| The ARCH(autoregressive conditional heteroscedasticity)-type models, proposed by Engle (1982), are parametric time series models and have been playing a primary role for modeling volatility of discrete time series. The model is particularly popular for financial time series, since the volatility of the series is considered as risk of investment and is a primary concern for investors. For this reason many versions of the model have been proposed especially in late 80s and 90s by econometricians. Due to its complicated probabilistic structure, asymptotic properties of estimators are still under investigation for many models.;This thesis investigates the asymptotic properties for a couple of important ARCH-type models. The first chapter gives an overview of the ARCH-type model from both modeling and theoretical standpoints, then relates our contribution to the literature. The second chapter develops consistency and asymptotic normality for the semi-strong GARCH model. While the conventional GARCH model assumes that standardized errors are IID, this model only assumes the errors are ergodic and stationary. Since there is some empirical evidence in foreign exchange market which shows dependence among standardized errors, the robustness of the estimator when the IID assumption does not hold is crucial for applying the model to some real data. The third chapter establishes consistency and asymptotic normality for the fractionally integrated power ARCH model. It is known that the fractional GARCH models are more important for high frequency financial market data, which get more popular in recent years. However, in many important cases, the existence of stationary solutions for fractionally integrated model has not been proven yet and there is not a complete asymptotic theory. Hence, our result is significant contribution to this literature. The paper is concluded with some concrete examples of future research topics in the field. |
