigh/low volatility. We also find that volatility shows high persistence and ispredictable. In addition, we find support for a fat-tailed conditional distribution ofreturns, which implies that large changes in speculative prices are expectedrelatively often. To examine the relation between expected returns and expectedvolatility, we fit the generalized autoregressive conditionally heteroskedastic in themean (GARCH-M) model. We do not find any relation between expected returnsand expected risk as predicted by asset pricing models. Finally, using daily tradingvolume as a proxy for information arrival time does not have significant explanatorypower for the conditional volatility of daily returns.

II. Background on China's Stock Markets

Capital markets in China have experienced tremendous development overthe past decade. The inception ofa shareholding system in China dates from 1979.In the early stages, stocks had many of the characteristics of bonds and littlesimilarity to the traditional concept ofstocks in Western capitalist economies. Theyusually were characterized by a fixed income with a fixed maturity and were issuedto raise funds for specific investments. Initially, these stocks were issued mainly tocorporate employees. The development of a primary share market began in 1984,when enterprises were first allowed to raise funds by issuing stocks. A secondarytrading market was initiated in 1986 but was not fully developed until 1988, whenthe government first allowed state treasury bills to be openly traded in five majorcities. Since 1988, there has been rapid progress in the development ofa diversifiedsecurities market. Between mid 1988 and late 1990, trading in government andcorporate securities became common, and the official secondary market for statetreasury bills was extended across China. By December 1997, 782 stocks were listedon the two exchanges, with a total market capitalization of more than RMB 1,200billion (Chinese currency), equivalent to about US$140 billion.

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We use daily returns from December 12, 1990, to December 31,1997, forthe Shanghai A Index; from February 21, 1992, to December 31, 1997, for theShanghai B Index; from September 30, 1992, to December 31, 1997, for theShenzhen A Index; and from October 6, 1992, to December 31, 1997, for theShenzhen B Index. The Shanghai and Shenzhen stock exchanges provide us with allthe data. The Shanghai Stock Exchange, founded on November 26, 1990, begantrading securities on December 19 of that year. The Shanghai Securities ExchangeIndex is a weighted-average, market-capitalization index. Its base date is December19, 1990, and its base value is 100. The index comprises all listed shares. TheShenzhen Stock Exchange has been operating since December 1, 1990. TheShenzhen Securities Exchange Index is also a value-weighted (VW) average marketcapitalization index. To control for the well-documented size effect andnonsynchronized trading, we also use an equally weighted (EW) index ofboth stockmarkets.

III. Empirical Design and Results

Descriptive Statistics

Table 1 provides summary statistics for all stock return series in both theShanghai and Shenzhen stock exchanges. The statistics show that returns arepositively skewed although the skewness statistics are not large. The positiveskewness implies that the return distributions of the shares traded on theseexchanges have a higher probability of earning positive returns. All the kurtosisvalues are much larger than 3. This shows that for all series, the distribution ofreturns has fat tails compared with the normal distribution. Cartwright and Lee(1987) investigate the effects oftemporal aggregation on the existence ofa dynamicmarket model and on the magnitude of beta estimates. They find that temporalaggregation does affect the specification ofthe market model and beta estimates. Tocontrol for the potential time aggregation effect, we also analyze the weekly andmonthly data.'

Autocorrelation coefficients are reported in Table 2. Autocorrelation reflectshow quickly and completely stock prices adjust to new information. Whereaspositive autocorrelation suggests a slow and partial price adjustment, negativeautocorrelation suggests overreaction. The autocorrelation coefficients for returnseries ofB-share stocks have higher values than the autocorrelation coefficients forthose ofA-share stocks. Significant autocorrelation is detected at a lag ofone periodfor return series of B-share stocks in the Shanghai and Shenzhen stock markets.Most of the dependence may be due to thin trading. B shares account for less than5 percent of the total market capitalization. Also, B-share markets have been

'The results for weekly and monthly data are available upon request.

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illiquid and less active than A-share markets, as shown by the average turnover ratein B-share markets hovering around one-tenth ofthat in A-share markets. Ljung-Boxstatistics provide evidence ofpossible dependence in the first and higher momentsofthe return distributions. Under the null hypothesis ofno serial correlation, the Qstatistics are distributed asymptotically chi-squared with m degrees offreedom. Theresults show that the independent and identically distributed hypothesis is rejectedfor all the stock return series in China. This is consistent with the results of thefollowing variance ratio test where the random walk hypothesis is rejected.

