Seminar in Financial Data Analysis

ARCH-GARCH modelling in Turkish, Greek and Russian Stock Markets

Heval YUKSEL Hakan BAYRAM

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ABSTRACT ................................................................................................................................................................. 3 1. INTRODUCTION.................................................................................................................................................... 4 2. TURKISH, GREEK AND RUSSIAN STOCK EXCHANGES ............................................................................... 5

2.1 ISTANBUL STOCK EXCHANGE (ISE) ...................................................................................................................... 5 2.2 ATHENS STOCK EXCHANGE (ASE)........................................................................................................................ 5 2.3 RUSSIAN STOCK EXCHANGE (RTS) ....................................................................................................................... 6

3. ARCH AND GARCH MODELS ............................................................................................................................. 7 3.1 CHARACTERISTICS OF ASSET RETURNS ................................................................................................................. 7 3.2 HETEROSKEDASTICITY ......................................................................................................................................... 9

4. STRUCTURE OF A MODEL................................................................................................................................ 11 4.1 THE ARCH MODEL............................................................................................................................................ 11 4.2 THE GARCH MODEL ......................................................................................................................................... 12

4.2.1 GARCH-M model (GARCH in Mean model) ............................................................................................... 12 4.2.2 The TARCH Model (also called Threshold-GARCH or Leverage GARCH).................................................. 13 4.2.3 The EGARCH Model (Exponential GARCH)............................................................................................... 13

4.3 TESTING FOR ARCH EFFECTS ............................................................................................................................. 14 4.3.1 Lagrange Multiplier Test for ARCH Effects................................................................................................. 14 4.3.2 Ljung-Box Statistics.................................................................................................................................... 14

5. MODELLING PROCEDURES AND RESULTS.................................................................................................. 15 5.1 THE DATA.......................................................................................................................................................... 15 5.2 THE EMPIRICAL STEPS ....................................................................................................................................... 15

6. LITERATURE RESEARCH ................................................................................................................................. 19 7. CONCLUSION ...................................................................................................................................................... 20 8. REFERENCES....................................................................................................................................................... 22 9. APPENDICES........................................................................................................................................................ 24

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Abstract In this paper, stock market volatility in Turkish, Greek and Russian stock markets was investigated using the total return indexes based on the domestic currencies of the corresponding countries. Turkish and Russian stock markets are examples of emerging markets whereas Greek stock market is considered as a developed market. Our main purpose was to analyze the volatility in emerging markets especially in Eastern Europe, but firstly because of the lack of data for most of the countries, and secondly because there were already studies done for Poland, Hungary, Czech Republic, we decided to analyze Greece together with Turkey and Russia. The data set covers a period from 1994-2004. For Turkish and Greek stock markets, we found consistent results but in modelling the Russian stock market, we had several problems in finding out a model which really explain the ARCH effect within the data. Finally, forecasts based on the best fitting models were performed. A number of explanations for the forecasts and results were proposed.

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1. Introduction The risk-return effects have primary importance in finance. Most investors don’t prefer to take risk and they expect risk premium for holding assets with risky payoffs. In other words, there is a trade off between risk and the returns. The risky assets have higher payoffs. For example, if there is a one risky asset in an economy, investor sells some of the asset when the volatility increases. In a fixed supply case, the prices fall sufficiently to attract buyers. At this low price level, the high risk is balanced by the high expected return. Therefore, it can be said that high volatility corresponds to the high expected returns. The standard deviation of return on an asset is the volatility and conditional volatility models are used for predicting risk. These models are composed in a way that they successfully characterize the volatility clustering behaviour of financial data. Predicting volatility has very important implications, for instance if there is high volatility in forecasted period, the investor can either leave the market or require a high premium in order to compensate risk. ARCH and GARCH models are used to model the volatility. They are also used to understand whether the volatility is transmitted across markets. Mansoon (1998) describe this transmission mechanism as spillovers. When a crisis from an emerging market affects the other emerging markets' macroeconomic fundamentals, such as price devaluation in one country reduces the competitive power of the other country in trade, it is called spillover effects1. ARCH and GARCH models estimate the variance covariance transmission mechanisms between the countries.

Engle(2003) mentions that the GARCH specification can be used to describe the volatility dynamics of almost any financial return series. This model is applied not only to stocks traded in most developed markets, but also to most stocks traded in emerging markets, and to most indices of equity returns. It can also be used to analyze the volatility of exchange rates, bond returns and commodity returns. In this paper, we analyze the Turkish, Greek and Russian stock exchanges. First of all, we obtained stationary data by taking logarithms and differences and then used ARMA modelling in order to have best fit of the data. We used ARCH Lagrange Multiplier test for detecting whether there were any ARCH effects. After obtaining significant results from the test, we decided to use ARCH and GARCH models in order to eliminate the ARCH effects. After applying the models, we applied the ARCH LM Residual test because we wanted to be sure that there is no ARCH effect left in residuals. Finally, we forecasted the returns for understanding whether our models fitted good or not.

1 Glick and Rose (1999) focus on the currency crises that affect clusters of countries tied together by

international trade. The scope for trade links and links through common macroeconomic fundamentals are

examined. According to, Glick and Rose (1999), countries that trade and compete with the targets of

speculative attacks are themselves likely to be attacked. Glick and Rose (1999) show that given the

occurrence of a currency crisis, the incidence of speculative attacks across countries is linked to the

importance of international trade linkages.

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This paper is organized as follows; Section 2 discusses the history of Turkish, Greek and Russian stock exchanges. Section 3 discusses why we need ARCH and GARCH models, characteristics of asset returns and expresses the definition of heteroskedasticity. Section 4 explains the structure of ARCH and GARCH models. Section 5 expresses our empirical researches about ARCH and GARCH modelling in Turkish, Greek and Russian stock exchanges. Section 6 explains the different empirical researches about ARCH and GARCH modelling in the literature. Section 7 concludes.

