Forecasting Volatility of EmergingStock Markets: Linear versusNon-linear GARCH Models

SULEYMAN GOKCAN*

Citigroup, USA

ABSTRACT

ARCH and GARCH models are substantially used for modelling volatilityof time series data. It is proven by many studies that if variables aresigni®cantly skewed, linear versions of these models are not su�cient forboth explaining the past volatility and forecasting the future volatility. Inthis paper, we compare the linear(GARCH(1,1)) and non-linear(E-GARCH) versions of GARCH model by using the monthly stock marketreturns of seven emerging countries from February 1988 to December 1996.We ®nd that for emerging stock markets GARCH(1,1) model performsbetter than EGARCH model, even if stock market return series displayskewed distributions. Copyright # 2000 John Wiley & Sons, Ltd.

KEY WORDS emerging markets; forecasting; volatility; GARCH

Recently many articles have been written about the volatility of stock markets. Most researchersagree that volatility is forecastable in many stock markets, but there are di�erences in the waythey model the volatility. Among these models di�erent versions of the GARCH (GeneralizedAuto Regressive Conditional Heteroscedasticity) model are the most successful. One of thereasons that the GARCH models are very popular is that it can e�ectively remove the excesskurtosis in return series.

Besides having excess kurtosis market returns may display seriously skewed distributions. Thelinear GARCH models cannot cope with such skewness, and therefore we can expect forecast oflinear GARCH model to be biased for skewed time series. To deal with this problem non-linearGARCH models are introduced, which take into account skewed distributions; for example,Quadratic GARCH model (QGARCH) introduced by Engle and Ng (1993) and Sentana (1995),the model introduced by Glosten, Jogannathan, and Rankle (1992) is the GJR model, and thatintroduced by Nelson (1991) the Exponential GARCH model (EGARCH).

Most research has been conducted for developed capital markets. One should raise thequestion about the usefulness of linear or non-linear GARCH models to explain the pastvolatility and forecast the future volatility for emerging stock markets. These markets have very

Received March 1998Copyright # 2000 John Wiley & Sons, Ltd. Accepted January 1999

Journal of Forecasting

J. Forecast. 19, 499±504 (2000)

* Correspondence to: Suleyman Gokcan, Citigroup, One Court Square, 32nd Floor/Zone 2, Long Island City, NY11120, USA.

di�erent risk and return characteristics from developed markets. The risk of investing in emergingstock markets has been greater than the risk in investing in developed stock markets; in otherwords, emerging stock market volatility has been larger than that of developed stock markets.Investors in emerging markets su�er from a lack of information as well as reliability; somecountries lack telecommunication and transportation means and others have di�erent accountingsystems. These factors could increase the risk of investment in these countries. According to assetpricing theories expected returns are related to volatility. Therefore, for portfolio management itbecomes critical to model and examine the volatility.

In this paper we extend the Franses and Van Dijk (1996) analysis to emerging stock marketsand compare the performance of linear and non-linear GARCH models for these markets. Intheir paper Franses and Van Dijk compared two di�erent non-linear GARCH models(EGARCH and GJR) with linear GARCH models. However, we compare the EGARCH modelonly with the linear GARCH model.

The plan of this paper is as follows. Data and descriptive statistics are in the next section.Linear and non-linear GARCH models and their formulation appear in the third section. In-sample estimation results are given in the fourth section. Forecasting appears in the ®fth sectionand conclusions are given in the ®nal section.

DATA AND DESCRIPTIVE STATISTICS

The data we analyse is the monthly stock market portfolio returns for seven emerging stockmarkets, from February 1988 to November 1997. Monthly return indexes of emerging marketsare obtained from annual emerging market fact books. Monthly returns are calculated by usingthe following formula:

rit � log�It� ÿ log�Itÿ1� �1�where It is the return index for country I at time t.

