時(shí)間序列參數(shù)估計(jì)

1、時(shí)間序列模型參數(shù)估計(jì)1理論基礎(chǔ)1.1 矩估計(jì)1.1.1 AR模型矩估計(jì)法參數(shù)估計(jì)的思路：即從樣本中依次求中rk然后求其對(duì)應(yīng)的參數(shù)k值方差：1.1.2 MA模型對(duì)于MA模型采用矩估計(jì)是比較不精確的，所以這里不予討論1.1.3 ARMA(1,1)矩估計(jì)法參數(shù)估計(jì)的思路：方差:1.2 最小二乘估計(jì)1.2.1 AR模型最小二乘參數(shù)估計(jì)的思路:E一產(chǎn)k如Yi產(chǎn)。十%。匕之1）或n可檢其費(fèi)或呈式y(tǒng)一為質(zhì)冽變量，k為啕應(yīng)變虻的回歸斑型.景小二熊也訃是通過對(duì)如F若分的平方和的最小化來遑行牯計(jì)的、（匕-Q-MY一"制為只有KE的理前值，觸只能從，=2到上工布求麗.令»SU機(jī)尸）=£

2、（匕一產(chǎn)）一期K1力門.2打bE豫文通常仰為臬件平方就國(guó)基.1稱為葩件的理由端-花后面說明.）職始最小二乘原則，培定疆酒疽打陽(yáng)使壽年浦般小化的瞽數(shù)取值分別擔(dān)為4和口船估計(jì).由上理3SJ日產(chǎn)3口，IfX2】'Yr_/1一如Ki-四）M_1+*=0現(xiàn)r-f或考，簡(jiǎn)化并求N，尸=77軍一心汨對(duì)于數(shù)值較大的E國(guó)蛇.無(wú)論。取何值,方程（7,工3*都可糖化為白4了（7.2.451一府再對(duì),應(yīng)小化£aY1有T今上式等于零.求解*得蚓除也浸效應(yīng)以外，有時(shí)可以說,2%£紅0-F）TOTi-TjZKYi,幻riX（Yt-YJ（Y,-i-Y）除了分母中少一頊,即（£¥

3、>,曲式g門相等.對(duì)平建過程累讖，這個(gè)殿項(xiàng)是可以融略的，區(qū)北他小二系F衽估計(jì)量用乎相等，恃別是對(duì)大樣本而言對(duì)于AR（P）而言也可以得到類似矩估計(jì)得到的方程，即最小二乘與矩估計(jì)得到的估計(jì)量相同2 MA模型最小二乘參數(shù)估計(jì)的思路:2 ARMA模型最小二乘參數(shù)估計(jì)的思路:1.3極大似然估計(jì)與無(wú)條件最小二乘估計(jì)2R中如何實(shí)現(xiàn)時(shí)間序列參數(shù)估計(jì)2.1對(duì)于AR模型ar(x,aic=TRUE,order.max=NULL,method=c("yule-walker","burg","ols","mle","yw&q

4、uot;),na.action,series,.)>ar(ar1.s,order.max=1,AIC=F,method='yw')#即矩估計(jì)Call:ar(x=ar1.s,order.max=1,method="yw",AIC=F)Coefficients:0.8314Orderselected1sigmaA2estimatedas1.382>ar(ar1.s,order.max=1,AIC=F,method='ols')#最小二乘估計(jì)Call:ar(x=ar1.s,order.max=1,method="ols&quo

5、t;,AIC=F)Coefficients:10.857Intercept:0.02499(0.1308)Orderselected1sigmaA2estimatedas>ar(ar1.s,order.max=1,AIC=F,method='mle')#Call:ar(x=arl.s,order.max=1,method="mle",AICCoefficients:10.8924Orderselected1sigmaA2estimatedas1.008極大似然估計(jì)=F)1.041采用自編函數(shù)總結(jié)三個(gè)不同的估計(jì)值>Myar(ar2.s,order.

6、max=3)最小二乘估計(jì)矩估計(jì)極大似然估計(jì)11.51371461.46944761.50613692-0.8049905-0.7646034-0.7964453對(duì)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

7、=list(),kappa=1e+06,io=NULL,xtransf,transfer=NULL)order的三個(gè)參數(shù)分別代表AR,差分MA的階數(shù)>arima(arma11.s,order=c(1,0,1),method='CSS')Call:arima(x=arma11.s,order=c(1,0,1),method="CSS")Coefficients:ar1ma1intercept0.55860.36690.3928s.e.0.12190.15640.3380sigmaA2estimatedas1.199:partloglikelihood=-

8、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:ar1ma1intercept0.56470.35570.3216s.e.0.12050.15850.3358sigmaA2estimatedas1.197:loglikelihood=-151.33,aic=308.65采用自編函數(shù)總結(jié)三個(gè)不同的估計(jì)值>Myarima(arma11.s,order=c(1,0,1)$coef條

9、件SS估計(jì)極大似然估計(jì)條件似然估計(jì)ar10.55858280.56474770.5647498ma10.36688140.35569650.3556973intercept0.39276540.32161660.3216152$log條件SS估計(jì)極大似然估計(jì)條件似然估計(jì)1,-150.984-151.3268-151.3268$sigma2條件SS估計(jì)極大似然估計(jì)條件似然估計(jì)1,1.1993781.1969841.196984$aic條件SS估計(jì)極大似然估計(jì)條件似然估計(jì)1,NA308.6537308.6537采用自助法arima.boot()此函數(shù)估計(jì)的是參數(shù)的取值置信區(qū)間，而不是指具體的某個(gè)值

