Stock Market Volatility in Saudi Arabia : An Application of Univariate GARCH Model

Study of stock market volatility has been the focus of financial economics. Modelling stock market volatility has great contributions to make in the areas of portfolio management, asset allocation, risk management, etc. We estimate the conditional volatility of Saudi stock market by applying AR (1)-GARCH (1, 1) model to the daily stock returns data spanning from August 1, 2004 to October 31, 2013. We show that a linear symmetric GARCH (1, 1) model is adequate to estimate the volatility of the stock market of the country. We find that Saudi stock market returns are characterised by volatility clustering and follow a non-normal distribution. Saudi stock market returns show a time varying volatility, show persistence and are predictable. Past volatility impacts the current period volatility and past returns play a role in determining the current period return. Saudi stock market is nervous in its reactions to market fluctuations. This finding of the study offer important input into the decisions relating to asset allocation and risk management strategies of investors and treasury managers in Saudi stock market.


Introduction
A stock's volatility gives the range of its outcomes.Markowitz (1952) defined risk as volatility.Volatility study has been at the core of financial economics research.Volatility arises due to the arrival of news in the market and the consequent trading activity by the traders.Dispersion of beliefs among the traders will cause a rise in the level of volatility.When volatility is high compared to the equilibrium values of the stocks, it impacts the return on the stock investment.Describing volatility and the ability to forecast it can provide valuable inputs for decisions relating to asset allocation, portfolio management and risk management.
There is a large number of research studies that examine stock market volatility carried out in the context of both developed and developing countries.However, such a study does not exist for Saudi Arabia.The review of literature on the stock market volatility studies shows that varied range of econometric models are employed by the studies and no single model has emerged superior till date.This requires the study of every single market to help the investors and the regulators as no general lessons can be drawn based on the studies carried out elsewhere.We try to fit an adequate model for Saudi stock returns.
The study of Saudi stock market volatility is of paramount importance to treasury managers and investors in the market.Hence, we carry out the study of stock market volatility in Saudi Arabia.The results of the study have important implications for asset allocation, portfolio construction, risk management and other allied areas related to investment management.Investors looking forward to reduce the uncertainty of their portfolios many reduce it by pulling down their exposure to stocks that are expected to face increased volatility of returns in the future or by adopting appropriate diversification strategy to hedge the risk of increased estimated future volatility.
Many earlier works (see for example Akgiray, 1989;Pagan & Schwert, 1990: Brailsford & Faff, 1996;Brooks, 1998) applied GARCH to US stock data and argue that GARCH models perform better than the competing models.Corhay and Rad (1994) find that GARCH (1, 1) model fits well to the stock market data from France, Germany, Netherlands and UK with the exception of Italy.Hsieh (1989) show that GARCH (1, 1) capture the stochastic dependencies of time series data well.This result is supported by the findings of Taylor (1994), Brook andBurke (2003), Frimpong and Oteng-Abayie (2006) and Olowe (2009).Tse (1991) and Tse and Tung (1992) show that the model that best fits the Japanese and Singaporean markets is the exponential moving average models.Frances and Van Dijk (1996) show that asymmetric GARCH models like GJR model do not perform better than standard GARCH while stock market indices' volatility is forecasted.Alexander and Lazar (2006) argue that GARCH models are at the core of dynamic volatility models.This is due to the ease of their estimation and the availability of diagnostic tests.(Drakos et al., 2010) However, GARCH model does not completely capture the skewness and leptokurtosis of the data.In case of non-normal distribution data, the need for introducing non-normal conditional densities arises.Bollerslev (1986) offers the student t-GARCH model.According to Alexander and Lazar (2006), if the data happens to be non-normal, normal GARCH model could not capture the entire leptokurtosis in the data, hence GARCH model that takes into account this characteristic is more appropriate.Some of the previous works find that asymmetric models do a better job in forecasting as a strong inverse association exist between volatility and shock.(See for example Lee, 1991;Cao & Tsay, 1992;Heynen & Kat, 1994) Poon and Granger (2003) review 93 papers, of which seventeen works evaluate alternative versions of GARCH.That GARCH dominates ARCH is clearly brought out.Models that include volatility asymmetry such as EGARCH and GJR-GARCH outperform GARCH.Bekaert and Harvey (1997) and Aggarwal et al. (1999) conclude that asymmetric GARCH models capture asymmetry in stock return volatility adequately from their study on emerging market volatility.Su (2010) applies GARCH and EGARCH models to assess the financial volatility of daily data of Chinese stock market.He shows that EGARCH model explain the volatility in the daily stock market data from January 2000 to April 2010 better than GARCH model.
The results of various studies carried out so far are not consistent.This can be attributed to the fact that the models, sampling period, frequency of data and forecast horizon used vary.(Vilhelmsson, 2006) We apply GARCH (1, 1) model which is considered as the most appropriate model for estimating conditional volatility by many studies.

