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Lithuanian stock market analysis using a set of Garch models.

1. Introduction

Generalized autoregressive conditional heteroskedasticity Autoregressive Conditional Heteroskedasticity (ARCH)

A nonlinear stochastic process, where the variance is time-varying, and a function of the past variance. ARCH processes have frequency distributions which have high peaks at the mean and fat-tails, much like fractal distributions.
 models (GARCH GARCH Generalized Autoregressive Conditional Heteroskedasticity ) are quite popular all over the world. These models can be used for stock, bond, indices, currency, derivative price volatility modelling and forecasting. GARCH models were applied in various areas, so the main point of the author is not to analyse the objects of the research but to find which models are the most popular.

There is not much research which analyses GARCH models in Lithuania. Girdzijauskas and Simutis (informatics) (2002), Danilenko (mathematics) (2007) applied GARCH (1,1) model for financial markets. Girdzijauskas and Simutis I2002) analysed S&P500 index's volatility so there is only one work of Danilenko (2007), in which the volatility of the OMXV index is modelled and forecasted. These researches are not carried out by economists so the main point of the articles is to present the main characteristics of the GARCH model and not the ability of modelling and forecasting volatility. Such authors as Leipus, Norvaisa (2004) and Klivecka (2007) analysed the GARCH models from a mathematical background. So there is no research in which some different GARCH type of models were applied to the Lithuanian stock market. There is a lack of such researches and economic view.

The aim of the research is: after analyzing stock price volatility factors and specifics of generalized autoregressive conditional heteroskedasticity models as a tool of volatility modelling, to create a classification system of stock price volatility factors and also practically to apply a set of "GETIP" models to the Lithuanian stock market. "GETIP" is an acronym for five GARCH type models which are applied to the OMXV index and to all corporations from the Main and Secondary lists. "GETIP" models are GARCH (1,1), EGARCH (1,1), TARCH TARCH Tape Archive  (1,1), IGARCH IGARCH Integrated Generalized Autoregressive Conditional Heteroskedasticity  (1,1) and PARCH parch  
v. parched, parch·ing, parch·es

v.tr.
1. To make extremely dry, especially by exposure to heat: The midsummer sun parched the earth.
 (1,1).

The tasks of the article are:

1. To inspect the reasons of stock return volatility.

2. To suggest a classification system of stock price volatility factors.

3. To apply GARCH models to the Lithuanian stock market: OMXV index and to all corporations from the Main and Secondary lists.

The object of the research is Lithuanian stock market (OMXV index and all stocks of corporations from the Main and Secondary lists).

In this article statistical, mathematical and econometric methods are used, i.e. correlation analysis, static and dynamic prognostication, various unit root tests (ADF (1) (Application Development Facility) An IBM programmer-oriented mainframe application generator that runs under IMS.

(2) (Automatic Document Feeder) A paper stacker that feeds one sheet of paper at a time into the unit.
, PP), ARCH-LM--heteroskedasticity test, autocorrelation Autocorrelation

The correlation of a variable with itself over successive time intervals. Sometimes called serial correlation.
, partial autocorrelation, ARMA (1,1), calculated "LADSH" model suitability selection criterions, various prognostication accuracy estimation parameters, applied set of general autoregressive conditional heteroskedasticity models "GETIP", descriptive statistics descriptive statistics

see statistics.
, regression analysis In statistics, a mathematical method of modeling the relationships among three or more variables. It is used to predict the value of one variable given the values of the others. For example, a model might estimate sales based on age and gender. , time series. Thus, qualitative and quantitative models are used. For applying the GARCH models Eviews6 software was used.

GARCH models are widely applied for modelling the volatility of various markets, but nobody tried to apply these models to the Lithuanian stock market. In this work for the first time various GARCH models are applied to the Lithuanian stock market and the best ones are chosen for modelling and forecasting. The research is carried out using not only the OMXV index but also all the corporations from the Main and Secondary lists. Such a wide investigation of GARCH models allowed us to find the most suitable GARCH model not only for the market as a whole, but also for every company separately and for different sectors.

2. Factors of stock return volatility

Macroeconomic mac·ro·ec·o·nom·ics  
n. (used with a sing. verb)
The study of the overall aspects and workings of a national economy, such as income, output, and the interrelationship among diverse economic sectors.
 variables play a key role in asset pricing theories. For this reason, many authors have empirically studied the link between macroeconomic variables and stock market volatility. A number of studies document that a relationship exists between macroeconomic variables and equity market returns. The APT literature suggests that macroeconomic variables may proxy for pervasive risk factors (Bilson et al. 2000: 30) Both macroeconomists and finance specialists are increasing their attention to the relationship between the stock market and the rest of the economy. There can be little doubt about the growing importance of the stock market from the point of view of the aggregate economy. It has always been recognized that the stock market reflects to some extent the goings-on in the rest of the economy, but recently there has been widespread recognition that the influence is also in the opposite direction--dramatic events in the stock market are likely to have an impact upon the real economy (Black et al. 2001: 35).

Beber and Brandt (2007) in their article described a negative relation between macroeconomic uncertainty and the reduction of open interest after the news release. This result was consistent with market participants closing out hedging positions and with the degree of hedging activity depending on the degree of macroeconomic uncertainty to be hedged.

