拟合garch模型及autoreg模型-预测

#第五章SAS作业学号：200930980106姓名：何斌年级专业：10级统计1班指导老师：肖莉问题1:试选择恰当模型拟合某种股票的价格数据，数据如下:10.510.449.9410.25119.8810.51213.9412.2512.6113.513.4412.4413.515.3915.7513.8814.515.516.1314.7511.7515.2517.1320.51921.520.2525.6326.8827.6323.8826.382424.38U利用拟合模型预测未来二期该股票的价格；2、按照书本相应例题的格式完成问题，并附上SAS程序。解:1.1建立数据集，绘制时序图datagupiao;inputvaluetim令二n;MMFcards;10.510.449.9410.25119.8810.51213.9412.2512.6113.513.4412.4413.515.3915.7513.8814.515.516.1314.7511.7515.2517.1320.51921.520.2525.6326.8827.6323.8826.382424.38procgplotdata=gupiao;plotvalue*time;symboli=joinv=starh=lci=redcv=blackw=2;run;L2输出时序图显示该序列非平稳。如图1-1所示。

图1-1股票价格序列的时序图3对该序列进行一阶差分运算，程序修改如下：datagupiao;inputvaluedifv=dif(value);cards;10510.449.9410.25119.881051213.9412.2512.6113.5134412.4413.515.3915.7513.8814515.516.1314.7511.7515.25171320.51921.520.2525.63268827.6323.8826.382424.38procgplotdata=gupiao;plotdifv*time;symboli=joinv=starh=lci=redcv=blackw=2;procarima;identifyvar=value(l)minicp=(0:5)q=(0:5);run;1.4考査一阶差分后序列的平稳性。所得时序图如图1-2所示。图1-2序列difv的时序图时序图显示差分后序列difv没有明显的非平稳特征。1.5序列d辻v的识别，如下图所示。CovarianceCorrelation-1987654321C)1234567891StdError3.5212571.00000床床床床床床床床床床床床床床******0-0.85681924333.*****0.1690310.2948010.08372♦*0.178759・1.476456・.41930********0.1798761.4684670.41703******林0.205917-0.57304716274***0.2287780.5835960.16574♦**0.232062-1.092783・.310340.2354200.0788680.022400.246832Autocorrelationsm^rkstwostandarderrorsIg012345678LInverseAutocorrelationsLagCorraction-1987654321012345678911-0.0088720.11143**30.26772*****・4-0.28641・**c*c*c*c*c50.1944560.04433*70.07494*S0.20365图1-3序列cHfv的自相关图.偏自相关图AutocorrelationCheckforWhiteNoise6ChL-ChiSqSquareDFAutocorrelations'19.2860.0037-0.2436ChL-ChiSqSquareDFAutocorrelations'19.2860.0037-0.2430.084-0.4190.417-0.1630.166图1-4残差图MA0FflA1MA2MA3MX4MA51230471.3251891.4143071.376S021.2328361.2&3827126874712838811.3256551.193191.2245111.325804135642713445541.4148541.287151.3110521.410&321204878110616S1.1933031.291SS81.3&1TT81.3917331.0948111672311.2686741.3884221.3&12011.42721T116411912654611.345S931.3143541.3931.4&5095riknLinujnIntormatLonCntenongs012345LaARARARARARARErrorseriesmodel：AR(5)MinimumTableValue：BIC(4,0)=1.09431图1-5最优模型选择识别部分的输出结果显示，1阶差分后，序列difv为平稳非白噪声序列，最优拟合模型为AR(4)模型。

