Measuring volatility persistence and long memory in the presence of structural breaks Evidence from African stock markets David McMillan School of Management, University of St Andrews, St Andrews, UK, and Pako Thupayagale Bank of Botswana, Gaborone, Botswana Abstract Purpose – The purpose of this paper is to estimate volatility in African stock markets (ASMs), taking account of periodic level shifts in the mean level of volatility, where the regime shifts are determined endogenously. Design/methodology/approach – Volatility estimates are incorporated into standard volatility models to assess the impact of structural breaks on volatility persistence, long memory and forecasting performance for ASMs. Findings – The results presented here indeed suggest that persistence and long memory in volatility are overestimated when regime shifts are not accounted for. In particular, application of breakpoint tests and a moving average procedure suggest that unconditional volatility displays substantial time variation. Practical implications – A modification of the standard generalised autoregressive conditional heteroscedasticity model to allow for time variation in the unconditional variance generates improved volatility forecasting performance for some African markets. Originality/value – This paper describes one of the first studies to incorporate endogenously determined regime shifts into volatility estimates and assess the impact of structural breaks on volatility persistence, long memory and forecasting performance for ASMs. Keywords Africa, Stock markets, Volatility, Variance Paper type Research paper 1. Introduction Reform of local equity markets and relaxation of capital controls in order to attract portfolio equity flows have become integral parts of the establishment or rehabilitation of African stock markets (ASMs). Against this background, these equity markets have experienced rapid and substantial growth as investors have taken the opportunity to diversify their portfolios internationally in search of the highest potential returns to their investment. Such portfolio investment inflows, in search of higher returns and diversification benefits have helped contribute to the growth and modernisation of ASMs. These developments have attracted further investor attention and raised the profile of ASMs. In addition, these opportunities have also motivated research into The current issue and full text archive of this journal is available at www.emeraldinsight.com/0307-4358.htm The authors gratefully acknowledge numerous useful comments and suggestions from the Editors (Suzanne Fifield and David Power) and two anonymous referees on an earlier draft. Measuring volatility persistence 219 Managerial Finance Vol. 37 No. 3, 2011 pp. 219-241 q Emerald Group Publishing Limited 0307-4358 DOI 10.1108/03074351111113298
Transcript
Measuring volatility persistenceand long memory in the presence
of structural breaksEvidence from African stock markets
David McMillanSchool of Management, University of St Andrews, St Andrews, UK, and
Pako ThupayagaleBank of Botswana, Gaborone, Botswana
Abstract
Purpose – The purpose of this paper is to estimate volatility in African stock markets (ASMs), takingaccount of periodic level shifts in the mean level of volatility, where the regime shifts are determinedendogenously.
Design/methodology/approach – Volatility estimates are incorporated into standard volatilitymodels to assess the impact of structural breaks on volatility persistence, long memory and forecastingperformance for ASMs.
Findings – The results presented here indeed suggest that persistence and long memory in volatilityare overestimated when regime shifts are not accounted for. In particular, application of breakpoint testsand a moving average procedure suggest that unconditional volatility displays substantial timevariation.
Practical implications – A modification of the standard generalised autoregressive conditionalheteroscedasticity model to allow for time variation in the unconditional variance generates improvedvolatility forecasting performance for some African markets.
Originality/value – This paper describes one of the first studies to incorporate endogenouslydetermined regime shifts into volatility estimates and assess the impact of structural breaks on volatilitypersistence, long memory and forecasting performance for ASMs.
1. IntroductionReform of local equity markets and relaxation of capital controls in order to attractportfolio equity flows have become integral parts of the establishment or rehabilitationof African stock markets (ASMs). Against this background, these equity markets haveexperienced rapid and substantial growth as investors have taken the opportunityto diversify their portfolios internationally in search of the highest potential returnsto their investment. Such portfolio investment inflows, in search of higher returnsand diversification benefits have helped contribute to the growth and modernisationof ASMs. These developments have attracted further investor attention and raised theprofile of ASMs. In addition, these opportunities have also motivated research into
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/0307-4358.htm
The authors gratefully acknowledge numerous useful comments and suggestions from theEditors (Suzanne Fifield and David Power) and two anonymous referees on an earlier draft.
Measuringvolatility
persistence
219
Managerial FinanceVol. 37 No. 3, 2011
pp. 219-241q Emerald Group Publishing Limited
0307-4358DOI 10.1108/03074351111113298
various aspects of stock-return behaviour in these markets. In particular, policy makers,regulators and investors may be concerned with the potential impact of a major event(e.g. a policy shift or a change in either the domestic or external economic environment)on stock-return volatility, and specifically, the question of whether or not suchepisodes precipitate sudden changes in stock-return volatility. Furthermore, in lightof the important role that shocks to volatility play in driving volatility persistence (i.e. theextent to which shocks to current volatility remain important for long periods intothe future), it follows that the identification of structural breaks (or regime shifts in thestock-return volatility profile may have important ramifications for portfolio and riskmanagement (Malik et al., 2005).
A great deal of research modelling financial time series has focused onestimating time-varying volatility. Indeed, an extensive literature has established thepresence of non-constant and time dependent volatility in high-frequency asset returnsdata[1]. The main representatives of this class of model are the autoregressiveconditional heteroscedasticity (ARCH) model (Engle, 1982) and its extensions includingthe generalised ARCH (Bollerslev, 1986) and the fractionally integrated generalisedautoregressive conditional heteroscedasticity (FIGARCH) (Baillie et al., 1996) models[2].These models explicitly recognise the difference between conditional and unconditional(or long run) variance, where the former is allowed to change over time and the latterremains constant.
