時(shí)間序列作業(yè)第五章

1、1、（1）判斷序列的平穩(wěn)性該序列時(shí)序圖如圖1所示：時(shí)序圖顯示該序列有顯著的變化趨勢(shì)，為典型的非平穩(wěn)序列。（2）對(duì)原序列進(jìn)行差分運(yùn)算：對(duì)原序列進(jìn)行1階差分運(yùn)算，運(yùn)算后序列時(shí)序圖如圖2所示時(shí)序圖顯示差分后序列在均值附近比較平穩(wěn)的波動(dòng)。為了進(jìn)一步確定平穩(wěn)性，考察差分后序列的自相關(guān)圖，如圖三所示：AutocorrelationPartialCorrelationACFACQ-StatProb匚1匚11-0.155-0.1552.60690.106111120.019-0.0052.64780.26611113-0.069-0.0693.17930.3651匸11匚14-0.D89-0114407400

2、.39611115-0.031-0.066418450.5231|11160.1050.0875.43770.4S911n70.18S0.2169.54090.21611110-0.056-0.0059.90330.2721111190.0480.04310.1710.3371匚11110-0131-0.07312.2290.27011111110.0630.07312.7080.31311111120.0100.04112.7220.39011111130.041-0.00212.9280.453匚1匚1U-0140-0.19915.370035S1|11150.1530.12518.326

3、0.246匚11匚116-0.143-0.09120.9190.132111117-0.041-0.06921.1S50.22011111180.065-o.ou21.6360.24611111190.0360.065213580.29111111200.0430.06522.1040335自相關(guān)圖顯示差分后序列不存在自相關(guān)，所以可以認(rèn)為1階差分后序列平穩(wěn)，從圖中我們還可以判斷差分后序列可以視為白噪聲序列。（3）對(duì)白噪聲平穩(wěn)差分序列擬合AR模型原序列的自相關(guān)圖和偏自相關(guān)圖如圖4：ACFACQ-StatProbAutocorrelationPartialCorrelation0.8350.885

4、86.2240.0000.0000.077157.360.0000.699-0.105212.210.0000.6290.D68256.970.0000.&830.104295.790.0000.5490.045330.650.0000.481-0.179S57.630.0000.4-01-0.118376.540.0000.309-0.075387.390.0000.227-0.035394.090.0000.1930_140398.640.0000.146-0.103401.250.0000.096-0.100402.330.0000.034-0.044402.530.0000.0040.

5、175402.530.0000.046-0.095402.300.0000.0590.003403.260.0000.0750.006403.990.0000.096-0.051405.210.0000.120-0.005407.150.000圖中顯示序列自相關(guān)系數(shù)拖尾，偏自相關(guān)系數(shù)1階截尾，實(shí)際上我們用ARIMA（10，0）模型擬合原序列。在最小二乘估計(jì)原理下，擬合結(jié)果為：x=0.888x+31.489+&tt1t對(duì)殘差序列進(jìn)行檢驗(yàn)殘差白噪聲檢驗(yàn)：AutocorrelationPartialCorrelationACFACQ-StatProb|匚11匚11-0113-0.1131.38550

6、.23911111120.0410.0291.56970.456111113-0.036-0.0291.71640.63311114-0.057-0.0662.07920.721111150.000-0.0112.07920.8381|1|60.1110.1153.4S970.7451n170.1990.227S.0S99032511113-0.054-0.0163.43510.39211111190.0460.0293.68240.4671匚11110-0.111-0.07310.1410.4281111110.0630.07210.6930.4691111120.0140.02310.72