2. Turkish, Greek and Russian Stock Exchanges

2.1 Istanbul Stock Exchange (ISE) Even though Istanbul Stock Exchange (ISE) was established 18 years ago in 1986, it has developed rapidly. It is mentioned that, as a leading emerging market, ISE’s progressive infrastructure and dynamism are attracting increasing international interest. According to Bildik and Gulay (2001), in average, foreign and international institutional investors own 50% of the free float of the shares at the ISE. Total market capitalization is approximately US$ 80 Billion whereas it is a highly active market with an average daily trading value of US$ 753 Million and 315 listed stocks at yearend of 2000. The ISE is an order-driven, multi-price, continuous auction market with no market makers or specialists. The trading is realized through the computerized trading system. There is no opening session or pre-open procedure at the ISE. The market is open Monday through Friday, (morning session) from 10:00 a.m. until 12:00 and after two hours lunch break, (afternoon session) from 2:00 p.m. to 4:00 p.m. The “National-100 Index” (ISE-100) which is the main market indicator of the Istanbul Stock Exchange is a market capitalization-weighted index. Bildik and Gulay (2001) state that it represents at least 75% of the total market capitalization, traded value, number of shares traded and number of trades realized in the market. ISE has also been calculating and broadcasting a new index since 1997 which is called ISE-30 that contains 30 the largest-market value stocks. There are no futures or derivatives trading on index on stocks in Turkish capital markets. However, the number of mutual funds and total asset value of mutual funds has been growing rapidly in recent years. Those are still very low relative to total market capitalization of ISE and to 200 Billion USD GNP of Turkish economy. Total asset value of 260 mutual funds as of September 2000 is only 3 billion USD and the share of stocks in funds’ portfolio is only 16.2% which is almost equal to half billion USD. (http://www.ise.org)

2.2 Athens Stock Exchange (ASE) The ASE was founded on September 1876 when the government granted the permission for its founding. It has been the only stock exchange of Greece. It has

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experienced a considerable growth since its establishment and played a major role in economical development of the country. According to Gounopoulos (2003), between 1991 and 1992, after many years of hardly any activity the Greek stock market finally started to play its role as a source of cheap capital for growing companies with strong potential. Period 1993-1996 is characterised by the influx – and excesses of construction companies to the stock exchange and by the great volatility in prices and indices. From the beginning of 1997, value of turnover showed signs of revitalisation and prices began tending upwards. During the period 1997-2000, he states that the Greek economy was characterised by its attempt at readjusting its macroeconomic achieving the criteria to become the 12th member of the ‘Euro Zone’. General Index jumped from 954 (2nd January 1997) to 5794 (3rd

January 2000). The Athens Stock Exchange consists of three markets: the official Stock Exchange (Main Market), a market for small caps (Parallel Market) and a market for mainly new technological companies (New Market). It is mentioned that the main difference between the parallel and the main market is that in parallel underwriter assumes responsibility for the full coverage of the issue and buys the shares that will not be covered by the investment public a the issue price. In order this provision to be also valid in main market there should be an agreement between the underwriter and the issuing company. Between 1989 and 2001, the number of companies traded on the exchange climbed from 119 to 349. Most of the firms (66%) are traded at the Main Market. The total market capitalization of the firms traded has increased from EU 938 million at the end of 1989, to EU 30.8 billion at the end of 2001. (http://www.ase.gr/default_en.asp)

2.3 Russian Stock Exchange (RTS) In 1992, Russia started the long and difficult path of transition towards a market economy. This process has resulted in a profound change in Russia's economy, even though the transition is far from complete. The Russian stock market influences significantly Russian economic development by providing mechanisms for resource re-allocation between different sectors of the Russian economy. As a rapidly developing emerging market, it also plays a significant role in the world-wide context by affecting international capital flows. Russian Trading System (RTS) was established in 1995 to act as a secondary market for the Russian equities. RTS is modelled after the NASQAQ market in United States and the trading on RTS is done electronically. Currently there are twelve stock exchanges in Russia and the dominant ones are Moscow Interbank Currency Exchange (MICEX) and RTS. The Federal Commission on the Securities Market (FCSM) and Central Bank of Russia regulates these equity markets. Foreign investors initially had considerable presence in the Russian capital market. The total capital flows into Russia was USD6 billion in 1994, USD13.5 billion in 1995, USD28 billion in 1996 and USD40 billion in 1997.(http://www.rts.ru/?tid=2) Jithendranathan and Kravchenko (2004) mention that the development of the market based economy in Russia suffered a serious set back in August 1998 when the Russian government defaulted on the domestic and external debt payments.

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On August 17, 1988 Russia abandoned the defence of the Russian rubble and placed a 90-day moratorium on commercial external debt payments. The value of Russian rubble plunged from USD1=RUR6.235 at the end of July 1998 to USD1=RUR16.064 by the end of September 1998. They claim that the direct cause of the crisis was the failure of Russian government in addressing the fiscal imbalance of the economy and falling oil prices, which was the main source of foreign exchange for Russia. As a consequence, it is stated that the crisis of 1998 had considerable adverse effect on the international investor confidence in Russia. The inflow of foreign capital went down to USD19.5 billion in 1998, USD10 billion in 1999 and USD11 billion in 2000 and thereby increased the equity returns.