Against this background, this paper examines the properties of stock-returnvolatility in ASMs (henceforth ASMs) with a view to characterising the behaviour of theconditional variance. This is important since potential gains from international portfoliodiversification have attracted investors to these markets yet little is known about thevolatility profile in ASMs. In particular, this paper investigates the effect of structuralbreaks on volatility persistence, long memory and the forecasting performance of ASMs.
These questions may provide investors with a better understanding of how shocksaffect volatility over time and the role that regime changes may play in this process.Indeed, Poterba and Summers (1986) show that the extent to which stock-returnvolatility is persistent is important since it affects stock prices through a time-varyingrisk premium. More specifically, in this paradigm, an increase in (expected) volatilitypersistence would imply a decline in the current stock price. Second, Andersen andBollerslev (1998) demonstrate that GARCH models provide accurate volatility forecastswhen volatility persistence is accurately estimated.
More recently, Starica and Granger (2005) examined US data and demonstrated thatthe incorporation of occasional breaks in the unconditional variance generates improvedforecasts in comparison to the standard GARCH(1,1) model. In the GARCH model thespeed of mean reversion is given by the parameters governing the magnitude andpersistence of shocks to the volatility process. That is, these models are based on theassumption that although volatility is persistent it ultimately reverts to a constantmean volatility[3]. However, the degree of persistence may be overstated if breaks in thedata are ignored. For example, Mikosch and Starica (2004b) and Malik et al. (2005)show that volatility persistence is overestimated if regime shifts are not accounted forin the standard GARCH model. Further, Mikosch and Starica (2004a) present evidenceindicating that the unconditional volatility of a GARCH model may not be constant;hence, leading to spurious evidence of persistence. In a more formal long-memorycontext, Diebold and Inoue (2001), Granger and Hyung (2004) and McMillan and
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Ruiz (2009) show that the failure to account for structural breaks can lead to spuriousevidence of long memory.
These issues have, to date, not been examined for ASMs, and this paper attempts to fillthat gap. Hence, our study makes a contribution to the extant literature in the followingways. First, there does not appear to be any tests examining the impact of structuralbreaks (or time variation in the unconditional volatility process) on measures of volatilitypersistence and long-memory behaviour in ASMs. Second, as is well-known accuratevolatility forecasts are essential in portfolio and risk management. Accordingly, weexamine forecasting performance in order to ascertain if the adjustment for structuralbreaks in standard volatility models generates improved forecast accuracy. Moregenerally, in order to determine future returns from risk taking or the need for policyintervention, it is important to forecast volatility (Loeys and Panigirtzoglou, 2005).Third, some of the techniques used in this paper represent relatively new innovations ineconometric analysis, and as such, may provide fresh perspectives for both policymakers and market participants. Fourth, this study will also use data from the USA (S&P500) and the UK (FTSE 100) for comparative purposes in order to highlight differences inASMs and developed country markets. This comparison may help assess if the degree ofmarket development is reflected in volatility dynamics, and as a result, it may potentiallyhave a bearing on trading and risk management strategies if significant differences existbetween ASMs and their developed country counterparts. Indeed, Domowitz et al. (1998)argue that emerging markets are typically much smaller and less liquid and as aconsequence are potentially more volatile than equity markets in industrialisedeconomies. Overall, our results are important for market participants as time-varyingvolatility is key for measuring risk (e.g. calculation of hedge ratios and value at risk) andthe formulation of trading strategies and pricing derivatives (Hull and White, 1987;Chesney and Scott, 1989). Volatility also provides market participants with informationon market sentiment, market beliefs about the future and the evolving attitude towardrisk (Krichene, 2003).
