Covariate-adjusted Regression Model On Dependent Data With Applications In Financial Time Series

Classical statistics base on the independence assumption. Limit theories of in-dependent random variables has get the perfect development in the30s-40s of20th century. These limit theories constitute the theory foundation of statistical inference and play an important role in statistics. Although the hypothesis of independence is sometimes rational. it is hard to check the independence of the samples. Moreover, in most of the practical problems, the samples are not independent observations. Since the50s of20th century, the dependence of random variable has attracted much inter-est of statistician. In some branch of probability and mathematical statistics, such as Markov chain, random field theory and time series analysis, the dependence concept has been proposed and developed. In the study of dependent data, mixing is a broadly applied concept. A mixing process can be viewed as a sequence of random variables for which the past and distant future are asymptotically independent.There exist a wide variety of data in economy and finance field, meteorology, hydrology, project technology and science, and so on. Most of these data appear in the form of time series, such as the stock closing price, annual turnover, the gross national product, and so on. Thus, analysis to time series is of growing importance to help participants make scientific decisionsCovariate-adjusted regression model is a new method which studied recently. As-sume X and Y are predictor and response, respectively. The traditional regression model study the relationship between X and Y based on the observations of (X,Y). In practical problems, however, variables X and Y might be distorted by other factors. If the distorting factors has not been considered, inaccurate statistic inference may oc- cur. Covariate-adjusted regression model precisely take into account the distortion of the other factors to analysis the relationship between X and Y. The distorting factor is called covariate.Covariate-adjusted regression model has get a lot of attention because of its im-portant practical significance and application value. The extent of covariate-adjusted regression model include data type and model type. Most extent of data type extend the model from independent data to longitudinal data, and the extent of model type include varying-coefficient model, nonlinear model, partial linear model, and so on. In this thesis, the extent of the model in both data type and model type are considered. In data type, we consider the covariate-adjusted regression model in dependent data with application in financial data. In model type, we study the covariate-adjusted parametric and nonparametric regression on dependent data.1. covariate-adjusted parametric regression model on dependent dataIn Chapter2, we discuss the covariate-adjusted parametric regression model on dependent data where Xio=1,Φo(·)=1. We assume the unobservable data{(Ui, Xi, Yi), i=1,2,..., n} is a jointly strictly stationary α-mixing sequence. The main objective is to estimate the unknown regression parameters γr(r=0,1,2,...,p) and to consider the asymptotic property based on the observable data{(Ui. Xi, Yi), i=1,2,...,n}. To achieve this goal, we propose a two-step procedure.Step1. Firstly, we transform the covariate-adjusted model into where This is a functional-coefficient time series model. In order to estimate coefficient func-tions βr(·)(r=0,1,...,p), a local linear smoothing method is employed. Set To minimize the sum of weighted squares it follows from the least square theory that the estimate of θ is The local least-square estimate of coefficient function βr(·) is given by where er.2p+2T is an unit vector of length2p+2with1at r-th position and0elsewhere.Step2. The estimates of γ0and γr are given by wherewe consider the asymptotic property of the proposed estimator. Theorem2.1proves that estimates are consistent and gives the consistency rate. The asymptotic normality of estimates are presented in Theorem2.2.Theorem2.1Under the technical conditions given in§2.6, it holds thatTheorem2.2Under the technical conditions given in§2.6. as n→∞, it holds that whereIn order to test the goodness-of-fit between the covariate-adjusted regression model and the general linear regression model, we propose a goodness-of-fit test. Suppose the covariate-adjusted regression model can be transformed into a functional-coefficient regression model. If βr(U)(r=0,1) are constant, that is βr(U)=βr{r=0,1), we get this implies that the simple linear regression model fit the data well. Otherwise, if βr(U)(r=0,1) are not constant, this implies that functional-coefficient regression model is more appropriate for the data.Consider the null hypothesis The test statistic is defmed as and we reject the null hypothesis for a large value of Tn. We propose a nonparametric bootstrap approach to evaluate the p-value of the goodness-of-fit test.To illustrate our methods, we consider the relationship between Copper Spot Price (CSP)(response) and Copper Futures Price (CFP)(predictor). A simple linear re-gression model would be Shanghai and Shenzhen300Stock Index Futures (IF), as a stock index futures, affects the relationship between CSP and CFP. Then we choose IF as the covariate U and consider the functional-coefficient regression model We apply the goodness-of-fit test described in§2.4. The conclusion indicate the functional-coefficient regression model is more appropriate for the data than the linear regression model. This provide the evidence that the relationship between CSP and CFP changes with IF.2. covariate-adjusted nonparametric regression model on dependent dataIn chapter3, we propose the covariate-adjusted nonparametric regression model on dependent data. The sample version is We assume the unobservable data{(Ui, Xi, Yi), i=1,2,..., n} is a jointly strictly sta-tionary α-mixing sequence. To estimate the regression function, a two-step estimate procedure is proposed as follows:Step1. The nonparametric estimators of ψ(U) and Φ(U) are proposed as We may construct the approximate formula of covariate-adjusted model asStep2. We propose the Nadaraya-Watson estimator of regression function as whereTheorem3.1proves the asymptotic convergence of the estimated regression func-tion m(·) and gives the convergence rate. Theorem3.1. If conditions (A3-1)-(A3-3) and (C3-1)-(C3-5) in§3.5are satisfied, the following result holds: Both real data and simulated examples are provided for illustration.3. Detecting Sparse Signal Segments by Local LRS MethodSparse signal detection is an important problem in signal processing. Two impor-tant challenges in detecting sparse signals are how to improve the detection accuracy and reduce the computational complexity. In Chapter4, we propose a Local LRS method. Compare with conditional LRS method, the proposed procedure can greatly reduce the computational complexity and improve the improve the detection accuracy.Suppose the data{Xi,i=1,2,...,n} satisfy the model where I1,I2,...,Iq are disjoint intervals which presenting signal segments with unknown locations,μ1,μ2,...,μq are unknown signal strength, q=q(n) is the unknown number of the signal segment, possibly increasing with n.{Zi,i=1,2,...,n} are noise data. Let Ⅱ={I1,I2,..., Iq} denote the collection of all the signal segments. Our goals are to detect whether signal segments exist and identify the locations of these segments when they do exist. The detection and identification of the signal segments can be regard as a statistical testing problem where Φ denote empty set. If H1is true, it means that there exist some signal segments, and then to identify the set of signal segments Ⅱ. The test statistic is proposed as where Ⅰ(?){1,2,...,n} is any interval, and|Ⅰ|denote the length of Ⅰ. The threshold for the test statistic can be set as Our procedure first finds all "important" points whose observed data greater than tln, and then considers the L-neighborhood of each selected point. The proper estimates of signal segments should be intervals whose corresponding test statistic is greater than t2n and achieve the maximum.Theorem4.1present the consistency of the proposed procedure.Theorem4.1Assume (C4-1) and (C4-2) in§4.3hold. Additionally, assume where Then if we haveThe simulation results indicate that the proposed procedure has high detection accuracy and computational efficiency.