3. ARCH and GARCH Models Volatility is a statistical measure of the tendency of a market or security to rise or fall sharply within a period of time. Modelling the volatility of asset return is an important aspect in financial area. There are several reasons for modelling and forecasting volatility. • Analyzing the risk of holding an asset or the value of an option • Eliminating time varying forecast confidence intervals and obtaining more accurate intervals by modelling the variance of errors • Obtaining more efficient estimators if heteroskedasticity in the errors are handled properly

3.1 Characteristics of Asset Returns

ARCH models are very successful in the financial applications because it can be applied for many statistical problems with time series data. This applicability is important because it gives the investor predictive power of risk in returns. In finance, to predict the returns is very difficult because they have large numbers of extreme values and these extreme values and quite periods are clustered in time. There are several characteristics of ARCH model such as unpredictability, fat tails and volatility clustering. Engle(2003) stated that people buy and sell financial assets because of the expected future payments. These payments are uncertain and depend on the future events that cannot be known today. For finding the fair price of the asset, the forecasts of the distribution of these payments based on our best information today is needed. When time passes, the more information is available and the assets are revalued according to this new information. Basically, financial price volatility is due to the arrival of new information. Engle (2003) mentioned that as news typically clustered in time, volatility clustering is simply clustering of information arrivals. For example, an event, which raises the value of a firm such as an invention, will have different effect on stock prices according to the economic conditions in the economy and in the firm. If the company is near bankruptcy, the effect can be very large, and if it is operating with full capacity, it may be small. . If the economy has low interest rates and surplus labor, it may be easier to develop this new product. With everything else equal, the response will be greater in a recession than in a boom period. Therefore, it is not surprising to find higher volatility in economic

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recessions although the arrival rate of new inventions is constant. This is a slow moving type of volatility clustering that can give cycles of several years or longer. ARCH-GARCH models are developed to account for empirical regularities in financial data. Many financial time series have a number of characteristics in common. 1) Time varying risk premia2 2) Heteroskedastic variance3 3) Thick tails-Leptokurtic distribution

• Fat tails and excess peakedness at the mean4 • Excess kurtosis decreases with aggregation

4) Volatility Clustering • Large changes followed by large changes; small changes followed by

small changes • News arrivals are serially auto correlated. News tends to cluster in time.

5) Leverage Effects-Asymmetric reactions5 • Changes in prices often negatively correlated with changes in volatility • Volatility reacts differently to a big price increase or a big price drop.

“People react more when prices fall.” 6) Non trading periods – Nonlinearity in the model

• Time deformation; economic activity does not match calendar time • Volatility is smaller over periods when markets are closed than when they

are open6 7) Forecastable events7

• Forecastable releases of information are associated with high ex ante volatility8

8) Volatility and serial correlation9 • Inverse relationship between volatility and serial correlation of stock

indices 9) Volatility co-movements

• Evidence of common factors to explain volatility in multiple series • Volatilities of different securities very often move together

2 Asset prices are generally non stationary. Returns are usually stationary. Some financial time series are fractionally integrated. 3 Not constant variance 4 Normality has to be rejected in favour of some thick tailed distribution. 5 The so-called "leverage effect" first noted by Black (1976) refers to the tendency for stock prices to be negatively correlated with changes in stock volatility. A firm with debt and equity outstanding typically becomes more highly leveraged when the value of the firm falls. This raises the equity return volatility if returns are constant. 6 Information that accumulates when financial markets are closed is re-flected in prices after the markets reopen. If for example, information accumulates at a constant rate over calendar time, then the variance of the returns over the period from Friday close to the Monday close should be three times the variance from the Monday close to the Tuesday close. 7 Patell and Wolfson (1979,1981) show that the stock return volatility of an individual firm is high around earning announcements. 8 Engle(2003) mentioned that the duration of forecasted period should be chosen properly because too long period can be irrelevant and the too short period can be very noisy. 9 Return series usually show no or little autocorrelation.

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In statistical terms volatility means conditional variance of the underlying asset returns. ARCH models are used to model and forecast conditional variances. The variance of the dependent variable is a function of past values of the dependent variable and independent or exogenous variables. Engle(2003) mentioned that it is logically inconsistent to assume that the variance is constant for a period such as one year ending today and also that it is constant for the year ending on the previous day but with a different value. ARCH models are designed to model the dynamic volatilities, in other words, the time varying variance. They are used to forecast volatility and risk over a long horizon. The uncertainty about future costs and prices avoids entrepreneurs to invest. As Engle (2003) stated, if uncertainty is changing over time, it is called heteroskedasticity.

3.2 Heteroskedasticity

In financial applications where the dependent variable is the return on an asset or portfolio, the key issue is the variance of the error terms, because variance of the error term represents the risk level of those returns. When we look at the financial data we can see that some time periods are riskier than the others, that is the expected value of error terms at those periods is greater than at other periods. If the values of error terms change in some points, it is nonetheless likely that heteroskedasticity is an issue. The least square model assumes that the expected value of all error terms is the same at any given point [linear model is: yt = α + βxt + et]. Therefore, expected value of squared error term is equal to its variance. This assumption of constant variance is called homoskedasticity, as shown in Figure 1. In opposite, if variance is different for each observation, it is called heteroskedasticity (see Figure 2 in Appendix 1). 1. [ ] 0=Ε iε

2. ( ) [ ] [ ]22iiiVAR εεε Ε+Ε= ( ) [ ] 22 σεε =Ε= iiVAR

3. ( ) 0, =jiCOV εε Heteroskedasticity creates problems in ordinary least square analysis. Because of the heteroskedasticity, OLS underestimates the variances of estimates. T-scores are biased upwards; variables seem significant although they are not significant. Heteroskedasticity assumes that variance is not constant such as:

( ) 22ii ZVAR σε =

Student’s T- test utilizes the test statistics which is used to make inferences about particular β parameters that have practical significance. Variance of the estimation of β and the estimated standard deviation of the model are given as follows:

10

( )( )∑ −

= 2

2ˆ

xxVAR

i

σβ

2

11ˆ iekn −−

=σ

The null hypothesis and the rejection regions are given: H0: β = 0 Rejection region: ( )1

2

−−> kntt α & ( )12

−−−< kntt α

T statistics is based on the assumption of normal error terms.