Fama and Schwert (1977) found mixed results with respect to the direction of causality causality, in philosophy, the relationship between cause and effect. A distinction is often made between a cause that produces something new (e.g., a moth from a caterpillar) and one that produces a change in an existing substance (e.g.  between return volatility and the volatility of macroeconomic and financial variables. They found that: (a) inflation volatility predicts stock volatility and stock volatility does not predict inflation volatility, (b) money growth volatility predicts stock volatility and stock volatility predicts money growth volatility (c) industrial production volatility weakly explains the volatility of stock returns while stock volatility helps to predict industrial production volatility. Overall, their results point to a positive linkage between macroeconomic volatility and stock market volatility, with the direction of causality being stronger from the stock market to the macroeconomic variables.

Empirically, the evidence of linking macroeconomic factors to the stock market is mixed at best. Chen, Roll, and Ross (1986) were among the first to explore the link between macroeconomic variables and stock prices, and using a multifactor model they found evidence that macroeconomic factors are priced. Pearce and Roley (1985) also conclude that stock prices respond to macroeconomic news.

Bordo and Wheelock (2006) found that many, but by no means all, U.S. and British stock market crashes of the 19th and 20th centuries were followed by recessions. A serious decline in economic activity was more likely, they concluded, if a crash was accompanied or followed by a banking panic.

2.1. Inflation's influence on stock market

The empirical relationship In science, an empirical relationship is one based solely on observation rather than theory. An empirical relationship requires only confirmatory data irrespective of theoretical basis.  between inflation and common stocks was first investigated by Jaffe and Mandelker (1976), Bodie (1976). Although employing different empirical approaches, these authors all concluded with a significant negative relationship between the proxies for inflation and stock returns. Following these pioneer studies, Fama and Schwert (1977) investigate the inflation effect on asset returns in a number of assets. They conclude that, similar to previous studies, common stocks seem to perform poorly as hedge against both expected and unexpected inflation. Since these earlier studies, the empirical literature on the Fisher Hypothesis The Fisher hypothesis is the proposition by Irving Fisher that the real interest rate is independent of monetary measures, especially the nominal interest rate. The Fisher equation is

 has been prolific, and the findings have been largely similar.

The relationship between stock returns and inflation has inspired both theoretical and empirical studies Empirical studies in social sciences are when the research ends are based on evidence and not just theory. This is done to comply with the scientific method that asserts the objective discovery of knowledge based on verifiable facts of evidence. . Most empirical research Noun 1. empirical research - an empirical search for knowledge
inquiry, research, enquiry - a search for knowledge; "their pottery deserves more research than it has received"
 employed exclusively the United States United States, officially United States of America, republic (2005 est. pop. 295,734,000), 3,539,227 sq mi (9,166,598 sq km), North America. The United States is the world's third largest country in population and the fourth largest country in area.  (US) data in the analysis. Some papers extended the investigation to other country samples, but only a few employed emerging markets data.

Stocks are a good hedge against inflation because, in theory, the company's revenues and earnings should grow at the same rate as inflation over time. Of course companies can react to inflation by raising their prices, but others who compete in a global market may find it difficult to stay competitive with foreign producers who do not have to raise prices due to inflation. More importantly, inflation robs investors (and everyone else) by raising prices with no corresponding increase in value. People pay more for less. This means company's finances are overstated o·ver·state  
tr.v. o·ver·stat·ed, o·ver·stat·ing, o·ver·states
To state in exaggerated terms. See Synonyms at exaggerate.



o
 by inflation because the numbers (revenues and earnings) rise with the rate of inflation in addition to any added value Added value in financial analysis of shares is to be distinguished from value added. Used as a measure of shareholder value, calculated using the formula:

Added Value = Sales - Purchases - Labour Costs - Capital Costs
 generated by the company. When inflation declines, so do the inflated earnings and revenues.

Inflation erodes purchasing power Purchasing Power

1. The value of a currency expressed in terms of the amount of goods or services that one unit of money can buy. Purchasing power is important because, all else being equal, inflation decreases the amount of goods or services you'd be able to purchase.

2.
 and retirees on fixed income. That is why financial advisers caution even retirees to keep some percentage of their assets in the stock market as a hedge against inflation (Liitle 2008).

Canto can·to  
n. pl. can·tos
One of the principal divisions of a long poem.



[Italian, from Latin cantus, song; see canticle.
 and Kraussl's (2006) main results can be summarized as follows: (i) there are clear short-term international dynamic interactions among the European stock futures markets; (ii) foreign economic news affects domestic returns; (iii) futures returns adjust to news immediately; (iv) announcement timing of macroeconomic news matters; (v) stock market dynamic interactions do not increase at the time of the release of economic news; (vi) foreign investors react to the content of the news itself more than to the response of the domestic market to the national news; and (vii) contemporaneous con·tem·po·ra·ne·ous  
adj.
Originating, existing, or happening during the same period of time: the contemporaneous reigns of two monarchs. See Synonyms at contemporary.
 correlation between futures returns changes at the time of macroeconomic releases.

It is now taken as a stylized styl·ize  
tr.v. styl·ized, styl·iz·ing, styl·iz·es
1. To restrict or make conform to a particular style.