Stylized Facts about Saudi Stock Market Prices and Returns
Before, we decide on the methodology employed by the study, we analyse the Saudi stock prices and returns to bring out a few stylized facts about them.Earlier works show that financial time series data is characterized by volatility clustering (Poon, 2005), leverage effects (Christie, 1982), deviate from the mean value (Dowd, 2005), have fat tails and a greater peak at the mean than normal distribution.(Brooks, 2008) Cont (2001) argues that the stylized facts about the financial time series data render the common statistical approaches to its analysis invalid.In order to freeze an appropriate methodology, we study if these characteristics are present in Saudi stock market prices and returns.Before we present this analysis, we present data on some indicators of the market.Jarque-Bera (1987) test checks if the Saudi stock market return is normally distributed.The test statistic rejects the null hypothesis of normality at 1% level.The mean is less than the median and the data series is skewed towards the left.The mean return is positive which indicates that the investors have earned a positive rate of return on their investments in TASI during the study period.The value of kurtosis is greater than 10 which mean that the Saudi stock market return series has a heavier tail than a standard normal distribution.In conclusion, we decide that Saudi stock market returns are non-normally distributed.
That Saudi stock returns are not normally distributed can be ascertained by looked at the figure 2 also that presents the histogram along with the normal distribution plot.The high peak and the fat tails of the Saudi stock market return distribution brings out the fact that it is leptokurtic.That stock market returns follow leptokurtic distribution is well established in finance literature.Figure 3 shows that Saudi stock returns show volatility clustering.Large price fluctuations are followed by large price fluctuations and small price variations are followed by small price variations, of both signs.Volatility clustering is explained by the theory by heterogeneous expectations.(Kirchler & Huber, 2007).Returns fall as traders learn from their past strategies.As a result, the market tends to remain in a state of partial equilibrium till new information arrives.Statistically speaking, volatility clustering stands for strong autocorrelation in squared returns.The findings about the Saudi stock market returns are in line with the suggestions of some of the earlier works.For example, Mandelbrot (1963) and Fama (1965) suggest, among others, that stock returns tend to follow non-normal distribution.Rydberg (2000) show that stock market returns are characterized by leptokurtosis, skewness and volatility clustering.GARCH model is most apt for data series with volatility clustering and tail behavior time series data.According to Bollerslev (1986) and Engle (1993), standard GARCH (1, 1) specification is adequate for modeling many high frequency time series data.Engle (2001) argues that higher order models are applicable only when the data span a long period of time or of a higher frequency of hourly data.Hence, the study adopts GARCH (1, 1) model.