10、，與arima是不同的。arim+boa；函數(shù)基于振合的4RFMA模型展開自助分析.第一個(gè)參數(shù)是擬合的&RIMA模型，即arima函數(shù)的輸出.及四個(gè)不同的自助法可以使用：自助序列可以用提供的數(shù)值初始化facnciboot=門城若不提供初始化效值（coe<l卻*"I用使用非參自助此&normal*F）或若假設(shè)正右筋息的叁數(shù)自助T）.對(duì)于條件自助*初研數(shù)值可以以向后方式提供（arima»boot南數(shù)將會(huì)使用提供的向陡中的數(shù)值作為初始優(yōu)L自助樣本容麻，比姆1骸心可以通過選項(xiàng)B=工0口。來設(shè)定.3m&*人口七函數(shù)輸出個(gè)矩陣，談降的每一行都昆對(duì)H助數(shù)器使

11、用極大似然估計(jì)法所褂到的AK1MA系數(shù)的自助沿計(jì)旗.因此*如果旦=:”匚井旦模型為4R33螂么精跑是個(gè)】。0。然以4的班陸，誣燉際的每一行按照孔，樂）的RA門319順序作白助ARC),AR(2)和素?cái)?shù)加上均值估計(jì)值組成.>res=arima(sqrt(hare),order=c(3,0,0),include.mean=T)>set.seed（12345）>#MethodI以最初三個(gè)觀測(cè)為條件，并假設(shè)誤差服從正態(tài)分布，得到95%的置信區(qū)間quantile用于計(jì)算置信區(qū)間值，signif類似于四舍五入函數(shù)，保留有效數(shù)值。>coefm.cond.norm=arima.boot

12、(res,cond.boot=T,is.normal=T,B=1000,init=sqrt(hare)>signif(apply(coefm.cond.norm,2,function(x)quantile(x,c(.025,.975),na.rm=T),3)ar1ar2ar3interceptnoisevar2.5%0.593-0.667-0.67405.120.54897.5%1.2800.244-0.01356.381.540>#MethodII假設(shè)誤差并不服從正態(tài)分布，而是需要從樣本抽樣中得到coefm.cond.replacenit=sqrt(hare)>=arima

13、.boot(res,cond.boot=T,is.normal=F,B=1000,isignif(apply(coefm.cond.replace,2,function(x)quantile(x,c(.025,.975),na.rm=T),3)ar1ar2ar3interceptnoisevar2.5%0.611-0.700-0.67204.980.51697.5%1.3000.241-0.04176.321.500>#MethodIII基于平穩(wěn)自助法的置信區(qū)間，且誤差服從正態(tài)分布>coefm.norm=arima.boot(res,cond.boot=F,is.normal=T,

14、ntrans=100,B=1000,init=sqrt(hare)>signif(apply(coefm.norm,2,function(x)quantile(x,c(.025,.975),na.rm=T),3)ar1ar2ar3interceptnoisevar%0.687-0.747-0.66004.990.508%1.3800.192-0.01686.331.500>>#MethodIV基于平穩(wěn)自助法的置信區(qū)間，且誤差不服從正態(tài)分布coefm.replace=arima.boot(res,cond.boot=F,is.normal=F,ntrans=100,B=1000

15、,init=sqrt(hare)>signif(apply(coefm.replace,2,function(x)quantile(x,c(.025,.975),na.rm=T),3)ar1ar2ar3interceptnoisevar2.5%0.70-0.715-0.66204.980.4797.5%1.360.183-0.01876.301.503附自編函數(shù)Myar#用于自回歸模型的參數(shù)估計(jì)，整合矩估計(jì)，最小二乘估計(jì)，以及極大似然估計(jì)#該函數(shù)用于對(duì)時(shí)間序列中心化數(shù)據(jù)(因此截距項(xiàng)一定為0)估計(jì)AR模型的參數(shù)，AIC為真時(shí)，滯后項(xiàng)根據(jù)AIC準(zhǔn)則確定，為假時(shí)則根據(jù)設(shè)置的order.max設(shè)

16、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<-ols2ywcoef<-yw2mlecoef<-mle2result=data.fra

17、me(olscoef,ywcoef,mlecoef)colnames(result)=c('最小二乘估計(jì),矩估計(jì)','極大似然估計(jì)')return(result)Myarima#用于自回歸模型的參數(shù)估計(jì)，整合矩估計(jì)，最小二乘估計(jì)，以及極大似然估計(jì)#該函數(shù)用于對(duì)時(shí)間序列中心化數(shù)據(jù)(因此截距項(xiàng)一定為0)估計(jì)AR模型的參數(shù)，AIC為真時(shí)，滯后項(xiàng)根據(jù)AIC準(zhǔn)則確定，為假時(shí)則根據(jù)設(shè)置的order.max設(shè)定Myarima=function(tsdata,order=c(0,0,0)library(TSA)result=NULLcss<-arima(tsdata,or

18、der=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(r