2. To represent conventionally; conventionalize.
 fact by market participants and academicians alike that returns on stocks suffer significantly in the face of high or rising inflation. This view contrasts starkly with the state of conventional wisdom and economic theory less than three decades ago, when an investment in equities was believed to be a relatively good hedge against inflation. That earlier perception was challenged by the drubbing equity investors took during the 1970s, and was refuted by several studies that offered compelling statistical evidence of inflation's negative effect on stock returns.

The other side of this coin, as some Wall Street investment strategists see it, is that the recent-years' decline in inflationary pressures and inflation expectations justifies the recent outsized stock returns, as well as the high current valuations as gauged by record-high price- earnings ratios or record-low dividend yields (Sharpe 1999).

Elaborating on Fama's work, Geske and Roll (1983) propose that, besides money demand, the money supply linkage may help explain the phenomenon. The authors propose a chain of macroeconomic events that leads to a "spurious" correlation between stock returns and inflation. They suggest that stock prices' reaction in anticipation of future economic activity (Fama's model) is highly correlated to government revenue, so that the government faces a deficit when economic output decreases. In order to balance the budget, the Treasury either borrows or issues money through the central bank, causing inflation. Thus, stock returns and inflation are negatively related due to a fiscal and monetary linkage--the Reverse Causality Hypothesis. The authors find some evidence in support of their framework, especially the signalling from stock returns to changes in nominal interest rates and changes in expected inflation.

Theoretically, inflation could be neutral with respect to stock prices. In such inflation--indexed world, news of higher-than-expected inflation is incorporated into the numerator numerator

the upper part of a fraction.


numerator relationship
see additive genetic relationship.


numerator Epidemiology The upper part of a fraction
 (higher cash flows as the price increases are passed through to the consumers) of equation 1, with an offsetting adjustment in the denominator (higher discount rates to compensate stockholders for losses in purchasing power).

[P.sub.t] = [[infinity].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  over ([tau]=1)] E[[D.sub.t+[tau]]/[[OMEGA].sub.t]]/1 + [r.sub.t+[tau]]. (1)

In the first equation [P.sub.t] is the price of the stock at time t, E [*/[[OMEGA].sub.t]] denotes the mathematical expectation conditional on information available at time t, [D.sub.t+[tau]] is the dividend paid at time t + r, and [r.sub.t+[tau]] is the stochastic By guesswork; by chance; using or containing random values.

stochastic - probabilistic
 risk-adjusted discount factor for cash flows that occur at time t + r (Funke, Matsuda 2002). Economic announcements influence stock price movements if the new information affects either expectations of future dividends, or discount rates, or both. The new information is represented by the difference in the announced value of inflation at time t + 1 and the expected value Expected value

The weighted average of a probability distribution. Also known as the mean value.
 as of time t.

The historical influence of inflation on stock prices is mysterious because stocks are claims to the profits generated by the corporate capital stock, and thus they are real assets that should not be directly vulnerable to inflation (Maghayereh 2002). It is obvious that when inflation increases the P/E ratio decreases and vice versa VICE VERSA. On the contrary; on opposite sides. . Stock prices are undervalued Undervalued

A stock or other security that is trading below its true value.

Notes:
The difficulty is knowing what the "true" value actually is. Analysts will usually recommend an undervalued stock with a strong buy rating.
 when inflation is high, and can become overvalued Overvalued

A stock whose current price is not justified by the earnings outlook or price/earnings (P/E) ratio and thus, expected to drop in price. Overvaluation may result from an emotional buying spurt, which inflates the market price of the stock or from a deterioration in a
 when inflation drops. When examining the links between the U.S. economy and the stock market, many investment professionals rely on what is known as the "Fed model". The model assumes that bonds and equities compete for space in investment portfolios; if bond yields increase, then stock yields must also rise in order to remain competitive (Li, Hu 1998).

Historically, the rate of inflation has been a major influence on nominal bond yields. Therefore, the Fed model implies that stock yields and inflation must be highly correlated.

Christiansen and Ranaldo (2005) wrote that for bonds, the relevance of inflation and risk premium varies across the time. In the present value model, inflation (real interest rate) changes make bond and stock returns move in opposite (same) directions. Changes in risk premium and term premium typically affect bonds and stocks differently. Although the bond-stock return correlation is generally positive, the relation might be negative in periods of "flight to quality". Li, Hu (1998) shows that real interest rates drive the bond and stock comovements and that inflation shocks make bond and stock returns move in opposite directions. Other drivers that decrease the bond-stock correlation are dividends and risk premium. Moreover, he finds that the bond-stock correlation mainly depends on inflation uncertainty.

Inflation or the central bank's response to inflation damages the real economy and, by extension, the profitability of corporations. Inflation also might make investors more risk-averse, thus driving up the risk premium.

Modigliani and Cohn (1979) contend that stock investors and not their bond counterparts are subject to "inflation illusion"; that is, they fail to understand the impact of inflation on nominal dividend growth rates Growth Rates

The compounded annualized rate of growth of a company's revenues, earnings, dividends, or other figures.

Notes:
Remember, historically high growth rates don't always mean a high rate of growth looking into the future.
 and extrapolate extrapolate - extrapolation  historical nominal growth rates even in periods of changing inflation. From a rational investor's viewpoint, then, stock prices are undervalued when inflation is high, and can become overvalued when inflation drops.