The autocorrelation coefficients of returns suggest a slowly decayingautoregressive effect. We test the unit root of returns and report the results in PanelB of Table 2. The null hypothesis of a unit root is rejected. On the basis of theresults ofthe unit root test, we fit returns using the ARMA(l,1) model."

Testing the Random Walk Model Using Variance Ratio Tests

To test for the random walk in returns on China's stock markets, we followthe procedure used by Lo and MacKinlay (1988) to test for an Ito process in stockreturns. The idea behind the variance ratio test is that if the return series is a purerandom walk, the variance of its n-differences grows proportionally with thedifference n. Lo and MacKinlay examine the random walk hypothesis by testing thenull hypothesis that the variance ratio is given by:

VR(n) = 1cr(n) 1,ncr(l)

(1)

where a2(n) is an unbiased estimator of the variance of the nth difference of returnr.; and a\1) is the estimator of the variance of the first difference of rr

We reject the hypothesis ofa random walk by using the autocorrelation testand the Dicky-Fuller unit root test, but these results are justified only if theunderlying variables are normally distributed. Table 1 documents that thedistribution exhibits excess kurtosis relative to a normal distribution. Lo andMacKinlay (1988) show that the variance ratio test is more powerful than the BoxPierce Q-test and the Dicky-Fuller unit root test against several alternativehypotheses. A variance ratio less than unity indicates the presence ofnegative serialcorrelation, which is consistent with mean-reverting behavior in the series. Avariance ratio greater than unity indicates the presence ofpositive serial correlationor an explosive function in the series. Lo and MacKinlay derive an asymptoticstandard normal test statistic, Z, that provides the statistical significance of the

4Theparameter estimates and the descriptive statistics for fitted values are available on request. We findthe fitted values are highly skewed and have large kurtosis coefficients.


TABLE 3. Results of Variance-Ratio Tests.

No. (n) ofMultiple Sampling PeriodsNo. (N) Used to Generate Variance Ratio

Returns ofObs. 2 4 8 16

Shanghai A 1767 0.0341 0.2047 0.1160 0.0625EW Return (20.1043) (16.3760) (12.2105) (8.9286)Shanghai A 1767 0.3344 0.1994 0.1119 0.0617VWReturn (17.2371) (16.3443) (12.1630) (8.9420)Shanghai B 1470 0.4726 0.2717 0.1532 0.0833EW Return (16.2966) (14.8470) (11.1014) (7.9333)Shanghai B 1470 0.4646 0.2669 0.1488 0.0809VWReturn (16.3018) (14.9944) (11.1045) (8.0900)Shenzhen A 1295 0.3395 0.1964 0.1129 0.0606EW Return (18.0585) (14.0286) (10.4537) (7.5750)Shenzhen A 1640 0.3339 0.1972 0.1086 0.0601VW Return (20.3598) (15.7760) (11.8043) (8.5857)Shenzhen B 1759 0.4041 0.2511 0.1466 0.0736EWReturn (17.8018) (13.7967) (10.2517) (7.5102)Shenzhen B 1406 0.4414 0.2812 0.1637 0.0834VWReturn (18.7830) (14.5699) (10.8411) (7.9429)

Note: The variance ratio is given by:

VR(n) = l<?(n) = 1,n<?(l)

where a\n) is an unbiased estimator ofthe variance ofthe nth difference ofreturn r.;and a2(1) is the estimatorof the variance of the first difference of rr

variance ratios, as well as an alternative statistic, Z*, that is robust toheteroskedasticity and nonnormal disturbances.

We calculate the standard VR(n) and Zen), where n represents multiples ofeach series. The results are presented in Table 3. The variance ratios are reported inthe main rows of the table, and the Z-statistics are given in parentheses below eachentry. The variance ratio estimates in Table 3 are less than unity for all n, and theratios decrease with increasing n. Under the hypothesis of homoskedasticity, therandom walk hypothesis is rejected for all stock return series. For example, the Zstatistics for the Shanghai VW returns for n = 2, 4, 8, and 16 are 17.2371, 16.3443,12.1630, and 8.9420, respectively. All four Z-statistics indicate the random walkhypothesis is rejected for the Shanghai VW return for all four intervals examined.By a similar analysis, the remaining Z-statistics in Table 3 present evidence rejectingthe random walk hypothesis in the remaining seven return series on the Shanghaiand Shenzhen stock exchanges.