7、30.553參數(shù)顯著性檢驗(yàn)：VariableCoefTicientStd.Errort-StatisticProb.C31.4S90511.176612.3174-060.00&8X-1)0.83343200(3940422.546730.0000圖中顯示：延遲6階和12階的P值均大于0.05，可以認(rèn)為該殘差序列即為白噪聲序列，系數(shù)顯著性檢驗(yàn)顯示兩參數(shù)均顯著。這說(shuō)明ARIMA(1,0,0)模型對(duì)該序列建模成功。(5)模型的預(yù)測(cè)：估計(jì)下一盤(pán)的收盤(pán)價(jià)為：xt=888x289+31.489=288.1212、(1)繪制時(shí)序圖：時(shí)序圖顯示該序列具有長(zhǎng)期遞增趨勢(shì)和以年為周期的季節(jié)效應(yīng)。（2）差分平穩(wěn)化對(duì)

8、原序列作1階差分，希望提取原序列的趨勢(shì)效應(yīng)，差分后序列時(shí)序圖考察差分后序列相關(guān)圖和偏自相關(guān)圖的性質(zhì)，進(jìn)一步確認(rèn)平穩(wěn)性判斷，并估計(jì)擬合模型的階數(shù)。AutocorrelationPartialCorrelationACFACQ-StatProbI111213U151617181920-0.523-0.528474380.0000.234-0.06256.8140.000-0.0500.06757.2490.000-0.164-0.21161.9270.0000.177-0.020673610.000-0.271-0.20730.2620.0000.136-0.075363790.000-0.224

9、-0.23995.2930.0000.032-0.27495.4790.0000.2SS0.151105.620.000-0.4&4-U414142.940.0000.7310.453240.230.000-0.491-0.004284350.0000.253-0.009296.660.000-0.0030.208296.660.000-0.1320.065299.920.0000.1620.060304.860.000-0.307-0.041322.700.0000.163-0.182327.780.000-0.1350.026334.390.000自相關(guān)圖和偏自相關(guān)圖顯示延遲12階自相關(guān)系

10、數(shù)和偏自相關(guān)系數(shù)大于2倍標(biāo)準(zhǔn)差范圍，說(shuō)明差分后序列中仍有非常顯著的季節(jié)效應(yīng)。延遲1階的自相關(guān)系數(shù)和偏自相關(guān)系數(shù)也大于2倍的標(biāo)準(zhǔn)差，這說(shuō)明差分后序列還具有短期相關(guān)性。根據(jù)差分后序列自相關(guān)圖和偏自相關(guān)圖的性質(zhì)，嘗試擬合ARMA模型，但擬合效果均不理想，擬合殘差均通不過(guò)白噪聲檢驗(yàn)。所以我們可以考慮建立乘積模型：ARIMA(11,1)x(0,1,1)12：Wx二(1-0B12)12t12t1（4）參數(shù)估計(jì)使用最小二乘法估計(jì)方法，得到該模型的估計(jì)方程為1+0.986VVx=（1-0.833B12）12t1-0.606Bt（5）模型的檢驗(yàn)對(duì)擬合模型進(jìn)行檢驗(yàn)，檢驗(yàn)結(jié)果顯示該模型順利通過(guò)了殘差白噪聲檢驗(yàn)（圖2

11、1）和參數(shù)顯著性檢驗(yàn)（圖22）。白噪聲檢驗(yàn)（圖21）AutocorrelationPartialCorrelationACPACCl-StatProb1|1|10.0490.0490.40710.52311112-0.009-0.0110.4204-0.310l11113-0.D50-0.0490.85060.S371l1|40.0680.0731.65340.799111150.001-0.0071.65360.3951|1|60.0810.0812.8174-0.S311|1|70.0300.0303.93620.787II1|10-0.086-0.1005.2374-0.73211119

12、0.1010.1267.07170.63011111100.0350.0137.29640.69711|111-0.031-0.0567.47370.760匚1匸112-0.271-0.255208200.053l1|113-0.D61-0.06S21.4900.0641111U0.0020.00721.4910.0901|111150.0470.02621.9010.1101111160.002-0.00021.9020.146參數(shù)顯著彳性檢驗(yàn)（圖22)|170.0090.04321.9170.188VariableCoefficientStd.Errort-StatisticProb.C2