2. Brief review of volatility breaks literatureTo investigate the impact of structural breaks on stock-return volatility, previousstudies have employed a variety of estimation procedures and data of encompassingvarious periods and spanning different durations. For example, Lamoureux andLastrapes (1990) examine the extent to which volatility persistence may be overstatedas a result of not incorporating deterministic structural shifts in the model. Their results,obtained from both Monte Carlo simulation and analysis of daily US stock-returndata indicate that GARCH measures of volatility persistence are sensitive tomodel misspecification of this form. Lobato and Savin (1998) test for the existence oflong-memory daily equity data from the US S&P 500, their empirical analysis revealsthat spurious long memory is a function of nonstationarity and aggregation in equitydata. They address these shortfalls by analysing sub-periods (i.e. stationary periods) ofequity returns and using disaggregated data (i.e. individual stocks). Diebold and Inoue(2001) use a range of methods, including a Markov-switching model and a stochasticpermanent break model to show that long memory and structural change may beconfused. In particular, they note that the finding of long memory may be spurious,resulting from a failure to incorporate (even small amounts) of structural change in theestimation procedure. Gourieroux and Jasiak (2001) demonstrate through a simulation
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exercise the manner in which stochastic processes may produce a long-memory effect(in the autocorrelation function (ACF)). Similarly, Granger and Hyung (2004) analysethe absolute equity returns of the S&P 500 and report that the finding of long memorymay in part be a spurious result emanating from the presence of neglected breaks in theseries. Mikosch and Starica (2004b) provide a theoretical basis as to why the findingof long-memory behaviour in volatility may be unauthentic. In particular, they showthat nonstationarity in the data, specifically, changes in the unconditional variance(or long-run volatility) may lead to the erroneously conclusion of long memory involatility. Mikosch and Starica (2004a) show that the assumption of a constantvolatility unconditional variance in GARCH modelling is too restrictive. Indeed, usingstock-return data from the S&P 500 they detect changes in the structure of the dataderiving from the existence of structural breaks in the unconditional variance. In total,both papers from Mikosch and Starica reinforce the finding that finding of volatilitypersistence may be overstated due to the failure to account for the existence of structuralbreaks. Malik et al. (2005) use weekly stock-return data from Canada and shows thatafter incorporating endogenously determined volatility shifts in the GARCH model, theextent of volatility persistence is significantly reduced. Finally, McMillan and Ruiz(2009) analyse daily equity market data from ten industrialised countries and find thatboth the degree of volatility persistence and long-memory behaviour are diminishedonce neglected breaks (or time variation in the unconditional variance) is included in theestimation methodology. Furthermore, they report that the adoption of a GARCH modelwhich incorporates mean variation delivers improved volatility forecastingperformance over long horizons. Despite the extensive research on the impact ofstructural breaks (or regime shifts) in stock-return volatility in developed financialmarkets, the extant literature does not appear to report any empirical evidence forASMs. As a consequence, this paper attempts to address these issues.
3. Stock markets in Africa: an overviewPrior to 1987, there were only eight stock markets in Africa, at the end of 2007, there were21 stock markets ranging from new markets such as that in Cape Verde and Libya(launched in 2005 and 2007, respectively) to the more established markets like those inSouth Africa and Egypt (founded in 1887 and 1890, respectively). From 1997 to 2007,African stocks markets increased their total market capitalisation from about US$320billion to approximately US$1,125 billion as these countries opened up to foreigninvestors. In particular, investors and investment funds have channelled capital intothese markets in order to take advantage of high-return prospects and concomitantdiversification benefits.
The establishment or revitalisation of ASMs took place in a context of major reforms,especially during the 1990s. These measures included the liberalisation of their financialsectors, privatisation of state-owned enterprises, improvement of the investment climate(through enactment of policies targeting the achievement of macroeconomic stability andbetter business environments), introduction of a more robust regulatory framework andimprovements in the basic infrastructure for capital market operations (de la Torreand Schmukler, 2005). These reforms set the stage for a significant market expansion, witha trend development in size and liquidity. New equity issues, volume and value of tradingand the number of traded companies all recorded significant progress. As a result, marketcapitalisation increased from a median of about 19.5 per cent of gross domestic product
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(GDP) to approximately 57.8 per cent of GDP from 1997 to 2007 for ASMs, and theturnover ratio rose from a median of 5.2-19.3 per cent. These policy-driven regime shiftshave been an important feature of the market environment in ASMs but their impact onstock-return volatility has received little attention.
The diversity of ASMs is illustrated in Table I, which presents calculations for thenumber of companies listed in each market, market capitalisation in US dollars, and otherkey market indicators. The Johannesburg Securities Exchange (JSE) in South Africa hasa market capitalisation of US$855.3 billion and is the largest on the continent. The JSE isan anomaly in several respects. First, it represents almost 70 per cent of Africa’s marketcapitalisation; second, while the other ASMs have a low correlation with other globalmarkets (Smith et al., 2002), it is integrated with the major international financial markets(for example, during the emerging market crises of 1998, the JSE overall share indexdeclined by 30 per cent in August 1998); third, it is, as a consequence, similar in characterto the larger emerging markets found in Latin America and Asia. The second and thirdlargest equity markets are those of Egypt and Nigeria, which have a marketcapitalisation of US$148.5 billion and US$131.1 billion, respectively. These three marketsaccount for almost 90 per cent of the market capitalisation of ASMs. In addition, thesemarkets dominate the number of listed firms in ASMs.
Although, there are marked differences in the size and number of listed companies,ASMs share a number of attributes. For example, low liquidity as measured by theturnover ratio range from a low as 3.3 per cent in Botswana to 47.5 per cent inSouth Africa. These measures are low relative to comparable figures from otheremerging markets. Indeed, the most liquid markets have turnover ratios in excess of 100per cent (Magnusson and Wydick, 2002). Another feature shared by ASMs pertains tothe presence of non-synchronous trading or non-trading effects which in turn reflectsmall market size and compound illiquidity (Yartey and Adjasi, 2007). In the case ofSouth Africa, Smith et al. (2002) suggest that the relative illiquidity of the JSE reflectsthe domination of share holdings by a small number of very large of corporations.This is further reinforced by a pattern of cross-shareholding among these companies
which further stifles liquidity. More generally, illiquidity implies that the cost of tradingremains high and illiquidity begets illiquidity by limiting the capacity of investors tounwind their positions; which in turn, may potentially stimulate further volatilitythereby deterring further market entrants on both buy and sell sides, which, in turn,perpetuates the cycle of illiquidity (de la Torre and Schmukler, 2005). Although keyindicators of market development point to underdeveloped equity markets, ASMs havebeen among the fastest growing in the world and have as a result attracted significantinvestor attention especially in light of the potential portfolio diversification benefitsthey offer. In addition, ASMs continue to perform well in terms of return on investmentrelative to other emerging markets and indeed the major international markets.For example, in 2004, the Ghana stock market recorded a growth rate of 144 per cent inUS dollar terms making it the world’s best performing equity market in that year; incomparison, the Morgan Stanley Capital International Global index appreciated by only30 per cent (Databank Group, 2004). Similarly, in Egypt, the Cairo and Alexandria StockExchange 30, which groups the stocks of the top Egyptian companies in a benchmarkindex, has risen more than five-fold since Egypt launched an economic reform drive inJuly 2004. The Zimbabwe Stock Exchange (ZSE) grew by almost 450,000 per cent in 2007compared to a year ago. According to our calculations, we find that even after strippingout the effects of hyperinflation, the ZSE is among the best performing stock market inthe world[4].
