Date of Award


Degree Type


Degree Name

Doctor of Philosophy


Statistics and Actuarial Sciences


Hao Yu


This thesis is motivated to investigate distribution theory of a quasi maximum likelihood estimator (QMLE) and test of goodness-of-fitting of an ARMA-(I)GARCH model. We obtain asymptotic consistency and normality of the QMLEs based on an arbitrary likelihood kernel. It shows that the moment conditions of errors in the ARMA part and innovations in the GARCH part depend on the choice of likelihood kernel. For example, the asymptotic normality of QMLEs based on student t likelihood kernel holds with arbitrary small positive moment on error term and 2 — L moment on innovation term, where 0 ≤ L < 1. It also shows that the asymptotic efficiency of QMLEs depends on the choice of likelihood kernel and the distribution of innovation. For the pure GARCH model with nonzero constant mean, we show that the common practice of using the sample mean to center financial data is workable if the error term has finite variance. Consequently, we study some processes based on residuals of an ARMA-(I)GARCH model. We show that the k-th power partial sum process converges to a Brownian process plus two correction terms, where the correction terms always depend on ARMA-GARCH parameters. We also show that the 111 CUSUM and the self-normalized processes (standardized by the residual sample mean and variance) behave as if the residuals were asymptotically IID. Finally, applications of these results are exhibited with numerical examples. Chapter 1 gives a brief introduction of financial return, econometric models such as ARMA, GARCH and their extensions, as well as model estimation and diagnosis. Chapter 2 focuses on the distribution theory of one step QMLEs and two step QMLEs of an ARMA-(I)GARCH model. Special cases like pure ARMA and pure GARCH are considered too. Three specific examples with varied kernels are presented. Chapter 3 deals with the high moment partial sum processes, the CUSUM and the self-normalized processes based on residuals of an ARMA-(I)GARCH model, originally proposed by Kulperger and Yu (2005) for a pure GARCH model. In Chapter 4, we present some numerical examples of the applications of Chapter 2 & 3, for instance, efficiency of QMLEs based on different kernels, CUSUM statistic for testing ARMA-GARCH model structural changes, Jarque-Bera omnibus statistic for testing normality of the unobservable innovation of an ARMA-GARCH model. Finally some conclusions and discussions are put forward. Keywords: ARMA-GARCH, ARMA-IGARCH, quasi-maximum likelihood estimation, two-step estimation, asymptotic consistency, asymptotic normality, asymptotic efficiency, residuals, high moment partial sum process, weak convergence, CUSUM, omnibus, skewness, kurtosis, vn consistency.



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