The rejection ofthe random walk for returns on China's stock markets maybe due to heteroskedasticity or serial correlation. To further investigate the timebehavior of returns, we employ a heteroskedasticity-consistent variance ratio testwith the statistics Z*(n). Because the results are identical to the homoskedasticity-


consistent results, we do not report them. This finding suggests that the rejection ofthe random walk may be due to serial correlation. The overall results from varianceratio and Box-Pierce tests provide evidence rejecting the random walk hypothesisfor stock returns in China.

Because the rejections obtained from the variance ratio test are robust toheteroskedasticity, they suggest autocorrelation. The existence ofautocorrelation infinancial assets does not necessarily imply market inefficiency. The reasons offeredfor such dependencies include information asymmetries and the competencedifficulty gap. Poterba and Summers (1988) suggest some alternatives to the randomwalk. The alternatives are a mean-reverting process, the sum ofa random walk, anda stationary mean-reverting process. These alternatives imply that prices and returnsare negatively serially correlated and that the serial correlation becomes morenegative as the holding period increases. If such processes govern returns, thevariance ratios should be less than unity for long horizons. Our results support themean-reverting hypothesis. The presence of negative serial correlation in returnseries may also be linked to thin trading.

Long Memory ofReturn Series

A common statistical alternative to the random walk is long-term memory.Long memory describes the correlation structure of that series at long lags. If aseries exhibits long memory, there is persistent temporal dependence amongdistance observations. The potential existence of long-term dependence in stockreturns has important consequences for financial theory. A growing body ofliterature explores the long-memory property of financial price series. For example,Poterba and Summers (1988) report that stock returns display positive correlationover short horizons and negative correlation over long horizons. The long-termmemory property of the mean-reverting model of stock returns implies that stockreturns are negatively serially correlated, and that the negative serial correlationbecomes more severe as the length of the holding period increases. Lo (1991)attributes this observation to the possible presence of long cycles and potentiallypredictable components in long-horizon stock returns. We employ the fractionaldifferencing test for long memory devised by Geweke and Porter-Hudak (1983) todetect long memory. The fractional differencing approach models long-memorydynamics parametrically. Under this approach, whether a series displays longmemory depends on a fractional differencing parameter, which is subject toestimation and hypothesis testing. A general class of the long-memory process isdescribed by an autoregressive fractionally integrated moving average (ARFIMA)model,

(2)

where B(L) and qL) are polynomials in the lag operator L, and (1 - L)d is thefractional differencing operator defined by:


(1 - L)d = L r(k - d)L k ,

k=O I'(-d)r(k + 1)(3)

where r is the Gamma function. The fractional parameter given by d assumes anyreal value. When d E (0, 0.5), the ARFIMA process is said to exhibit long memory;when d E ( -0.5, 0), the process exhibits intermediate memory; and when d = 0, theprocess exhibits short memory. When d E (0.5, 1), the process is mean revertingbecause there is no long-run effect ofan innovation on future values ofthe process.A spectral method suggested by Geweke and Porter-Hudak (1983) can be used toestimate the fractional parameter d. The spectral regression is defined by:

(4)

where

s = 21tAA T

denotes the harmonic ordinates of the sample, T is the number of observations, andn = T ~ for 0 < 11 < 1 is the number oflow-frequency ordinates used in the regression.In the spectral procedure, the number of low-frequency ordinates, n, involvesjudgment. Although a large value of n will cause contamination of the d estimatebecause of medium- or high- frequency components, a small value of n will lead toimprecise estimates because of limited degrees of freedom in estimation. Weestimate the fractional parameter d for 11 = 0.5, 0.55, and 0.6.

We report the estimates for the fractional parameter d in Table 4. Almost allof the d estimates are within the range of 0 and 0.5, suggesting long memory. Fourestimates are within the range of -0.5 and 0, implying intermediate memory. Theevidence supports long memory in both the Shanghai and the Shenzhen markets.

