Factor analysis of the Lithuanian equity market indices

Statistical measures that can reproduce the state of the stock market and the tendencies of its change dynamics are the stock indexes. Having in mind the more complicated state of the finance system it is important to answer the question of what impacts the fluctuations of the stock prices. The article discusses various factors that impact the fluctuations of the Lithuanian stock index OMXV ; also stock index factor analysis is performed. Factors are determined using the main components method.


Introduction
Instability of the share prices is influenced by many factors: both clearly measurable quantitative (microeconomic, macroeconomic and other indicators) and clearly not defined qualitative aspects (social, political, psychological and other). Arbitrage theory describes that a limited risk component quantity exists [6]. Ross suggested that the components represent the fundamental macroeconomic and financial indicators. Which means that the existing indicators fairly accurately describe the economic situation, which had formed in the country at a certain point in time, and can be analysed as factors influencing the share prices. It has been shown that at least three factors influence the asset prices [5].
The article analyses the influence of sector indices on the Lithuanian share index OMXV return during years 2000-2008, and factor analysis of the share indices is performed. The factor analysis method is one of the most commonly known classical multi-measure methods [2,7,8]. The purpose of factor analysis -to replace the set of characteristics of the observed phenomenon by a combination of several factors while losing minimal information.

Research method
Variable description. The article analyses the impact of sector indices on the Lithuanian share index OMXV return during the years 2000-2008. With the help of factor analysis method factors are determined (the main components method is used) that describe the main Lithuanian share sectors, using the Granger causality method the causation relationship is determined.
Normally share price changes are researched since they reflect the changing economic environment. Let us assume P (t) share index at time. The return for the same period is the share index change logarithm r(t): .
The Lithuanian share index return is calculated according to the indicated formula rOMXV .
Preparation of data for the factor analysis to be performed. The preparation of data for the factor analysis had been performed in two stages [4].
First stage -ensuring stationarity. The process is stationary if its mean and dispersion are constant values, i.e., they do not depend on the time variable. The existing data is not stationary, it needs to be transformed. One of the methods applied is to differentiate the process, i.e., every value of the time series is substituted with the difference of the existing and preceding values. To achieve stationarity second-order differentiation is applied to the transformation of the primary data.
Second stage -standardisation of the time series. Since the existing data is not standardised it needs to be transformed. The data standardisation is performed by deducting the mean of the time series and dividing by the variance, thus the mean of all of the standardised data is zero, and the variance -one. The data must be standardised so that the variables with a large deviation would not dominate and would not alter the results.
Description of the mathematical model. Let us assume that we are observing s variables X 1 , X 2 , . . . , X s . The model is based on the assumption that behaviour of every variable X i is caused by m general latent factors F 1 , F 2 , . . ., F m and a specific latent factor e i . There are less latent factors than there are variables, i.e., m < s. Let the variable X i be linearly dependent on the factors. The mathematical model is: Multiples λ ij are called the factor weights.
Externally the factor analysis model resembles linear regression -knowing F j and λ ij values we could predict the values of X i . However, the purpose of factor analysis is the opposite -only X i values are known, and we want to estimate the general factors F j .
When applying factor analysis the similarity in observed variables is sought. When the variables are not correlated the factor analysis would be purposeless, therefore, it needs to be ensured that the variables observed are correlated. From existing data only the correlated should be selected, and the uncorrelated should be eliminated from the factor analysis data list.
The primary factor analysis data -observation correlations (or covariance) matrix. It can be seen from its form which variables are correlated.
Suitability of data for factor analysis can be evaluated using the Kaizer-Meyer-Olkin (KMO) adequacy coefficient. It is the empirical correlation coefficient and partial correlation coefficient magnitude comparative index where r ij is correlation coefficient of X i and X j , r ij -X i and X j partial correlation coefficient. If KMO value is low then the factor analysis of the observed variables is inconclusive. When KMO value is less than 0.6, then the correlation of variable pairs cannot be explained by other variables, and the factor analysis of the observed variables is not acceptable. Suitability measure of the observations of every variable can be calculated using the following formula The variable which has the smallest MSA value is removed from the primary variables list and then the measure of suitability of data for factor analysis is calculated KMO. The procedure is repeated until KMO value exceeds 0.6.
Granger causality test. When using the Granger causality test [1] we will research the observed variables' causation relationship. Granger causality test for time series is based on the assumption that: if x causes y then before y changes, the x must change, but not vice versa.
In other words, two conditions must be met: (a) x must input a statistically significant contribution to the prediction of y; (b) y must not input a statistically significant contribution to the prediction of x. When verifying the Granger causality the following regression equations are formed: where ε t and u t are uncorrelated random errors. The null hypothesis that the coefficients are statistically significant is tested for every equation Let us highlight that the fact that x causes y shows only that earlier values of x explain further values of y, i.e., shows the possibility of causation. If the null hypothesis that x does not cause y is not rejected, it means x does not cause y [3].
We select 5% confidence intervals for testing the null hypothesis.

Review of results
All the variables are used for composing the initial factor model. sector index general variance. The general position of the Lithuanian sector index market can be represented using three components. The first component describes the energy, production, consumption of goods and services, healthcare as well as finance sector indices. The second component describes the consumption of essential goods and services, telecommunication services and utility service sector indices. The third component describes the commodity and information technology sector indices.
The second stage analyses the causation relationship of the primary variables and the factor analysis method factors (F 1 , F 2 , F 3 ). After applying the Granger causality test it has been determined that the returns of the Lithuanian share indices rOMXV are caused by: the consumption of goods and services, healthcare, finance sector as well as the first component expressed using the main component method (see Table 4).

Conclusion
For years the share return and the country fundamental indicator relationship has been researched in various academic articles and works. Different publications describe the share return relationship with the economic, financial and political indicators.
To research the impact of sectors to Lithuanian share market 10 main sector indices have been analysed.
It has been determined that the observed Lithuanian share market indices can be expressed using three factors.
With the help of Granger causality test the causation relationship has been discovered between the Lithuanian share index return and the sector share index indicators. It has been determined that Lithuanian share index return cause can be put down to three sector indices from the 10 used in this paper as well as one of the found factors.