4时间序列参数估计

时间序列模型参数估计1理论基础1.1矩估计1.1.1AR模型矩估计法参数估计的思路：即从样本中依次求中r k然后求其对应的参数Φk值方差：1.1.2MA模型对于MA模型采用矩估计是比较不精确的，所以这里不予讨论1.1.3ARMA（1，1）矩估计法参数估计的思路：方差：1.2最小二乘估计1.2.1AR模型最小二乘参数估计的思路：对于AR（P）而言也可以得到类似矩估计得到的方程，即最小二乘与矩估计得到的估计量相同。

1.2.2MA模型最小二乘参数估计的思路：1.2.3ARMA模型最小二乘参数估计的思路：1.3极大似然估计与无条件最小二乘估计2R中如何实现时间序列参数估计2.1对于AR模型ar(x, aic = TRUE, order.max = NULL,method=c("yule-walker", "burg", "ols", "mle", "yw"),na.action, series, ...)> ar(ar1.s,order.max=1,AIC=F,method='yw')#即矩估计Call:ar(x = ar1.s, order.max = 1, method = "yw", AIC = F)Coefficients:10.8314Order selected 1 sigma^2 estimated as 1.382> ar(ar1.s,order.max=1,AIC=F,method='ols')#最小二乘估计Call:ar(x = ar1.s, order.max = 1, method = "ols", AIC = F)Coefficients:10.857Intercept: 0.02499 (0.1308)Order selected 1 sigma^2 estimated as 1.008> ar(ar1.s,order.max=1,AIC=F,method='mle')#极大似然估计Call:ar(x = ar1.s, order.max = 1, method = "mle", AIC = F)Coefficients:10.8924Order selected 1 sigma^2 estimated as 1.041采用自编函数总结三个不同的估计值> Myar(ar2.s,order.max=3)最小二乘估计矩估计极大似然估计1 1.5137146 1.4694476 1.50613692 -0.8049905 -0.7646034 -0.79644532.2对于ARMA模型arima(x, order = c(0, 0, 0), seasonal = list(order = c(0, 0, 0), period = NA), xreg = NULL, include.mean = TRUE, transform.pars = TRUE, fixed = NULL,init = NULL, method = c("CSS-ML", "ML", "CSS"), n.cond, optim.control = list(), kappa = 1e+06, io = NULL, xtransf, transfer = NULL)order的三个参数分别代表AR，差分MA的阶数> arima(arma11.s,order=c(1,0,1),method='CSS')Call:arima(x = arma11.s, order = c(1, 0, 1), method = "CSS")Coefficients:ar1 ma1 intercept0.5586 0.3669 0.3928s.e. 0.1219 0.1564 0.3380sigma^2 estimated as 1.199: part log likelihood = -150.98> arima(arma11.s,order=c(1,0,1),method='ML')Call:arima(x = arma11.s, order = c(1, 0, 1), method = "ML")Coefficients:ar1 ma1 intercept0.5647 0.3557 0.3216s.e. 0.1205 0.1585 0.3358sigma^2 estimated as 1.197: log likelihood = -151.33, aic = 308.65 采用自编函数总结三个不同的估计值> Myarima(arma11.s,order=c(1,0,1))$coef条件SS估计极大似然估计条件似然估计ar1 0.5585828 0.5647477 0.5647498ma1 0.3668814 0.3556965 0.3556973intercept 0.3927654 0.3216166 0.3216152$log条件SS估计极大似然估计条件似然估计[1,] -150.984 -151.3268 -151.3268$sigma2条件SS估计极大似然估计条件似然估计[1,] 1.199378 1.196984 1.196984$aic条件SS估计极大似然估计条件似然估计[1,] NA 308.6537 308.65372.3采用自助法arima.boot()此函数估计的是参数的取值置信区间，而不是指具体的某个值，与arima是不同的。

> res=arima(sqrt(hare),order=c(3,0,0),include.mean=T)> set.seed(12345)> # Method I以最初三个观测为条件，并假设误差服从正态分布，得到95%的置信区间quantile用于计算置信区间值，signif类似于四舍五入函数，保留有效数值。

