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第二章习题答案2.1（1）非平稳（2）0.01730.7000.4120.148-0.079-0.258-0.376（3）典型的具有单调趋势的时间序列样本自相关图AutocorrelationaCorreiaticin2.21.000000.700000.412120.14848-.07879-.25758-.37576脚illillillillill|IJIIlllIll Ilillillillillillillill |111Iipllll111illillillilliliillillillallillilill11illalaillalaillilillI|ii|lipI|H|ii|iTiT!TT!illillillillillillillillallillilaill11ill1|ii|11|11|11|111H|i111|1JTT111illillillillillillillillTTTTT精品.资料关图AutocarrelationsGerre1ation-1887654321012345678811.00000illillillillillillillillillillillillillillillillillillillilli|ii|ii|upipi|ilUITITllUIT0.907E1illiliilullilnhiliillilliInlullilnhiliillillill1|11|ll|11|11|11|111H|11|11|ll|11|11|11|111H|11111110.72171illillillillill1IIillillillillillillill1II|lip|0.512E2illillillillill1II111illillill|1111110.34982illillillillillillillTTTiT.0.24C90iiiiiiiiiiiiiiiTTiT.0.20809轉榊.0.21021栉榊0.2G429illiliilullillTTiT.0.36433iiiiiiiiiiiiiiiiiiiii0.48472illillillillillillillillillillininin|ii|n|ii|n|i111111.0.58456iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii|ii|ii|upipi|ii|iTii|n0.G0198illillillillillillillillillillillill1|11|ll|11|11|11|111H|11|11|ll|11|10.51841illillillillillillillillillillIJII|lipijll|IIJIIJI11|0.36056illillillillill1II111|10.20671O.OS1380.00135-.03248:十;-.02710出0.011240.08275艸：0.17011笊栅i.0.24320illillillillillIjll11111!1|10.25252illillillillill|IJImarkstwowtandarderrors2.3（1）自相关系数为：0.20230.013-0.0940.0248-0.068-0.0720.0070-0.0250.075-0.141-0.139-0.0340.206-0.0100.042-0.043-0.179-0.2510.0140.1090.2170.316-0.204-0.2450.0660.00620.0800.1182）平稳序列3）白噪声序列2.4LB=4.83,LB统计量对应的分位点为0.9634,P值为0.0363。显著性水平=0.05，序列不能视为纯随机序列。2.5（1）时序图与样本自相关图如下2)非平稳3)非纯随机2.6(1) 平稳，非纯随机序列(拟合模型参考：ARMA(1,2)(2) 差分序列平稳，非纯随机第三章习题答案3.1解：E(x)二0.7-E(x)+E(s)tt1t(10.7)E(x)=0tE(x)=0ttt(10.7B)xx=(10.7B)-is=(1+0.7B+0.72B2+)sttt1Var(x)=c2=1.9608c2t10.49ssp=e2p=0.490=0210223.2解：对于AR(2)模型：p=0p+0P=0+0P二0.5J11021121p=0p+0p=0p+0=0.321120112解得：0=7/15J0=1/1523.3解：根据该AR(2)模型的形式，易得：E(x)=0t原模型可变为：x=0.8x0.15x+stt1t2tVar(x)=t102c2(1+0)(100)(1+00)21212(1+0.15)(10.15)(10.8+0.15)(1+0.8+0.15)c2=1.9823c2p=0/(10)=0.6957112Jp=0p+0p=0.406621120p=0p+0p=0.2209312210=p=0.6957111J0=0=0.152220=0333.4解：原模型可变形为：(1-B-cB2)x=tt由其平稳域判别条件知：当10i1,e+e1且e-e1时，模型平稳。22121由此可知C应满足：丨c11,c-11且c+11即当一1c1，2模型非平稳；九二1.37381九=-0.873622)|e|=0.31，2e+e=0.81，21=-1.41,模型平稳。九二0.61九=0.523)|0|=0.31，20+021=0.61，0-02=-1.21，1模型可逆。