Financial Time series including ARCH-Garch models - Research Paper Example

?Running head: Financial Time Series Including ARCH-GARCH Models Financial Time Series Including ARCH-GARCH Models Insert Insert Insert Instructor’s Name 22 September 2011 OUTLINE Introduction Objectives the Research ARCH Methodology GARCH Methodology Comparison between ARCH and GRACH Conclusion Financial Time Series Including ARCH-GARCH Models Introduction Financial time series forms the basis of financial and macroeconomics where model builders use stochastic processes to test and construct equations of economic variables. Time varying volatility and non-stationarity has largely contributed to the understanding and applicability of financial time series. Economic variables are referred as non-stationary when there is no tendency of being linear or constant and assume stochastic trends. Empirical research is often conducted in macroeconomics to estimate variable relationships and test hypothesis of the theories of macroeconomics. Empirical financial models are formulated based on cointergration concept that forms the basis of the major breakthrough in macro and financial economics. The aspect of volatility is important to financial economics such as the stock exchange and capital markets. Indeed, financial managers and analyst have repeatedly used the time series volatility model to forecast volatility and make relevant decision concerning future and current investments in the financial markets1 Objectives the Research Financial time series research aims at shifting finance and investments from the normal risk and anticipatory processes to an optimistic framework that uses mathematics and modeling as an empirical attempt to understand and plan collective elements of the capital markets. Empirical attempts seek to approach and understand financial markets based on facts and manage the risk that faces it. The processes enable financial analysts and managers to model investment probabilities and make estimations with regards to pricing and risk options. Huge corporations and banks have encountered financial crisis due to financial risks involved in the markets in which they operate. Financial time series plays a big role in modeling and forecasting the financial operations of such corporations for countermeasures and avoidance of future financial crisis. The dynamics of corporate returns and stock prices can be effectively managed through application of financial time series and forecasting. Fluctuations of returns and the speculative prices in the stock markets are presented and modeled by time series volatility, which helps in decision making in investment. Time series models such as the autoregressive conditional heteroscedasticity are used by financial analyst to determine the relationship between returns and risks levels in investments. Volatility of the sequence of the returns in money markets, foreign exchange markets, and stock markets are best described by the autoregressive conditional heteroscedasticity model in financial time series. Advance usage of the autoregressive conditional heteroscedasticity model is also used in the futures markets and by profession and executives in the stock markets to enforce counteractive measures to stabilize the markets. China stock market is a good example of a market that has successfully used the autoregressive conditional heteroscedasticity model to describe the market2 Arch Methodology This financial time series model is used to forecast random variables from information of past variable trends and linearity through specific universal assumptions on the conditional macroeconomic and financial economic variables. Forecast of variables depends on past information with regards to conditional and random variables assumption although conditional variance in conventional models does not depend on the random variables past information but on estimations and test. Econometric use of the Autoregressive Conditional Heteroskedasticity Model focuses on the limitations of forecasting in resultant prediction of the future that varies from one period to another. The uncertainties that involve forecasting can be covered by the Autoregressive Conditional Heteroskedasticity Model that takes into account the periodic variance in forecasts and the errors that are involved with forecasts of the past Portfolios of investments in the monetary theory are considered as functions variance and expected mean of return rates. Changes in the rates of returns and expected mean relates to variation in demand of financial assets. Variance is assumed to be constant from one period to other when the mean follows the Autoregressive Conditional Heteroskedasticity time series model3 The following ARCH formula can be used to model time series: Where denotes error term split into stochastic and standard deviation piece that equal 1 and being modeled by:- The Autoregressive Conditional Heteroskedasticity Model is applicable as an estimation of complex regression models that does not put into consideration the effect of structural change and variable. The Autoregressive Conditional Heteroskedasticity Model is a more relevant and reliable estimation of an ideal situation where the aspect of structural change and variables is not omitted through universal assumptions of the errors. The model’s empirical attempts in time series use the method of ad hoc to allow for changes in time variance such as moving averages and moving mean. The Autoregressive Conditional Heteroskedasticity Model has and continues to play a big role in modeling investors