13、.1990260.9086702.4200500.0167X-1)0.9231160.03004629.926610.0000AR1)0.6056760.0788297.6834270.0000SAR(12O.0S294-O0.04705517.701230.0000MA(1)-0.9362380.013336-73.953100.0000（6）模型預(yù)測(cè)下一年度該城市月度嬰兒出生率預(yù)測(cè)如下表月份123456789101112預(yù)27.6127.6027.8927.7627.8827.8027.8427.8327.827.8427.7427.78測(cè)值1452158612883、（1）展開(kāi)原模型，等

14、價(jià)形式為：（1-B）x二（1-0.3B）tt即x二x+80.3sx=50,x(1)=51tt1tt1100100所以x100(1)=x1000.38100Ax100(2)Ax100(1)=51x101=x100(1)+81018101=1x101(1)=x1010.38101=51.7從該序列時(shí)序圖中可以看到該序列為非平穩(wěn)序列。（2）模型擬合：ITI/1LU*LXmLJ,7L.IKJhJTVariableCoefficientStd.Errort-StatisticProb.X(-10-9955140.007552131.S1770.0000MA(1)0.5970580.0073926.793

15、0850.0000R-squared0.88S485Meandependentvar185Z634AdjustedR-squared0.886S69S.D.dependentvar221.9333S.E.ofregression74.66405Akaikeinfocriterion1149164Sumsquaredresid334655.7Schwarzcriterion1155538Loglikelihood-4-05.9532Hannan-Quinncriter.11.51699urbin-Watsonstat2.050394-ARCH過(guò)程檢驗(yàn)：HeteroskedasticityTest

16、:ARCHF-statistic0.027076Prob.F1,6S)0.S69SObsR-squared0.027362Prob.Chi-Square(10.8674-異方差懷特檢驗(yàn)：HeteroskedasticityTest:WhiteF-statistic0.137351Prob.F(3,67)0.9374ObsR-squared0.433933Prob.Chi-Square(30.9331ScaledexplainedSS0.3S2676Prob.Chi-Square(S)0_953fiDW=2.05序列中殘差不存在自相關(guān)；懷特檢驗(yàn)之后也不存在異方差；ARCHLM檢驗(yàn)之后也不存在AR

17、CH過(guò)程。所以確定該模型為：x二0.9955x+stt1t二卩+0.597卩ttt1（3）預(yù)測(cè)：19391945年英國(guó)綿羊的數(shù)量預(yù)測(cè)如下表年份1939194019411942194319441945預(yù)測(cè)值16521645163716301623161516085、（1）考察該序列的方差齊,性：該序列時(shí)序圖顯示序列顯著非平穩(wěn)，如圖所示：對(duì)序列一階差分之后殘差進(jìn)行懷特檢驗(yàn)，檢驗(yàn)結(jié)果如下：HeteroskedasticityTest:WhiteF-statistic7.506816Prob.F(1,305)0.0065ObsR-squared7374535Prob.Chi-Square(10.006

18、6ScaledexplainedSS26.29378Prob.Chi-Square(10.0000結(jié)果說(shuō)明序列殘差存在異方差，（2）但殘差序列異方差時(shí)，我們需要對(duì)它進(jìn)行進(jìn)一步的處理，由于我們不知道異方差的具體函數(shù)，所以擬合條件異方差模型。x=PX+我們模擬的方程形式為：GARCH(2,1)即tt-itG2=0+02t1t12t2t1采用ARCH方法得到的擬合結(jié)果為：GARCH=C(2)+C(3fRESID(-12+C(4fRESID(-22+C(5)*GARCH(-1)VariableCoefficientStd.ErrorStatisticProb.X(-1)1.0002220.001554643.81730.0000VarianceEquationC0.0076010.0014575.2159080.0000RESID(-12-0.0297910.001566-19.019760.0000RESID(-220.1219520.0179696.7866780.0000GARCH-1)0.8935050.0166S953538030.0000R-squar