九=0.45+0.2693i1九=0.450.2693i24)|0|=0.41，20+021=-0.91，21模型不可逆。九=0.25691=-1.55695)|e|=0.71，1模型平稳；九=0.71|0|=0.61，1模型可逆；九二0.616)|e|=0.51，2e+e=-0.31，2121模型非平稳。九=0.41241九=-1.2124210|=1.11，模型不可逆；尢=1.1。113.12解法1：G=1，G=eG-0=0.6-0.3=0.3，01101GGk-1G=0.3X06k-1,k2k1k-111所以该模型可以等价表示为：X=+左0.3X0.6kttt-k-1k=0解法2：(1-0.6B)x=(1-0.3B)8ttx=(1-0.3B)(1+0.6B+0.62B2+)&tt=(1+0.3B+0.3*0.6B2+0.3*0.62B3+)&t=s+兰0.3*0.6j-18G二1,G=0.3*0.6j-10j3.13解：E(B)x=E3+0(B)8n(1-0.5)2E(x)=3tttE(x)二12。t113.14证明：已知0二,0二，根据ARMA(1,1)模型Green函数的递推公式得:1214G=1，G=0G-0=0.50.25=02，G=0G=k-1G=协+1,k2011011k1k-1111Pk3.153.16艺GG02+艺0jj+1114=0=-艺G21+艺02(j+1)j1j=0j=1艺GG艺G(0Gjj+kj1j+k-1j=0j=01)解：成立1)x艺G2jj=02)成立-10=0.3*(x11-0202-04+051=114041-02+041+1111-021)艺GGjj+k-1-=0j=01%j=01Pk-1j=03)成立4)不成立-10)+8，t-1txT=9.67=0.2726=9.88T+1(1)=E(x)=E10+0.3*(x-10)+8t+1Tx(2)=E(x)=E10+0.3*(x-10)+8=9.964Tt+2T+1T+2x(3)=E(x)=E10+0.3*(x-10)+8=9.9892Tt+3T+2T+3已知AR(1)模型的Green函数为：G=0j，j=1,2,j1e(3)=Gs+Gs+Gs=w+028T0t+31t+22t+1t+31t+21t+1Vare(3)二(1+0.32+0.092)*9二9.8829Tx的95%的置信区间：9.9892-1.96*9.8829,9.9892+1.96*.98829t+3即3.8275,16.1509(2)8二x-X(1)二10.5-9.88二0.62T+1T+1TX(1)二E(x)二0.3*0.62+9.964二10.15T+1t+2X(2)二E(x)二0.09*0.62+9.9892二10.045T+1t+3Vare(2)二(1+0.32)*9二9.81T+2x的95%的置信区间：10.045-1.96乂哂981,10.045+1.96911t+3即3.9061,16.1839。3.17 (1)平稳非白噪声序列(2) AR(1)(3) 5年预测结果如下：FareceistsforYarieiblexbsForecastStdError35SConfidenceLimits6490.156322.729445.6075134.7050S583.888223.886337.1698130.60656681.908323.944034.9789128.837667SI.82923.354734.33S5128.332S881.085323.955834.1329128.03773.18 (1)平稳非白噪声序列2)AR(1)(3) 5年预测结果如下：ForecastsforvariablexObsForecastStdError750.70460.2771760.7956D.2957770.82950.2981780.8421D.2985790.84600.298595XConfidenceLimits0.16150.21610.24520.25710.28171.247G1.37511.41391.42711.43193.19 (1)平稳非白噪声序列(2)MA(1)(3)下一年95%的置信区间为(80.41,90.96)3.20 (1)平稳非白噪声序列(2)ARMA(1,3)序列WORD格式分享第四章习题答案4.1解：1x=(x+x+x+x)T+14TT-1T-2T315551(x+x+x+x)=x+x+x+x4T+1TT-1T216T16T-116T216T-35八,以，在t+2中t与t1前面的系数均为16。4.2 解由x=ax+(1tt)xttt1x=ttx+(1tt)xt+1t+1t代入数据得x=5.25tt+5(1tt)t5?26=5.5tt+(1tt)xt解得x=5.11的情况)4.3 解：(1)11x=(x+x+x+x+x)=_(13+11+10+10+12=11.221520191817165精品资料WORD格式.分享精品.资料x=(x+x+x+x+x)=!(11.2+13+11+10+10=11.0422521201918175利用Xt=0.4Xt+0,6Xt-1且初始值%0=%1进行迭代计算即可。另外，X22=X21=X20该题详见Excel。11.792773)在移动平均法下:X二1X+-昱X215205ii=16X=-X+-X+-昱X225215205ii=171116a=+x=在指数平滑法中:x2255525=x=x=0.4x+0.6x21202019b=0.4=0.16。