Volatility Estimates Using GARCH Models

Security returns in China's stock markets do not follow random walks. Thismay be due to the strong assumption that the entire underlying probabilitydistribution of returns remains stationary through time. It is reasonable to believethat because of changes in the risk of a firm, the variance of stock returns willchange over time. Financial economists show the autoregressive conditionalheteroskedasticity (ARCH) process provides a good fit for many financial returntime series. The ARCH model allows the conditional variance to change over timeas a function ofpast squared errors. The strength ofthe ARCH technique is that theconditional means and variances can be estimated jointly using traditional specified


TABLE 4. Results of Fractional Differencing Analysis.

No. (N) n = To.5 n = T°.55 n = To.6

Returns ofObs. (d) (d) (d)

Shanghai A 1767 0.017 0.053 0.011EWReturn (0.157) (0.579) (0.151)Shanghai A 1767 0.076 0.081 0.081VWReturn (0.666) (0.886) (1.086)Shanghai B 1470 0.019 -0.020 0.048EWReturn (0.159) (-0.208) (0.612)ShanghaiB 1470 0.064 -0.022 0.059VWReturn (0.528) (-0.235) (0.746)Shenzhen A 1295 0.111 0.003 0.032EWReturn (0.892) (0.003) (0.395)Shenzhen A 1640 0.152 0.133 0.101VWReturn (1.301) (1.413) (1.319)Shenzhen B 1759 0.119 0.103 0.105EW Return (0.935) (1.007) (1.251)Shenzhen B 1406 -0.078 -0.038 0.003VWReturn (-0.637) (-0.382) (0.035)

Note: The spectral fractional test for long memory suggested by Geweke, Porter, and Hudak (1983) isperformed on the stock return series for both Shanghai and Shenzhen stock markets. The figures inparentheses are the z-statistics for the corresponding fractional parameter d-estimates.

models for economic variables. ARCH imposes an autoregressive structure onconditional variance, allowing volatility shocks to persist over time. This persistencecaptures the propensity of returns of like magnitude to cluster in time and canexplain the well-documented nonnormality and instability ofempirical asset returndistributions.

Engle, Ito, and Lin (1990) provide two possible explanations for volatilityclustering. First, if information arrives in clusters, returns may exhibit clustering.Nominal interest rate, dividend yield, money supply, oil price, margin requirement,business cycles, and information patterns are the sources of volatility clustering.Second, if participants have different prior beliefs and take time to digest theinformation shocks and to resolve their expectation differences, market dynamicscan lead to volatility clustering.

Bollerslev (1986) extends the ARCH process to GARCH, which allows fora more flexible lag structure. He shows that the extension of the ARCH process ismuch like the extension of the standard time-series process to a general ARMAmodel. The GARCH(1, 1) model for daily stock return is given below:

rt = a + brt _1 + Et , where

(5)

534 The Journal ofFinancial Research

In (5), r/ represents the rate of return, 11- 1 is the set of information available at thebeginning oftime t, and the conditional density function is modeled as a generalizederror distribution (GED). The brl _1 component is included in the mean equation toaccount for the autocorrelation potentially induced by nonsynchronous trading in theassets that make up a market index. This problem can be particularly severe inemerging markets, such as China, given their low level of liquidity. The choice ofGED density is dictated by the inability of Gaussian GARCH processes to accountfor leptokurtosis in most return series, an issue that is likely to be even more relevantwhen using emerging market data. The GED distribution is written as follows:

vexp[ -(1I2)!etht-1I2/1,r J -112 [2(-2/v)f(lIV)]1I2I(e) = h ,where 'A =

t 'A2[(v+1)Jv1r ( lIv) t r(3/v)(6)

In (6), I'(.) is the gamma function and v is a measure oftail thickness, which is equalto 2 for the normal density and less than 2 for the leptokurtic distribution. If theparameters of the lag polynomials a and ~ are positive, shocks to volatility persistover time. The degree of persistence is detennined by the magnitude of theseparameters. If the magnitude of this sum is close to unity, the process is said to beintegrated in variance, where the current information remains important for theforecasts of the conditional variance for all horizons.

Table 5 contains parameter estimates. At first glance, the results areconsistent with those of other empirical works on time-varying volatility. First, thelikelihood ratio statistics are large, which implies the GARCH model is an attractiverepresentation of daily return behavior, successfully capturing the temporaldependence of volatility. Second, the GARCH parameterization is statisticallysignificant. Third, most of the estimated ~ coefficients in the conditional varianceequation are considerably larger than the a coefficients. This implies that largemarket surprises induce relatively small revisions in future volatility. Finally, thepersistence ofthe conditional variance process, measured by a +~, is high and oftenclose to the integrated GARCH model ofEngle and Bollerslev (1986). This impliescurrent information is also relevant in predicting future volatility at a long horizon.

