Bayesian Quantile Regression Analysis For Three Classes Of Autoregressive Time Series Models

In statistics and data analysis,regression model is the most common statistical modeling method,and its most typical estimation method is the least squares method.However,least squares regression is essentially mean reversion,which is unable to capture the characteristics of the data distribution at different quantiles.In this case,quantile regression was developed.It breaks through the limitation of least squares regression by analyzing the linear relationship between different quantile points of the whole population and provides a unique perspective for the comprehensive description of the distribution characteristics of the data.Nevertheless,quantile regression mainly relies on the sample information to make inferences in practical applications,ignoring the importance of a prior information.The emergence of Bayesian quantile regression fills the gap in this aspect,which combines the prior information with the sample information through Bayesian theory,improves the accuracy of inference,and shows significant advantages in solving complex data analysis problems.Based on the Bayesian quantile regression method,this paper investigates the estimation and order shrinkage of three types of autoregressive time series models.Specifically,the research work mainly includes the following three aspects:To better fit the effects of internal and external factors on the trends and changes of stock and other financial data returns,we investigates the Bayesian quantile regression estimation problem of the quantile autoregressive(QAR-X)model with explanatory variables.Firstly,under the condition that the random errors following the asymmetric Laplace distribution,the latent variables are introduced to decompose the asymmetric Laplace distribution.Without considering the effect of scale parameters,MCMC(Markov Chain Monte Carlo)sampling technique is used to estimate the parameters of the model,and based on this,a revised estimate of the posterior variance is given.In order to further optimize the model,a scale parameter is introduced,the asymmetric Laplace distribution is re-decomposed,and the parameters are estimated using MCMC sampling.In addition,in order to solve the problem of increasing model complexity caused by too many lagged dependent and explanatory variables,three different Bayesian variable selection methods are proposed based on the Laplace prior and the spike-and-slab prior,and the MCMC sampling algorithm is used for the model ordering and variable selection.Finally,based on a large number of numerical simulations,the model is applied to the empirical analysis of gold price data and bike sharing data,which provides investors with reasonable references for their analyses and forecasts,and helps them better understand the market dynamics and make more informed decisions.To better capture the nonlinear relationship among the data and reflect the dynamic changes of the financial market,we considers the Bayesian empirical likelihood inference problem of threshold quantile autoregressive model.The empirical likelihood method is used to construct the nonparametric likelihood function from the nonparametric perspective.Then,under the assumption that the autoregressive parameters obey the normal prior and the threshold parameters obey the uniform prior,the model parameters are estimated by the MCMC sampling algorithm,and the asymptotic nature of the posterior estimates is proved.Next,we consider the order shrinkage problem of the sparse threshold quantile autoregressive model,and use the spike-and-slab a priori to estimate the nonzero parameters while determining the order of the model.Finally,based on a large number of numerical simulations,the model is applied to the actual data of the American consumer price index to provide researchers and practitioners in the field of economics with a more flexible and accurate method of analysis and forecasting,and to provide more reliable information for economic policy and market decisions.To capture the dynamic relationship and nonlinear characteristics of different quantiles in time series data,we studies the Bayesian quantile regression estimation and order shrinkage of hysteretic quantile autoregressive model.Firstly,under the condition of fixed order,by introducing the “working likelihood” and normal prior of asymmetric Laplace distribution,the autoregressive parameters and hysteretic parameters of the model are estimated by MCMC sampling method.This method can better understand the structure and parameters of the model,so as to better describe the dynamic relationship of time series data.Secondly,for sparse hysteretic quantile autoregressive model,the order of the model is determined by introducing spike-and-slab prior,and the non-zero coefficient are estimated.Finally,based on a large number of numerical simulations,the model is applied to the empirical analysis of the American gross national product data,which provides strong support for decision makers and helps them to formulate more effective economic policies.