zr.b一a=0.4254.4解：根据指数平滑的定义有（1）式成立，（1）式等号两边同乘（1Q）有（2）式成立x=tu+(t一10(1a)+(t一2)a(1a)2+(t2)a(1a)3+(1)t(1a)x=ta(1a)+(t1)a(1a)2+(t2)a(1a)3+(2)t-得ax=taa(1a)a(1a)2tx=t(1a)(1a)2t1a=taIt-匕则limt=limII=1otT8ttT8ItI4.5该序列为显著的线性递增序列，利用本章的知识点，可以使用线性方程或者holt两参数指数平滑法进行趋势拟合和预测，答案不唯一，具体结果略。4.6 该序列为显著的非线性递增序列，可以拟合二次型曲线、指数型曲线或其他曲线，也能使用holt两参数指数平滑法进行趋势拟合和预测，答案不唯一，具体结果略。4.7 本例在混合模型结构，季节指数求法，趋势拟合方法等处均有多种可选方案，如下做法仅是可选方法之一，结果仅供参考列：x=T+S+1。（注：如果用乘法模型也可以）tttt首先求季节指数（没有消除趋势，并不是最精确的季节指数）0.9607220.9125751.0381691.0643021.1536271.1165661.042920.9841620.9309470.9385490.9022810.955179消除季节影响，得序列y=X-Sx，使用线性模型拟合该序列趋势影响（方法不唯一）:tttT=97.70+1.79268t，t二1,2,3,t注：该趋势模型截距无意义，主要是斜率有意义，反映了长期递增速率）得到残差序列1=xSx=yT，残差序列基本无显著趋势和周期残留。tttttX302010010-20-30-aiJAN48aiJAM50O1JAN5201JAN5401JAN5601JAN58YesirJANFEBMARAPRJUN196260S.307608.002809.997612.172814.4226ie.62S1963613.380620.197622.556626.132630.059638.2731964645.771649.468852.4866E4.405S55.241655.5521365672.427673.544673.923673.773673.316672.7211966604.132689.462894.546699.150703.20470S.735196772S.303727.237728.134729.363730.844732.615196S740.451741.853743.154744.442745.822747.51&1969751.978752.171753.571756.280759.856763.5181970771.350771.557772.252773.571776.OSS773.771脱690.900692.614634.518636.588688.761700.926is1.159CurveRatioD1213-termFinalTrendCycle-HendersonMovingAveragertppliedI/CYeeirJULmSEPOCTmDECTotal1962618.744620.335621.173621.202620.591619.7717389.341963635.SGI636.446637.006637.7G9839.393642.1557579.731964656.079657.423659.903663.336667.078670.3107887.051965672.119671.764671.959673.029S75.423679.2108053.21isee709.932713.173716.547719.787722.667724.8508484.13iae?734.302735.554736.4&9737.289738.073733.105S795.351968743.5W751.47D752.880753.495753.296752.5818976.471969766.763763.256770.681771.206771.284771.2889157.851970784.011788.01D791.379793.GG3795.195795.97G9392.84Avs702.980704.826706.444707.864709.222710.584I/CRatio13-termMovingAverageis1.159Total：7574GMean：701.35S.D.：56.792DIEFinalTrendCycle-HendersonCurveAppIied预测1971年奶牛的月度产量序列为x=T+Sx,t=109,110,ttmod(/12),120得到771.5021739.517829.4208849.5468914.0062889.79893）839.9249800.4953764.3547772.0807748.4289787.3327该序列使用x11方法得到的趋势拟合为趋势拟合图为4.8 这是一个有着曲线趋势,但是有没有固定周期效应的序列,所以可以在快速预测程序中用曲线拟合（stepar）或曲线指数平滑（expo）进行预测（trend=3）。具体预测值略。第五章习题5.1拟合差分平稳序列，即随机游走模型X=x+，估计下一天的收盘价为289t-1t5.2拟合模型不唯一，答案仅供参考。