应用时间序列分析第4章答案解析

河南大学:姓名:汪宝班级:七班学号:1122314451班级序号:685:我国1949年－2008年年末人口总数（单位:万人）序列如表4－8所示（行数据）．选择适当的模型拟合该序列的长期数据，并作5期预测。解：具体解题过程如下：（本题代码我是做一问写一问的）1:观察时序图:datawangbao4_5;inputx@@;time=1949+_n_-1;cards;5416755196563005748258796602666146562828646536599467207662076585967295691727049972538745427636878534806718299285229871778921190859924209371794974962599754298705100072101654103008104357105851107507109300111026112704114333115823117171118517119850121121122389123626124761125786126743127627128453129227129988130756131448132129132802Iprocgplotdata=wangbao4_5;plotx*time=1;symbol1c=blackv=stari=join;run;10000SOOijO500001Adio1A7H1AAI17汕『I2(111120000ODCIOO9000080000?0LlflUrun;10000SOOijO500001Adio1A7H1AAI17汕『I2(111120000ODCIOO9000080000?0LlflU4Q0i血30000分析:通过时序图,我可以发现我国1949年－2008年年末人口总数（随时间的变化呈现出线性变化.故此时我可以用线性模型拟合序列的发展．-Xt=a+bt+Itt=123,...,6OE(It)=O,var(It)=6其中，It为随机波动；Xt=a+b就是消除随机波动的影响之后该序列的长期趋势。2:进行线性模型拟合：procautoregdata=wangbao4_5;modelx=time;outputout=outp=wangbao4_5_cup;procgplotdata=out;plotx*time=1wangbao4_5_cup*time=2/overlay;symbol2c=redv=nonei=joinw=21=3;run;

TheSriSSystem15:28Wednesday,December7,20131TheAUTOREGProcedureDependentVariablexClrdineiryLeastSquarusEstimaitesSSEMSESBCRegressR-SquareDurbin-Watson26223660745223551085.889830.99810.0724DFERootMSEAICTotalR-Sc1081luare582127.711140.9981VariableDFEmtimateStandardErrortValueApproxPr>111Intercept1-2770828313S6-88.34<.000111me11449is.esisC.OOOI分析：由上面输出结果可知：两个参数的p值明显小于0.05，即这两个参数都是具有显著非零，4:模型检验又因为RegressR-square=totalR-square=0.9931,即拟合度达到99.31%所以用这个模型拟合的非常好。5:结论所以本题拟合的模型为：Xt=-2770828+1449t+Itt=123,...,60E(It)=O,var(It)=66：作5期预测

procforecastdata二wangbao4_5method二stepartrend二2lead=5out=outoutfulloutest=est;idt;varx;procgplotdata=out;plotx*time=_type_/href=2008;symbol]i二nonev=starc二black;symbol2i二joinv二nonec=red;symbol3i二joinv二nonec=blackl=2;symbol4i二joinv二nonec=blackl=2;run;5D0D01B4DI5DQD0I4.DQD0MDflDflI2DQD0I1DQD0IflDaDflaDQDO9DQD0TDflDfl6DODO:MM13P196nIA?01BQO138020005D0D01B4DI5DQD0I4.DQD0MDflDflI2DQD0I1DQD0IflDaDflaDQDO9DQD0TDflDfl6DODO:MM13P196nIA?01BQO13802000MM6：爱荷华州1948-1979年非农产品季度收入数据如表4——9所示（行数据），选择适当的模型拟合该序列的长期趋势。解：具体做题过程如下：（本题代码我是做一问写一问的）1、绘制时序图datawangbao4_6;inputx@@;time=_n_;cards;TOC\o"1-5"\h\z60160462062664164264565568267869270773675376377577578379481382382682983183083885487288290391993792796297599510011013102110281027104810701095111311431154117311781183120512081209122312381245125812781294131413231336135513771416143014551480151415451589163416691715176018121809182818711892194619832013204520482097214021712208227223112349236224422479252825712634268427902890296430853159323733583489358836243719382139344028412942054349446345984725482749395067523154085492565358285965;procgplotdata=wangbao4_6;plotx*time;symbolc=blackv=stari=join;run;500030QDIDO[|id2D304D6D70EDmo分析;可知时序图显示该序列有明显的曲线递增趋势。尝试使用修正指数型模型进行迭代拟合：2、拟合模型procnlinmethod=gauss;modelx=a+b*c**time;parametersa=0.1b=0.1c=1.1;outputpredicted=xhatout=out;NLIN过程输出以下六方面信息：1）迭代过程x=a+bet+$,

