Research Methodology And The Garch Type Model Finance Essay - Free Essay Example

This chapter introduces the research design which has been used for the completion of this research work. Research methodology is the most important aspect of a research as it regulates the flow of emulation. A research design should be selected carefully because it helps in many important steps of a research mainly that of collecting data and its analysis. Research methodologies are advantageous and at the same time have some limitations, which must be carefully focused and understood by the researchers. In this chapter, the methods used for data collection and the data type will be looked upon, as well as the calculations that might have been applied to this type of data. This chapter will also examine the GARCH type models, and the reasons for choosing this model over other available ones. 3.4 GARCH Model: For the purpose of this research, GARCH model is being used for asymmetric distinction between positive and negative shocks. GARCH is a volatility based model introduced by Bollerslev in the year of 1989. This model is being widely used for reversion and conational volatility. GARCH model helps in appointing a symmetric response of volatility to positive and negative shocks. A number of extensions have been made in the GARCH model from the time it had been developed. GARCH-type models are the most appropriate for series which do not have constant error terms and for classic regression model which are not likely to be satisfied when working with financial type of data. GARCH models are able to account for auto correlated volatility clustering in the financial data. The limitation of GARCH model is that it does not allow any feedback for conditional variance and conditional mean (Brooks 2008). The approach to the present study is quantitative in nature; that involves collection of numerical data from reliable sources and then applying GARCH Model for assessing stock sensitivity to changes in the foreign exchange rates and interest rates. The GARCH model was preferred over the ARCH models because according to Brooks (2002), the former is more of a complete model and thereby avoids over fitting, as the current conditional variance can be influenced by an infinite number of past squared errors. In addition, the GARCH-type models are less likely to breach non-negativity constrains. Though GARCH-type models are more appropriate for the type of data and estimations in this research, when compared to traditional linear regression or ARCH models, they do have a few restrictions as well. A concept of symmetric response to shocks, contradicts a theory of leverage effects which suggests that a negative shock is more likely to have a greater impact on volatility as opposed to the positive shock of the similar size. 3.6.1 Descriptive Statistics The descriptive statistics is very important as it gives a number of statistical information about the data. Descriptive statistics are simply used to describe the sample that one is concerned with. They are used in the first instance to get a feel of the data; in the second, for the use in statistical tests; and in the third, to indicate the error associated with results and graphical output. 3.6.1.1 Testing the Significance of the Mean The mean is the average value of the series which is obtained by adding up the series and dividing by the number of observations. The standard deviation (Std. Dev.) is a measure of the dispersion of the series. The Standard deviation is calculated as †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. (1) Where n is the number of observations, is the i-th observations and y is the mean of the series. Regarding the mean test, it is defined as the mean of the series, y which is equal to zero with the null hypothesis. 3.6.1.2 Skewness Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right from the center point. The skewness value can be positive or negative, or even undefined. It is the extent to which the distribution is asymmetric. The equation of a series is calculated by using E-views as: S= 3 †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. (2) Where is an estimator for the standard deviation that is based on the biased estimator for the variance, and can be expressed as = for the normal distribution or any other symmetric distribution, the skewness is either negative or positive. Positive skewness means that the distribution has skewed to the right. A positive skewed distribution differs from the log- normal distribution; that the bank price changes will increase by a larger amount than the log-normal distribution. Negative skewness means that the distribution is skewed to the left. 3.6.1.3 Kurtosis Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution, that is; the data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case. Measuring kurtosis of the series enables to capture another possible factor away from normality. Kurtosis can be calculated as: K=4 †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. (3) Where , is the standard deviation, i based on the biased estimator for the variance. A normal distribution has a kurtosis of 3. Compared with the normal, if the kurtosis is larger than 3, the distribution is peaked, with larger probability of values close to or far away from the mean. If the number is smaller than 3, the distribution is flatter than the normal distribution, with larger probability of values intermediately distant from the mean. 3.6.1.4 Jarque-Bera Statistic The Jarque-bera  is a  goodness of fit  measure of departure from  normality, based on the sample  kurtosis  and  skewness. Jarque-Bera is a statistic test for measuring the returns that are normally distributed within the null hypothesis. The test measures the difference of the skewness and kurtosis of the series and compares it with those from the normal distribution. Jarque-Bera is defined as {} †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. (4) Where k is the number of estimated coefficients used to create the series and N is the number of observations. Under the null hypothesis of a normal distribution, the Jarque-Bera statistic has an asymptotic chi-squared or X2 distribution and two degree of freedom. 3.7 Data Collection The data collected for this study is taken from all the leading banks in the US and is adjusted to remove discrepancies related to the no-existence of data and holidays. The reason for taking daily data instead of monthly data, is that it would allow much better reflection of the movements in the FX rate and the interest rate and thus the results are expected to be more accurate and complete. The data is obtained for 60 banks operating in US and sampling is done to ensure that all three categories of banks including Money Centered Banks, Large Banks and Regional Banks have fair representation in the sample data. The data collected is for a period of 10 years starting from January 1st 2000 to 31st December 2010. The stock prices are adjusted for dividends so as to avoid a systematic bias in relation to ex-dividend days. The  data collected comprises of US index- trade weighted exchange rate, US 3-month Treasury bill  rates, US total market  price index  and share  pr ices  for US banks.  For the US market, the FX rate is dollar trade weighted exchange rate, provided by the Bank of England. The interest rate is the rate used in daily returns of US three-month treasury bills. US total stock market index calculated by Datastream are employed as US market return indexes. For the estimation purposes several modifications are applied on the data in order to stabilize the variance and to induce stationary. The data was transformed to natural logarithms. The value of 3-month logarithms were calculated as In(1+it/100). It was further required to compute first differences due to the series in natural logarithms being non-stationary. Consequently, the changes of interest rate is computed as ?= In(/100+1) In(/100+1). For any other time series, the transformation = In(Yt)- In(Yt-1) is used. The stationary of the series is important when applying statistical regression models as non-stationary series may lead to biased estimates. 3.8 Model Specification To measure the interest rate sensitivity and the foreign exchange rate sensitivity, we use the following equation ?Pi,t = a0,i + ?0,i?Xt + ?0,i?It + ?0,i?M + ?i,t.. †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦. (5) Here, ? denotes the difference operator (log changes).Pi,t which is the ith stock price at time t. This variable is assumed to capture all financial and economical factors that are specific to the bank iMt , which is the price of the all-share market index which is considered to reflect economy-wide factors; It denotes a risk-free interest rate or bond index; and Xt denotes an foreign exchange rate or foreign exchange index. In Equation (5), a0,t is the intercept term while ?i,t is a white noise error term that is assumed to follow the iid condition. Equation (5) is an extension of Sharpes single index model which assumes the relationship between any two securities, that is completely explained by their relationship to the market index. So the covari ance among securities returns should be zero once the market influence has been removed. Equation (5) is also consistent with Mertons (1973) inter-temporal asset pricing model (Joseph and Vezos 2006).