ARIMA預(yù)測(cè)原理以及SAS實(shí)現(xiàn)代碼

ARIMA定義ARIMA的完整寫(xiě)法為ARIMA(p,d,q)其中p為自回歸系數(shù)，代表數(shù)據(jù)呈現(xiàn)周期性波動(dòng)d為差分次數(shù)，代表數(shù)據(jù)差分幾次才能達(dá)到平穩(wěn)序列q為移動(dòng)平均階數(shù)，代表數(shù)據(jù)為平穩(wěn)序列，可以用移動(dòng)平均來(lái)處理獲得觀察值序列差分

運(yùn)算irN擬合ARMA獲得觀察值序列差分

運(yùn)算irN擬合ARMA平穩(wěn)性檢測(cè)方法?方法一：時(shí)序圖序列始終在一個(gè)常數(shù)值附近隨機(jī)波動(dòng)，且波動(dòng)范圍有界，且沒(méi)有明顯的趨勢(shì)性或周期性，所以可認(rèn)為是平穩(wěn)序列。下圖明顯不是一個(gè)平穩(wěn)序列procgplotdata=gdp;plotgdp*year=1;symbolc=redi=joinv=star;run;

??方法二：自相關(guān)圖自相關(guān)系數(shù)會(huì)很快衰減向0所以可認(rèn)為是平穩(wěn)序列。procarimadata=gdp;?adf單位根檢驗(yàn)（精確判斷）三個(gè)檢驗(yàn)中只要有一個(gè)Pr<Rho小于0.05即可認(rèn)定為平穩(wěn)序列，主要是stationarity

=(adf=3)起作用procarimadata=gdp;identifyvar=gdpstationarity=(adf=3)nlag=12;run;Dick&y-Fuller單位根檢蜀濡后RhoPr<RhoTauPr<TauFPr>FZ&roMean03.96450.99939710.9999112.27E50.99991.S20.56U23.74340.99912.as0.990033.57650.99002.230.9924SingleMean03.B9520.99976.45D399945.65&.001&117.06320.99991.M-19陰91.19[).77&423.93620.99972.770.99993.000.156333.63570.99962.222.46&.40MTrend03.D2090.99981.&&D.999920.23&.001&137.Z6670.99991.1&D-99991.19D.92M2411990.9刪1.713.653415090.99991.51D29992.34&.7Z37■白噪聲檢驗(yàn)Pr>卡方<0.05即可認(rèn)定為通過(guò)白噪聲檢驗(yàn)。procarimadata=gdp;identifyvar=gdpstationarity=(adf=3)nlag=12;run;■非平穩(wěn)序列轉(zhuǎn)換為平穩(wěn)序列方法一：將數(shù)據(jù)取對(duì)數(shù)。方法二：對(duì)數(shù)據(jù)取差分dif函數(shù)datagdp_log;setgdp;loggdp=log(gdp);cfloggdp二dif(loggdp);run;/**對(duì)數(shù)數(shù)據(jù)散點(diǎn)圖**/procgplot;plotloggdp*year=l;symbolc二blacki=joinv=star;run;/*一階差分對(duì)數(shù)數(shù)據(jù)散點(diǎn)圖*/procgplot;plotcfloggdp*year=l;symbolc=greenv=doti=join;run;