( )( )β

βσβ

ˆˆ

ˆˆ

2

SExxi

=

−

is t-distributed with n-k-1 degrees of freedom.

ββ σβ

σββ

ˆˆ

0ˆˆ −=−=t because of the null hypothesis we assume that the value of β is

zero. Model misspecification can cause bias. Heteroskedasticity does not cause bias but it causes OLS to no longer have the minimum variance property. So impure heteroskedasticity caused by a model misspecification causes OLS to be both biased and not minimum variance. The estimators of coefficients, β , are still unbiased but inefficient and inconsistent. Because variance of β is no longer equal to the OLS estimation value.

( ) ( )( )( ) ( )∑∑

∑−

≠−

−= 2

2

22

22

ˆxxxx

xxVAR

ii

ii σσβ

The variance is incorrect, i.e. biased. Standard error of coefficients is incorrect. Therefore, significance test, confidence intervals etc. cannot be used. In the presence of heteroskedasticity, the OLS regression is still unbiased but the standard error estimates and confidence intervals are too narrow. Therefore, the results give false sense of precision. ARCH - Autoregressive Conditional Heteroskedasticity- and GARCH - Generalized Autoregressive Conditional Heteroskedasticity- models are used for dealing with time series heteroskedastic models. These models provide a volatility measure, such as standard deviation, which can be used financial decisions such as risk analysis, portfolio selection and derivative pricing.

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4. Structure of a Model The dependent variable, rt, is the return on an asset or portfolio. The mean value is µt and the variance is σt

2, which is defined relative a past information set. The conditional mean and the conditional variance of rt given Ft-1which denotes that the information set available at time t-1. Ft-1 consists of all linear functions of the past returns.

( )1−Ε= ttt Frµ

( ) ( )[ ]12

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−− −Ε== tttttt FrFrVar µσ rt follows a simple time series model such as a stationary ARMA (p,q) model. We consider the model

ttt ar += µ ∑ ∑= =

−− −+=p

i

q

iitiitit ar

1 10 θφφµ

for rt, where p and q are non negative integers. The order of an ARMA model may depend on the frequency of return series. For instance, when daily returns of market index include some minor correlations, the monthly returns may not show any significant correlation. GARCH model concerns with the evolution of σt

2 because the behaviour σt2

distinguishes one volatility model from another. GARCH model uses the exact function to describe σt

2.

( ) ( )112

−− == ttttt FaVarFrVarσ at denotes the shock or mean corrected return of en asset return at time t. ARCH model is the systematic framework for volatility modelling. The basic idea of ARCH models is that:

• The mean corrected asset return at is serially uncorrelated but dependent, • The dependence of at can be described by a simple quadratic function of

its lagged values

4.1 The ARCH Model ARCH(m) model assumes that

ttta εσ= 22

1102

mtmtt aa −− +++= ααασ L where εt is a sequence of independent and identically distributed random variables with mean zero and variance 1, ,00 >α 0≥iα for i > 0.

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From the structure of the model, we could easily see that, if there is a large shock in previous periods, in other words if there is large variance in previous periods, it makes the present period’s variance higher10. In other words, large shocks tend to follow by another large shock. This property of ARCH, accounts for the volatility clustering in financial time series11.

4.2 The GARCH Model GARCH model is the generalized ARCH model. For log return series rt, the mean equation of the process is assumed to be adequately described by an ARMA model.

ttt ra µ−= is the mean corrected log return. At follows a GARCH (m, s) model if

ttta εσ= , ∑∑=

−=

− ++=s

jjtj

m

iitit a

1

2

1

20

2 σβαασ

where εt is a sequence of independent and identically distributed random variables with mean zero and variance 1, ,00 >α 0≥iα , 0≥jβ , and ( ) .1),max(

1<+∑ =

sm

i ii βα

• The mean of unconditional variance (long-run average variance): α0 • News about volatility from the previous period, measured as the lag of the

squared residual from the mean equation: at-i2

(The ARCH term) ARCH term represents the new information that was not available when

previous forecast was made. • Last period’s forecast variance: σt-1

2 (The GARCH term) GARCH (1, 1) model is:

( ) .1,1,0 1111 <+≤≤ βαβα σt

2 increases due to an increase in at-12 or σt-1

2. In other words, a large at-12 tends to

be followed by another large at2. This approach is similar to the volatility clustering

behaviour of asset returns.

4.2.1 GARCH-M model (GARCH in Mean model)

In finance the return of an asset may depend on its volatility. This case is modelled by GARCH-M specification by adding cσt

2 in the mean equation. The GARCH (1, 1) –M model is:

tttt acr ++= 2σµ

10 Last past squared shocks { } m

iita 12

=− imply a large conditional variance σt2 for the mean corrected

return at. 11 For details see Engle(1982) and Engle and Susmel (1994).

211

2110

2−− ++= ttt a σβαασ

211

2110

2−− ++= ttt a σβαασ

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The parameter c is called the risk premium parameter. If c>0, the return is positively related to its past volatility. If c>0 and it is significant (t-test), there is a trade-off between the mean (return) and the conditional variance (time varying risk). The more risk means the higher return.