Data and Methodology
To ascertain this stylized fact about Saudi stock market volatility, daily stock returns on the Saudi stock market composite index, Tadawul All-Share Index, TASI, for the period from August 1, 2004 through October 31, 2013 is used.Daily stock returns (r t ) at time t is defined as the logarithm of TASI indices.
where, r t is the logarithmic daily return at time t.P t-1 and P t are the daily values of TASI at two consecutive days, t-1 and t.
A dummy is inserted for the days of market downturn during 2006 and 2008.
Since volatility of asset returns are serially correlated, to capture the serial correlation characteristic of volatility, Engle (1982) presented a new approach for the analysis of time-series data that demonstrates time-varying variance called autoregressive conditional heteroskedasticity (ARCH).ARCH model writes conditional variance as a distributed lag of past squared innovations: α(L) is a polynomial in the lag operator.In order to have the conditional variance positive, ω and the coefficients in α(L) must be non-negative.
However, Fan and Yao (2003) argue that ARCH (p) model is apt for financial time series only with a large number of lags.This warranted the extensions of the ARCH model.Generalized autoregressive conditional heteroskedasticity (GARCH) model is the extension of the ARCH (ρ) model offered by Bollerslev (1986).This approach models the conditional variance as an ARMA process where it is determined by the innovations and its own lags.GARCH captures the serial dependence of the volatility.On account of its conditional property, GARCH draws on immediate past observations to model future observations.
The simplest GARCH (1, 1) model can be specified as follows: Mean equation: Variance equation: Where, ω 0; α 1 ≥ 0; β 1 ≥ 0; r t = return at time t; μ = average return; ε t = σ t z t; z t = standardized residual returns; σ t 2 = conditional variance GARCH (1, 1) requires the constraints, α 1 ≥ 0 and β 1 ≥ 0 to ensure σ t 2 is strictly positive.The coefficient α 1 measures the degree to which volatility shock that occurs now feeds through into next period's volatility.If β 1 coefficient is high, it can be inferred that the shocks to conditional variance dies after a long time.Volatility is said to be persistent in such a case.If α 1 is high, it can be inferred that the responses of volatility to market movements is intense.If α1 is high and β1 is low, volatilities are sharp.The sum of α 1 and β 1 measures the rate at which this effect dies out over time.If the sum of α 1 and β 1 is close to one, a shock at time t will persist for a long time in the future.If the sum of α 1 and β 1 is high, it indicates a long memory.If the sum of these two variables is equal to one, it implies that any shock will lead to a permanent change in all future values.GARCH captures the serial dependence of the volatility.On account of its conditional property, GARCH draws future observations based on the immediate past observations.Variance of the current error term is modeled as a function of the previous period's error terms' variances.GARCH, thus, depends on the time-varying variance.

Results
In order to determine the dynamics of the conditional mean, we look at the Schwarz information criterion (SIC) to decide on the ARMA specification appropriate for modelling Saudi stock market returns.Enders (2010) argues that SIC always select a model better than Akaike information criterion (AIC) as the penalty for the inclusion of more repressors is higher with SIC than with AIC.We estimate many combinations of ARMA (p, q) models up to 5 lags.

* indicates optimal lag
We find that ARMA (1, 0) is the most appropriate model for our data.This is also confirmed by the results shown in table 7. The results of AR (1)-GARCH (1, 1) model is presented below.
Mean equation estimated is where DUM is the dummy variable.Coefficients of the constant and the ARCH term are higher than zero and the GARCH term is positive that confirm that the conditional variance is non-negative.The coefficients of both the lagged squared residual, α 1 , and lagged conditional variance terms in the conditional variance equation, β 1 , are highly statistically significant.This can be inferred as the news regarding the volatility of the previous periods can explain the current volatility.The sum of both α 1 and β 1 indicate the short run dynamics of the resultant volatility time series.β 1 at 0.85 shows that shocks to conditional variance take a long time to fade out.Volatility is persistent.α 1 show the intensity of the volatility to market fluctuations.α 1 of 0.14 shows that the Saudi stock market is quite nervous in its reactions to market variations.Since α 1 and β 1 is close to one, a shock that occurs at a particular time will prolong for many periods in future.Since the sum of α 1 and β 1 is less than one, it also indicates that the unconditional variance of ε t is stationary.α 1 is less than β 1 .It can be inferred that the volatility of the stock market is affected by past volatility more than by related news from the previous period.The dummy variable included is significant and bears an expected negative sign.Significance of AR (1) term means that the past period returns have a crucial role to play in deciding the current period returns.
We applied a set of tests to check the adequacy of the fitted model.ARCH-LM test checks if the standardized residuals exhibit ARCH behaviour.If the variance equation is specified correctly, there should be no ARCH effect left in the standardized residuals.Ljung-Box Q-test is to check the presence of autocorrelation of order m in the residuals.This test is applied to check for the adequacy of the fitted model.(Tsay, 2005).Q 2 test checks for the presence of autocorrelation of order m in squared residuals.This test checks for the presence of ARCH effect that remains in the variance equation.The results of the BDS test shown in the table above indicates that AR (1)-GARCH(1, 1) model is adequate and we are not required to specify a non-linear model.Our results are in alignment with the results of many of the earlier works.Anderson and Bollerslev (1998) argue that the GARCH models give the most accurate forecasts.

