| High dimensional and high frequency time series provide new challenges for researchers and practitioners, especially in financial applications. To analyze high frequency time series with possibly significant exogenous variables, connections between the target time series and exogenous variables should be considered. Therefore, there is an urgent demand for new statistical methodology to be developed to deal with high dimensional and high frequency inference. For this purpose, we provide a general framework on high dimensional and functional time series analysis.;Chapter 1 provides a bird's eye view of the state-of-the-art and challenges of the high dimensional and high frequency time series. An introduction to the nonlinear spline smoothing, GARCH-family models, functional time series, variable selection is also provided.;In Chapter 2, we develop a new model for financial volatility estimation and forecasting by incorporating exogenous covariates in a semi-parametric log-GARCH model. We propose a quasi maximum likelihood procedure via spline smoothing technique. Consistent estimators and asymptotic normality are obtained under mild regularity conditions. Additionally, an application to SPY index data demonstrates strong competitive advantage of our model comparing with GARCH(1,1) and log-GARCH(1,1) models.;In Chapter 3, for functional time series with physical dependence, we construct confidence bands for its mean function. We estimate functional time series mean functions via a spline smoothing technique. Confidence bands have been constructed based on a long-run variance and a strong approximation theorem, which is satisfied with mild regularity conditions. Additionally, an application to S&P 500 index data demonstrates a non-constant volatility mean function at a certain significance level.;In Chapter 4, we consider a varying coefficient regression model for dense functional time series. For response variable depending linearly on some time independent covariates with coefficients as functions of time-dependent covariates, we propose an estimation procedure via polynomial splines to estimate the nonparametric components. In addition, we conduct variable selection through penalized spline regression to ensure parsimonious models simultaneously. With proper regularization condition, we show that the proposed varying-coefficient estimator is consistent and enjoys the oracle property for the non-zero nonparametric components for high-dimensional data. |