L6“estimatep=4;"对序列<Hfv拟合AR(4)模型。输出结果显示mu.ARI,1>ARI,2均不显著，修改估计命令如下：estimatep=(3,4)noint;拟合结果显示模型显著且参数显著。如图1-6所示。ParajneterEstimateStaxilarAErrortValueApproxPr>|t|LagARI,1-0.354230.14518-2.440.02023ARI,20.542810.154813.510.00134CoitiitionalLeastSquaresEstimation.VarianceEstimate2.44562StAErrorEstimate1.563848AIC132.5667SBC135.6774ffujnberofResiAu.als35*AICandSBCAonotin.clu.AelogAeiermiitajit.AutocorrelationCheckofResidualsToChi-Pr>LagSquareDFChiSqAutocorrelat1ons61.4840.83010.1180.0140.0460.090-0.1060.026125.20100.8773-0.034-0.199-0.1230.050-0.0840.095186.43160.9828■0.024■0.0240.1010.0490.054■0.0412412.35220.94960.118-0.042-0.0420.1450.110■0.096ModelforvariablevalueFeriod(s)ofDifferencing1Nomeanterminthismodel.AutoregressiveFactorsFactor1：1+0.35423B**(3)-0.54281B**(4)图1-6序列difv模型拟合结果输出结果显示,序列value的拟合模型为ARIMA((3,4),1,0)模型，模型口径为:一f1+0.35423B3-0.54281B4或等价记为：Vt=-0・35423叫_3+0・89704匕_4-0.54281Ft.5+st(7表示股价价格)1.7^forecastlead=2id=time;”，利用拟合模型对序列乞作2期预测。如图1-7所示。1.7^forecastlead=2图1-737图1-737期、38期股票价格预测(Forecast值)Fcr^c^stsf-orv^ri=ibl^v=alu^OBsForecastSidError95%ConfidexterLimits3721.45891.563818.393824.52403823.85902.211619.324327.99371.8拟合图:vaIue40-30・2030・20・10・0・010203040timePLOT**vaIueForecastforvaIue■—Lower95HConfidenceLimiF—Upper95%ConfidenceLimitSAS程序：datagupiao;inputvaluedifv=dif(value);time=n;cards;10.510.449.9410.25119.8810.51213.9412.2512.6113.513.4412.4413.515.3915.7513.8814.515.516.1314.7511.7515.2517.1320.51921.520.2525.6326.8827.6323.8826.382424.38procgplotdata=gupiao;plotdifv*time;symboli=joinv=starh=lci=redcv=blackw=2;procarima;identifyvar=value(l)minicp=(0:5)q=(0:5);estimatep=(3,4)noint;forecastlead=2id二timeout=results;run;procprint;run;procgplot;plotvalue*time=lforecast*time=2195*time=3u95*time=3/overlaylegend;symbol1c=blacki=nonev=star;symbol2c=redi=joinv=nonew=2;symbol3c=greeni=joinv=none1=32;run;问题2：1867-1938年英国绵羊数量如下所示:2203236022542165202420782214229222072119211921372132195517851747181819091958189219191853186819912111211919911859185619241892191619681928189818501841182418231843188019682029199619331805171317261752179517171648151213381383134413841484159716861707164016111632177518501809165316481665162717911＞选择恰当模型，拟合该序列的发展；2、利用拟合模型预测1938-1945年英国绵羊的数量；3、按照书本相应例题的格式完成问题，并附上SAS程序。解.;建立数据集，绘制时序图datamianyang;inputx輕；t=n;cards;220323602254216520242078221422922207211921192137213219551785174718181909195818921919185318681991211121191991185918561924189219161968192818981850184118241823184318801968202919961933180517131726175217951717164815121338138313441384148415971686170716401611163217751850180916531648166516271791procgplotdata=mianyang;plotx*t;

symboli=joinv=starci=redcv=blackw=2;run;1.2输出时序图显示该序列非平稳。如图1-1所示。图1-1绵羊数量序列X的时序图1・3时序图显示序列具有显著线性递减的趋势，且波动幅度随时间递增，所以考虑使用AUT0REG过程建立序列｛俎｝关于时间t的线性回归模型，并检验残差序列的自相关性和异方差性。执行程序：procautoregdata=mianyang;modelx=t/nlag=5dwprobarchtest;run;得到结果如下：DW检验结果显示，残差序列具有显著的正自相关性，如图1-2所示。Durbin.-Wa.tson.Sta.tisticsOrderDWFr<DWFr>DW10.3613<00011.0000图1-2普通最小二乘估计DW检验结果

残差序列5阶延迟自相关图显示残差序列至少具有2阶显著自相关性，如图1-3所示。EstimatesofAutocorrelationsLagCoxrarianceCorrelation-198765432101234567831LagCoxrariance019123.019123.11.000000115280.70.799068294&4.90.49494634&16.70.24141942722.2040.145449WWW图1-3残差序列5阶延迟自相关图参数估计结果显示回归模型参数显著，如图1-4所示。Varia*bleHFEstimateStandardErrortValueApproxFr>111Intercept1216433.404164.78<0001t1-8.38860.7953-10.56<0001图1-4线性回归模型参数估计结果异方差检验结果显示残差序列具有显著的异方差性，且具有显著的长期相关性，如图1-5所示。Ord^r123456789W1112QOrd^r123456789W1112QFr>QUliFr>Uil35.4104<.000134.9710<.000144.8342<.000138.1644<.000145.2763<.000138.4478<.000146.8955<.000139.3154<.000150.9033<.000139.3477<.000154.6288<.000139.3710<.000157.8390<.000139.6152<.000159.5575<.000139.6340<.000159.5628<.000140.4606<.000160.8433<.000140.6282<.000163.7334<.000140.6339<.000166.0093<.000140.7443<.0001Q^ndUilTestsforARCHDisturbances图1-5异方差检验结果1.4“modelx=t/nlag=2garch=(p=l,q=l)综合考虑残差序列自相关性和异方差性检验结果，尝试拟合AR(2)-GARCH(1,1)模型。模型最终拟合结果如图1-6所示。