2.2. Interest rates and stock market

The tool for inflation reduction of Central bank is short-term interest rates. By making money more expensive to borrow, the Central bank effectively removes some of the excess capital from the market. So it is essential to analyse the influence of interest rates on stock market.

The unexpected increase in inflation generates temporary variations in real and nominal interest rates. The real interest rate on loans, which affects investment decisions through the real cost of borrowing to firms, increases temporarily, generating a drop in the growth rate of investment (Dos Santos, Zezza 2004).

High energy prices, rising unit labour costs and pressure on supplies of key resources such as steel and cement guarantee that the Central bank will continue raising short-term interest rates. High interest rates and companies raising prices do not add up to an investment profile most investors enjoy.

When the next Central bank meeting is expected to bring interest rate cuts or increases, it is wise, as for a stock investor, to be aware of the potential effects behind such decisions. Although the relationship between interest rates and the stock market is fairly indirect, the two tend to move in opposite directions.

Stock valuations and interest rates are directly linked. Businesses and the stock market pay close attention to what the Central bank decides because interest rates are so important. There are obviously some very practical concerns about interest rates, such as the cost of borrowing, the effect on consumer spending Consumer demand or consumption is also known as personal consumption expenditure. It is the largest part of aggregate demand or effective demand at the macroeconomic level.  and so on. There is also a fundamental consideration that this is the basis for beginning any process that leads to the valuation of a stock.

A decrease in interest rates means that those people who want to borrow enjoy an interest rate cut. But this also means that those who are lending money, or buying securities such as bonds, have a decreased opportunity to make income from interest. If we assume investors are rational, a decrease in interest rates prompts investors to move money away from the bond market to the equity market (Walti 2005). At the same time, businesses enjoy the ability to finance expansion at a cheaper rate, thereby increasing their future earnings potential, which, in turn, leads to higher stock prices. Investors and economists alike therefore view lower interest rates as catalysts for expansion.

Overall, the unifying effect of an interest rate cut is the psychological effect it has on investors and consumers; they see it as the benefit to personal and corporate borrowing, which in turn leads to greater profits and an expanding economy.

A rise in bond yields (as a result of monetary policy measures or alterations in inflation expectations), for example, will trigger a move out of stocks, as a result of which share prices and P/E ratios fall and the earnings yield rises in parallel to the bond yield (Bulthaupt, Hofmann 2004).

A rise under the interest rates affects the valuation of the stocks. The rise in the interest rates raises the expectations of the market participants, which demand better returns commensurate with the increased returns on bonds. The above relationship of market P/E and 10-year bond Yield of the US treasury gives a very good understanding of, how over the long-term the stock markets are impacted by the change in interest rates.

Moreover, under a low interest rate regime, corporates are able to increase profitability by reducing their interest expenses. However, under a rising interest rate regime since interest expenses rise, profitability is hit. Besides that, when one calculates the inherent value of a company by the cash flow discounting model, there is a twofold impact. First, there is a reduction in the cash flows due to lower profitability, second, there is a higher discounting rate due to higher interest rate regime. This leads to a relatively lower intrinsic value Intrinsic Value

1. The value of a company or an asset based on an underlying perception of the value.

2. For call options, this is the difference between the underlying stock's price and the strike price.
 of the company.

Due to these reasons there may be a change in asset allocation Asset Allocation

The process of dividing a portfolio among major asset categories such as bonds, stocks or cash. The purpose of asset allocation is to reduce risk by diversifying the portfolio.
 among equities and debt. Investors, who are averse to risk, tend to move funds from one asset class to another. When interest rates rise, investors move from equities to bonds and vice versa. Having said that it does not mean that all the funds move from one asset class to another, but it happens that the marginal shift of funds does change valuations to some extent (Kim et al. 2005). Whereas interest rates drop, returns on bonds drop while the returns on equities tend to look relatively more attractive and the migration of fund from bonds to equities takes place, increasing the prices of equities.

Samitas and Kenourgios (2005) in their findings noticed that generally domestic industrial production was more prominent than domestic interest rates, while US interest rates were more prominent than US industrial production. Furthermore, a number of short run causal relationships were also found giving different implications for policy makers interested in the long run and short run contagion Contagion

The likelihood of significant economic changes in one country spreading to other countries. This can refer to either economic booms or economic crises.

Notes:
An infamous example is the "Asian Contagion" that occurred in 1997 and started in Thailand.
. The main findings strongly suggest that the emerging CE European capital markets are macroeconomically co-integrated with a German economic influence, but less or not influenced by the American global factor.

[FIGURE 1 OMITTED]

Viale (2006) wrote that the well-documented nonlinear relation between macroeconomic news and stock market returns depends on the quality of the information disclosed by what a priori a priori

In epistemology, knowledge that is independent of all particular experiences, as opposed to a posteriori (or empirical) knowledge, which derives from experience.
 investors believe to be vague news, i.e., ambiguous and noisy announcements.

2.3. Three-dimensional system of stock volatility factors

Factors of stock price volatility are analysed by various authors. Usually, in the literature not classified different factors in various situations are mentioned. According to according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
 the author, for the stock price volatility factors analysis a principal of "percolator percolator

a vessel used in percolation.
" can be applied. The "percolator" principal can be explained as a classification system when all stock price volatility factors are divided into three levels, which are inside, outside factors and investors' psychology as the main factor in all the system. The author of this work thinks that investors sift all the information that they have and choose the main moments according to their psychological state, and after that they take an investment decision. So, in such a way they form supply and demand and also stock price volatility.

