Determination Of The Number Of Change Points In The Regression Mode

In the era of big data,high dimensionality is one of the characteristics of large datasets.However,high-dimensional data are often strongly dependent.New methods are called for analyzing the high-dimensional data.The high-dimensional factor model is useful for analyzing high-dimensional data.However,research on high-dimensional fac-tor models typically assumes constant factor loadings,while empirical studies have shown that factor loadings may undergo structural changes over time.Multiple structural changes may occur when the time span is large enough.Determining the number of change points is crucial to multiple change-point analysis.Various methods are proposed to determine the number of change points.However,most methods involve pre-determining the tun-ing parameters.As is known to all,the optimal tuning parameters usually vary from the error distribution,sample size,model complexity,etc.In practical applications,it is necessary to adjust the tuning parameters to adapt to different contexts of data.In addi-tion,the estimation results are largely affected by the tuning parameters.If data-driven methods such as the cross-validation method are used to determine the number of change points,the estimation process will be more convenient.However,there is currently no research using the cross-validation method to determine the number of change points in high-dimensional factor models.In the big data era,the data are not all point-valued as in traditional datasets.Interval-valued data can summarize the dataset to a manageable size while retaining as much of the information in the original dataset.Researchers propose a theoretical framework for processing interval-valued time series data by treating intervals as an inseparable whole.Subsequently,they explore the nonlinear features of the interval-valued time series and introduce a threshold autoregressive interval model.However,they only consider the case of a single threshold.Based on this background,this article will propose a series of cross-validation methods to determine the number of change points in high-dimensional factor models and interval-valued time series regression models.Chapter 3 explores how to use cross-validation method to determine the number of change points in high-dimensional factor models.Due to the unobservability of re-gressors in high-dimensional factor models,the cross-validation method based on the order-preserved sample-splitting strategy is not available here.Chapter 3 proposes to transform the multiple change-point problem in high-dimensional factor models to esti-mate the structural changes in a multivariate time series,and then use the cross-validation method to estimate the number of change points.Under certain conditions,Chapter 3proves the asymptotic properties of the estimator and verifies the theoretical properties through Monte Carlo simulations.Chapter 4 introduces how to use the cross-validation method with matrix completion to determine the number of change points in high-dimensional factor models.Chapter3 proposes using the cross-validation method to estimate the number of change points in high-dimensional factor models.The key to this method is converting the multiple change-point problem of high-dimensional factor models into estimating the change points of multivariate time series models.Researchers point out that the transformed time se-ries model cannot use the cross-sectional units effectively,so efficiency loss may occur.Even if the common factors can be obtained through certain methods(such as principal component analysis),the time-varying property of common factors makes the common factors obtained from the training set unsuitable to substitute for the common factors of the validation set.Chapter 4 takes a different perspective,and treats the validation set data as”missing”data from the training set data,then proposes a cross-validation method with matrix completion to determine the number of change points in high-dimensional factor models.This method can be directly applied to the original model and make full use of the information in the original data.The theoretical result indicates that the estimator estimated by the cross-validation method with matrix completion is consistent.Chapter 5 discusses the estimation of interval-valued time series regression models with multiple structural changes.The information in interval-valued data is richer than point-value data.The information advantage of the interval-valued data may lead to more effective estimation and inference.Most studies related to interval-valued time series do not consider the interval as a whole,so the information within the interval is not fully utilized and may introduce more parameters to be estimated.Chapter 5 is based on the theoretical framework which considers the interval as a whole to explore the estimation of interval-valued time series regression models with multiple structural changes,includ-ing the estimation of the number of structural changes,the location of change points,and the regression coefficients.The statistical properties of the proposed estimators are stud-ied.In addition,the number of change points is determined through the cross-validation method based on D_K-distance,and the regressors are obtained by a sequential conditional estimation method based on D_K-distance.The Monte Carlo simulation study verified the theoretical properties of the estimators and provided an estimation method for empirical research based on interval data.