4. DataThe data for this study are obtained from Bloomberg and consists of data for 11 Africancountries (Table II). Daily observations on stock returns for the following countries:Egypt, Kenya, Morocco, Namibia, Nigeria, South Africa, Tunisia and Zimbabwe.For Botswana, Ghana and Mauritius, the data distribution is uneven; hence, the two-dayholding return is calculated. In addition, daily data from UK and US stock indices areincluded for comparative purposes. These indices are all denominated in local currencyand refer to end-of-day quotes. The equity return (rt) is defined as ð pt 2 pt21Þ £ 100,where pt is the log of the equity price at time t.
Country Index Start date of data Sample size
Botswana Domestic Companies Index 4 April 2001 1,237Egypt Hermes Financial Index 27 July 1995 3,243Ghana Ghana Stock Exchange All Share Index 20 September 2002 1,377Kenya Nairobi Stock Exchange 20 14 May 1992 4,073Mauritius Stock Exchange of Mauritius All Share Index 14 August 1998 2,447Morocco Casablanca Stock Exchange Most Actives Index 22 May 2002 1,464Namibia Namibia Overall Stock Exchange Index 31 January 2003 1,275Nigeria Nigeria Stock Exchange Index 21 January 1999 2,333South Africa FTSE/JSE Africa All Share Index 9 January 1996 3,125Tunisia Tunisia Stock Market Index 15 March 1999 2,296Zimbabwe ZSE Industrials Index 4 October 1994 3,455USA Standard & Poor’s 500 Index (S&P 500) 9 August 1990 4,538UK FTSE 100 9 August 1990 4,538
Table II.Stock market data
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5. Empirical analysisA GARCH model is defined by its conditional first and second moments, which aretypically referred to as the mean and variance equation, respectively. The mean equationis given by:
rt ¼ mþ 1t; where 1tjI t21 , N ð0; htÞ ð1Þ
wherem is the conditional mean, which may include autoregressive and moving averageterms, and an error term 1t, which follows a conditional normal density with a zero meanand a conditional variance h2
t , where the information set available to investors up to timet 2 1 is given by It21. The specification of the volatility equation is consistent with aforecast of the variance at time t ðh2
t Þ on the basis of a long-term average (the constantunconditional mean value, v), the volatility forecast from the previous period ðh2
t21Þ andinformation about volatility in the last period ð12
t21Þ:
h2t ¼ vþ a12
t21 þ bh2t21: ð2Þ
Furthermore, the inequality restrictions v . 0 and a, b $ 0 are imposed to ensure thatthe conditional variance is strictly positive.
In the GARCH(1,1) model the sum of a and b measures volatility persistence (i.e. theextent to which shocks to current volatility remain important for long periods into thefuture). As this sum approaches unity, the persistence of shocks to volatility becomesgreater. However, when a þ b ¼ 1 then any shock to volatility is permanent and theunconditional variance is infinite. In this case, the process is denoted an IGARCH (Engleand Bollerslev, 1986) and implies that volatility persistence is permanent; hence, pastvolatility is significant in predicting future volatility over all finite horizons. When thesum of a and b is greater than unity, then volatility is explosive, i.e. a shock to volatilityin one period will result in even greater volatility in the next period (Chou, 1988).
As a simplification, the standard GARCH model restricts the dynamics of theunconditional variance to an arbitrary constant in order to focus on the conditionalvariance. This paper will relax this assumption in order to examine the implicationson the behaviour of the conditional variance; especially, the ARCH and GARCHparameters. In particular, the unconditional variance (s 2) in this model is equal tov=ð1 2 a2 bÞ. This specification allows us to investigate the possibility of structuralbreaks in the variance constant, v, which in turn imply shifts in the unconditionalvariance; and hence, may lead to spurious evidence of volatility persistence in a standardGARCH model.
To provide another perspective on the estimates of volatility persistence, the half-lifeof volatility shocks are also presented. The half-life measures the number of days overwhich a shock to volatility decays to half its original size and is calculated asl ¼ logð0:5Þ= logðaþ bÞ.