We report point estimates of the tail-thickness parameter v and compute aone-sided test against the alternative hypothesis that the parameter is less than 2, thebenchmark value for normal density. In all instances the null hypothesis ofnormality is strongly rejected. This implies China's stock markets are more likelyto be affected by big surprises, conditional on information available at any time.China's stock markets are some of the world's most volatile markets, with sharesroutinely moving by their 10 percent daily limit.

We report the Ljung-Box statistics for 24th-order serial correlation in thelevel and squared standardized residuals as well as the asymmetry test statistics (signbias, negative size, positive size, andjoint tests) developed by Engle and Ng (1993).The latter is included because the findings by Engle and Ng suggest the Ljung-Box

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test may not have much power in detecting misspecifications related to asymmetriceffects. Both the Ljung-Box and asymmetry checks indicate the estimated modelsfit the data well.

Nelson (1991) develops the EGARCH model. EGARCH has certainadvantages over GARCH. First, by using the exponential formulation, therestrictions of positive constraints on the estimated coefficients in ARCH andGARCH are no longer necessary. Second, a weakness ofthe GARCH model is thatthe conditional variance depends on the magnitude of the disturbance term, but notits sign. GARCH fails to capture the negative asymmetry apparent in many financialtime series. The EGARCH model ameliorates this problem by allowing for thestandardized residual as a moving average regressor in the variance equation, whilepreserving the estimation ofthe magnitude effect. Additionally, the ARCH/GARCHapproach to modeling changing volatility precludes the testing ofthe leverage effect.The tendency for negative shocks to be associated with increased volatility iscaptured in the ARCH/GARCH class of models.

If the sum of the parameters of the lag polynomials a and b equals 1 in theGARCH(1,I) process, the model is known as integrated GARCH, or IGARCH,which implies persistence in the forecast of the conditional variance over all futurehorizons and implies an infinite variance for the unconditional distribution. Thepresence of the near-integrated GARCH being close to but slightly less than unityis found in several financial market series (e.g., Bollerslev (1986)). We employ thefollowing EGARCH(1, 1) model to estimate stock return volatility:

lnh = w+ a( leH I + YeH ] + A lnh a + A = 11 1/2 I-' 1-1' I-' •

hl

_1

(7)

The results of the EGARCH model are reported in Table 6. The leveragefactory is positive for EW returns ofShanghai A-share stocks, whereas the leveragefactors yare negative for both EW and VW returns ofShenzhen B-share stocks. Theleverage factors for other return series are mixed and not statistically significant.The results do not support the negative leverage factors suggested by Nelson (1991).The likelihood ratio ofthe EGARCH model is not larger than that ofGARCH. Thisresult shows that the EGARCH model with a leverage factor does not produce abetter description of the data than GARCH.

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Relation Between Expected Returns and Expected Conditional Variance in theGARCH-M Models

The capital asset pricing model and arbitrage pricing theory provide atheoretical foundation for the positive relation between risk and return. In theory,risk is measured by the conditional covariance of returns with the market. Mostprevious empirical studies use the actual return and risk based on the unconditionaldistribution of return. We employ GARCH-M to investigate the effect ofvolatilityon the return-generation process. Application of the GARCH-M model to testingcapital asset pricing theories presents an improvement in specification of assetpricing models because it allows measurement ofthe conditional variance ofreturnsas the measure of risk. This framework discards the restrictive assumptions oflinearity, independence, and constant conditional variance. The volatility measuredefined by the conditional variance in the GARCH model is an expectationformulation. This volatility measure may be influenced by it own past values andpast values of the return series. If these forecasts of variance can be used in anefficient market to predict expected returns, we should expect c in the followingGARCH-M formulation to be positive for a risk-averse investor:

(8)

We study whether investors in China's stock markets are rewarded for theirexposure to high levels ofvolatility by using the GARCH-M model. The results areshown in Table 7. The coefficient estimates that link first and second moments canbe interpreted as a measure of the price of risk. We expect a positive relationbetween conditional expected returns and conditional market volatility. Instead, wefind a negative relation between expected returns and conditional volatility inShanghai A-share stocks, and the coefficient estimates are significant. The relationsbetween expected return and conditional volatility in the Shanghai B-share andShenzhen A-share and B-share markets are mixed and not significant. To capturethe negative asymmetry, we employ the EGARCH(l,I)-M model to estimate therelation between expected returns and expected conditional variance. We observethe same results as those for the GARCH-M model.'