The explanation of stock return volatility is quite a difficult process. There is not any special formula for exact return determination. After analysing various factors of stock price volatility (according to information flows) they can be divided into inside and outside factors. All information from corporations' data sources can be assigned to the inside factors and all information, which influences stock price from corporations to the market, can be assigned to outside factors. As an additional group of factors that influence stock price can be investor's psychology. The author has noticed that there is no research about all three groups of factors together. Usually, inside and outside factors are analysed without investor's psychology and investor's psychology is analysed separately. The main idea of this three-dimensional system is shown in Fig. 1.

3. Understanding volatility

When we talk about asset pricing we usually use the term 'volatility'. Volatility is described as a parameter of stochastic processes that are used to model variations in financial asset prices. Two kinds of volatility can be found in the literature: implied volatility Implied volatility

The expected volatility in a stock's return derived from its option price, maturity date, exercise price, and riskless rate of return, using an option pricing model such as Black-Scholes.
 and statistical volatility (Table 1). Both of them normally refer to the same process volatility, but volatility estimates often turn out to be quite different and because volatility can only be measured in the context of a model, it is very difficult to assess the accuracy of estimates and forecasts (Alexander 2005).

Volatility models can be divided into two groups: constant and time-varying volatility models.

Constant volatility models only refer to the unconditional volatility of a return process. These models have a finite constant a, the same throughout the whole data generation process. Time-varying volatility models describe a process for the conditional volatility. The conditional distribution is a distribution that governs a return at a particular instant in time and the conditional volatility at time t is the square root of the variance of the conditional distribution at time t.

The majority of time-varying volatility models assume that returns are normally distributed, in which case each conditional distribution is completely determined by its conditional mean and its conditional variance In statistics, conditional variance is a special form of the variance. If we have a conditional distribution Y|X the conditional variance is defined as



where
. The conditional mean and variance could change at every time period throughout the process, but for the purposes of estimating and forecasting conditional volatility it is often assumed that the conditional mean is a constant. (Alexander 2005).

Volatility over sufficiently long periods of time reverts back to a constant mean. However, volatility may depart from this mean for extended periods of time. This process is called "volatility clustering In finance, volatility clustering refers to the observation, as noted by Mandelbrot, that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes. ". Volatility clustering is one of the most important "stylized facts In social sciences, especially economics, a stylized fact is a simplified presentation of an empirical finding. While results in statistics can only be shown to be highly probable, in a stylized fact, they are presented as true. " in financial time series data. Whereas price changes themselves appear to be unpredictable, the magnitude of those changes. As measured, e.g. by the absolute or squared returns, it appears to be predictable in the sense that large changes tend to be followed by large changes--of either sign--and small changes tend to be followed by small changes (McQueen, Vorkink 2004).

Asset price fluctuations are thus characterized by episodes of low volatility, with small price changes, irregularly interchanged by episodes of high volatility, with large price changes. This phenomenon was first observed by Mandelbrot (1963) in commodity prices. Since the pioneering papers by Engle (1982) and Bollerslev (1986) on autoregressive conditional heteroskedastic (ARCH) models and their generalization to GARCH models, volatility clustering has been shown to be present in a wide variety of financial assets Financial assets

Claims on real assets.
 including stocks, market indices, exchange rates and interest rate securities.

Stylized facts about volatility clustering include the following:

* both good news (positive shocks) and bad news lead to higher levels of volatility;

* bad news tends to increase future volatility more than good news;

* the effect of news on volatility has a transitory (rapid decay) and more permanent (slow decay) component;

* volatility appears to have an effect on the risk premium. (McQueen, Vorkink 2004).

Volatility is a very important parameter in financial risk management. Daily volatility can be calculated using such a formula:

[[sigma].sub.daily] = [square root of [T.summation over (t=1)][([X.sub.t] - [bar.X]).sup.2]/T - 1, (2)

where [X.sub.t] = 100ln [Y.sub.t]/[Y.sub.t-1], t = 1, 2,..., T. and

[bar.X] = [T.summation over (t=1)[X.sub.t]/T, (3)

[[sigma].sub.period] = [[sigma].sub.daily] x [square root [[tau].sub.period]]. (4)

4. GARCH models as volatility valuation measure

General autoregressive conditional heteroskedasticity models (GARCH) today are the most widely used models for risk management in finance. The autoregressive conditional heteroskedasticity (ARCH) models were introduced by Engle in 1982 and their generalization, the so-called GARCH models by Bollerslev in 1986. It has been the most commonly employed class of time series models in the recent finance literature. These models have been very successful in describing the behavior of financial return data. Their appeal comes from the fact that they can capture both volatility clustering and unconditional return distributions with heavy tails--two stylized facts associated with financial return data.

GARCH models can be applied for stock and index trading, risk management, portfolio management and asset allocation, option valuation, etc. Analysing time series it is often used for volatility clustering.