Some of the descriptive statistics for monthly returns are displayed in Table I. Sample size isthe number of the months for the sample. Mean returns of the emerging markets range from0.28% (Malaysia) to 2.52% (Argentina). Volatility (measured as a standard deviation) ranges

Table I. Descriptive statistics

CountrySamplesize

Mean(%)

Standarddeviation

(%) Skewness Kurtosis Q(4)aNormality

testbARCHtest

Argentina 118 2.52 21.29 ÿ0.12 13.15 20.48 507.56 17.03Brazil 118 1.87 18.04 ÿ0.88 7.00 3.17 94.26 0.03Colombia 118 1.91 8.11 1.21 5.91 15.97 70.94 1.90Malaysia 118 0.28 8.24 ÿ1.36 6.80 15.95 107.65 0.19Mexico 118 1.88 10.49 ÿ0.96 6.60 26.57 82.35 28.99Philippines 118 0.56 9.24 ÿ0.39 5.48 4.86 33.36 3.55Taiwan 118 0.72 12.91 ÿ0.07 4.19 10.29 7.08 0.49

a Ljung±Box Q statistics at lag 4.bNormality of return series are tested by using Jarque±Berra statistics.

Copyright # 2000 John Wiley & Sons, Ltd. J. Forecast. 19, 499±504 (2000)

500 S. Gokcan

from 8.11% (Colombia) to 21.29% (Argentina). All the emerging stock markets, including worldstock market returns, are leptokurtic in the sense that kurtosis exceeds positive three (kurtosis fornormal distribution should be positive three). All the kurtosis estimates are signi®cant as thestandard error under the null hypothesis of normality is 0.450. Since there are 118 observationsthe standard error under the null hypothesis of normality is 0.225, therefore all the countries'return series except Argentina display signi®cant skewness, and six out of seven country returnsare negatively skewed (skewness for normal distribution should be zero). Negative skewnessshows that the lower tail of the distribution is thicker than the upper tail, in other words, marketdeclines occur more often than market increases.

We report the Ljung±Box Q-statistics for squared residuals. Except Brazil and the Philippinesfor all the countries at lag 4 we reject the hypothesis of no autocorrelation at the 5% signi®cancelevel. Autocorrelated squared residuals are indications of GARCH type of heteroscedasticity.Also in Table I, according to Jarque±Berra statistics normality is rejected for all the return series.

MODEL

Our basic model of the return series for each country is:

rit � a0 �Xpi�1

airtÿi �Xqj�1

bjetÿj �2�

where rit is the return on country i's portfolio at time t, and the a0 ai and bj are the constantparameters. For each country we select the best-®tting AR, MA or ARMA model. For most ofthe countries the ARMA(1,1) model was the best ®tting. To measure the volatility we considerthe following linear and non-linear GARCH models.

Linear GARCH modelMost studies that examine time-varying volatility of stock market returns conclude that GARCHmodels perform well in explaining and forecasting volatility. This suggests that the GARCHmodel is useful to model volatility. The linear GARCH model can be formulated as:

ht � a0 �Xpi�1

aie2tÿ1 �

Xqj�1

bjhtÿj �3�

where a0 , ai , bj are constant parameters. a0 , a1 and b1 are non-negative and a1 � b15 1. Theserestrictions on the parameters prevent negative variances (see Bollerslev, 1986). Among all thedi�erent linear GARCH models, the GARCH(1,1) was found to be the most popular.

Non-linear GARCH modelA number of the researchers have found asymmetry in stock market return series; negative returnshocks seem to increase volatility more than positive return shocks of the same size (seeBollerslev, Chou and Kroner, 1992; Engle and Ng, 1993; Pagan and Schwert, 1990). Despite thesuccess of the linear GARCH model, it cannot capture the asymmetry and skewness of the stockmarket return series. Among the number of non-linear GARCH models the Exponential

Copyright # 2000 John Wiley & Sons, Ltd. J. Forecast. 19, 499±504 (2000)