tt……，128IlerIterati\■已PhasebSumofcSquares2）收敛状况3）估计信息摘要123415161718NOTE：0.10001126.21123.11119.01114.011C7.61098.81079.01065.71035.3.8.65844?33197COCO.1605.96D4.8604.86D4.8Convergence0.10001.10009.5216ES0.39561.06781.8825E80.52481.06E81.8SO5ES0.7M81.06251.8803E80.9S5&1.05961.8S01ES1.41121.05681.8773E82.02031.0535I.SCSSES2.89911.05051.8489E84.13611.04761.815E85.82241.04501.7634E810.25071.04031.74E821.07871.08471.6792E861.53051.02811.3238E8101.81.033222849327107.01.0312543005111.71.0807895973112.21.0307395624112.21.0307395624112.21.0307395624criterion（本次迭代收敛）EstimationNummaryMethodGauss-MevftonIterations18SubiteratIonsViAverageSubiterations1.222222R5.563E-7PPC(b)82.26EE-7RPC(b)5.627E-6Object1.99E-10Objective395624□bservationsRead128Obser^ationsUsed128□bservaticinmMissing04）主要统计量Source[JFSumofSquaresMeanSquareFValueModel22.4608E81.2304E838S75.9Error1253956243165.0UorrectedToteil1272.4648E8ApprcxPr>F<.00015）参数信息摘要ParameterEatimsiteApproxStdErrorApproximate95^ConfideneeLimits604.813.0152579.0630.E112.24.0939104.1120.31.03070.0002981.03011.0312得到的拟合模型为：x二604.8+112.2x1.03071+et=1,2,tt6）近似相关矩阵ApproximateCorr&lionMatrizabca1.0000000-0.87589790.8427539b-0.8758373I.ODOOOOO-0.9351736c0.8427533-0.93517361.00000003、拟合效果为了直观看出拟合效果，我们可以将原序列值和拟合值联合作图：procgplotdata=out;plotx*t=1xhat*t=2/overlay;symbollc=blackv=stari=join;分析：由上图图我们可以看出，原序列值和拟合值很接近，拟合效果较好。综合以上的分析，我们可以选择模型：x-604.8+112.2x1.03071+e来拟合该序列的长期趋势。拟合效果很不错。tt8:某城市1980年1月至1995年8月每月屠宰生猪的数量（单位：头）如表1—11所示（行数据），选择适当的模型拟合该序列的发展，并预测1995年9月至1997年9月该城市的生猪屠宰量。解：具体解题过程如下：（本题代码我是做一问写一问的）atawangbao4_8;inputx@@;time=_n_;cards;7637871947338739642810508495741110647100311941331030559059510145776889812919164396228102736100264103491970279524091680101259109564768928577395210937719820297906100306940891026807791993561117062812258835710617591922104114109959978801053869647997580109490110191909749898110718894177115097113696114532120110936071109251033121201841030691033511113311061611115909944710198785333869701005618954389265827197949874846738197702978446869787587869571757226418277357632925938078332723815597169750854727013379125858058177886852690697955688174666987225873445761318608275443739697813978646662697377680034706948182375640755408222975345770347858979769759827807477588841009796689051935038474774531919008163589797810227826577271850439541879568103283957709129710124411452510113993866951711001831039261026431083879707790901903368873283759992677329278943943999293790130910551060621035601040751017839379110231382413835341090119649910243010300291815990671100671015999764610493088905899361067238430711489610674987892100506procgplotdata=wangbao4_8;plotx*time=1;symbollc=redi=joinv=star;run;procarimadata=wangbao4_8;identifyvar=x;1：时序图与平稳性判别屠宰量大约在80000上下波动。