n.no1995cfloggdpyear從上圖中可以看出，一階差分后序列已經(jīng)變成平穩(wěn)的了，因此，數(shù)列需要做一階差分0.30n.250.20一0.15-0.10-0.0519B019B5znri5199nn.no1995cfloggdpyear從上圖中可以看出，一階差分后序列已經(jīng)變成平穩(wěn)的了，因此，數(shù)列需要做一階差分0.30n.250.20一0.15-0.10-0.0519B019B5znri5199nznnn■專換完畢后再驗(yàn)證下面代碼中的(1)就代表1階差分，adf=3則代表平穩(wěn)性檢驗(yàn)0-3,/*一階差分對(duì)數(shù)數(shù)據(jù)的自相關(guān)圖、偏自相關(guān)圖、純隨機(jī)性檢驗(yàn)、單位根檢驗(yàn)*/procarimadata=gdp_log;identifyvar=loggdp(l)stationarity=(adf=3)nlag=12;run;曰嗓尸的目相關(guān)檢查至耶后卡萬(wàn)目由度618.81G<0.0045}0.4790.1G1-0.1G1-0.293-0.309-0.3471220.4512CLOSSff-0.1390.0440.1070.068-0.017-0.046増廣Dickey-Fuller單位根檢驗(yàn)類型滯后RhoPr<RhoTauPr<TauFPr>FZeroMean0-1.83960.3437-0.850.33791-1.54520.3B15-0790.36492-1.06100.4567-0.520.47933-0.44950.5685-0.280.5744SingleMean0-12.53320.0434-2.700.08773.670.20461-15.19870.0166-2.53丄12丄13.200.31122-39.7723^.0001-2.93/0.057^U.370.0S813-15B.1750.0001-2.90M&4/4.340.0097Trend0-12.48910.1999-2.610.27303.490.52281-14.971B0.1004-2.420.35983.030.59442-40.3B09<.0001-2.860.19204.090.41713■166.0470.0001-2.910.17864.300.3817用Qlb統(tǒng)計(jì)量作的2檢驗(yàn)結(jié)果表明：對(duì)數(shù)差分后的GDP序列的Qlb統(tǒng)計(jì)量的P值為0.0045(<0.05)，故序列為非白噪聲序列。單位根檢驗(yàn)結(jié)果表明：對(duì)數(shù)差分后的GDP序列有常數(shù)均值、無(wú)趨勢(shì)的二階自回歸模型的Tau統(tǒng)計(jì)量的P值小于0.0573,故序列基本可以確定為平穩(wěn)序列，并可初步考慮用ARMA(2,g)模型對(duì)它們進(jìn)行擬合。嚴(yán)型定階/**定階**/procarimadata=gdp_log;identifyvar=loggdp(l)nlag=6minicp=(0:2)q=(0:4);run;/*minic為一定范圍模型定階相對(duì)最優(yōu)模型識(shí)別*/曰嗓戸的目相關(guān)檢查至師后卡萬(wàn)目由度Pr>卡力目相關(guān)G18.8160.00450.4790.161-0.161-0.293-0.309-0.347MinimumInformationCriterionLagsMAOMA1MA2IUIA3MA4AR0-5.50235-5.92E5B-5.91546-6.03307-6.03373AR1-5.9469-5.07077-6.01308-6.36963-6.376AR2-6.20883-6.09519-5.97303-6.3134-6.26134誤差序歹I晦：AR(EJ采用相對(duì)最優(yōu)模型識(shí)別，根據(jù)上述分析及序列的自相關(guān)和偏自相關(guān)圖，適當(dāng)選擇加=4,n=2,使用indentify命令中的minicp=(0:n)q=(0:m)短語(yǔ)進(jìn)行相對(duì)最優(yōu)模型定階。結(jié)果顯示(圖6.10),在p=1,q=4時(shí)，BIC函數(shù)值最小。執(zhí)行ARIMA過(guò)程的Estimatep=1q=4命令做參數(shù)檢驗(yàn)，結(jié)果未能通過(guò)參數(shù)檢驗(yàn)。讓q在0?3之間取值，通過(guò)反復(fù)測(cè)試,只有ARMA(1,3)模型與ARMA(1,0)模型通過(guò)參數(shù)檢驗(yàn)及模型檢驗(yàn),其檢驗(yàn)結(jié)果及參數(shù)估計(jì)如圖6.11所示。■參數(shù)估計(jì)/**參數(shù)估計(jì)**/procarimadata=gdp_log;identifyvar=loggdp(l);estimatep=1q=4;run;/*SAS支持三種估計(jì)，默認(rèn)為條件最小二乘估計(jì)，要制定可增加選項(xiàng)：METHOD=ML極大似然估計(jì)METHOD=ULS最小二乘估計(jì)METHOD=CLS條件最小二乘估計(jì)輸出項(xiàng)的含義見(jiàn)王燕P104*/;條件最乎二乘估訐1滲數(shù)怙計(jì)標(biāo)準(zhǔn)誤差t值近似Pr>|t|I滯后］:MU0.157770.043643.610.00170：：MA1J0.423525.231530.000.93631:0.030B42.941690.010.9917:MA1730.435272.7S7040.160.87753'1MAI,40.1102G0.551330.200.84354；;AR1,10.965595.117540.190.84921：從上面可以看出，在p=1q=4時(shí)，通不過(guò)檢驗(yàn)祭件最小二乗估計(jì)參數(shù)估計(jì)標(biāo)淮誤差t值近似Pr>|t|滯后MU0.163580.026796.11<.00010MA1J-1.481360.36408-4.070.00061MA1,2-1.084490.34753-3.120.00522MA1.3-0.637390.21520-2.600.02093AR1,1-830000.3S787-2.U0.04431常數(shù)估計(jì)0.299354萬(wàn)差估計(jì)標(biāo)準(zhǔn)誤差估計(jì)0.0041530.064441AIC-64.3524SBC-5e.0G19殘羞數(shù)26AIC和SBC不旦括對(duì)數(shù)行列式