The author accomplished comprehensive research of generalized autoregressive conditional heteroskedasticity models' appliance to the Lithuanian stock market. Five heteroskedasticity models were applied to the national market: GARCH (1,1), EGARCH (1,1), TARCH (1,1), IGARCH (1,1), PARCH (1,1) from which the best models are selected for each company from the Main and Secondary lists and for OMXV index. Also checked in this work is the exponential autoregressive conditional heteroskedasticity model, which was most suitable for the OMXV index, ability to forecast for short and long periods. The results of the research will help financial analysts and investors to properly value Lithuania's stock market tendencies and replenish a set of market analysis tools in Lithuania's conditions. The estimation of the GARCH model involves the joint estimation of a mean and a conditional variance equation (Lildholdt 2002).

In the standard GARCH (1,1) specification:

[[sigma].sup.2.sub.t] = [omega] + [alpha][[epsilon].sup.2.sub.t-1] + [beta][[sigma].sup.2.sub.t-1], (5)

[[sigma].sup.2.sub.t] is the one-period ahead forecast variance based on past information, it is called the conditional variance. The conditional variance equation specified in formula (5) is a function of three terms:

* The mean: [omega].

* News about volatility from the previous period, measured as the lag of the squared residual from the mean equation: [[epsilon].sup.2.sub.t-1] (the ARCH term).

* Last period's forecast variance: [[sigma].sup.2.sub.t-1] (the GARCH term).

Nelson (1991) created the model which values the leverage effect and called such a model exponential GARCH (EGARCH). Nelson argued that the non negativity constraints in the linear GARCH model were too restrictive. The GARCH model imposes the non-negative constraints on the parameters [[alpha].sub.i] and [[gamma].sub.j], while there are no restrictions on these parameters in the EGARCH model. In the EGARCH model, the conditional variance, [[sigma].sub.t] is an asymmetric function of lagged disturbances [u.sub.t-i]:

log ([[sigma].sup.2.sub.t]) = [[alpha].sub.0] + [[alpha].sub.1] [absolute value of [u.sub.t-1]]/[[alpha].sub.t-1]] + [[beta].sub.1] log ([[sigma].sup.2.sub.t-1]) + [gamma][u.sub.t-1]/[[sigma].sub.t-1]. (6)

The other model which analyses the leverage effect of new information on stock volatility is TARCH. This model was analysed by such authors as Zakoian (1994) and Glosten et al. (1993). This model can be explained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ], (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

In this model "good news" [u.sub.t-i] > 0 and "bad news" ut_i < 0, have different influence on conditional variance. When [[gamma].sub.k] = 0 for all k, then the TARCH model is adequate for the GARCH model. The difference between TARCH and EGARCH is that in the first model leverage effect has expression of quadratic quadratic, mathematical expression of the second degree in one or more unknowns (see polynomial). The general quadratic in one unknown has the form ax2+bx+c, where a, b, and c are constants and x is the variable.  and in the other one--exponential. In TARCH and EGARCH models persistence of volatility is very long.

When [alpha] + [beta] = 1 and [beta] = [lambda], the main GARCH expression can be rewritten as follows:

[[sigma].sup.2.sub.t] = [omega] + (1 - [lambda]) [[epsilon].sup.2.sub.t-1] + [lambda][[sigma].sup.2.sub.t-1], 0 [less than or equal to] [lambda] [less than or equal to] 1. (9)

There is no defined non-conditional variance and time forecasts do not converge in this model. So, in this situation the process of variance is not stationary, therefore

such a model is called integrated GARCH (IGARCH).

Most GARCH models analyse conditional variance. Analysing volatility researches often have a mean standard deviation. But some researches offered GARCH model which uses standard deviation. Such class of models was defined by Ding, Granger (1996) and this idea was named power ARCH model (PARCH). This model can be expressed in such a way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where [sigma] > 0, [absolute value of [[lambda].sub.i]] [less than or equal to] 1, when i = 1, 2,..., r and [[lambda].sub.i] = 0, when i > r and r [less than or equal to] p. PARCH model when [gamma] [not equal to] 0 has a leverage effect. PARCH model can be explained as GARCH when [sigma] = 2 and [[gamma].sub.i] = 0 for every i.

When two or more models are taken for analysis which have the same number of the parameters, then for suitability selection log likehood function can be used. But when the models have a different number of parameters, Akaike information criterion Akaike's information criterion, developed by Hirotsugu Akaike under the name of "an information criterion" (AIC) in 1971 and proposed in Akaike (1974), is a measure of the goodness of fit of an estimated statistical model. It is grounded in the concept of entropy.  is used. If the number of model parameters is signed P, then AIC can be expressed as follows:

AIC (P) = 2 ln(maximum likelihood) - 2P. (11)

5. The practice of a set of ,,GETIP" models in the Lithuanian stock market

"GETIP" is an acronym of five GARCH type models which are applied to OMXV index and to all corporations from the Main and Secondary lists. "GETIP" models are GARCH (1,1), EGARCH (1,1), TARCH (1,1), IGARCH (1,1) and PARCH (1,1). The period for analysis is from 2000-01-01 till 2008-01-18. All the results of the research are shown in Table 2. Estimated "GETIP" models' coefficients:

* GARCH

[[sigma].sup.2] = 1.59 E - 05 + 0.156801[[epsilon].sup.2.sub.t-1] + 0.606602[[sigma].sup.2.sub.t-1].

* EGARCH

log ([[sigma].sup.2.sub.t]) = -0.169412 + 0.987085 log [[sigma].sup.2.sub.t-j] + 0.068175 [absolute value of [[epsilon].sub.t-i]/[[sigma].sub.t-1]] - 0.002287 [[epsilon].sub.t-k]/[[sigma].sub.t-k].