所以由该序列图我可以认为它是个平稳的数列。即可以用第三章的AR模型或MA模型或ARMA模型进行拟合。但是为了稳妥起见，我还需要利用自相关图进一步辅助识别。具体如下:自相关图：1/:斗HWednesday,December1/:斗HWednesday,DecemberHU1H1The^RIMAProcedureNameofVariabIe-xMeanofWorkinsSeries90640.34StandsrdDeviation13880.89NumberofObccrvationc188NumberofObccrvationc188AutocorreIations&o1234567—no9o123&o1234567—no9o1234R-678:H.1—1—f—1—1—1—1—-a—1—LCovarianceCorreIation-1907654.9210294567891StdError19230119611462298610316775111Pfinanis0239461433C357CC88735393315703967970123630632668CCC51C7775911183ddbeitia^I57138675572040075127-13263711895052085763343414931.000000.594210.53432n.RfliRn0.430240.45C730.46000U.4Zi!bb0.413590.418000.045520.39352U.4Bb4I0.296210.296550.265810.192420.2700I0.17303iji丁订・・丁订・・iji订・iE・T"F>!>>!■：：：：：：0.0729320.095264n.11fin胎0.1255250.13313C0.1422520.1499560.15S17C0.1613910.1C750斗0.171282U.I沁氏0.1830070.1855400.1S80440.1910320.1910660.193035分析：由上图可知：样本自相关图中的自相关系数在延迟4阶之后几乎全部落入2个标准差范围之内，并且向零衰减的速度还是比较快的。所以我认为该序列是平稳的序列。由时序图与自相关图可知其是平稳序列。故可以用第三章的AR模型或MA模型或ARMA模型进行拟合lhe淞AsternV：42:edneW?December打HTheARIMAProcedureAutocorreationCheckforWhiteNoiseToChi-Pr>LagSquare[)FChiSq6311.336<.05010.53412519.0612<.00010.42818599.1818<.08010.29824634.1624<.00010.179-Autocorrelations----0.5350.5850.4300.4860.4600.4140.4180.3460.3940.4850.297o.2ee0.1920.2700.1780.1820.1200.1980.1780.106分析：由上面数据可知：由于p值显著小于0.05，故可以否定原假设（H。），接受备选假设旧丿，即我可以认为该序列是平稳的非纯随机性序列。这样就说明了我们可以根据历史信息预测未来月的生猪屠宰量。PairtialAutocorrektionsLagCorrelation-13876543210123456783112U045678901234567009012-■--------12U045678901234567009012-■---------.・■23240.594210.20098U.y1//B-O.OC3240.182930.03637O.O8G09-0.015560.10226-0.091360.137670.20160-0.20333-0.09154-0.116450.004490.05737-0.097120.00789-0.03669-0.017040.047190.058900.03542■H；艸.*!lL***：+：：+：：+：：■+：.!+!!4!■啊-Hi.「!+!!4!-:!+:半・分析：观察自相关图和偏自相关图，从这两图来看，偏自相关图是拖尾，而自相关系数是拖尾的。因而我们可以用ARMA模型进行拟合。但是为了稳妥起见，我还需要利用计算机进行相对最优定阶。2:相对最优定阶：identifyvar=xnlag=18minicp=(0:5)q=(0:5);run;2d1316SAS:玉h杳"l亡inTHeARIMA尸ldc=©«zI«_i厂□MA1MA3MA5MA-4MAZman5B12CSE^-BESlHTlnn^^EETls7221-1----5-曰曰曰曰「5HH7BES吕口1BED日却曰口73SB2B<1

