Estimation Technique Of Copula Based Garch Models And Its Application To Bangladesh Stock Market

The theory of Copula is a very powerful tool for modeling joint distributions because it does not require the assumption of joint normality and allow us to decompose any n-dimensional joint distribution into n marginal univariate distributions and a copula function. Consequently, we can construct a wide range of multivariate distributions by choosing the marginal distributions and a suitable copula. So, with a set of copula function and a set of univariate distribution we can construct better multivariate models than other existing multivariate distributions. The parameter estimation from copula based model becomes a major problem due to the curse of dimensionality. For this reason the Exact Maximum Likelihood method sometimes become harder for estimation. Thus, a two step optimization method called the Inference Function for Margins is broadly adopted to estimate Copula based models. However, this IFM method shows the problem of loss of efficiency in estimation if the estimating functions of the approximate model deviate from the exact score function.In this dissertation we propose to estimate the copula based models by applying Maximization by Parts, a multi-step optimization method. This method decomposes and optimizes a very complicated likelihood function. The first part includes the likelihood function containing only the marginal distribution function, while the remaining part includes the dependence parameters from a multivariate likelihood and is used to update the estimates from the first part.According to our study the background, aim and importance as well as organization and innovation of the study are given in Chapter 1. It indicates that creating and estimating copula based GARCH models by the efficient and easily computational method are too much interesting for financial time series analysis and also estimating Value at Risk for any stock market is always necessary.Chapter 2 contains all the previous study which is related to our study. According to the aim of our dissertation we have several options such as creating new copula based GARCH types model, their efficient estimating technique and application of these models to estimate the Value at Risk for stock markets. So in this chapter we discussed some previous study which is related to different copula based GARCH models, different estimation technique, estimation VaR and previous study about two Bangladeshi stock markets.Chapter 3 contains all the definition about copula, their properties, families and some statistical tools which are useful for our study. It also presents the introduction about two Bangladeshi stock markets and some VaR estimation techniques.Chapter 4 presents the details estimation procedure by the MBP method for all of our models which are used here. In our study we use two elliptical copula such as Bivariate Gaussian and bivariate T copula. Here we use four GARCH type models such as GARCH, EGARCH, TGARCH and APARCH models with Gaussian and t-marginal density.Chapter 5 contains the comparison study based on simulation as well as empirical study. In our simulation study we examine the speed of convergence of MBP algorithm to EML methods for all of our models with different parameter combinations and different sample sizes. For empirical study we estimate all of parameters by these three EML, IFM and MBP methods. In the case of convergence of MBP to EML method we consider the IFM estimation value as the initial value of MBP methods. The empirical study also compares the mean and variance of the fitted conditional covariance matrix among these three methods considering the EML method as a benchmark.Value at risk is probably the most popular risk measure, having a central role in the risk management. So estimation VaR is always important for any stock market. Here we use all of our models to compare the performance to estimate the VaR by using the efficient MBP methods. Then the best performing models is to use to compare the performance of other different traditional approach to estimate VaR for two Bangladeshi stock markets (DSE and CSE). All of these procedures are includes in Chapter 6.Finally Chapter 7 presents the Summary, Conclusions and Recommendations about the study which indicates that the MBP method is more efficient than IFM and easier than EML method. To compare the performance of copula based GARCH models to estimate VaR for two Bangladeshi stock markets we found that T copula GARCH with t-marginal density give most successful result.