Contributions to Conditional Heteroscedastic Models: M-Estimation and Other Methods.

This research makes contributions to conditional heteroscedastic models in financial time series. A class of M-estimators for time series models with asymmetric form of heteroscedasticity are developed. A weighted resampling method is used to approximate the sampling distribution of M-estimators. The primary finding is that there are estimators in this class that can perform better than the widely-used quasi-maximum likelihood estimator (QMLE) and even outperform the least absolute deviation estimator. The asymptotic distributions of the squared and absolute residual autocorrelations for generalised autoregressive conditional heteroscedastic (GARCH) models estimated by M-estimators are derived. Diagnostic tests based on M-estimators are developed to check the adequacy of GARCH-type models. The performance of M-estimators in the estimation and prediction of value-at-risk (VaR) is investigated. A wide range of summary statistics is used to evaluate and compare M-estimators in estimating the in-sample and predicting the out-of-sample VaR of three well-known stock indices. Some of the M-estimators are observed to show better performances in predicting the one-day-ahead VaR than the commonly-used QMLE. The Linear Estimator (LE) for ARCH models is explored and results show that this estimator provides good estimates for the parameters of the ARCH model and also predicts the volatility better than the QMLE. Using a class of weighted resampling schemes, it is found that there are schemes that can match and even perform better than the commonly-used paired bootstrap scheme. Bootstrap prediction intervals for returns, volatilities and value-at-risk in ARCH models are also developed. A weighted linear estimator (WLE) for the multivariate ARCH parameters is proposed. This estimator involves solving sets of linear equations and hence is very easy to compute. A weighted resampling method for multivariate ARCH models is also discussed. The accuracy of this estimator is compared with the QMLE in estimating the parameters of multivariate ARCH models. The WLE is also applied to real data sets and forecasts of volatilities and value-at-risk are obtained. Our study indicates that the forecasting performance of the WLE is not inferior to the QMLE and one-day-ahead risk estimates are found better. M-estimators for multivariate GARCH models are discussed. Two different methods for the estimation of multivariate GARCH models using univariate GARCH specifications are proposed. These methods are easy to apply as these require several univariate GARCH estimations to estimate the full multivariate GARCH model. Results of Monte Carlo simulations and application to real data sets show that our methods provide better results in terms of estimating and predicting the conditional correlations and value-at-risk