The GARCH model estimates are reported in Table III. The estimates of volatilitypersistence for the 11 ASMs vary considerably. In Nigeria and Zimbabwe, volatilitypersistence is explosive and equal to 1.045, and 1.109, respectively. For all other marketsa þ b , 1, for example, among ASMs, Kenya exhibits the greatest persistence at 0.991,while Botswana registers the lowest level of volatility persistence at 0.796. Meanwhile,evidence of high-volatility persistence is also found in the UK and the USA at 0.984and 0.994, respectively. Kenya’s high-volatility persistence translates into a half-lifeof 74 days while Botswana’s relatively lower degree of volatility persistence is equivalent
Measuringvolatility
persistence
225
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Table III.GARCH and fractionalintegration estimates
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to a half-life of only three days. South Africa and Egypt, are Africa’s largest equitymarkets, and have half-lives of 29 and 35 days, respectively. This means that shocks tovolatility will taper off with a half-life of about four weeks and five, respectively. Incomparison, it takes 122 days for a shock to volatility to diminish to half its original sizein the USA[5]. The generally smaller degree of volatility persistence (and hence half-life)of ASMs in comparison to the developed markets in our sample may reflect differencesin the structural composition of these markets. In particular, ASMs are permeated by thepresence of nonsynchronous trading (or non-trading) effects which may imply thatinformation and hence volatility shocks dissipate quicker given the fragmented natureof trading in these markets. This in turn may also be a manifestation of the illiquiditycharacterising these markets (Magnusson and Wydick, 2002).
ACF and fractional integrationIn the time domain, long memory is characterised by a very slow mean-revertinghyperbolically decaying ACF. To investigate this asymptotic property in ASMs, wereport the sample ACF for absolute index returns and for up to 100 lags for each series inFigure 1, together with the 5 per cent critical value based upon 2=
ffiffiffiffiffiffiffiðTÞ
p. The graphical
evidence shows that in ASMs most autocorrelations are not significant at all lags exceptfor Egypt. For Ghana, there is evidence of long memory with all autocorrelations beingsignificant, except at points around lag 45. Botswana, Namibia and Zimbabwe are theextreme examples, in that, their ACFs are mostly insignificant at all lags, a patterninconsistent with the existence of long memory. In total, the ACFs of the majority of theASMs do not suggest the presence of long memory. In comparison, the ACFs from theUSA and the UK suggest the existence of long memory in volatility.
In the frequency domain, a long-memory process is revealed by the behaviour of itsspectral density function f(lj) estimated at the harmonic frequencies lj ¼ 2pj=T ,where j ¼ 0, 1, 2, . . . ,m defines the set of harmonic frequencies. Indeed, a stationaryprocess is defined to have long memory when:
f ðljÞ < cl�2dj as lj !1 ð3Þ
where c . 0 and d [ (0,0.5) is the long-memory parameter.Geweke and Porter-Hudak (1983) proposed a semi-parametric procedure to obtain an
estimate of the fractional differencing parameter d based on the slope of the spectraldensity function around the angular frequency lj ¼ 0. In particular, let I(lj) denotethe sample periodogram at the jth Fourier frequency, lj ¼ 2pj=T , j ¼ 1, 2, . . . , [T/2].The estimator of the parameter of fractional integration, d, is then based on theleast-squares regression:
logðI ðljÞÞ ¼ b0 þ b1logðljÞ þ 1j ð4Þ
where j ¼ 1, 2, . . . ,m, and d ¼ 21=2b1 provides the estimated long-memory parameterusing the conventional truncationm ¼ T 0.5, for the equity returns, presented in the finalcolumn of Table III. The results highlight the prevalence of long memory in ASMs. Thisresult may reflect the structural features of ASMs – notably small size and illiquidity.Nagayasu (2003) argues that stock markets in developing countries are likely to displaya long-memory component because of the shallowness of their markets coupled withtheir less mature institutional and regulatory environment. Our results do suggest long
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memory with the size of d ranging from 0.25 for Botswana to 0.54 in Zimbabwe. Further,all these results are statistically different from zero. However, we observe that the ACFprovides generally ambiguous evidence with respect to the existence of long memorycompared to the Geweke and Porter-Hudak (GPH) estimates. This may be because ourchoice of volatility proxy jrtj is a “noisy” proxy for the true volatility process (Poon andGranger, 2003).
6. Evidence of structural change in volatilityIn order to test for potential breaks within the conditional mean of the volatility process(proxied by absolute returns), we perform the breakpoint tests of Bai and Perron (1998,2003a, b)[6]. This method allows us to identify shifts in volatility endogenously incontrast to methods where regime shifts are imposed on a priori grounds.
The results of the breakpoint tests are plotted in Figure 2, and indicate differentpatterns and levels of volatility in the ASMs. More specifically, these graphs showthat the variance process is indeed time varying and characterised by distinct regimes.The number of regime changes in ASMs is generally more numerous than those found inthe UK and the USA and thus reflecting a variety of country-specific developments.For example, break point tests show that Namibia is characterised by only one point ofsudden change in volatility and therefore two volatility regimes; while, Egypt and SouthAfrica both exhibit six well-defined episodes of variability in the mean level of theirrespective unconditional variance processes.