Forecasting Volatility of Emerging Stock Markets 501

GARCH (EGARCH) model is the most commonly used. The EGARCH model can be writtenas:

log�ht� � a0 �Xpi�1

aijetÿij���������htÿ1

p ÿ��������2=p

p" #� g

etÿi��������htÿi

p �Xqj�1

bj log�htÿj� �4�

In equation (4) a0 , ai , g and bj are the constant parameters. Unlike the linear GARCH modelthere are no restrictions on the parameters to ensure non-negativity of the conditional variances.The EGARCH model allows good news (positive return shocks) and bad news (negative returnshocks) to have a di�erent impact on volatility, where the linear GARCH model does not(see Engle and Ng, 1993). The parameter g would cause the asymmetry. If g � 0, then a positivereturn shock has the same e�ect on volatility as the negative return shock of the same amount. Ifg5 0, a positive return shock actually reduces volatility, if g4 0 a positive return shock increasesvolatility. Previous studies have viewed this coe�cient as typically negative, therefore positivereturn shocks generate less volatility than negative return shocks.

IN-SAMPLE ESTIMATION

In Table II we report the parameter estimates and the value of the Akaike Information Criterion(AIC) for the GARCH(1,1) model and estimates of parameter g and AIC values for theEGARCH(1,1) model. We use AIC values to compare between models in equations (3) and (4).From Table II it is clear that the b parameter is usually signi®cant at the 5% level. For all thecountries the a and b parameters are positive and also their sums are less than unity. Also inTable II we report the estimation results for the EGARCH(1,1) model. We show estimates of theg parameter along with AIC values. The g parameter is signi®cant only for three countries out ofseven and negative for only four countries. When we compare the results of the EGARCH(1,1)model with the GARCH(1,1) model, for all countries the linear GARCH(1,1) model produceslower AIC values than the EGARCH(1,1) model. According to AIC values, for all the countriesthe GARCH(1,1) model outperforms the EGARCH(1,1) model. Overall, the GARCH(1,1)

Table II. Parameter estimates and AIC values

Country

Parameter estimates AICa values

GARCH (1,1) EGARCH

g

GARCH(1,1)

EGARCH

a0 a1 b1

Argentina 0.0011 (0.88) 0.25 (1.05) 0.73 (4.97) 0.3 (1.6) ÿ2.98 ÿ2.97Brazil 0.0004 (0.43) 0.14 (1.45) 0.84 (8.06) 0.005 (0.06) ÿ3.34 ÿ3.3Colombia 0.0002 (2.27) 0.07 (0.91) 0.87 (9.68) 0.22 (3.7) ÿ5.15 ÿ5.14Malaysia 0.0007 (1.66) 0.52 (1.91) 0.49 (3.71) ÿ0.14 (ÿ0.74) ÿ4.86 ÿ4.85Mexico 0.004 (1.24) 0.12 (0.59) 0.37 (0.67) ÿ0.27 (ÿ4.55) ÿ4.54 ÿ4.5Philippines 0.001 (0.95) 0.12 (1.01) 0.71 (2.8) ÿ0.11 (ÿ0.78) ÿ4.75 ÿ4.73Taiwan 0.003 (1.07) 0.13 (1.34) 0.62 (2.91) ÿ0.37 (ÿ4.5) ÿ4.05 ÿ4.03aAkaike Information Criterion.b t-statistics are in parenthesis.

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502 S. Gokcan

model seems to outperform the EGARCH(1,1) model in capturing the dynamic behaviour ofemerging stock market returns.

FORECASTING

In order to evaluate the forecasting power of the di�erent GARCH models we must have a truemeasure of the volatility (see Day and Lewis, 1992; Pagan and Schwert, 1990; Franses and VanDijk, 1996). We use the following formula to ®nd the true so-called unconditional volatility:

s2t � �rt ÿ �r�2 �5�where s2t is the unconditional volatility, rt is the actual monthly return for month t, and �r isexpected return for month t. The expected return in July 1997 is measured by calculating thearithmetic average of monthly returns from February 1988 to June 1997; the expected returnin August 1997 is measured by calculating the arithmetic average of monthly returns fromMarch1988 to July 1997. This is repeated for the ®ve months July 1997 to November 1997. The squareof the di�erence between actual returns and moving average returns would give us the impliedvolatility as in equation (5).