GAHCHEstimatesSSE344169.24901>servations72MSE4780UncondVar5316.72882LogLikelihood.-407.25362TotalR-Sq*uar€0.9036SBC840.167247AIC826.50725MAE52.735066AICC827.799558MAPE2.89335488formalityTest0.0876Pr>ChiSq0.9572VariableDFEstimateStandardErrortValueApproxFr>111Intercept1213047.912744.46<.0001t1-7.38651.2637-5.85<.0001ARI1-1.261701203-10.49<0001kR210.52140.12274.25<0001ARCKO15182127S4.05<0001ARCK110.02540.18480.140.8908GARCK11-S.26E-231.454E-15-0.001.0000图1-6普通最小二乘估计输出结果参数估计结果显示该模型拟合较成功。最终模型口径为：(xt=2130-7.3869(+utut=1.2617血“—0.5214uf_2+st£t=密N(0»4780)/if=5182+0.0254^-!1.5最终输出拟合效果图如图1-7所示。

图1-7拟合效果图2.1预测1939-1945年英国绵羊的数量，如下表2-1所示。表2-11939-1945年英国绵羊的数量19391820.2819401769.5919411688.3919421610.519431552.719441518.519451503.562.2预测效果图，如图2-2所示。

图2-21939-1945年英国绵羊的数量预测效果图3.SAS程序：datamianyang;inputx@@;t=intnx(*year','ljanl867,d,_n_-l);formattyear4.;cards;22032360225421652024207822142292220721192119213721321955178517471818190919581892191918531868199121112119199118591856192418921916196819281898185018411824182318431880196820291996193318051713172617521795171716481512133813831344138414S415971686170716401611163217751850180916531648166516271791procgplotdata=mianyang;plotx*t;symboli=joinv=starci=redcv=blackw=2;run;procautoregdata=mianyang;modelx=t/nlag=5dwprobarchtest;modelx=t/nlag=2garch=(p=l,q=l);outputout=outp=xp;procgplotdata=out;plotx*t=2xp*t=3/ove:rlay;symbol2i=nonev=starc二black;symbol3i=joinv=nonec=redw=2;run;procprintdata=outnoobs;run;问题3,使用Auto-Regressive模型分析例5.9序列。（要求只对变量为延迟序列值进行模型拟合，作业格式参照书“例5.6续”）1962-1991年德国工人季度失业率序列行数据（％）1.01.51.01.01.08.07.07.48.59.010.08.7&8&99.010.2&&7.6&6.86.06.26.2

解：1・1建立数据集，绘制时序图datashiyelv;inputx輕；t=intnxfquarter，,'ljanl962,d>_n_-l);formattyear4.;cards;1.01.51.01.01.08.07.07.48.59.010.08.7&88.910.48.9&910.28.6&&4&28.87.67・57.68.1■6.86.06.26.2rprocgplotdata=shiyelv;plotx*t;symboli=joinv=starci=redcv=blackw=2;run;1・2输出时序图。如图1-1所示。

图1-1德国工人季度失业率时序图1・2建立延迟因变量回归模型执行语句：procautoregdata=shiyelv;modelx=lagx/lagdep=lagx;run;输出结果如下图所示TheAUTOREGProcedureDependentVari^blexESESEBCAEAP63.ESESEBCAEAP63.788407DFE1170.54520RootNSE0.73836273.062927AIC267.504680.56825477AICC267.60812827.1498286RegressR-Squwr电0.3489TotalR-Sqaar电0.S483OrdinaryLeastSquaresEstimatesStatisticValuerrobLatelDurbinh-2.02450.0215Fr>hSt^nd^rdApproxVariableDFEstimateErrortV£LnePr>ItlIntercept10.16560.11231.470.1430lagx10.97150.020846.62<.0001MiscellaneousStatistics图1-2延迟因变量回归分析结果检验结果显示,Durbinh统计量的值为-2.0245,P值为0.0215,故拒绝原假设，认为残差序列高度自相关。参数检验结果显示，截距项不显著(P值大于0.1430)。1.3拟合无截距项，并检验残差的自相关性。执行程序：procautoregdata=shiyelv;modelx=lagx/lagdep=lagxnlag=4noint;run;输出结果如下图所示VariableDFEstimateStandardErrortValueApproxPr>|t|lagx10.99600.012678.92<.0001CovarianeeEstimatesofAutocorrelationsCovarianee00.54601.000000******************林1■0.1019■0.1865802・0.1233■0.225901*****3■0.1172■0.21458540.47770.8748505・0.1334■0.244299*****Correlation-198765432101234567891图1-3参数检验及残差的自相关性检验检验结果显示,参数显著，拟合残差的AR检验结果显示,参数显著，拟合残差的AR(4)模型。1.4残差的AR1.4残差的AR(4)模型执行程序：procautoregdata=shiyelv;modelx=lagx/1agdep=1agxn