Statistical Inference Of Two Kinds Of Time Series Models

Abundant evidences indicate that volatility of returns on financial assets often takes on the feature of sharp peak and heavy tail, and conditional heteroscedasticity. The clas-sical assumption of traditional Econometrics violates these properties. Autoregressive conditional heteroskedasticity (ARCH) models, introduced by Engel (1982), reflect the change of variance over the past and can fit these features of financial time series well. However, ARCH models are only suitable for short term autocorrelated data, because higher autoregressive order may arise the curse of dimensionality. Bollerslev (1986) put foward the generalized autoregressive conditional heteroskedasticity (GARCH) model to improve the ARCH model. He used few lags to replace the multiple hysteresis val-ues, which avoid the the curse of dimensionality. The periodic GARCH model, whose parameters are periodic, can not only describle the feature of sharp peak and heavy tail, and conditional heteroscedasticity, but also reflet the periodicity of financial data. In this thesis, we are concerned with the estimation for the parameters of periodic GARCH model. For general periodic GARCH model, we propose a robust estimator based on quantile regression. Under certain mild conditions, we establish the strong consistency and asymptotic normality of the quantile regression estimator. Compared to the tradi-tional Gaussian quasi maximum likelihood estimator(QMLE), the proposed estimator only needs the existence of the seconder order for the asymptotic normaliy. Besides, the quantile regression estimators have good performance on heavy-tailed distribution. They are robust to heavy-taild distribution. However, the quantile regression estimator may not be efficient at times, especially when the quantiles are close to zero, because the asymptotic relative efficiency of the quantile regression estimator can be arbitrarily small with respect to the QMLE. In order to improve the efficiency of QRE, we apply the composite quantile regression method to estimate the parameters. The composite quan-tile regression estimatror (CQRE), combining the the strength across multiple quantile regression models, keeps the advantages of QRE and greatly increases the efficiency of QRE. We can obtain the conditional quantiles of this model through both QRE and CQRE, from which we achieve the estimation for value at risk (VaR) of this model.Moreover, many economic activity varies with time continuously. Hence, finan-cial data are generally continuous data. Discrete financial data will inevitably lead to the lack of information. With the development of technology, many data are collected and saved on more and more concentrated time scales. These data present clear trends of function. Functional data analysis, based on the data fitting theores, is a powerful tool to solve these problems. Functional data analysis treat these data as a whole, guarantee- ing the integrity of the information. In traditional statistics, the linear model is perhaps the most widely considered and used statistical model. It is also the same to functional linear model in functional data analysis. Therefore, it is of significance to combine the period GARCH model with the functional linear model. However, in general case, er-rors are considered independent and identically distributed in functional linear model. But the period GARCH model is a kind of auto-correlated time series. When errors are serial correlated, many statistical methods no longer work. Strong serial correlation even means the lack of fit of the model. Therefore, we should test the serial correlaiton of the errors first in functional linear model. In this thesis, we consider the serial corre-lation test of the error in functional linear model with scalar response. We introduce the empirical likelihood ratio test according to the empirical likelihood ratio approach pro-posed by Owen (1988). Through the"least square estimator", we obtain the least square residuals and then construct the autocovariance vectors. On this basis, the test statistic is proposed via the empirical likelihood ratio function of the autocovariance vectors. Under null hypothesis and certain regularity conditions, the test statistic is asymptotic χ2 distributed. Numerical simulation results show that our test has good power.