Applications of Quantile Regression Under Asymmetric Laplace Distribution and Copula Based Returns and Risk Measures to Financial Econometrics

Abstract

The research work presented in this dissertation provides some novel contributions to financial econometrics. Essentially, we investigated two main important issues in financial econometrics, namely portfolio optimization in high dimension of stock returns, and evaluation of portfolio returns in several models (Fama-French model, and Capital Assets Pricing Model/ CAPM). While these issues have been previously investigated, we improved upon previous results by using more modern methodologies, namely quantile regression and copulas. Specifically, we used quantile regression models with error terms having asymmetric Laplace distributions and model more realistic dependence structures by copulas. Our main results are summarized as follows. In determining the factors influencing the prediction of portfolio returns and risk in CAMP and Fama-French three factor model, we found that quantile regression can explain the entire conditional distribution of the outcome variable, and is more robust to outliers and heteroskedasticity without the need to specify the error distribution (i.e., we carry out a semi-parametric statistical analysis). When data reveal information for specifying error distribution, a fully parametric statistical analysis is desirable as it should be more efficient. In contrast to mean linear regression where (symmetric) normal distribution for error leads to the use of Maximum Likelihood Estimators (MLE) with asymptotic properties such as asymptotic efficiency, asymptotic normality and consistency, the error distribution in a quantile regression model is an asymmetric Laplace distribution (ALD). This is consistent with conditional quantile estimators obtained as minimizers of corresponding loss functions for quantiles in the general theory of quantile regression. Our empirical results show that quantile regression under ALD can capture the stylized facts in financial data to describe the stock returns under various quantile levels. Furthermore, since there is no monotonic relationship between risk and stock returns, our analysis reveals the ability to increase the returns of portfolios beyond risk exposure, which can be achieved by using quantile regression for risk measurement. The incorporation of copulas into our analysis improved previous similar works. Specifically, we used t-copulas to model dependence structures of portfolio risks. Multivariate t-copula based on GARCH models were used to explain portfolio risk structure in high dimensional asset allocation. Within this method, we applied Monte Carlo Simulations to estimate the expected shortfall of the portfolios. Finally, we obtained the optimal weighted conditional Value-At-Risk. The optimization technique was used to allocate risk in the portfolios.