To illustrate the nature of the breakpoints further, Kenya displays four distinctregimes: the first is from the start of the sample until 1995; the second is from 1995 to 1996where volatility spikes substantially; the third is from 1996 to 2003 where the level ofunconditional volatility is considerably lowered; and finally, volatility rises to a higherregime in 2003 to the end of 2007. The first period encompassed the period of stockexchange modernisation; the second period, coincides with a relaxation of exchangecontrol; the third period was characterised by the adoption of international accountingstandards and, the fourth coincided with economic and political uncertainty[7]. As such,breakpoints appear country specific, in comparison, the USA and the UK exhibit threevolatility regimes these appear to be synchronised and similar in timing and duration.In particular, heightened volatility from 1997 to 2003 coincide with a period of globalfinancial crisis. While, the breakpoints in the USA and the UK correlate with majorinternational financial events, those in the ASMs coincide with country-specificeconomic or political developments. This in turn may reflect that ASMs are notintegrated into the wider international financial markets. This may be because thesemarkets are illiquid which may deter entrants on both buy and sell sides. In addition,restrictions on foreign participation in some ASMs (e.g. Zimbabwe) may also explain thedisconnection with the major international equity markets.
Breakpoint tests imply that the volatility processes are marked by discrete shifts inthe level of fluctuations. However, Mikosch and Starica (2004a) demonstrate that theunconditional volatility may exhibit a more gradual pattern of time variation. In orderto investigate this aspect, we follow the procedure considered by Mikosch and Starica(2004a) derivation time variation of the unconditional variance through recursiveestimation of the GARCH(1, 1) model.
The time-varying nature of the unconditional volatility is shown in Figure 3. Theseresults indicate that the unconditional variance process is not constant as the GARCHmodel assumes but rather exhibits wide fluctuations and in some cases very abruptchanges. Furthermore, the form and timing of time variation of the ASMs volatilityseries differ considerably and appear to reflect country-specific developments. Again,in comparison, the broad thrust of time variation in the unconditional variance observedin the USA and the UK appears synchronised perhaps reflecting that these markets aredriven by common events (e.g. international financial crisis). In addition, ASMs, withthe exception of South Africa, are isolated from the major global equity markets
and are therefore driven more by domestic economic fundamentals (Smith et al., 2002;Irving, 2005)[8].
In order to examine the impact of removing the time-varying unconditional variancecomponent on the long-memory property of the equity volatility, we examine thebehaviour of the ACF. In particular, Figure 4 shows the ACF for 100 lags after accountingfor breaks in the volatility data[9]. In particular, from these graphs it is apparent that
after adjusting for time variation in the unconditional volatility process the ACF decaysquickly and erratically. For example, Egypt’s ACF decays quickly and becomesinsignificant in at around lag 20 and crosses zero at lag 35. Similarly, Morocco’s ACFdecays quickly and fluctuates in a choppy manner below zero. In sum, comparingFigure 1 with Figure 4, these results show that after accounting for time variation in theunconditional mean variance the ACF decays quickly and for the most part suggest thatthe long-memory component in the data is diminished. Nonetheless, for some markets,such as Botswana and Tunisia, the ACF remains erratic (before and after accounting
for structural change in the data) and does not offer any support in favour of a hyperbolicdecay structure.
To further evaluate the extent to which structural change has a bearing on the degreeof long memory, we re-estimate the fractional integration parameter, with the findingsreported in the final column of Table V. The results show that the extent of long memoryin volatility is reduced for all the markets considered in this study. For example, thelong-memory parameter for Kenya and Mauritius decline to 0.24 and 0.08 from 0.40 and0.35, respectively. These results suggest that the long-memory parameter is overstated ifstructural breaks are not accounted for.
While the results presented so far suggest possible bias introduced by structuralbreaks, Perron and Qu (2006) argue that while a short-memory process with breaks willbias upward the persistence estimate of a short-memory process; equally, the persistenceof a long-memory process is biased downward after filtering structural break tests. Sincethese two processes may be confused, Perron and Qu propose a test which distinguishes along-memory process from a short-memory process with breaks[10]. This test stipulatesthat a genuine long-memory process is such that the estimates of the fractional integrationparameter d should be invariant to the choice of the bandwidth m, while the test statisticshould not be statistically different from zero. This test is performed and the resultsare presented in Table IV. The results across all markets show an inverse relationshipbetween d and m. These test results suggest that volatility is these markets are notintrinsically long memory but rather short-memory processes subject to level shift in theunconditional variance.
Time-varying mean-adjusted GARCH modelThe empirical results of the previous section, and particularly Figure 3, show that thestandard GARCH assumption of a constant unconditional variance is not tenable.Accordingly, this assumption is relaxed, in order to account for time-variation in themean process of the unconditional variance. Hence, we follow McMillan and Ruiz (2009)
Truncation value, mT 1/3 T 1/2 T 2/3 T 4/5 Perron-Qu
Notes: Entries are estimates of the fractional integration parameter over different truncation values,see equation (4); the final column is the Perron-Qu test (Perron and Qu, 2006), which is normallydistributed
Table IV.Fractional d estimatesfor across different m
Measuringvolatility
persistence
233
and estimate a GARCH model with a rolling widow to capture time variation in theunconditional mean variance. In particular, we modify the GARCH model of equation (2)to allows the unconditional mean to vary, as such:
h2t ¼ vm21
X260
w¼1
jrt212wj þ a112t21 þ b1h
2t21 ð5Þ
where m is the window length of the moving average[11]. The results of this model arepresented in Table V. The results generally point to a lower degree of volatilitypersistence in the rolling GARCH model in relation to the standard GARCH model. Forexample, the level of volatility persistence in Egypt and Kenya decrease to 0.976 and0.953 from 0.980 and 0.991, respectively. This is further illustrated by the half-life decayperiods which are generally lower. For instance, before allowing for structural breaksthe half-lives, in ASMs ranged from just over three days (in Botswana) to 74 days(in Kenya); while the UK and the USA registered 43 and 122 days, respectively.In contrast, after incorporating breaks into the model, the half-lives, ranged from justunder three days (in Botswana) to 28 days (in Egypt). Similarly, half-lives were reducedto 37 and 109 days in the UK and the USA, respectively. Further, where volatilitypersistence was explosive as in the case of Nigeria and Zimbabwe; the use of the rollingGARCH reduced (but did not change) the fact that volatility persistence in these marketswas still explosive. However, there are two exceptions to this pattern of results whichare represented by the cases of Morocco and Tunisia where the extent of volatilitypersistence rises. Notably, the graph for these markets in Figure 3 suggests fairly stableunconditional variance with a few extreme outliers. This means that a moving averageapplication may not be the appropriate method to address the phenomena of structuralchange in these markets.