'The results of the EGARCH-M model are available on request. There is a negative relation betweenexpected return and conditional volatility in Shanghai A-share stocks, and the estimated coefficients arestatistically significant. The relations between expected return and conditional volatility in Shanghai B-shareand Shenzhen A-share and B-share stocks are mixed and not significant. The leverage factory is positive forEW returns of Shanghai A-share stocks and negative for both EW and VW returns of Shenzhen B-sharestocks. The leverage factors for other return series are mixed and not significant.

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It appears that an increase in the conditional volatility of returns is notimportant in explaining returns in China. This finding is consistent with the resultsof Poon and Taylor (1992) for U'K, data and of Baillie and DeGennaro (1990) forU.S. data, where evidence suggests that variables other than volatility estimates areused when formulating expected returns.

Information Flow and Volatility

To examine the hypothesis that the flow of information to the market helpsexplain the volatility of returns, we use trading volume as a proxy for informationinnovations. Autocorrelation in the time-varying rate of information arrival leads totime-series dependencies in conditional volatility that are accounted for by GARCHmodels. This explanation is rooted in a class of theoretical models where tradingvolume and price volatility are driven by exogenous information innovations.Lamoureux and Lastrapes (1990) used this framework to test whether GARCHeffects remain after the conditional volatility specification expands to includecontemporaneous trading volume as a proxy for information arrival. They find thatfor individual stocks, volatility persistence falls significantly once contemporaneoustrading volume is included.

We choose daily trading volume as a measure of the amount of dailyinformation that flows into the market. The model to be estimated is given by thefollowing equations:

(9)

The mixture model predicts that X> O. Furthermore, in the presence ofvolume with X> 0, a and p will be small and not significant if daily volume isserially correlated. In particular, the persistence ofvariance as measured by (a + P)should become negligible ifaccounting for the uneven flow ofinformation explainsthe presence of GARCH in the data.

The GARCH coefficients a and p, shown in Table 8, are significant forreturn series in China's stock markets. The sums (a + P) are fairly close to 1,indicating the persistence ofpast volatility in explaining current price volatility. Weexamine the X coefficients and compare the results with the output from ourconstrained model where conditional variance is modeled as a function ofonly pasterrors and past variances. If observed GARCH effects tend to disappear whenunexpected innovations in market value are included in the model, this suggestsobserved persistence in returns might be at least partially explained by information

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arrival. The GARCH effect remains significant when volume is included in themodel; however, the persistence in volatility as measured by (a + P) is marginallysmaller when volume is included in the model. These results may suggest thatlagged squared residuals still contribute much when there is additional informationabout the variance of the stock return process after accounting for the rate ofinformation flow, as measured by contemporaneous volume.

Najand and Yung (1991) and Bessembinder and Seguin (1993) also test forthe relation between a proxy for information arrival and GARCH effects. They notean important specification issue is implicit in including the contemporaneousvolume variable in the conditional volatility specification of a GARCH model.Strictly speaking, inferences can be made only if volume is exogenous. However,in the mixture of distribution model, volume and price changes are a joint functionof information arrival. Whether this is the case, estimates based on such aspecification could have an unquantified bias.

Najand and Yung (1991) address this simultaneity problem by using laggedvolume, which may be treated as a predetermined variable. They find that withlagged volume the GARCH effect is consistently significant in all calendar periods,and they observe a statistically meaningful positive correlation between pricevariability and volume for their sample. Bessembinder and Seguin (1993)demonstrate, over a range of different futures contracts, that the conditionalvolatility exhibits strong persistence even when unexpected and expected volumeand open interest are included in the specification. These results are at odds withthose of Lamoureux and Lastrapes (1991).

To overcome the simultaneity problem, we reestimate (9) with lagged valuesof volume (Vt _1).6 We find that the estimated GARCH coefficients a and parestatistically significant for return series in China's stock markets. The sums (a + P)are fairly close to 1, indicating the persistence ofpast volatility in explaining currentreturn volatility. The GARCH effect remains significant when lagged volume isincluded in the model; however, the persistence in volatility as measured by (a + P)is still marginally smaller when volume is included. Trading volume as a proxy forinformation innovations does not help explain the volatility of returns.