4.2.2 The TARCH Model (also called Threshold-GARCH or Leverage GARCH)

In this case, there are two types of news. There is a squared return and there is a variable that is the squared return when returns are negative, and zero otherwise. The specification for the conditional variance is given by:

where It = 1 when at<0, and 0 otherwise. In this model, good news (at>0), and bad news (at<0), have differential effects on the conditional variance—good news has an impact of α1, while bad news has an impact of (α1 + γ). If γ>0, the leverage effect exists in that bad news increases volatility. If 0≠γ , the news impact is asymmetric. In EViews, the leverage effect term, γ, represented by (RESID<0)*ARCH(1) in the output. If it is not significantly positive (even with a one-sided test), we conclude that there does not appear to be an asymmetric effect.

4.2.3 The EGARCH Model (Exponential GARCH)

This model allows the asymmetric effects between the positive and negative asset returns. The specification for the conditional variance is:

The left-hand side is the log of

the conditional

variance. This implies that the leverage effect is exponential, rather than quadratic, and that forecasts of the conditional variance are guaranteed to be nonnegative. The presence of leverage effects can be tested by the hypothesis that γ>0. The impact is asymmetric if 0≠γ . The leverage effect term, γ, denoted as RES/SQR[GARCH](1) in the output of EViews. If γ is negative and statistically different from zero, it indicates the existence of the leverage effect in future bonds returns during the sample period.

2111

21

2110

2−−−− +++= ttttt Iaa σβγαασ

211

1

1

1

110

2 loglog −−

−

−

− +++= tt

t

t

tt

aa σβσ

γσ

αασ

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4.3 Testing for ARCH effects

4.3.1 Lagrange Multiplier Test for ARCH Effects

This is a Lagrange multiplier (LM) test for autoregressive conditional heteroskedasticity (ARCH) in the residuals (Engle 1982). The test’s null hypothesis is there is no ARCH up to order q in the residuals. We use ARCH LM test in order to understand whether the standardized residuals exhibit additional ARCH. If the variance equation is correctly specified there should be no ARCH effect left in the standardized residuals. The squared series at

2 is used to check for conditional heteroskedasticity, where ttt ra µ−= is the residual of the ARMA model. For checking heteroskedasticity,

the Lagrange multiplier test is used. This test is equivalent to usual F statistics test. The null hypothesis is H0: 021 ==== mααα LL In the linear regression

tmtmtt eaaa ++++= −−22

1102 ααα L ,,,1 Tmt K+=

where et denotes the error term, m is a specified integer and T is the sample size. F-statistics is asymptotically distributed as chi-square distribution with m degrees of freedom.

( )

( )121

10

−−

−

=

mTSSR

mSSRSSR

F

( )2

10 ∑

+=−=

T

mtt waSSR ∑

+=

=T

mtteSSR

1

21 ˆ

where w is the sample mean of at

2 and te is the least squares residual of the linear regression. In EViews, F-statistic and an Obs*R-squared statistic are reported. Obs*R-squared: TR2 statistics is the number of observations times the R2. The TR2

statistic has an asymptotic 2χ distribution under the null hypothesis.

4.3.2 Ljung-Box Statistics Ljung-Box statistics and their p values are used to check the adequacy of the mean equation. The null hypothesis is there is no autocorrelation up to order k. If the series are white noise, Q statistics should not be significant.

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We use the Ljung-Box statistics of at2 in order to check the validity of the volatility

equation. The validity of the distribution assumption is checked by the skewness, the kurtosis and quantile-to-quantile plot (i.e. QQ plot) of at. Kurtosis is the measure of the magnitude of the extremes. If returns are normally distributed, then the kurtosis should be three. If the kurtosis is high, there is strong evidence that extremes are more substantial than would be expected from a normal random variable. If the returns are normally distributed, the QQ plot should lie on straight line and will have an s-shape if there are more extremes.

5. Modelling Procedures and Results

5.1 The Data Total return indexes for each country are retrieved from Thomson Data Stream in domestic currencies based on S&P. The period analyzed starts from 29.12.1995 and ends in 31.12.2004.

5.2 The Empirical Steps

1. Initial observations

After importing the data into E-Views, we wanted to see the Line graphs of the data for each country in order to have a first impression. As it can be seen in Appendix 2, Figures 3-4-5, each country has different volatility structures. Turkey seemed to be the most volatile among all. Here, we got the first clue that our series might have heteroskedasticity as there are sudden ups and sudden downs in the line graphs.

2. We took the logarithm and the first difference of the initial series to get a stationary series. To be able to continue our analysis we had to have stationarity in the data. To attain the mean stationarity, we took differences and for the variance stationarity we took the logarithms. Then we applied the ADF-Test to check the stationarity of the final series. dlturkey=dlog(turkey) dlgreece=dlog(greece) dlrussia=dlog(russia) ADF test statistics has to be rejected in order to have stationary process because null hypothesis of Augmented Dickey Fuller test assumes the presence of unit root in time series data. After taking differences and logarithms, the ADF statistics increased significantly and the probability of accepting null hypothesis becomes zero at 1% critical level. Therefore, we rejected the null hypothesis which implied that there was no unit root and the all series were stationary (Appendix 5). Then we analyzed the histograms of the data for each country. For any series to be normally distributed, it should have Jarque-Bera test statistic less than the critical value. The test statistic measures the difference of the skewness and kurtosis of the series with those from the normal distribution. The statistic is computed as:

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Jarque-Bera= ( )

−+−4

36

22 KSkN

where S is the skewness, K is the kurtosis, and k represents the number of estimated coefficients used to create the series. Jarque-Bera statistics should be less than 9,21034 (1% critical level for Chi-square distribution with 2 degrees of freedom) in order to accept the null hypothesis of normal distribution. For normal distribution we expect mean and skewness to be around 0 and kurtosis to be 3. After taking differences and logarithms we looked at the line graph of the data as shown in Appendix 6, Figures 12-13-14 and the histograms of the data as shown in Appendix 4, Figures 9-10-11. From the line graphs, we saw that large values are followed by the large values and smaller values are followed by the smaller ones. In other words, we detected the volatility clustering behaviour of the data. From the histograms, we detected the high excess kurtosis-leptokurtosis. After observing volatility clustering and leptokurtosis, we thought that ARCH model might provide a good approximation for the structure of conditional variance and time series characteristics of the daily returns.. As shown in Table A, Jarque-Bera statistics is very high which tells us that there is sharp volatility in each stock market. Moreover, there exists leptokurtosis for each country. Because of high kurtosis and nonnormality of the data, we think that there is ARCH effect in the data. Moreover, in the following steps, we looked at the autocorrelation in squared residuals and than we applied Lagrange Multiplier test for proving the existence of ARCH effect.

