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

51 Pages·2005·1.22 MB·English

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: y = α + βx + e]. Therefore, expected
t t t
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).

[ ]
Ε ε =
1. 0
i [ ] [ ]
( ) [ ] ( )
2. VAR ε = Εε2 +Ε ε 2 VAR ε = Εε2 =σ2
( i ) i i i i
3. COV ε,ε = 0
i j

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:

( )
VAR ε =σ2Z 2
i i

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:
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( ) σ2
VAR βˆ =
( )
∑ x − x 2
i
1
σˆ = e 2
n−k −1 i

The null hypothesis and the rejection regions are given:

H : β = 0
0
( ) ( )
Rejection region: t >t n−k −1 & t < −t n−k −1
α α
2 2
T statistics is based on the assumption of normal error terms.

βˆ βˆ
= ( ) is t-distributed with n-k-1 degrees of freedom.
σˆ SE βˆ
( )
x −x 2
i

βˆ −β βˆ −0
t = = because of the null hypothesis we assume that the value of β is
σ σ
βˆ βˆ
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,βˆ, are still unbiased but inefficient and inconsistent.
Because variance of βˆ is no longer equal to the OLS estimation value.

( )
( ) ∑ x − x 2σ2 σ2
VAR βˆ = (∑(xi − x)2)i2 ≠ ∑(x − x)2
i i

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.