The Estimation Theory And Application Of Two Kinds Of Financial Time Series Model

Generalized autoregressive conditional heteroskedasticity (GARCH) models, o-riginally introduced by Bollerslev (1986), have been successfully used in modeling fi-nancial volatility. Thanks to the automatic trading system, the intraday high-frequency data can be recorded conveniently nowadays. Analysts and traders always wish to make efficient improvement using such data. Visser (2011) proposed a technique to embed intraday log-return processes into GARCH models and generate volatility proxies. The maximum likelihood estimation (MLE) is widely used to estimate the GARCH parame-ters based on the Gaussian assumption (see Straumann and Mikosch,2006). The likeli-hood function is obtained by assuming the conditional distribution of the log-returns is normal. Visser shows that based on suitable proxies, MLE can outperform the method based on daily close-to-close data. However, embedding intraday log-return processes into the GARCH models also brings several problems. First, MLE performs unstable when the proxy distribution departures from Gaussian. Second, high-frequency data brings microstructure noises. MLE suffers much from the outliers and microstructure noises due to the quadratic form of the objective function. Besides, the asymptotic nor-mality of MLE necessarily needs the finite fourth moment of the innovations, which is not always satisfied in the financial data.Quantile regression (QR) is considered as a robust alternative to MLE and gradu-ally develops into a comprehensive procedure for regression. To overcome the afore-mentioned problems, we first propose the quantile regression (QR) method for suitable proxies to estimate the parameters in the GARCH models with high-frequency data. Note that the choice of quantile impacts the performance of quantile regression. When we know little information about the conditional distribution of proxy, it is a problem how to choose a suitable quantile. Considering that composite quantile regression (C-QR) can fully use the information at different quantiles, we propose CQR for intraday log-return processes, which is both more accurate and robust to extreme shocks. For both QR and CQR on proxies, the consistency and asymptotic normality of the pro-posed estimators is investigated. We also provide the asymptotic relative efficiency (ARE) between the regression based on different proxies.Functional Data Analysis (FDA), which regards the observations as functions de-fined over some set, has received a amount of attention in past years. As an extension from ordinary linear models, the functional linear regression model, which is focused on analyzing relationship between a functional explanatory variable and a scalar response variable, has been extensively studied. Further, some new regression models have been proposed for two or more functional covariates, such as non-parametric additive mod-el (Ferraty and Vieu (2009)), multiple functional regression model (Lian (2013)). The above-mentioned models usually assume the random error sequence is independent. Sometimes, this condition is unreasonable, especially when the observations are col-lected sequentially over time. To tackle this problem, autoregressive errors term is as-sumed in regression model. Unlike the linear regression model, functional parameters in functional linear regression model are infinite dimensional. The general estimation pro-cedure is to expand functional parameters by some basis functions, like spline functions or eigenfunctions of covariance operators of functional variables. Then the infinite di-mensional estimation problem turns into finite dimensional estimation problem and the estimates of functional parameters can be obtained by minimizing an objective function. In this thesis, an estimation method based on eigenfunctions of covariance operators of functional variables is proposed. Under certain regular conditions, we study the conver-gence rates of estimators of functional regression parameter and scalar autoregressive parameter, and also establish the asymptotic normality of an estimator of error variance. Moreover, because there is no manifest solution of our objective function, we provide a iterative estimation algorithm to deal with the computation problem.