R语言实现ARMA模型的估计

基于R的ARMA模型的估计首先，我们给出一个ARMA模型：yt=_0,6yt_L+St_0.8备」随机生成一组含200个观测值的时间序列，代码如下：#ARMA(1,1)y[t]=-0.6y[t-1]+x[t]-0.8x[t-1]set.seed(10)x<-rnorm(200)y<-vector(length=2)y[1]=x[1]for(iin2:200)(y[i]=-0.6*y[i-1]+x[i]-0.8*x[i-1])y事实上，在R中有更简单的语句可以生成ARIMA时间序列，以上述ARMA(1,1)模型为例:set.seed(10)y<-arima.sim(list(order=c(1,0,1),ar=-0.6,ma=-0.8),n=200)在本次实验中，我们采用第一种方法生成的时间序列做估计。时间序列图如下：ts.plot(y)ACF和PACF0如下：acf(y,xaxp=c(0,20,20),yaxp=c(-1,1,10))pacf(y,xaxp=c(0,20,20),yaxp=c(-1,1,10))Sbriasyfieriasy下面给出三个模型的估计：模型1：yt=ayt」.；t模型2：yt=：优乂」,飞,b；t」模型3：yt=ayt」,azyt/【模型1]a<-1;b<-0,c<-0ARMA<-arima(y,order=c(a,b,c),method="ML")ARMAcall;ar-iria(x=order=c(l,0,Q)(method="ml")coefficients:arlinxercept-0.8114-0.0174S.e.0,04050.0429signal?esTimaredas1.2;log1ikelihood=7。？,*aic=611.2SBC准贝U:#SBC=-2ln(模型中极大似然函数值)+ln(n)(模型中未知参数个数)loglike<-ARMA$loglikSBC<--2*loglike+log(200)*1SBC>SBC[1]6164g5残差平方和：residual<-ARMA$residuals#残差ssr<-0for(iin1:200)(ssr=ssr+(residual[i]A2))ssr#残差平方和>ssr>♦残差平方和[1]2430993Ljung-Box检验：Box.test(residual,type="Ljung",lag=8,fitdf=1)Box.test(residual,type="Ljung",lag=16,fitdf=1)Box.test(residual,type="Ljung",lag=24,fitdf=1)Box^test(residual,type=11Ljung'*,1ag=8,-Fitdf=1)BOX-Ljungdata:residualX-squared=34.61,df=7,p-v^lu&=1.324@-05box.tesi(residual,Type='LjungP,,189=16^11df=1)bax*Ljungrestdata:residualx-5quared=42.857,df=15,p-value=0.0001657box.tesr<res_idua1rType="Ljupgri11ag=24,fitdf=1)Box-Ljungxesxdata：residualX-squared=55.Bfi2tdf=23Tp-value=0.000147【模型2](R语言代码类似模型1,以下代码省略)上为之-[；bv-0；C-^—j_aARMA<-arima(y,order=匚(a,b】c),method='*ML0)>ARMAcall:ar1=ytorder=c(a,b,c),method="ml")coefficients.:arLintercept6267-0.715。-O.O145S.Q.0,05^90,06040,0119siescimatedas0.R9土：1og1iIcelihood=-273.45)aic=554.9#5BC-一£ln[僵整甲假大僚鸿图数恒)+ln【ri"模型申水都秀却个数)一loglikeK-ARMASloglikSBC<--2Aloglike+log(200)*2a5BC[1]557,5011

>residual<-ARMAiresiduals单残差>ssr<-0>far(1Ini:200)+ssr=3sr-f-(resIdual['i]A2)+?>35「春茏差平方和fl]176*6069ii>eox.t«st(residualTtypei="Ljung"5fitdf=2)Box-ljurgTestdaxa:residualX-squar€d-2.0744,df-61p-VAlue-0,-9127>Box.teat(residual,type=1lLjun^11,1ag=1611£|^=2)box^ljungTestdata:residualx-squarcd=11.407,df=14,p-value=0.6538>eo^+T€st(residualtxype^'Ljung"Box-Ljungtestdata;r-esidu-alX-squared=2G.051,df=22fp-value=0.5799【模型3]>a<-2;b<-0;c<-0>ARMA<-arima(y,orderc),mert*iod="bMLlh3AARMAcall：coefficients:■arl-1.12035,e.Q.0651arimafx=y,order=c(acoefficients:■arl-1.12035,e.Q.0651ar2intercept-0.3771—0.01690.06500.02Sfi□Igma^?esxlmatedas1,027:log1ikelihood=-2Q7,11,aic=53?,22ASBC[1]584.8185assr校I差平方和[1]205,5252eox.test(residual,typ电="Ljung"Jag=8,fitdf=2)BOX-LjungTestdata;residualx-squared-20.65Tdf-6,p-value=0.00212BOX.test(residual,type="LjLing,1,1ag=16,f1tdf=2)Box-LjungTestdata:residualx-squarcd=31.305,df=14,p-value=0,005024Boxiest(residualhrype="Ljung,T11ag=24,fntdf-2)Box-Ljungtestdata；re5idualx-squared=43,5^3tdf=221p-valu€=0.003578yt=a〔yt」tyt=a乂」th「yt三研1」azyt/；t残差平方和240.10178.61205.334的倩计值-0.8114-0.6267-1.1203（标准差）(0.0405)(0.0599)(0.0651)a2的倩计值-0.3771（标准差）(0.0650)b的估计值-0.7150（标准差）(0.0604)AIC;SBC611.2;610.5554.9;557.5582.2;584.8Ljung-BoxQQ(8)=34.610(0)Q(8)=2.074(0.9127)Q(8)=20.650(0.0021)统计量（显著Q(16)=42.857(0.0002)Q(16)=11.407(0.6538)Q(16)=31