p=1q=3和條件最小二秉佶計(jì)蓼數(shù)怙計(jì)標(biāo)準(zhǔn)誤差t值近似Pra|t|滯后MU0.165960.023596.61<.00010AR1,10.49B530.131732.740.01131靜估計(jì)0.078207：;萬(wàn)差估計(jì);標(biāo)準(zhǔn)誤差估計(jì)0.00403G：0.06353：:AIC-67.6200iSBC-65.1046i謹(jǐn)差數(shù)r眄乂AIC和SBC不包括對(duì)數(shù)行列式op=1q=0時(shí)能通過(guò)檢驗(yàn)從上面2個(gè)模型的檢驗(yàn)結(jié)果可以看到，它們均為有效模型，但ARMA(1,0)模型的AIC為-67,SBC為-65均比ARMA(1,3)的AIC與SBC小，根據(jù)AIC準(zhǔn)則和SBC準(zhǔn)則,ARMA(1,0)應(yīng)該更有效，所以應(yīng)選擇前者作為預(yù)測(cè)模型?！觥鯝IL怕SDU已列STI」o裁數(shù)估計(jì)相關(guān)性裁救MUAR1JMU1.0000.069AR1J0.0691.000至滯后殘卷的目相黃檢查卡萬(wàn)目由廢Pr＞卡萬(wàn)目申目關(guān)66.0750.29970.0540.062-0.109-0.191-0.001■0.301127.22110.7806-0.0640.0660.0880.080-0.0140.0681811.99170.8006-0.087-0.1890.0390.0460.144-0.0622415.94230.85770.0000.0260.103-0.075-0.0660.026|佶計(jì)均值0.1659661―I1'-I目回歸因子因子1：1-0.49053曠⑴GDP對(duì)數(shù)序列模型的口徑為：￡Vlogx=0.155955+-t1-0.49853B其中，xt表示GDP序列，模型可寫(xiě)為：logx=1.49853logx一0.49853logx+0.078207+￡tt-1t-2t■預(yù)測(cè)/**參數(shù)估計(jì)及預(yù)測(cè)**/procarimadata=gdp_log;identifyvar=loggdp(l)nlag=16;estimatep=1q=0;forecastlead=4id=yearout二results;run;@year|@leccO圃TDUCA5T|@i5T>LK|@U95|@KE5I1TALAlOTB5.D9325926E01OTD5.￡4740764425.24S2L4T6750.Q6353D235T\5.1函詣祁3龍5.373T31G814HO.bbLT^dGSi13305.134420^4955.4025^706^60.0&3530E35T〔5”2700^0390?55ZT]]^C^50.D3L0Z3305；1?15.5E022014B4E.G0SBL996^G0.06353023575”40129701D75.7903309995-0.DB559犯至ii10825.57329^6655.641200QTD30.0B3530235I5E16BE3曲弗57651179442<1.0BT90E9M:1?35.T9￡06115775.0779601660.06353023575”5534431號(hào)L5.00247713990.114900^161?45.Diaeiiiorra5.0SOE28TS770.Q6353D235T[5.85G011S1376.L050^57816-0.D6&91768!13355.113L0579316.O5Z015&132CL0&3530E35T[552749S&3956.17&53255710.0510?11799IT19EIB6.220411227G6.2907602&250.06353023576.1GG25129B6G.4L520523&4-O.070357035108?6.4t2S03flM4B.322tL2T6520.0B3530235IB227Effi7Qt264T品西7生10.06069023?|7F1314151?BG.&iei：i5'313e5B.506S229T550.0635302357&.462^6131&6.7L143^94946.D3L93Q1B1iB.74607129406.?7QLiT320.Q6353D235T!B.6752741QD3"6.9243031471^O.05371flarei19905.BWLIZIZTT6.35汁盯旳囲0.0&3530E35T[6T631EO410970LZZ]^1^5TF04T5Z52E5[1991|952410^6516.9S529L29020.06353023576.8^077432437.0896062721^O.0129200131M20B3530235IB9621440598T2L]]7a6O7S0.0677583917：LBITIB192D1993T.41623^15137.33330401050.06353023577.20076733&&7.4518217&440.0匪號(hào)陰4C61OT47.7072550627.624D63]1G^0.Q6353D235T[7.5QQ44B14257.74949009=130.OB22929GSS!1995S.DDTZ0O^S417.33054580560.a&353DZ35T[T.0O6OZS535T60550627535CLOT6T3467?199B9.ED554Q77E86.S3SD5839G70.06353023578.11054142268.35957537OS-0.DE9517G1S8.3136103784S.3325S63I120.0B3530235I￡!2501^3373850T]fl334S2-0.0BS9]5^93iZ122Z32425199B3.37944721560.44576403320.06353023576/321267059￡a.5703Ulrnri-q.DB&33aai0iS.428602368：3.4?445829^0.Q6353D235T!9.36503^545'8.&L49G28023^O.DB1343^42!300DS.5443530019S.5313L40T350.063&3DE35T:0406TOT9D566.65503105340.0130301224；2CO19.B9765004706.693265190^0.06353023578.55574022458.8O4T021723P.D4260G3512OJ287272^l82filS.7&23SLBQ520.0B3530235I￡!637E647212888B&gS6Bgi-0.035L33^34^za27ZB293D￡DD33.BDO5920ra0.650LL63020.06353023570.125031^0L3.97463533S9O.O1D46451302CO4Q.DB42203O46■3.CO526L19a^0.Q6353D235T[9.83074421B5|12977316430.07B9591142!ZCO59.2T39LT112^0.Q&353DZ35T[9.1^3?013B59:亍站4兇涵了.i300B9.44SB995&570.1144S296949”22236KST7社671017SS3S2021?1|9.Bl1034T463|0.159431m892935542038Q3235155833.；3L￡DDB9.7711rows'0.1??3B&9Tg^QKOJTSOID.L&i：i'33633T■還原預(yù)測(cè)數(shù)值并畫(huà)圖/**繪制預(yù)測(cè)圖**/dataresults;setresults;y=exp(loggdp);estimate1=exp(forecast);el95=exp(l95);eu95=exp(u95);run;procgplotdata=results;ploty*year=lestimate1*year=2el95*year=3eu95*year=3/overlay;symbol1c=blacki=nonev=star;symbol2c=redi=joinv=none;