As a final exercise, we consider the forecasting performance of the rolling GARCHmodel relative to the standard GARCH model and a FIGARCH model[12]. This isrelevant since it would confer a benefit to investors in applications where volatilityforecasts are necessary. To gauge forecasting power, we simply start by splitting ourrespective datasets in half and we estimate each model for all series from the samplespanning the first half; then, we use those estimates to forecast volatility over thesecond part of the sample. This strategy is motivated by the simple fact that ourrespective samples are of differing time span; hence, by splitting our datasets in halfallows for consistency in terms of enhancing comparability of results. Against thisbackground, forecasting power is then evaluated using the following regression:
r2t ¼ aþ bh2f
t þ 1t ð6Þ
where r2t denotes squared daily returns and we use this measure as a proxy for the
“actual” volatility, while h2ft represents the obtained volatility forecast. We then use the
coefficient of determination, R 2, after estimating equation (6) above, as a criteriaagainst which to evaluate the forecast performance of the alternate models. Theresults of this exercise are presented in Tables VI and VII. At the daily level, the resultsindicate that the standard GARCH model generally delivers the best forecasting resultsfor most ASMs, with the exception of Egypt, Namibia, Nigeria and Tunisia. For thesemarkets, the rolling GARCH is superior. For Morocco and the USA, the results of theGARCH and rolling GARCH are almost the same. In contrast, the FIGARCH model
MF37,3
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fractional integrationestimates
Measuringvolatility
persistence
235
delivers the weakest results. However, this is not entirely surprising because thismodel explicitly captures long-term volatility.
Since volatility forecasts are additive, it is straightforward to derive the monthlyforecasts. Over the longer horizon, the results are mixed, with no overwhelming supportfor any single model. The standard GARCH generally delivers superior forecastingperformance for five of the ASMs (Botswana, Ghana, Kenya, Mauritius and Morocco),while the rolling GARCH model produces better results for Tunisia and Namibia.Meanwhile, in the case of Zimbabwe, South Africa, Nigeria and Egypt the long-memoryFIGARCH model performs the best. These results suggest that despite their beingevidence of time variation in mean volatility for ASM the rolling GARCH model may notbe suitable across the range of markets, perhaps due to the idiosyncratic nature of the
Note: R 2 values are from equation (6)Table VII.Forecast R 2
MF37,3
236
volatility breaks in these markets. As such, more country-specific measures may berequired. In contrast, the results for the UK and the USA are broadly consistent with theearlier work of McMillan and Ruiz (2009), indicating a slight preference for the rollingGARCH model.
7. ConclusionThe aim of this paper is to examine the behaviour of stock-return volatility in ASMs,especially since volatility is an important driver of active investment returns and riskpremia in the market (Loeys and Panigirtzoglou, 2005). Furthermore, accurate volatilityestimates have become a basic input for shaping hedging strategies and managing risks.As such, these findings may have potential value for market participants in portfoliomanagement.
Against this background, the empirical analysis undertaken here found that bothpersistence and long-memory estimates are biased upwards when structural breaks arenot accounted for in the standard GARCH models and fractional integration parameterestimates. In particular, our results indicate time variation in the unconditional mean ofthe volatility series, which in turn, is shown to bias upward the finding of volatilitypersistence. That is, these results are consistent with the notion that misspecification ofthe GARCH model due to ignored structural breaks in the unconditional variance canlead to an overstatement of the extent of volatility persistence in equity data (Mikoschand Starica, 2004a; McMillan and Ruiz, 2009). Furthermore, adjusting the GARCH andfractional integration estimates for regime shifts typically result in lower degrees ofpersistence. Hence, failure to account for such breaks can lead to incorrect inference.With respect to forecast performance, this study finds that accounting for time variationin the unconditional volatility provides useful information which improves forecastingperformance in some cases. However, the standard GARCH model is generally found todeliver superior forecasting performance at the daily level. This is not unexpected asdaily values are unlikely to be affected by structural breaks. At the monthly level, theresults are more mixed, but generally supportive of longer memory models.