TABLE A Turkey Greece Russia Mean 0.001885 0.000586 0.001728 Standard Dev. 0.031031 0.000120 0.032094 Skewness 0.036820 -0.045444 0.584223 Kurtosis 6.795759 6.995054 32.4873 Jarque-Bera 1411.294 1563.603 85272.29

3. We took the square of the stationary series we obtained in the previous step.

squddlturkey=dlog(turkey)ˆ2 squddlgreece=dlog(greece)ˆ2 squddlrussia=dlog(russia) ˆ2 To estimate an ARMA model, we checked the correlogram of these series. Autocorrelation and Partial autocorrelation values were investigated to estimate the values of the ARMA (p, q) process. Autocorrelations are the measure of correlations between the value of a random variable today and its value some days in the past. If there are significant autocorrelations in returns, we may predict the returns. If there are significant

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autocorrelations in squared or absolute returns, it means there is volatility clustering. In our research, the return autocorrelations are almost all insignificant while the autocorrelations of squared returns are significant. Therefore, we can say that there are both unpredictability of returns and volatility clustering. As you can see from the PACF values in Appendix 8, we thought that for Turkey 3 lags were significant. For Greece and Russia the significant number of lags was 4. 4. We estimated the ARMA models. An ARMA model for observed time series is necessary to remove any serial correlations within the data. In Appendices 9-10-11, the steps to obtain the ARMA model for each country are shown clearly. From the observations made in previous step, for Turkey, the analysis started with an initial model: Dlturkey c Ar(1) Ma(1) Ar(2) Ma(2) Ar(3) Ma(3) As you can see in Appendix 9, Table 13; the p-values of Ar(1) Ma(1) Ar(2) Ma(2) Ar(3) Ma(3) are all “0” the null hypothesis that these coefficients are “0” is rejected and continued the analysis with this model The same procedure was applied to Greece and Russia with an initial model including a constant and “Ar” and “Ma” till 4. The steps are summarized in Appendix 10 and 11. The mean equation of Greece consisted of only constant whereas that of Russia included a constant, Ar(1) Ar(2), Ma(1) and Ma(2).

5. We looked at the Residual LM test.

After estimating the correct ARMA model, ARCH-LM test was applied to see whether there exists any conditional heteroskedasticity (ARCH effect) within the series. From Table 14, it can be seen that the F statistic for Turkey is significantly high (113.4075) so that we rejected the null hypothesis that there exists no ARCH effects within the data and went on modelling the appropriate GARCH model. The ARCH-LM tests for Greece and Russia implied that the series still had ARCH effects as the F-statistics for these two countries were also significantly high (95.57139, 229.0122 respectively). (Tables 18 and 21)

6. The GARCH Modelling In all cases, we observed ARCH effect and proceeded with estimating a GARCH model. To decide on the right GARCH model, we compared the “Akaike’s Information Criterions (AIC )” in the estimation outputs, “F-statistics” and “the probabilities to accept that there is no ARCH effect any more” from the LM-test. We looked at the Residual LM test to check whether any information left in variance. We selected the model which has the lowest value of AIC and F-statistic and the one with the highest probability.

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The results are as follows for the three countries:

Turkey GARCH(1,1) TARCH(1,1) EGARCH(2,2) GARCH-M(1,1) AIC -4.259769 -4.260074 -4.256666 -4.249502 F-Stat(LM-Test) 2.226497 1.938754 3.202612 0.350978 Probability 0.108134 0.144114 0.040834 0.704036 Refer to Tables 22-23 Tables 24-25 Tables 27-28 Tables 30-31

Greece GARCH(1,1) TARCH(1,2) EGARCH(1,1) GARCH-M(1,1) AIC -5.660794 -5.665152 -5.652252 -5.660306 F-Stat(LM-Test) 2.433669 0.975125 3.779714 2.948697 Probability 0.087936 0.377298 0.022968 0.052602 Refer to Tables 32-33 Tables 35-36 Tables 37-38 Tables 39-40

Russia GARCH(2,1) TARCH(1,1) EGARCH(1,1) GARCH-M(2,1) AIC -4.667728 -4.661447 -4.571464 -4.664552 F-Stat(LM-Test) 4.782797 3.797105 5.106400 4.439985 Probability 0.008454 0.022574 0.006125 0.011895 Refer to Tables 41-42 Tables 45-46 Tables 47-48 Tables 50-51

We tested the data for each country by using GARCH, TARCH, EGARCH and GARCH-M models. We tried to find out which model was the best for each stock market. We collected all the AIC values and LM-Test statistic values up to third lag for each country. For Turkey, we decided that the GARCH-M(1,1) model is the best as it has both the lowest F-statistic and the highest probability to accept the null hypothesis that there is no ARCH effect any more in the model. We also tested it by Ljung-Box test statistics. As it can be seen from the Q statistics in Tables 55-57(Q statistics for Tarch ,Egarch and Garch-M respectively), Tarch(1,1) has the best values in the sense of higher p-values. But still we thought that the F-statistic of Garch-M model is comparatively better. In the case of Greece, TARCH(1,2) is selected as it has the best values among the other models for each 3 criteria. For Russia, we chose the TARCH(1,1) model because it has the lowest F-statistic and the highest probability as in the case of Turkey. Yet, the probability to accept the null hypothesis fails, so we thought this model is still not satisfactory, but still we couldn’t manage to achieve a better result. The Q-statistics in Tables 61-63(Q statistics for Tarch ,Egarch and Garch-M respectively) also showed that our decision was correct as Tarch(1,1) has the best p-values among the other statistics.