>coefm.cond.norm=arima.boot(res,cond.boot=T,is.normal=T,B=1000,init=sqrt(ha re))> signif(apply(coefm.cond.norm,2,function(x){quantile(x,c(.025,.975),na.rm=T)}),3) ar1 ar2 ar3 intercept noise var2.5% 0.593 -0.667 -0.6740 5.12 0.54897.5% 1.280 0.244 -0.0135 6.38 1.540>> # Method II假设误差并不服从正态分布，而是需要从样本抽样中得到coefm.cond.replace=arima.boot(res,cond.boot=T,is.normal=F,B=1000,init=sqrt( hare))>signif(apply(coefm.cond.replace,2,function(x){quantile(x,c(.025,.975),na.rm=T)}),3) ar1 ar2 ar3 intercept noise var2.5% 0.611 -0.700 -0.6720 4.98 0.51697.5% 1.300 0.241 -0.0417 6.32 1.500> # Method III基于平稳自助法的置信区间，且误差服从正态分布>coefm.norm=arima.boot(res,cond.boot=F,is.normal=T,ntrans=100,B=1000,init=s qrt(hare))> signif(apply(coefm.norm,2,function(x){quantile(x,c(.025,.975),na.rm=T)}),3) ar1 ar2 ar3 intercept noise var2.5% 0.687 -0.747 -0.6600 4.99 0.50897.5% 1.380 0.192 -0.0168 6.33 1.500>> # Method IV基于平稳自助法的置信区间，且误差不服从正态分布coefm.replace=arima.boot(res,cond.boot=F,is.normal=F,ntrans=100,B=1000,init =sqrt(hare))> signif(apply(coefm.replace,2,function(x){quantile(x,c(.025,.975),na.rm=T)}),3) ar1 ar2 ar3 intercept noise var2.5% 0.70 -0.715 -0.6620 4.98 0.4797.5% 1.36 0.183 -0.0187 6.30 1.503附自编函数3.1Myar#用于自回归模型的参数估计，整合矩估计，最小二乘估计，以及极大似然估计#该函数用于对时间序列中心化数据（因此截距项一定为0）估计AR模型的参数，AIC为真时，滞后项根据AIC准则确定，为假时则根据设置的order.max设定Myar=function(tsdata, order.max = 1,AIC = F){library(TSA)ols<-ar(tsdata,order=order.max,AIC=AIC,method='ols')yw<-ar(tsdata,order=order.max,AIC=AIC,method='yw')mle<-ar(tsdata,order=order.max,AIC=AIC,method='mle')olscoef<-ols[[2]]ywcoef<-yw[[2]]mlecoef<-mle[[2]]result=data.frame(olscoef,ywcoef,mlecoef)colnames(result)=c('最小二乘估计','矩估计','极大似然估计')return(result)}3.2Myarima#用于自回归模型的参数估计，整合矩估计，最小二乘估计，以及极大似然估计#该函数用于对时间序列中心化数据（因此截距项一定为0）估计AR模型的参数，AIC为真时，滞后项根据AIC准则确定，为假时则根据设置的order.max设定Myarima=function(tsdata, order=c(0,0,0)){library(TSA)result=NULLcss<-arima(tsdata,order=order,method='CSS')ml<-arima(tsdata,order=order,method='ML')cssml<-arima(tsdata,order=order)result$coef=cbind(css$coef,ml$coef,cssml$coef)result$log=cbind(css$log,ml$log,cssml$log)result$sigma2=cbind(css$sigma2,ml$sigma2,cssml$sigma2)result$aic=cbind(NA,ml$aic,cssml$aic)colnames(result$coef)=c('条件SS估计','极大似然估计','条件似然估计') colnames(result$log)=c('条件SS估计','极大似然估计','条件似然估计') colnames(result$aic)=c('条件SS估计','极大似然估计','条件似然估计') colnames(result$sigma2)=c('条件SS估计','极大似然估计','条件似然估计') return(result)}。