Dependent Functional Data Analysis And Its Applications In Finances

With the development of science and technology and the improvement of information storage technology,it is easier to collect functional data.But most literatures of functional data analysis(FDA)assume that the observed functional data are independent and identically distributed(i.i.d.).And statistical analysis is carried out under this condition.However,data in fields such as finance are not only natural functional data but also have the important feature is that these data are dependent,that is,they have strong memory.If the FDA method under i.i.d.is used to directly analyze these dependent functional data,it will inevitably lead to bias in relevant statistical analysis.The high frequency transaction data of the financial market reflects the operating laws and affects the behavioral decisions of governments,enterprises and individuals.How to use the FDA method to objectively and accurately characterize the change patterns of intra-day price and volatility,timely and accurately discover the internal structural changes,and accurately predict in short-run are all very important.This paper proposes two representation methods of dependent functional data,and applies these two function representation methods to functional hypothsis test and the estimation of functional data linear regression model and analysis of mixed data.Specifically,the main contents and innovations are summarized as follows:(1)The first task of FDA is to represent the observed discrete data as a function curve.For the dependent function data,two methods of function representation are proposed.Existing literature proposed to use long-run covariance function to replace the covariance function under i.i.d.to modify the function representation,but the estimation of long-run covariance function faces the choice of kernel function and window width,this paper proposes to use Bartlett kernel without truncation to estimate long-run covariance function,avoid choosing kernel functions and window widths,and reduce function representation errors.The second method is to propose a function representation method based on the residual covariance function,avoiding the use of long-run covariance function,that is,to use the autoregressive model to fit dependent function data to obtain the i.i.d.residual function.Then use the i.i.d.residual function to estimate the unbiased covariance function and function principal components.Numerical simulations show that the two function representation methods proposed in this paper have smaller function representation errors than the existing long-run covariance function estimation methods and the covariance function method under i.i.d.Based on the empirical analysis of 1-minute high frequency Stock Index data of the CSI 300 from 2016 to 2018,the error of the function representation method based on the residual covariance function proposed in this paper is the smallest.(2)The existing literature only considers using the long-run covariance of the functional principal component scores to modify the change point test statistics of mean functions under i.i.d.,without considering the estimation bias of the functional principal component itself.This paper proposes to use the long-run covariance function two-step estimation method to modify the test statistic under i.i.d..Firstly estimate the sample long-run covariance function to obtain the more accurate functional principal components,and then use the long-run covariance of the dependent functional principal component scores to modify the statistical test under i.i.d..Numerical simulations show that the test size of the proposed method is closer to the nominal level,and the test power is higher than other methods.When conducting an example analysis of 1-minute high frequency Stock Index data of SSE 180 in 2013,The method proposed in this paper can correctly identify the change point of mean function,and the existing test method obviously misses two change points of mean function.(3)It is proposed to estimate the long-run covariance function by using the Bartlett kernel without truncation and the Newey-West estimation formula,so as to improve the statistical statistic of the equality function under i.i.d..Numerical simulation results show that the test size based on the Newey-West estimation method proposed in this paper is closer to the nominal level,and the test power is higher than the test method under i.i.d.and the existing long-run covirance test method.The test power of the test method based on the Bartlett kernel without truncation is higher than the test method under i.i.d..When testing the cumulative returns of CSI 300 and SSE 180 in 1-minute and 5-min high frequency stock index in 2018,the test result shows that the mean curves of the cumulative returns of the CSI 300 and SSE 180 stock indexes in 1-minute and 5-min are equal,that is,the levels of investment income in the both markets are comparable.(4)Functional data linear regression model is an important tool for studying the relationship between functional variables.Aiming at the dependent function data,this paper proposes a dependent functional linear regression model,and extends the estimation method to the binary and multivariate situations.The two numerical simulation results show that the estimation method based on the residual covariance function proposed in this paper is smaller than other estimation methods in both the regression coefficient function estimation error and the out-of-sample prediction error.The 1-minute data of CSI 300 from 2016-2018 are used for out-of-sample prediction.The results show that the out-of-sample prediction error based on the residual covariance function estimation method proposed in this paper is much smaller than other methods.(5)The Mixed Data Sampling(MIDAS)model can be used to model mixed frequency data,and the FDA is also suitable for analyzing mixed frequency data.Therefore,this paper proposes a univariate and multivariate partial dependent functional linear regression model,and provide a model method for the analysis of mixed frequency data.Numerical simulation results show that the out-of-sample prediction error of FDA method is smaller than that of MIDAS and MIDAS-AR(1)models.The monthly M2 and PPI,and the daily crude oil prices are selected for out-of-sample prediction of the monthly CPI.The results show that the out-of-sample prediction error of FDA method is smaller than MIDAS and MIDAS-AR(1).