attitudes towards their expectation of returns and the risk involved in their investments. This model has widely contributed to financial econometrics through predictions of volatility changes and the use of non linear dependence unlike variable exogenous structural changes 4 The autoregressive conditional heteroscedasticity model is an applicable financial model though it has limitations to its application in macroeconomics and financial economy. Financial modeling time series has proved to have a nonlinear dynamic feature and the autoregressive conditional heteroscedasticity model offering financial framework that can be adapted to fit specific financial problems in the market. This perspective of the autoregressive conditional heteroscedasticity model in statistical theory and financial market models can be applicable with regards to interest of practitioners. With respect to statistical theory, the autoregressive conditional heteroscedasticity model can be considered to a time series model that is specific and nonlinear and therefore allows for a thorough analysis of the financial asset return dynamics. The financial market time series models can possibly be reexamined with regards to a number of issues such as the random walk hypothesis, variability and prediction ranges so as to confirm the validity and reliability of previously findings. The reexamination can be done in two main applicable ways, which include the testing of various financial and economic theories that are concerned with capital markets, foreign exchange markets and most certainly to examine the relationship between the long run and the short run. The other approach is concerned with the interventions of financial institutions such as the central bank on market issue trading periods, optimal portfolio choice, and portfolio size and hedging against risks. Garch Methodology The Generalized Autoregressive Conditional Heteroskedasticity model has been extended from the Autoregressive Conditional Heteroskedasticity model, which allows a lag structure that is more flexible. Stationarity of the Generalized Autoregressive Conditional Heteroskedasticity model is derived to check conditional variance and the model’s behavior through partial autocorrelation and autocorrelation of the processes. The mean estimates and variability independence is carried forward from the Autoregressive Conditional Heteroskedasticity model to the Generalized Autoregressive Conditional Heteroskedasticity model although this tests on the models may not be feasible. However, the Generalized Autoregressive Conditional Heteroskedasticity model is considered as an improvement since it clearly outlines the uncertainties involved with econometric variables. This model has been widely used by financial analysts and managers in the financial world since it compresses the infinite parameters that are estimated to a controllable few. Few estimate parameters is significant to the models application in financial markets since this enables the recognition and consideration of volatility that is unusual. The model has been assessed and confirmed to be adequate and fit for modeling the financial markets5 6 Corporations and individual investors aim at simulating price trajectory of their financial assets for a particular period in a manner that captures specific conclusions with regards to returns of assets with respect to their volatility, heavy collaborations, and correlation. A Generalized Autoregressive Conditional Heteroskedasticity model is one of the methods that can be used to capture these specific conclusions. The Generalized Autoregressive Conditional Heteroskedasticity model is a more advanced method that has been developed as an improvement of the autoregressive conditional heteroskedasticity model. Financial practitioners model the financial time series that is regarded as a log-return model through a formula that contains an independent variable and a sequence that is identically distributed with the Gaussian random numerals that are well scaled. The variable volatility is dynamic and continuously governed by an equation that is referred to the state memory factor, which is also referred to as the variance memory factor in some cases. The variable contains discrete numbers one and two consecutively, then there are no changes made to the variance, which then obtains a discrete constant numeral. Financial analyst and managers who model the financial markets using the Generalized Autoregressive Conditional Heteroskedasticity model can manipulate the equations initial variance. The following GARCH formula can be used to model financial time series: Where p follows GARCH terms while q follows ARCH terms. The Generalized Autoregressive Conditional Heteroskedasticity model has three options that are available to individual who is modeling the financial market. The first option is the series of volatility series, which is not constant and is regarded as the most valuable feature of the Generalized Autoregressive Conditional Heteroskedasticity model. This aspect, which creates the impression that the variance memory factor is tiny, has the zero effect tendencies of the volatility series thus coming up with returns of real assets as those of an unrealistic model. The other option is the log-return series feature, which is used to build the third option asset price. There are arguments against the option, which claims that there exists a poor selection of parameter thus causing the state memory factor to become excessively large. There are various examples of state memory, which bring about assets