拟合ARIMA(1,1,O)模型，五年预测值为：FcirecastsforbsForecastStdError35蛊ConfidenceLimits61341444.41267372.7986326993.3929355894.832362349S21.846315413.404323632.0566$76211.C3&0G3357040.971118817.529320159.2928393922.649464363454.004023594.255317210.1143409697.893865369499.988127836.917314940.G341424059.3421讥iriatilex5.3 ARIMA(l,l,O)x(1,1,0)125.4 (1)AR(1)，(2)有异方差性。最终拟合的模型为x=7.472+8tt8=-0.55958+vJtt-1tv=th=11.9719+0.4127v2tt-15.5(1)非平稳(2)取对数消除方差非齐，对数序列一节差分后，拟合疏系数模型AR(1，3)所以拟合模型为lnxARIMA(1,3),1,0)3)预测结果如下：bs85ConfidenceLimits737.52140.0405747.54010.0682757.51450.0908767.49490.1029777.48160.110178?.48550.1153797.49350.1209ForecastStdError7.44207.40647.33637.29327.26587.25957.25657.60077.67887.69267.69687.69747.71157.73055.6原序列方差非齐，差分序列方差非齐，对数变换后，差分序列方差齐性。第六章习题6.1 单位根检验原理略。例2.1原序列不平稳，一阶差分后平稳例2.2原序列不平稳，一阶与12步差分后平稳例2.3原序列带漂移项平稳例2.4原序列不带漂移项平稳例2.5原序列带漂移项平稳(=0.06)，或者显著的趋势平稳。6.2 (1)两序列均为带漂移项平稳(2) 谷物产量为带常数均值的纯随机序列，降雨量可以拟合AR(2)疏系数模型。(3) 两者之间具有协整关系(4) 谷物产量二23.5521+0.775549降雨量tt6.3 (1)掠食者和被掠食者数量都呈现出显著的周期特征，两个序列均为非平稳序列。但是掠食者和被掠食者延迟2阶序列具有协整关系。即y-0x为平稳序列。tt-2(2)被掠食者拟合乘积模型：ARIMA(0,1,0)x(1,1,0)，模型口径为：1WX=5t1+0.92874B5t拟合掠食者的序列为：y=2.9619+0.283994x+-0.47988tt-2tt-1未来一周的被掠食者预测序列为：ForecastsforvariablexObsForecastStdError95%ConfidenceLimits4970.792449.4194-26.0678167.652650123.835869.8895-13.1452260.816751195.098485.596827.3317362.865152291.637698.838797.9173485.357953150.0496110.5050-66.5363366.63555463.5621122.5322-176.5965303.72085580.3352133.4800-181.2807341.95115655.5269143.5955-225.9151336.96905773.8673153.0439-226.0932373.82795875.2471161.9420-242.1534392.64755970.0053189.8525-302.0987442.109460120.4639214.1559-299.2739540.201761184.8801235.9693-277.6112647.371462275.8466255.9302-225.7674777.4606掠食者预测值为：ForecastsforvariableyObsForecastStdError95%ConfidenceLimits4932.769714.72793.903661.63585040.179016.33818.157072.20115142.334621.8052-0.402885.07215258.299325.98327.3732109.22545378.970729.542121.0692136.872254106.596332.709042.4879170.70475566.483635.5936-3.2787136.24585641.968138.6392-33.7634117.69965746.754841.4617-34.5085128.01825839.720144.1038-46.7218126.16195944.934246.5964-46.3930136.26146045.328648.9622-50.6356141.29286143.841156.4739-66.8456154.52796258.172563.0975-65.4964181.84136.4 (1)进出口总额序列均不平稳，但对数变换后的一阶差分后序列平稳。所以对这两个序列取对数后进行单个序列拟合和协整检验。(2)出口序列拟合的模型为InxARIMA(1,1,O)，具体口径为:tVInx=0.14689+t11-0.38845Bt进口序列拟合的模型为InyARIMA(1,1,0),具体口径为:tVlny=0.14672+t1-0.36364B(3)Iny和Inx具有协整关系tt4) 协整模型为:lny=0.99179lnx+-0.69938tttt-15) 误差修正模型为:Vlny=0.97861Vlnx-0.22395ECMttt-1
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