Applied Quantitative Methods in Finance - Essay Example

APPLIED QUANTITATIVE METHODS IN FINANCE By of the of the School Essay on ARCH and GARCH models This essay is divided into two parts. Part one regards to the use of ARCH and GARCH models and their applications in finance studies. Part two on the other hand concerns the estimation of models of determinants of banks’ profitability by using pooled OLS (POLS) and panel data analysis. ARCH and GARCH Models These models are important in the estimation of time-varying volatility. Time varying volatility refers to fluctuations or variations in volatility over time. Financial instruments and other stocks exhibit periods of low and high volatility at different points in time. Thus in this part, we are not concerned with stationary time series but rather we are concerned with conditional variances that change over time. Financial time series have features that are represented well by models with dynamic variances. In this part, we focus at modeling the financial time series of the Italian Stock market index as well as the Spanish stock market index. We thus intend to examine the volatility in the Italian Spanish markets by understanding the change in commodity prices over a period of time. Finally we aim to present the various processes through which financial decisions are taken by aid of volatility modeling. ARCH model The ARCH stands for Auto-regressive conditional heteroskedasticity. Consider a model with an AR(1) error term; where If we use successive substitution we get; where is approximately zero (negligible). i) Unconditional mean of the error It describes a situation when we have no information. We have; Since ii) Conditional mean of the error It describes a situation when we have information up to some point in time. We have; The information set at time and includes knowledge of iii) Unconditional variance of the error iv) Conditional variance of the error Conditioning improves precision. It is worth noting that the conditional mean of the error varies with time while the conditional variance does not. The model can also be written for ARCH (q) as; where We define Testing for the ARCH effects We estimate the mean equation, which probably could be the regression of the variable on a constant or may sometimes include other variables. The null hypothesis is; In the presence of ARCH effects the magnitude of would depend on its lagged values and would relatively be high. The test statistic is; where is the number of terms, is the coefficient of determination and is the number of complete observations. If , we reject the null hypothesis and conclude that ARCH effects are present. Estimating ARCH models i) Feasible GLS Uses regression to provide possible estimates of the disturbance variances at each sample point and the original relation is then re-estimated by the weighted least squares procedure that corrects for the heteroskedasticity. ii) MLE This is the asymptotically efficient estimator that gives rise to non-linear equations that require iterations. ARCH Model Extensions (ARCH(q) Model) ARCH(q) model has a conditional variance function that is given by In this case the variance or volatility in a given period depends on the magnitudes of the squared errors in the past q periods. Testing and estimation are natural extensions of ARCH(1). The GARCH Model – Generalized ARCH It is a generalization of the ARCH model and its equation can written as; Where, and The GARCH-IN-MEAN Model The equations for a GARCH-in-mean model are shown below: This particular model postulates that the conditional variance affects by a factor Data The data is a time series data that includes 4255 daily market index prices for the Italian stock market spanning over the period 12/31/1997-22/01/2014 we also have the Spanish stock market data that includes 7122 daily market index prices spanning from 1/5/1987 to 4/22/2014 Analysis of data and results Time series trend Figures 1A and 1B below shows the plot for the daily price indexes for Italian and Spanish markets respectively. The series are characterized by random, rapid changes and are said to be volatile. The volatility seems to change over time as well. There has been upward and downward effects in volatility for the Italian market while the time series plot for Spanish market shows somehow an upward increase with slight variations (decreases) in between the years. The variations in volatility could be as a result among other factors, political factors or the international market changes. The histograms of the empirical distribution of the series are given below. For both the two market indexes, we observe that the series are leptokurtic. That is, they have lots of observations around the average and a relatively large number of observations that are far from average; for the Italian market index, the centre of the histogram has a high peak and the tails are relatively heavy compared to the normal. Descriptive Statistics Table 1A and table 1B shown below reports a summative descriptive statistics for daily daily market prices both for the Italian index and Spanish index. The mean market prices are positive for both the Italian and Spanish market indexes (28825.87 and 7158.402 respectively). In both cases the returns are also approximately symmetric with slightly right skewed (0.159 and 0.183 for the Italian and Spanish markets respectively) and the kurtosis for both the market indexes are all higher than the standard Gaussian distribution. Table 1B Testing for stationary We investigated the stationarity of the market prices for both the Italian index and the Spanish index. We used Augmented Dickey-Fuller (ADF) which involves testing the null hypothesis whether it includes a drift term. Table 2A In table 2A above, the results shows that there is no constant and nor lagged terms. The p-value Read More