7. Forecast We looked at the Theil inequality coefficient to check whether we had a good fit of the model or not. The Theil inequality coefficient always lies between zero and one, where zero indicates a perfect fit. In our forecast, the Theil inequality coefficients are 0.910103, 0.966619, and 0.893729 for Turkey, Greece and Russia respectively.

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These coefficients showed that we obtain the best fit for Russia even if the TGARCH model does not eliminate the all ARCH effects from the model.

In the forecast, the bias proportion shows that how far the mean of the forecast is from the mean of the actual series and the variance proportion shows that how far the variation of the forecast is from the variation of the actual series. These should be as small as possible. In our results, as shown in Appendix 24, bias proportions are 0.004282, 0.000135, and 0.002313, the variance proportions are 0.982842, 0.999865 and 0.997687 for Turkey, Greece and Russia respectively. The bias proportions are very low, indicating that the mean of the forecasts does not track the mean of the dependent variable. The bias proportions are very good in this respect. However, the variance proportions are very close to 1. On the other hand, the covariance proportion measures the remaining unsystematic forecasting errors. Our results in covariance proportions are very low, they are around zero.

6. Literature Research

Hornikova (2003) examines the behaviour of the Prague Stock-Exchange Index, PX-50, which includes 50 leading Czech companies. In the paper, it is observed that there are nonnormality, volatility clusters, negative skewness, large kurtosis, and autocorrelation in the financial time series data. Therefore, Generalized ARCH model was applied to the index. It is found that the best fitting model is GARCH(1,1). Berument et al. (2001) analysed the dynamics of inflation uncertainty in Turkey. They used EGARCH method to model inflation uncertainty in Turkey. They used EGARCH model because this model enables the separate treatment of the negative and positive shocks to inflation. It does not impose the nonnegative constraints on parameters and avoids the effects of outliers on the estimation results as is specified in logarithms. They found that the effect of positive shocks to inflation uncertainty was greater than the effect of negative shocks to inflation. Hytinnen (1999) investigated the evolution of the conditional volatility of returns on three Scandinavian markets (Finland, Norway and Sweden) over the unstable period of the past decade. For Finland because of the availability constraints, the data starts from the first week of January 1987. For Sweden and Norway the sample is from the first week of January 1983 and ends in the 22nd week of 1997 for each country. The data set includes for each country a weekly value-weighted price index series for the entire stock market and the banking sector. It is found that in the unstable periods volatility can be modelled by symmetric GARCH process. For the market returns, it was found that GARCH(1,1) specification was appropriate for all the countries. Although, there was a turmoil period, the symmetric GARCH model, which is a function of past volatility shocks and of past conditional variances from the previous week, still captures the market volatility. For bank return series, it is found that the best model is TGARCH(1,1) according to SIC (Schwarz Information Criterion) for Sweden. For Finland and Norway, the best specification were found to be GARCH(1,1). As a result, it was shown that during the banking crisis periods, there is spillover effects evidence for cross-country volatility spillovers during the banking crisis episodes.

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7. Conclusion Especially in finance and risk management, volatility is a concept which is of great importance. As mentioned in the introduction part of the paper, conditional volatility models are used for predicting risk. As this is considerably a new subject, we preferred to start with introducing the basic terms used in this paper. We also briefly explained the models that were used in the empirical part of the work. The empirical analyses showed that, it is not that easy to obtain a model which fits the data together with removing the ARCH effects. However, in Turkish and Greek stock markets, we got GARCH-M(1,1) and TARCH(1,2) models respectively, without facing too much difficulties. There were cases we had a good fit but which failed the ARCH-LM test with very low probabilities to accept the null hypothesis which is basically that there exists no ARCH effects any more. The Russian case was more challenging as whatever model with a good fit we obtained; it failed the ARCH-LM test in. Finally, we had to choose the best, so we decided on TARCH(1,1) model, but still it didn’t seem to be a satisfactory result considering the Turkish an Greek cases. For Turkey, the best model is GARCH-M (1,1) as shown in Table 30. The total of the α1(0.142872) and β1(0.754030) is 0.896902. The coefficient of SQR(GARCH) term is 0.227654, which is significant and positive. Therefore, we understood that the return is positively related to its past volatility. In addition it showed us that there is a trade-off between the return and the risk. The more risk means the higher return because a rise in variance increases the mean of return. When we look at the Table A, we see that the skewness is negative which is due to asymmetry in the modelled data of Greece. Therefore, we can say that negative shocks cause higher volatility in the near future than positive shocks. The same result was obtained when we modelled the data by ARCH and GARCH modelling techniques. We found that the best model for Greece is TARCH(1,2). As shown in Table 35, the coefficient of (RESID<0)*ARCH(1) is 0.059187. The γ value is positive and significant which also shows us that leverage effect exists, bad news increases volatility. When we sum up the arch (0.052783, 0.044156) and garch values (0.867438) we get 0.964319, which is very close to 1. That means that the shock is persistent (is dying off slowly). For Russia, as shown in Table A, we have the highest skewness, kurtosis and the Jarque-Bera statistics. In addition, its QQ plot has s-shape as shown in Appendix 28. Therefore, we concluded that the extremes are more substantial than would be expected from a normal random variable. The best model is TARCH(1,1) for Russia. As shown in Table 45, the coefficient of (RESID<0)*ARCH(1) is 0.028327. The γ value is positive but it is relatively small and the probability of z statistic is 0.0141, so it is not significant at 5% critical level. Therefore, we can say that leverage effect exists, bad news increases volatility. The total of the α1(0.106889) and β1(0.849902) is 0.956791 which is close to 1. However, this model could not completely eliminate ARCH effects because its probability of F-statistic is very low.

