基于残差函数主成分的相依函数型回归模型估计及金融应用

Estimation of Dependent Functional Regression Model Based on Residual Functional Principal Component and Its Financial Application

本文研究具有相依特征的函数型数据的函数表示方法及其回归模型的估计方法。首先基于分整理论提出基于残差函数主成分的函数表示方法,然后利用残差函数主成分对回归系数函数正则化表示,最后把函数曲线和回归系数函数代入相依函数型回归模型进行估计,并通过蒙特卡洛模拟和金融实例分析来评估参数估计和样本外预测的准确性。研究发现,相比现有的基于协方差函数和长期协方差函数的函数主成分估计方法,本文提出的基于残差函数主成分的估计方法具有良好的有限样本性质,回归系数函数估计更准确、样本外预测效果更好;基于高频数据的股市开盘价预测实证研究表明本文方法的样本外预测精度最高。与现有方法相比,本文提出的基于残差函数主成分的估计方法,既考虑函数型数据的相依特征,又避免长期协方差函数估计时面临的核函数和窗宽选择问题,为经济金融等领域具有相依特征的函数型数据提供一种函数表示方法,丰富函数型回归模型理论,为其他函数型回归模型的拓展提供借鉴。

With the progress of science and technology and the improvement of storage technology,more and more high-frequency data,which are called functional data,appear in the form of function curves. The primary task of a functional data analysis is to express those functional data in the form of function curves,which directly relates to the effectiveness of model estimation and the effect of out-of-sample prediction. Previous research has generally assumed that functional data are independent and identically distributed(IID). The principal components of the function are obtained by calculating the covariance function and applying the spectral decomposition method.Then,the observed data are expressed as function curves based on the Karhunen-Loève(K-L)expansion. Finally,the function curves are substituted into the functional regression model for estimation. For functional data with dependent characteristics,such as high-frequency stocks,if the method under IID is still used to calculate the sample covariance function,the sample covariance function obtained is no longer the consistent estimator of the population covariance function. This can lead to inaccurate principal components of the function,an inaccurate function representation,and a poor out-of-sample prediction effect. To address this,Hormann and Kokoszka(2010),Horvath et al.(2016),and Rice and Shang(2017) proposed correcting the estimation deviation using a long-run covariance function to replace the covariance function. However,these methods of estimating long-run covariance function of samples face the problem of selecting kernel function and optimal bandwidth.Wang et al.(2021) studied the cross section correlation of financial panel data,finding that the sample covariance of different fractional integration stationary time series did not converge to the population covariance. As such,they proposed a correlation coefficient test method based on residual covariance. This paper extends the idea of Wang et al.(2021) to the case of functional data,proposing a functional representation method based on the principal component of residual function. Then,the regression coefficient function is regularized by using the principal component of residual function. The function curves and regression coefficient function are substituted into the dependent functional regression model for estimation. Monte Carlo simulation results show that,from the different parameter settings of the fractional and integral stationary data generation method,compared with the existing functional principal component estimation methods based on covariance function and long-run covariance function,the residual functional principal component estimation method proposed in this paper has good limited sample properties. Further,the regression coefficient function estimation is more accurate, the out-of-sample prediction effect is better, and the conclusion is more robust. Empirical research predicting the opening price of the stock market based on the CSI 300 high-frequency stock index data in three different time periods also shows that the proposed estimation method has a high prediction accuracy.In this paper,we propose a function representation method based on the principal component of the residual function for dependent functional data in financial and other fields. This method considers the dependent characteristics of functional data,does not need to estimate the long-run covariance function, and does not face the problem of selecting kernel function and optimal bandwidth. The proposed method is used to estimate the dependent functional regression model,which provides another general estimation framework for the estimation of the dependent functional regression model,enriches the theory of functional linear regression model,and also expands the estimation idea of full functional linear regression model,nonlinear functional regression model and other models. The study also provides a reference for analyzing functional data with dependent characteristics such as meteorological data and air quality index data. The accurate prediction of future economic and financial indicators may grasp the changes of future economic situation in advance,provide investors with more reasonable investment suggestions,and provide reference for financial institutions or relevant government departments to formulate scientific macroeconomic policies in advance.