Techniques for approximating probability distributions like the Edgeworth expansion have a long history in time series models. The purpose of this thesis is to give a detailed study of the asymptotic properties of the Moving Average (MA) and the Exponential GARCH (EGARCH) models. Extending the results in Sargan (1976) [80] and Tanaka (1984) [87], we derive the asymptotic expansions of the distribution, the bias and the mean squared error of the MM and QML estimators of the first order autocorrelation and the MA parameter for the MA(1) model. It turns out that the asymptotic properties of the estimators depend on whether the mean of the process is known or estimated. A comparison of the moment expansions, either in terms of bias or MSE, reveals that there is not uniform superiority of neither of the estimators, when the mean of the process is estimated. This is also confirmed by simulations. In the zero-mean case, and on theoretical grounds, the QMLEs are superior to the MM ones in both bias and MSE terms. The results are important for bias correction and increasing the efficiency of the estimators. Next, we derive the bias approximations of the ML and QML estimators of the EGARCH(1,1) parameters and we check our theoretical results through simulations. With the approximate bias expressions up to O(1/T), we are then able to correct the bias of all estimators. To this end, a Monte Carlo exercise is conducted and the results are presented and discussed. We conclude that, for given sets of parameters values, the bias correction works satisfactory for all parameters. The results for the bias expressions can be used to formulate the approximate Edgeworth distribution of the estimators. Moreover, the asymptotic properties of EGARCH models are still largely unexplored and are considered difficult tasks (see e.g. Straumann and Mikosch, 2006) [83]. There is still no complete answer to the following questions: under which conditions do EGARCH processes have bounded first and second order variance derivatives? And under which conditions is the expectation of the supremum norm of the second order log-likelihood derivative finite, in a neighborhood around the true parameter value? These questions are important because the existence of such moment bounds permits the establishment of large sample statistical properties, such as the asymptotic normality of the QMLEs.
Asymptotic expansions of the mean values of DirichletL-functions