* TARCH

[[sigma].sup.2.sub.t] = 1.46 E - 05 + 0.619615[[sigma].sup.2.sub.t-j] + 0.120691[[epsilon].sup.2.sub.t-1] - 0.083507[[epsilon].sup.2.sub.t-k][I.sub.t-k].

* IGARCH

[[sigma].sup.2.sub.t] = 0.975629[[sigma].sup.2.sub.t-j] + 0.024371[[epsilon].sup.2.sub.t-i].

* PARCH

[[sigma].sup.1.sub.t] = 9.79E - 05 + 0.964464[[sigma].sup.1.sub.t-j] + 0.035655 [([[epsilon].sub.t-i]] - 0.053604[[epsilon].sub.t-i]).sup.1].

According to the author the models can be ranged as follows:

1. EGARCH

2. PARCH

3. TGARCH

4. GARCH

5. IGARCH

A set of GARCH models was applied to all sectors classified by GICS GICS Government Information and Communication Service (UK)
GICS German Internet Chess Server
GICS Global Industry Classification Standards
GICS Grant Information and Control System
. The results of suitability of different GARCH models for separate sectors are shown in Figures 2-5.

For those investors who use general autoregressive conditional heteroskedasticity models the author of this article offers to use different models for separate sectors:

* OMXV index--EGARCH;

* Health Care--PARCH;

* Energy--TARCH;

* Materials--LFO--EGARCH; GRG--GARCH;

* Industrials--EGARCH;

* Consumer Discretionary--VBL, KBL--PARCH, SNG SNG
abbr.
1. substitute natural gas

2. synthetic natural gas
, APG--TARCH, UTR UTR Untranslated Region (genetics)
UTR Unicode Technical Report
UTR Unique Taxpayer Reference (UK Inland Revenue)
UTR Unable to Reach
UTR Unable to Reproduce
UTR University Technical Representative
, LNS--GARCH;

* Consumer Staples--EGARCH;

* Financial--TARCH;

* Telecommunication Services--PARCH;

* Utilities--EGARCH.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The best GARCH model for the Lithuanian stock market is EGARCH (for OMXV index and most sectors), so it is important to analyse the ability to forecast by this model. In Table 3 the relationship between forecast measures and time horizon is placed using exponential GARCH model.

After analysing EGARCH model's abilities to forecast in different periods, the conclusion can be drawn that there is not any tendency for which time horizon this model forecast is the best (Fig. 6). At a first glance it can appear that in short time (5, 10 days) EGARCH model forecasts better than in long period, but the correlation coefficient Correlation Coefficient

A measure that determines the degree to which two variable's movements are associated.

The correlation coefficient is calculated as:
 is only 0.24 which is not significant. After doing the research, the conclusions can be made that for the Lithuanian stock market the most suitable model is EGARCH. This model has a strong leverage effect which means that bad news has more effect on stock price volatility than good news. Thus, according to such an idea all investors tend to remember bad events longer than good information. Therefore, according to the author when bad information appears in the stock market it is quite difficult to value the flows of positive information.

[FIGURE 6 OMITTED]

6. Conclusions

1. Analysing generalized autoregressive conditional heteroskedasticity models the author excludes the three main characteristics. That is, the principal of heteroskedasticity, volatility clusterization (including excess, asymmetry and Jarque-Bera test In statistics, the Jarque-Bera test is a goodness-of-fit measure of departure from normality, based on the sample kurtosis and skewness. The test statistic JB is defined as

) and also leverage effect. The origin of the heteroskedasticity idea in finance gave bigger opportunities for investors in forecasting stock price volatility. In homoskedastic processes when the variance is determined and static, such a way is not suitable because the results of research would be wrong. For more accurate return forecasts volatility clusterization should be estimated. Analysing stock price dynamics volatility clusters can usually be noticed, which periodically repeats. According to the author there is a direct dependence between information flows and volatility clusters. It is important to underline that the influence of information on stock price volatility depends on its type (bad or good news) though clusters form in both situations. The other essential characteristics of heteroskedasticity models is a leverage effect when bad news has bigger impact on stock price than good news.

2. After the investigation of various GARCH models, all of them can be classified into univariate and multivariate models. Because of the big variety of GARCH models, a problem of suitable model selection appears. For this purpose it is essential to use model selection criterions which help to value the level of model suitability. There are various types of model selection criterions but not all of them are suitable for different situations, others are very similar and give the same results. According to the author's empirical researches the best results were achieved using Akaike in formation criterion. Using "LADSH"--a set of model selection criterions the author noticed that only in some situations the results of range were different.

3. The research of the Lithuanian stock market showed that the market is stationary and the data is not distributed normally. GARCH type of models can be applied to the Lithuanian stock market because for analysed returns homoskedasticity is not suitable. After using a set "GETIP" of GARCH models, the author noticed that the most suitable model for OMXV index is EGARCH. From the research results the conclusion can be drawn that there is a leverage effect in the Lithuanian stock market. So this means that investors react more to bad news than to good news. In this way the situation of nowadays in stock market can be explained. Moreover, bad news from USA and Europe dwarfed good news about separate corporation financial data. According to model selection results the hypothesis can be rejected that the best model for the Lithuanian security market is GARCH (1,1).