The results obtained in this study highlight some contrasts in stock-return volatilitybehaviour between the more developed equity markets and ASMs. For instance, thedegree of volatility persistence and long memory generally tend to be higher in thebenchmark comparators (i.e. US and UK markets) compared to ASMs, even, afteraccounting for structural breaks. Furthermore, the modification of the GARCH model toallow for mean variation (i.e. the rolling GARCH model) generally delivers more accuratevolatility forecasts for the benchmark comparators, while this innovation leads doesnot generally improve forecast performance in ASMs. Indeed, the simple GARCHmodel is generally superior, at both the daily and monthly levels. These findings arepertinent since an important and topical application of volatility modelling concernsthe calculation of value at risk estimates that rely on the derivation of accuratevolatility forecasts. More broadly, our findings suggest the possibility of a differentlong-term stochastic behaviour between developed markets and ASMs – perhapsrelated to the degree of market development. From a portfolio and risk managementperspective investment strategies focusing on ASMs should be based on a completecharacterisation of stock returns in these markets. Our findings suggest that accountingfor structural breaks in ASM volatility modelling represents an important aspect of thatcharacterisation.
Measuringvolatility
persistence
237
Finally, there are several implications of our study. While we implemented a rollingGARCH model to account for time variation in the unconditional variance, future researchmay wish to explore more subtle ways to capture breaks in the unconditional meanprocess, especially since in some markets (Morocco and Tunisia) the implementation of arolling GARCH model resulted in an increase in the level of volatility persistence. Anotherpossibility would be to use nonlinear models to allow for an asymmetric reaction ofvolatility to good and bad innovations (Bollerslev and Mikkelsen, 1996). This is especiallyrelevant since many of the break points identified in ASMs coincide with shifts in policiesor changes in the regulatory environment, which in turn may affect information flows inthese markets and hence the behaviour of investors and policymakers.
Notes
1. Bollerslev et al. (1992) provide an extensive survey of the literature.
2. Stochastic volatility models (SVMs) are a well-known alternative specification (Ghysels et al.,1996). However, SVMs are less tractable than observation-driven models; hence,concentration and application in this paper is on the GARCH models and its extensions.
3. This is because GARCH models are effectively moving average models that calculate thatvolatility will eventually pull back to some form of long-term average.
4. Despite ongoing macroeconomic instability, the Zimbabwe stock market continues toperform well primarily due to the lack of alternative investment opportunities – mostprominently, the relative unattractiveness of the money market given Zimbabwe’shyperinflationary environment and the use of official exchange rate (which is grosslyovervalued) in calculating market performance. For more details, the reader is referred toIrving (2005).
5. However, in the case where volatility is explosive, as in the case of both Nigeria andZimbabwe, the half-life cannot be interpreted (since the half-life approaches infinity asa þ b ! 1; hence, persistence should be considered large for these indices).
6. The breakpoint estimators correspond to the global minimum sum of squared residuals.The testing procedure aims to identify the number of breaks m (i.e. m þ 1 regimes). Thisprocedure estimates the following equation:
xt ¼ bj þ 1t ; t ¼ Tj21 þ 1; . . . ;Tj
For j ¼ 1, . . . ,m þ 1, where xt is the variable of interest and bj ( j ¼ 1, . . . ,m þ 1) is the meanlevel in the jth regime. The m-partition represents the breakpoints for the different regimesand are treated as unknown. Each partition is estimated by OLS with the estimate ofbj ( j ¼ 1, . . . ,m þ 1) generated by the minimisation of the sum of squared residuals. Furtherdetails of the Bai and Perron procedure can be found in a sequence of papers, Bai and Perron(1998, 2003a, b).
7. For a more detailed overview, visit: www.nse.co.ke/newsite/pdf/factbook_07.pdf
8. However, for ASMs, this paper makes no attempt to absolutely identify the causes of regimeshifts and instead focus on identifying the time periods of sudden changes themselves sinceinformation regarding the sources of these shifts cannot readily be imputed.
9. In particular, we subtract the changing mean from the volatility series.
where: 0 , a , b , 1 (a ¼ 1/3; b ¼ 4/5) and d is the long-memory parameter. For moredetails, see Perron and Qu (2006).
11. A window length of 260 corresponds with a five-day trading year and also seems reasonablesince investment managers commonly rebalance their portfolios on an annual basis.
12. Estimation results for the FIGARCH models are available upon request.
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Further reading
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Diebold, F.X. (1986), “Comment on modelling the persistence of conditional variance”,Econometric Reviews, Vol. 5, pp. 51-6.
Ding, Z., Granger, C.W.J. and Engle, R.F. (1993), “A long memory property of stock marketreturns and a new model”, Journal of Empirical Finance, Vol. 1, pp. 83-106.
Engle, R.F. and Lee, G.G.J. (1999), “A permanent and transitory component model of stock returnvolatility”, in Engle, R.F. and White, H. (Eds), Cointegration, Causality and Forecasting:A Festschrift in Honour of Clive W.J. Granger, Oxford University Press, Oxford.
Glosten, L.R., Jagannathan, R. and Runkle, D.E. (1993), “On the relation between the expectedvalue and the volatility of the nominal excess return on stocks”, Journal of Finance, Vol. 48,pp. 1779-801.
Granger, C.W.J. and Joyeux, R. (1980), “An introduction to long-memory time series models andfractional differencing”, Journal of Time Series Analysis, Vol. 1, pp. 15-29.
Lastrapes, W.D. (1989), “Exchange rate volatility and US monetary policy: an ARCHapplication”, Journal of Money, Credit and Banking, Vol. 21, pp. 66-77.
Corresponding authorDavid McMillan can be contacted at: [email protected]
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