that are realistic. The Generalized Autoregressive Conditional Heteroskedasticity models are in most cases sensitive to choices of parameter selection which an issue of concern when real data are used to estimate Generalized Autoregressive Conditional Heteroskedasticity model parameters. Other than having the current return variance being dependent on previous returns variance and squared previous return, the model can be allowed to use current return variance depending on the squared returns variance from previous multiple days. In other words, when the modeling process is dependent on squared returns of the previous days then the process is referred to as the Generalized Autoregressive Conditional Heteroskedasticity process. On the other hand, when we want to make the process less complex we only simulate the related asset prices and log-returns of the Generalized Autoregressive Conditional Heteroskedasticity model process. Comparison between ARCH and GRACH The comparison between Generalized Autoregressive Conditional Heteroskedasticity model and the Autoregressive Conditional Heteroskedasticity model is based on estimated parameters and modeling of volatility. 7The comparison between Generalized Autoregressive Conditional Heteroskedasticity model and the Autoregressive Conditional Heteroskedasticity model have been based on bivariate stock return index. Both models also capture varying uncertainties on returns that directly impact on other parameter returns. Stochastic volatility models bring about a variety of modeling approaches in discrete periods at the face value of financial information unlike data in the process of diffusion. The Generalized Autoregressive Conditional Heteroskedasticity model and the Autoregressive Conditional Heteroskedasticity model converge on a process of diffusion with regards to continuous periods on the basis of the Black-Schole model. Models of financial markets estimate the parameters reversely with regards to a continuous time where financial analysts approximate the processes of diffusion and parameter estimation through Generalized Autoregressive Conditional Heteroskedasticity model. Variables distribution convergences only indicate Generalized Autoregressive Conditional Heteroskedasticity model diffusion processes. None asymptotically equivalent likelihood processes have however been proved that they are not essential concerning approximation and estimation. Discrete time series are divergent to estimation parameters with regards to the highest likelihood estimator in the diffusion process. In financial time series models, many parameters in large orders undergo restrictory estimation for a more or less constant valuation. Parameters that are restricted are generally regarded as stationery parameter condition. The maximum likelihood methods that are categorized as the methods that are efficient for estimation methods have the challenge of being numerically complicated due to the aspect of wide space dimension parameter. Both Generalized Autoregressive Conditional Heteroskedasticity model and Autoregressive Conditional Heteroskedasticity model have significant disadvantage since it is not possible to model volatility asymmetries with regards to the aspect of financial shocks indications of the past during financial crisis. The financial shocks results from the lagged shocks that are squared, but do not consequently have an impact on the degree of the shocks but not on the indication of financial shocks. The explanation of this is the fact that negative news recognized as indicators have the same impact on volatility as the situation of positive indicators where the absolute values are similar. Empirically, there are assumptions that negative indicators have a more serious impact on volatility than the impact caused by positive indicators. However, with regards to this conclusion, there are still more findings that bring standard models to various conclusions. Conclusion Financial time series models are used to solve the issue huge fluctuations in financial returns and their dependency on econometric variants. The financial time series models such as the Generalized Autoregressive Conditional Heteroskedasticity model and Autoregressive Conditional Heteroskedasticity model have however not been proved with regards to the estimate parameters degree of distribution. There are no standard methods or consistent estimators that have been proved to approximate heavy tailed errors and likelihood of estimate parameter distribution. However, alternative solutions that have been put in place include the approximations of sub sample bootstrap and percentile–t whose applicability can be proved both numerically and theoretically. Bibliography Bollerslev, Tim. Generalized autoregressive conditional heteroskedasticity. Denmark, University of California, 1985. Cheng-yu, LI & Fang-yuan, LU. Impact on the Results of ARCH Models with different Distribution of the residuals: Based on Comparative Analysis of Shanghai Stock Market Index. China, 2007. Jurgen, Franke & Wolfgang, Karl. Christian Matthias Hafner, Statistics of Financial Markets: An introduction. NY: Springer, 2010. Nasser M. & Hossain, Altaf. Comparison of GARCH and Neural Network Methods in Time Series Prediction. Department of statistics, Rajshahi University. 2008. Nobel, Alfred. Time series Econometrics: Cointergration and Autoregressive conditional Heteroskedasticity. Sweden, 2003. Perrelli, Roberto. Introduction to ARCH and GARCH models. University of Illinois, 2001. Robert, F. & Engel. Econometrica: The Econometric society, 1982. Read More