4. After the application of GETIP- a set of GARCH models, to all sectors of the Lithuanian stock market some tendencies can be excluded. Exponential generalized autoregressive conditional heteroskedasticity model (EGARCH) is the most suitable for industrials, consumer staples Consumer Staples

The industries that manufacture and sell food/beverages, tobacco, prescription drugs, and household products.

Notes:
Proctor and Gamble would be considered a consumer staple company because many of its products are household and food related.
, information technology and utilities sectors. TARCH is the best for the energy and finance sectors. A power generalized autoregressive conditional heteroskedasticity model (PARCH) is best reflected in health care and telecommunication services sectors. Consumer discretionary sector did not distinguish any tendencies because three models PARCH, GARCH and TARCH gave the same results. According to the author it is quite difficult to forecast the volatility of this sector and herewith here·with  
adv.
1. Along with this.

2. By this means; hereby.


herewith
Adverb

Formal together with this:
 there is a risk to invest in it. For the materials sector the most suitable are GARCH(GRG GRG - A computer algebra system for differential geometry, gravitation and field theory. Version 3.1 works with PSL-based REDUCE 3.3 or 3.4.

E-mail: V.V. Zhytnikov <vvzhy@phy.ncu.edu.tw>.
) and EGARCH(LFO LFO Low Frequency Oscillator
LFO Lyte Funky Ones (dance band)
LFO Legal Framework Order (Pakistan)
LFO Light Finding Operation (anime, Eureka seveN)
LFO Last Flight Out
) models.

5. The researches of volatility forecasting using EGARCH model should be used for static forecast method because of more accurate prognosis. Analysing various statistical methods of forecasting accuracy it has been noticed that there is not a significant relationship between EGARCH model accuracy and the period of forecast. The correlation coefficient between the mentioned parameters is 0.237781 which rejects the strong dependence.

DOI (Digital Object Identifier) A method of applying a persistent name to documents, publications and other resources on the Internet rather than using a URL, which can change over time. : 10.3846/1611-1699.2009.10.349-360

Received 19 September 2008; accepted 28 August 2009

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A theory describing the long-run relationship between inflation and interest rates.

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Deimante Teresiene

Vilnius University History
Polish-Lithuanian Commonwealth
In 1568, the local nobility (szlachta) asked the Jesuits to open an academy either in Vilnius (Wilno) or Kaunas.
 Kaunas Faculty of Humanities,

Muitines g. 8, 44280 Kaunas, Lithuania

E-mail: deimante.teresiene@vukhf.lt
Table 1. Comparison of implied and statistical volatility

                     Implied                Statistical

Definition   The volatility forecast   The volatility
             over the life of an       estimate (or
             option that equates       forecast) that is
             an observed market        obtained using a
             price with the model      statistical model
             price of                  of the real world
             an option.                distribution of asset
                                       returns.

Data         Risk neutral option       Historical time
             prices and investor       series on underlying
             expectations.             asset prices.

Model        Option pricing model      Moving averages,
             (Black-Scholes)           GARCH.

Source: (Alexander 2005)

Table 2. Suitability of "GETIP" models for OMXV index

                                 GARCH      EGARCH      TGARCH

[alpha]                        0.156801    0.068175    0.120691
[beta]                         0.606602    0.987085    0.619615
[omega]                        1.59E-05    -0.169412   1.46E-05
[gamma]                                    -0.002287   0.083507
LL                             7938.516    7954.939    7942.262
AIC                            -6.922788   -6.936247   -6.92518
Durbin-Watson stat             1.804854    1.808738    1.808568
Schwarz criterion              -6.910273   -6.921228   -6.91016
Hannan-Quinn criterion         -6.918225   -6.930770   -6.91970
Suitability according to AIC       4           1           3

                                IGARCH       PARCH

[alpha]                        0.024371    0.035655
[beta]                         0.975629    0.964464
[omega]                                    9.79E-05
[gamma]                                    0.053604
LL                             7911.981    7954.289
AIC                            -6.901380   -6.935679
Durbin-Watson stat             1.811963    1.808517
Schwarz criterion              -6.893870   -6.920661
Hannan-Quinn criterion         -6.898641   -6.930203
Suitability according to AIC       5           2

Table 3. A relationship between forecast measures and time horizon

          5          10         20         40         50

RMSE   0.003929   0.004051   0.005551   0.006659   0.010110
MAE    0.003271   0.003539   0.004621   0.005287   0.007317
MAPE   54.69204   101.9385   113.0571   108.6843   105.4331
TIC    0.467739   0.559675   0.688790   0.780713   0.861805
BP     0.492965   0.007013   0.001774   0.028096   0.032032
VP     0.479679   0.947198   0.930804   0.886159   0.895631
CP     0.027356   0.045789   0.067422   0.085745   0.072336

         100        150        200        255

RMSE   0.008294   0.008380   0.008541   0.009324
MAE    0.005953   0.006196   0.006220   0.006703
MAPE   107.9143   103.9420   102.1564   101.8848
TIC    0.871881   0.890858   0.905589   0.922204
BP     0.016650   0.000021   0.000807   0.003409
VP     0.844829   0.837958   0.843370   0.869464
CP     0.138521   0.162021   0.155823   0.127127
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