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As a consequence, when we look at all the countries we can say that, Turkish returns are more affected by the news about volatility from the previous period, than the other countries, because their total α value (ARCH term) is bigger than Greece and Russia. 0.142872 > 0.09693456 and 0.142872 > 0.0106889. On the other hand, Greek returns are affected more by the last period’s forecast variance (σt-1

2) than the other countries, because their β value (The GARCH term) is bigger than the other countries’ β values. 0.867438 > 0.754030 and 0.867438 > 0.849902. This seems reasonable because Greece has more stable economic environment than Turkey and Russia. Therefore, they can trust the forecasted information. However, in unstable economies, people can not know what happen in the future. So, they can be affected more by the new information that was not available when previous forecast was made.

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8. References

Berument, H., Kivilcim, M. Ö. and Neyapti, B. (2001), “Modelling Inflation Uncertainty Using EGARCH: An Application to Turkey”, Contemporary Economic Policy.

Bildik, R. and Gulay, G. (2001), “Effects of Changes in Index Composition on Stock Market: Evidence from Istanbul Stock Exchange “, EFMA 2001 Lugano Meetings. Black, F., 1976, “Studies of Stock Price Volatility Changes,” Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economics Section, 177-181. Bollerslev, T., Chou, R., Nelson, D. (1994), “ARCH Modelling in Finance”, Journal of Econometrics, 69, 5-59. Engle, R. F. (1982), “Autoregressive Conditional Heteroscedasticity with Estimates of The Variance of United Kingdom Inflation”, Econometrica, 50, No. 4, 987-1007. Engle, R. F. (2003), “Risk and Volatility: Econometric Models and Financial Practice”, Nobel Lecture. Engle, R. F. and Susmel, R. (1994), “Hourly Volatility Spillovers between International Equity Markets”, Journal of International Money and Finance, 13, pp. 3-25. EViews 4 User’s Guide, 2002. Glick, R. and Rose, A. (1998), “Contagion and Trade: Why Are Currency Crisis Regional?”, Journal of International Money and Finance, 18:603-617. Gounopoulos D. (2003), “The Initial and Aftermarket Performance of IPOs: Evidence from Athens Stock Exchange”, EFMA 2003 Helsinki Meetings. Hornikova, M. (2003), “Modeling the Behavior of Prague Stock Exchange Index (PX-50),” Econometrics, 0304001, Economics Working Paper Archive at WUSTL. Hyytinnen, A. (1999), “Stock Return Volatility on Scandinavian Stock and the Banking Industry: Evidence from the Years of Financial Liberalisation and Banking Crisis” Bank of Finland Discussion Papers, 19/99. Jithendranathan T. and N. Kravchenko (2004), “Integration of Russian Equity Markets with the World Equity Markets- Effects of the Russian Financial Crisis of 1998”, Working Paper, University of St. Thomas. Masson, P. (1998), “Contagion: Monsoonal Effects, Spillovers and Jumps between Multiple Equilibria” IMF Working Paper 98/142.

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Patell, J. and M. Wolfson, 1979, “Anticipated Information Releases Reflected in Call Option Prices,” Journal of Accounting and Economics, 1, 117-140. Patell, J. and M. Wolfson, 1981, “The Ex Ante and Ex Post Price Effect of Quarterly Earnings Announcements Reflected in Option and Stock Prices,” Journal of Accounting Research, 19, 434-458. Scheaffer R. L. and McClave J. T. (1995), Probability and Statistics for Engineers, Duxbury Press, California. Tsay, R. S. (2002), Analysis of Financial Time Series, Wiley.

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9. APPENDICES APPENDIX 1

Figure 1

Figure 2

25

APPENDIX 2

26

APPENDIX 3

27

APPENDIX 4

28

APPENDIX 5

29

APPENDIX 6

30

APPENDIX 7

31

APPENDIX 8

32

APPENDIX 9

Table 13

Table 14

33

APPENDIX 10

Table 15

Table 18

Table 16

Table 17

34

APPENDIX 11

Table 21

Table 20

Table 19

35

APPENDIX 12

Table 22

Table 23

Figure 15

36

APPENDIX 13

Table 24

Table 25

37

APPENDIX 14

Table 26

Table 27

Table 28

38

APPENDIX 15

Table 29

Table 30

Table 31

39

APPENDIX 16

Table 32

Table 33 Table 34

40

APPENDIX 17

Table 35

Table 36

41

APPENDIX 18

Table 37

Table 38

42

APPENDIX 19

Table 39

Table 40

43

APPENDIX 20

Table 41

Table 42

Table 43

44

APPENDIX 21

Table 44

Table 45

Table 46

45

APPENDIX 22

Table 47

Table 48

46

APPENDIX 23

Table 49

Table 50

Table 51

47

APPENDIX 24

Table 52

Table 53

Table 54

48

Table 55

Table 56

Table 57

APPENDIX 25

49

Table 58

Table 59

Table 60

APPENDIX 26

50

Table 61

Table 62

Table 63

APPENDIX 27

51

APPENDIX 28