Research On Random Coefficient Autoregressive Models Driven By Covariants And Observations

In the past few decades,many mathematicians and statisticians have found that random coefficient autoregressive models have important applications in the analysis of social and economic activities,air quality monitoring,solar activity and infectious disease dynamics.The traditional random coefficient models concerned by scholars assume that the regression coefficients are independent and identically distributed random variables.However,in real life,researchers have found that random coefficients may depend on some covariates and observations,so that the sequence of independent and identically distributed random variables cannot flexibly and accurately describe the dynamic properties of random coefficients.For example,the daily stock price will be affected by the past stock price and crude oil price.The PM 2.5 concentration in the air will be affected by the PM 2.5 concentration in the past period and other covariates,such as local air temperature,air pressure,sulfur dioxide concentration in the air,ozone concentration.The crime data of a certain region in a certain period of time(the number of car accidents,shootings,robberies,etc.)will be affected by the past crime data and other covariates,such as national economic level,unemployment rate.Hence the traditional random coefficient autoregressive model is no longer suitable for analyzing such problems.In order to accurately describe such data dynamics,Zheng et al.(2007)proposed an observation driven integer-valued random coefficient time series model based on the logistic structure.Yang et al.(2021)combined the Logistic structure with the threshold model,and proposed an integer-valued random coefficients threshold autoregressive model driven by covariate.Therefore,we use Logistic structure and consider incorporating the covariates and observations into the random coefficients to propose the covariates and observations driven random coefficient autoregressive model.On the other hand,due to the widely existing segmented structure of time series data,threshold autoregressive model is one of the effective models to describe segmented data.According to the value of threshold variable,it uses segmented function to model the asymmetric series.Different segments in the function can describe different states to describing data flexibly.But Wu et al.(2007)found that the threshold model performs poorly at the boundary of regime transition.The reason of this defect is the sudden change of the probability structure during the regime transition.In order to solve this problem,Li et al.(2015)used a more flexible regime transition mechanism and proposed a buffered autoregressive model to accurately analyze the multiple dynamic characteristics of the data.Based on the above discussion,we introduce the main results of our studies,as follows:1.The random coefficient autoregressive model driven by covariates and observations(OD-RCAR).Because the random coefficient will be affected by many factors,for example,the PM 2.5 concentration in the air will be affected by the PM 2.5 concentration in the past period and other covariates,such as local air temperature,air pressure,sulfur dioxide concentration in the air,ozone concentration.,while the traditional random coefficient models rarely take into account the influence of covariates and observations.Hence,inspired by Zheng et al.(2007)and Yang et al.(2021),we use the Logistic transformation to make covariates and observations included in the random coefficients and propose the OD-RCAR model.The parameter estimation and hypothesis testing of the model are studied by empirical likelihood method.At the same time,multiple covariates in random coefficients are selected by penalty empirical likelihood inference.Under some assumptions,the consistency and asymptotic normality of the two estimators are proved,and the asymptotic distributions of hypothesis test statistics are given.The consistency of estimators and the accuracy of model selection are verified by numerical simulation.Finally,a real data example of the PM2.5 concentration in Changping,Beijing is analyzed to illustrate our model.2.The bivariate random coefficient autoregressive model driven by covariates and observations(BOD-RCAR).In real life,we will encounter bivariate time series problems,such as analyzing the relationship between snowfall and air pressure in the same area,and studying the rise and fall relationship between two stocks over a period of time.Although bivariate time series models have attracted a large number of researchers’ attention in recent years,most bivariate continuous time series models rarely take into account the effects of covariates and observations on random coefficients.So we propose the BOD-RCAR model to extend the OD-RCAR model in Chapter 2,and discuss the consistency and asymptotic normality of the conditional least squares estimator,conditional maximum likelihood estimator and maximum empirical likelihood estimator of the model parameters.At the same time,based on the empirical likelihood theory,the hypothesis test is carried out to determine whether there are covariates and observations in the random coefficients.In the numerical simulation,we investigate the accuracy of the estimators and study the efficacy of empirical likelihood test,and finally we verify the superiority and flexibility of BOD-RCAR model compared with other models through stock data.3.The random coefficient buffered generalized autoregressive conditional heteroscedasticity model driven by covariates(BRC-GARCH-X).The threshold autoregressive model can capture the nonlinear characteristics in real data,but because the financial time series generally have hysteresis,so only using the threshold model can not effectively describe the volatility of financial data(Zhu et al.2017;Chen et al.2019;Chen et al.2020).Therefore,based on the Logistic structure and the ideas of GARCH,we propose the BRC-GARCH-X model.Because the quasi likelihood method is inaccurate in analyzing asymmetric heavy tailed data,we introduce the quasi exponential maximum likelihood estimation method,which reduces the conditions of establishing the asymptotic normality and improves the accuracy of heavy tailed data estimation.At the same time,because the threshold variables of the buffered model may be affected by other factors,we set the threshold variables defined by a weighted average structure to improve the sensitivity of the model.And we propose a threshold search algorithm to improve the estimation efficiency.The simulations show that the quasi exponential maximum likelihood method is superior to the quasi Likelihood method.Finally,the BRC-GARCH-X model is verified to be superior to the existing models by S&P500 data.