Financial Econometrics - Testing for Unit Roots and Cointegration - Essay Example

FINANCIAL ECONOMETRIC a) Descriptive statistics is used to analyze and explain the basic characteristics of a data in a research or study. It allows an insight into the summary of the sample and measures of the data and in combination with graphics the descriptive statistics takes the form of visual quantitative analysis of data. However, the descriptive statistics is different from the inferential statistics. Normally speaking, the descriptive statistics gives a keen glimpse on the data alone. According to Larson(2006,pg76-81) ) “Descriptive statistics for continuous variables fall into 3 general classes, namely: location statistics (eg, mean, median, mode, quantiles), dispersion statistics (eg, variance, standard deviation, range, interquartile range), and shape statistics (eg, skewness, kurtosis)”.The descriptive statistics gives the overall description of the data by providing the measures of central tendency, and the measures of dispersion. When it comes to descriptive statistics, log prices and log returns has important roles to play. Log price and log returns forms an important aspect when it comes to descriptive statistics. According to Petty( 2015) “A logarithmic price scale is plotted so that the prices in the scale are not positioned equidistantly; instead, the scale is plotted in such a way that two equal percent changes are plotted as the same vertical distance on the scale. The log returns basically is auto – correlated while the case is different with log prices. The log return are usually preferred in quantitative analysis as it gives a better insights in to aspects like normalization and classical statistics. Here the data stream for the 20-year period of January 1995 to December 2014 is used to calculate the log price and log return. log prices log return Mean 334.934 5.740008 Standard Error 8.558659 0.024659 Median 298.85 5.699916 Mode 237 5.46806 Standard Deviation 132.5902 0.382016 Sample Variance 17580.16 0.145936 Kurtosis -0.30671 -1.02302 Skewness 0.785038 0.242708 Also a graph representation for the log price and the log returns is formulated in order to deliver a quantitative analysis of the specific data of the company. Here, histogram is used to represent the graph of the statistical data in a visual form Fig 1: Log Price and Log return graph The above analysis explains the descriptive statistics of the log prices and the log return of the data. From the log return, we can observe that the mean value of log return is 5.740008, with a standard deviation of 8.558659. The mean of log prices is 334.934 with the standard deviation of 0.024659. The skewnes of the variable log return is equal to 0.2427 which is a negative value implying that the value of the log return is negatively skewed. The value of kurtosis of the log return is equal to -1.02302 which is a negative value. This implies that the data of log return has a low distributed or low peaked about the mean. The skewnes of the variable log price is equal to -0.785038 which is a negative value implying that the value of the log return is negatively skewed. The value of kurtosis of the log return is equal to -1.02302 which is a negative value. This implies that the data of log return has a low distributed or low peaked about the mean. b) Here, it is possible with the help of a graph to understand that the variables log return and log prices, are not stationary. These two variables, the log return and the log price are transformed to make them be stationary in the time series aspect. If one look at the figure below , we can analyze that the log price variable seems to be a typical time series of the prices. Fig 2 : Log Price time series Here, the test used to determine whether each log return series is stationary is the Dickey Fuller Test. Even though there are many test available to determine this fact ,the most common test method used is Dickey Fuller Test method. This test basically focuses on determining the value ρ, whether it is equal to one, or it is less than one. According to Bo Jo(2008 ) “The most common test for testing I(1) versusI(0) is the Dickey-Fuller test. This test has as the null that the series isI(1), which in general might be hard to reject”. To evaluate the stationary aspect, a time series plot is done on the series to indicate if they are stationary or not. It also help in identifying which of the method can be used to transform the model in to a stationary model. Fig 3 : Time Series Plot c) GARCH model for the estimation of each return series i) Model Identification: The univariate model chosen for the estimation of each return series is GARCH Model. The GARCH model is designed to overcome problems in relation to computing .The documentation with regard to GARCH models can be constructed for many return series. Normally, the GARCH model is highly efficient when it comes to the description of the volatility clustering in financial time series. According to Princ(2012,pg.1-5 ) “Generalized ARCH model GARCH) proposed by Bollerslev is based on the infinite ARCH specification term that reduces the number of estimated parameters from infinity to two”. ii) Model Estimation; The estimation of the GARCH model has gained much attention due to its maximum likelihood estimation. This means that when there is normal error assumption there is a maximum likelihood estimation functioning. This model has gained popularity due to the fact that there is a presence of consistency and asymptotical normal distribution along the calculation process. iii) Diagnosis Checks ; Box and Jenkins suggest some diagnostic checks to help determine if an estimated model is statistically adequate. At this stage the residual of the fitted models are examined to see if the selected model is adequate. A model that fails these diagnostic tests are rejected (Conway, 2004). We repeat the cycle of identification, estimation, and diagnostic checking until we find a good final model. Here the volatility of the stock can be assumed with GARCH model and the following equation is used for the same: 1. yk = σϵk, −∞ Read More