IV. Conclusion

We find that the variance ratio test rejects the random walk hypothesis.Stock returns are not independent and identically distributed in China. Thealternative to the random walk is a mean-reverting process, which implies that pricesand returns are negatively serially correlated and that the negative serial correlationbecomes more severe as the holding period increases. If such processes governreturns, the variance ratios should be less than unity for long horizons.

6The results are available on request.


Our results support the mean-reverting hypothesis. The presence ofnegativeserial correlation in return series may also be linked to thin trading. To test for longmemory in stock returns in China's stock markets, we employ the fractionaldifferencing test for long memory devised by Geweke and Porter-Hudak (1983).There is some evidence of long memory, which suggests possibilities forconstructing nonlinear econometric models for improving price forecastingperformance. GARCH and EGARCH models are used to obtain appropriate seriesof conditional variances that can be used as expected volatility estimates. We findstrong evidence oftime-varying volatility. We also find that periods ofhigh and lowvolatility tend to cluster. Also, volatility shows high persistence and is predictable.Evidence in support of a fat-tailed conditional distribution of returns is found,implying that large changes in speculative prices are expected relatively often. Weexamine the relation between expected returns and expected volatility by fitting theGARCH-M model and do not find a relation between expected returns and expectedrisk as predicted by asset pricing models. This suggests variables other thanvolatility estimates should be used when formulating expected returns in China. Wedo not find daily trading volume used as a proxy for information arrival time hassignificant explanatory power regarding the conditional volatility of daily returns.Most of the findings are consistent with those documented for mature markets.

References

Baillie, R. T. and R. P. DeGennaro, 1990, Stock returns and volatility, Journal ofFinancial and QuantitativeAnalysis 25, 203-14.

Bessembinder, H. and R. J. Seguin, 1993, Price volatility, trading volume, and market depth: Evidence fromfutures markets, Journal ofFinancial and Quantitative Analysis 28, 21-39.

Bollerslev, T. R., 1986, Generalized autoregressive conditional heteroskedasticity,Journal ofEconometrics31,307-27.

Cartwright, P. A. and C. F. Lee, 1987, Time aggregation and the estimation ofthe market model: Empiricalevidence, Journal ofBusiness and Economic Statistics 5, 131--43.

Engle, R. F. and T. Bollerslev, 1986, Modelling the persistence ofconditional variance, Econometric Review5, I-50.

Engle, R. F., T. Ito, and w. L. Lin, 1990, Meteor showers or heat waves? Heteroskedastic intra-dailyvolatility in the foreign exchange market, Econometrica 58, 525--42.

Engle, R. F. and V. K. Ng, 1993, Testing for common features, Journal ofFinance 48, 1749-78.Fuller, W., 1976, Introduction to Statistical Time Series (Wiley, New York).Geweke, J. and S. Porter-Hudak, 1983, The estimation and application oflong memory time series models,

Journal ofTime Series Analysis 4, 221-38.Harvey, C., 1995, Predictable risk and returns in emerging markets, Review ofFinancial Studies, 773-816.Lamoureux, C. G. and W. D. Lastrapes, 1990, Heteroskedasticity in stock return data: Volume versus

GARCH effects, Journal ofFinance 55, 221-29.Lo, A. W., 1991, Long-term memory in stock market prices, Econometrica, 59, 1279-1313.Lo, A. W. and C. A. MacKinlay, 1988, Stock market prices do not follow random walks: Evidence from

simple specification test, Review ofFinancial Studies 1, 41-66.Najand, M. and K. Yung, 1991, A GARCH examination of the relationship between volume and price

variability in futures markets, Journal ofFutures Markets 11,613-21.Nelson, D. B., 1991, Conditional heteroskedasticity in asset returns: A new approach, Econometrica 59,

323-70.Poon, S. H. and S. 1. Taylor, 1992, Stock returns and volatility: An empirical study ofthe UK stock market,

Journal ofBanking and Finance 16,37-59.Poterba, J. M. and L. H. Summers, 1988, Mean reversion in stock prices: Evidence and implications, Journal

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stock returns and volatility on china's stock markets

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