Conclusion
Modelling and forecasting volatility of stock market returns is an important area of research in financial economics.Understanding stock market volatility offers important input for portfolio management, risk management, treasury management, asset allocation decisions and many related areas related to investments.We find that Saudi stock market is characterized by volatility clustering.Market dynamics results in volatility clustering.Volatility clustering is due to leptokurtosis in the returns distribution.It refers to the fact that large price changes are followed by large changes and small price changes are followed by small changes in either direction.Gallant et al. (1991) argues that this is due to the quality of information arriving in the market in clusters.According to Engle et al. (1990) this is mainly due to two reasons; first, when the information reaches the market in clusters; second, when the market participants have varied beliefs about the market and they take time to absorb the information shocks and resolve the differences in their expectations.Saudi stock market returns are found to be non-normal and follow leptokurtic distribution.Application of AR (1)-GARCH (1, 1) shows that past returns play an important role in determining the current period return.Saudi stock market volatility is found to be persistent.Past volatility affects the current period volatility.Investors in Saudi stock market can greatly benefit from the findings of this study.

Figure 2 .
Figure 2. Histogram and normal distribution of Saudi stock returns

Table 1 .
Saudi stock market indicatorsSource: Saudi Arabian Monetary Agency Annual Report, various issues Existence of Saudi stock exchange can be traced back to 1970s, when it came into being informally.Saudi stock exchange is known as Tadawul.As at the end of 2012, Saudi stock exchange has shares listed from companies that span 15 sectors.Table1presents data on some of the indicators of Saudi stock market during the study period, 2004 to 2012.The number of listed companies during the period increased from a mere 73 in 2004 to 158 in 2012 which is an increase of around 207%.As of 2012, the market capitalization of the shares issued by the listed companies stand at around 1,400 billion Saudi riyals which is around 52.5% of the country's gross domestic product.TASI, Saudi stock market composite index, was around 6,100 points on August 1, 2004.It registered a steady increase during the year and closed at around 8,200 at the year end.This increasing trend persisted and the index breached the 20,000 level on February 2006 and touched its zenith at 20,634 on February 25, 2006.However, from February end, TASI started sliding and fell below 10,000 on 31 st October 2006.Before the onset of the impact of global financial crisis on the Saudi stock market, TASI was at 8,700 points at the close of August 2008.During September 2008, the index further fell and reached around 7,400 on 28 th September 2008.Though the index showed trend reversal, it could never reach its historic highs again.Index was at 8,400 on the 31 st October 2013.During our period of study, Saudi stock index suffered two major downturns, one in 2006 and another in 2008.

Table 2
presents the descriptive statistics for the Saudi stock market return series.

Table 3 .
Ljung-box Q-statisticsfor Saudi stock market returns Table4gives the results of ARCH-LM test which shows the existence of ARCH effect in the residuals of mean equation of TASI return.

Table 4 .
ARCH-LM test for residuals of share prices on Saudi stock market ADF test results shows that returns are stationary in level.Hence, we can now apply the ARCH family models.

Table 6 .
Sic for the mean equation for Saudi stock market returns

Table 7 .
Serial correlation LM test and Heteroskedasticity tests for the estimated residuals of the ARMA (1, 0) model

Table 9 .
Residual diagnostic fitsDiagnostic test results show the absence of ARCH effect and autocorrelation in the standardized residuals.This can be inferred to mean that the variance equation specified is adequate.To check if the Saudi stock market return data requires a non-linear specification during the sample period, we apply the test developed byBrock, Dechert and Scheinkman (1986)to the residuals of the AR (1)-GARCH (1, 1) model to detect remaining dependence and the presence of omitted nonlinear structure.If null hypothesis is not rejected, then the original linear model is adequate.If null hypothesis is rejected, then the fitted linear model is not adequate and this can be inferred as indicating the presence of nonlinearity.