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AAAARRAAAAAAEr-rizirNarI0aEr-rizirNarI0aMiriimljhiTutjaVa.Ii_i曰zBICC-4t.E>=,5)，因而我们接下来分析：从上图可以看出，在众多模型中，ARMA模型的BIC信息量是最小的是ARMA(4会采用ARMA模型来进行拟合分析，这和我们人工预测的相吻合。

,5)，因而我们接下来3:参数估计：estimatep=4q=5;run;具体输出结果如下图:TheARIMAProcedureCondiConditianaILeastSquaresEstimationParameterEstimateStandardErrortValueApproxPr>|t|7G42G.570G4.410.82<.00010.844110.239053.530.0005-0.480980.23838-2.020.Q4E1-0.424420.19931-2.130.03460.GG3130.09581G.94<.0001-0.204330.11103-1.840.06671.214E70.237455.11<■0001■0.702280.35137-2.000.04720.049$50.308&7o.ie0.S72O0.412430.189542.180.0309ConstsintEstimateVarianceEstimate1344.097991G4C17StdErrorEstimate9358.143AIC4004.756SBC4037.12NumberofResiduals188mrilCandSBCdonotinclude1理determinant.Paramet亡rMU脚，2昭1,3M闭，4胎1,5AR1J聃1,5Paramet亡rMU脚，2昭1,3M闭，4胎1,5AR1J聃1,5-0.021-0.791on-42o?56801-Mh-QuCxionl—ooo1ortR2-u5057-H--H--H--H-oAR1,2-0.016-0.S850.934-0.8190.2820.641-0.951■10

Ari7667080-H--M.M-_111Qu-H-1—4-H-TI—oono4_h-COooooooo-0.024-O.S010.798-0.8750.3830.569-0.844ParameterMUMA1n1MAI,2MAI,3渦1,4MU1.0000.033-0.0460.012-0.036MALI0.0331.000-0.8940.754-0.097-0.046-0.8341.000-0.8650.3660.0120.754-0.8651.000-0.612MAI,4-0.036-0.0870.366-n.m1.000舶1,5-0.021-0.7910.658-0.364-0.282胡1,1-O.OD20.950-0.8750.754-0.192胡1,2-0.016-0.8950.934-0.8190.282胡1,3O.OD70.786-0.8760.887-0.400ARI,4-0.024-0.8010.798-0.8750.383CorrelationsofFnrameterEstimsit亡s

1U：4UWednesday,December/,illlJIhe沁冷悴1U：4UWednesday,December/,illlJTheARIUAProcedureCorrelationsofParoet&rEstimatesAR1,3宜R1,4ARI,20.641-0.9511.000ARIJ-a.5190.840-0.934師1,4a.569-0.8440.857ParameterMAI,5ARIJARI,24023fl48080.857-0.9421.000riutocorrelationCheckofResiduaIsToChi-Pr>LagSquareDFChiSqAutocorrelationsoJI7453089131-H-7—-U-M-JI-H_1—039517JI22<.00010.00050.0044O.OOS40.00670.00330.0030.008C.109-0.0310.0730.0280.0270.0280.0290.0140.0840.D620.0330.0320.020-0.068-0.043-0.136-0.029-0.02&-0.0840.005-0.0960.0970.0.7—-H--H--H-7—1—1-25-634oooooo44846SCO£-£-H--n.LJIJ11—ooooo0.080ModeIfarvariablekEstimatedMean7642S.47AutcifEgr-EggiveFsictorsFactar1：1-1.21457时机1)+0.7D228帥觥2)-0.04985即狀3)-0.41243M⑷该输出形式等价于xt=(1-1.21457B+0.70228B2-0.04985B3-0.41243B4)€t故该模型为：xt=p+(e(B)®(B))€t=⑶序列预测(1995年9月至1997年9月)forecastlead=24id=timeout=wangbaoyc;run;

TheSASSystem18:43?ednesdaj/,December7,201317TheftRIMfiProcedureForecastsforvariableObs901200456789012845670Q801s3sss3s3s3-9o・u-DOOOOOOO111111111111JI2-22222222222212Forecast9G315.903436460.143296995.951897636.395496736.895935486.302094853.762595182.133495B92.D65095312.250484439.755433732.426183641.183333888.203493812.323893290.069092667.539992389.E04592395.845592890.489