We ®nd one-period-ahead forecasting errors for di�erent GARCH models as follows:

ut�1 � s2t�1 ÿ h^t�1

where ut�1 is the forecasting error of the GARCH models, and h^t�1 is the forecasted variancewhich is generated by using equations (3) and (4). In order to ®nd the one-month-ahead forecastof the variance for July 1997, we use equations (3) and (4) to run the regressions by using the datafrom February 1988 to June 1997 and obtain the constant parameters. Then these parameters areentered into equations (3) and (4) to ®nd forecasted variances. In order to forecast the variancefor August 1997, we use the data fromMarch 1988 to July 1997 to obtain constant parameters inequations (3) and (4); this is done repeatedly for the July 1997±November 1997 period.

In Table III we report the mean squared errors obtained from GARCH(1,1) andEGARCH(1,1) models. The results indicate that for all the countries except Brazil, theGARCH(1,1) model produces smaller forecasting errors than the EGARCH(1,1) model.

Table III. Mean squared error terms

Country Error terms (�104)

GARCH(1,1) EGARCH(1,1)

Argentina 5.12 5.6Brazil 17.35 12.88Colombia 1.58 3.21Malaysia 45.99 55.08Mexico 4.45 4.73Philippines 13.5 13.55Taiwan 4.69 2.6

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Forecasting Volatility of Emerging Stock Markets 503

CONCLUSION

In this paper both linear and non-linear GARCH models are applied to seven emerging stockmarkets. Even if return series are signi®cantly skewed, linear GARCH models are very helpful inexplaining the volatility of the time series. In our comparisons in Tables II and III for all thecountries in the sample the linear GARCH model produced better results than the non-linearGARCH model. We ®nd within-sample evidence that conditional estimates of linear GARCHmodel outperformed the conditional estimates of non-linear GARCH models. We also ®nd out-of-sample evidence that monthly volatilities are predicted better when the linear GARCH modelis used.

REFERENCES

Bollerslev, T. `Generalized autoregressive conditional heteroskedasticity', Journal of Econometrics, 31(1986), 307±27.

Bollerslev, T., Chou, Y. R. and Kroner, F. K. `ARCH modeling in ®nance', Journal of Econometrics, 52(1992), 5±59.

Day, T. E. and Lewis, C. M. `Stock market volatility and the information content of stock index options',Journal of Econometrics, 52 (1992), 267±87.

Engle, F. R. `Autoregressive conditional heteroskedasticity with estimates of variance of United KingdomIn¯ation', Econometrica, 50(4) (1982), 987±1008.

Engle, F. R. and Ng, K. V. `Measuring and testing the impact of news on volatility', The Journal of Finance,XLVIII(5) (1993), 1749±78.

Franses, P. H. and Van Dijk, R. `Forecasting stock market volatility using (non-linear) Garch models',Journal of Forecasting, 15 (1996), 229±35.

Glosten, L., Jagannathan, R. and Runkle, D. `On the relation between the expected value and the volatilitynominal excess return on stocks', Journal of Finance, 46 (1992), 1779±801.

Nelson, B. D. `Conditional heteroskedasticity in asset returns: a new approach', Econometrica, 59(2) (1991),347±70.

Pagan, A. R. and Schwert, W. G. `Alternative models for conditional stock volatility', Journal ofEconometrics, 45 (1990), 267±90.

Sentana, E. `Quadratic ARCH models', Review of Economic Studies, 62 (1995), 636±61.

Author's biography:After receiving his Ph.D. degree in Financial Economics from the Graduate Center of the City University ofNew York in 1997, Dr Gokcan started teaching at Saint Peter's College as an Assistant Professor ofEconomics. He is currently working as a quantitative risk analysis for Citigroup in NY. His researchinterests are time series forecasting, and equity investments in emerging stock markets. He has published anarticle in the Journal of Istanbul Stock Exchange.

Author's address:Suleyman Gokcan, Citigroup, One Court Square, 32nd Floor/Zone 2, Long Island City, NY 11120, USA.

Copyright # 2000 John Wiley & Sons, Ltd. J. Forecast. 19, 499±504 (2000)

504 S. Gokcan