symbol3c=greeni=joinv=nonel=2;run;Iyear3DDDD197519BD19B5199019952DDD2DC520101DODD5000”nnn2DDDDIyear3DDDD197519BD19B5199019952DDD2DC520101DODD5000”nnn2DDDD15DDD■^一種確定p、q的方式procarimadata=gdp;identifyvar=gdpstationarity=(adf=3);run;直接對(duì)gdp求arima模型，可已看出acf是拖尾，而pacf是1階截尾，所以最好是p=1,q=0SDOD-&O0Ddg4D0D-0/M-o-a-cgdpSDOD-&O0Ddg4D0D-0/M-o-a-cgdp,B的趨勢(shì)和相黃分折2D0D--1.0-■確定p、q的方式理論由于ARMA(p,q)模型可以轉(zhuǎn)化為無(wú)窮階移動(dòng)平均模型，所以ARMA(p,q)模型的自相關(guān)系數(shù)不截尾。同理，由于ARMA(p,q)模型也可以轉(zhuǎn)化為無(wú)窮階自回歸模型，所以ARMA(p,q)模型的偏自相關(guān)系數(shù)也不截尾。總結(jié)AR(p)模型、MA(q)模型和ARMA(p,q)模型的自相關(guān)系數(shù)和偏自相關(guān)系數(shù)的規(guī)律，見(jiàn)表6.1所示。模型自相關(guān)系數(shù)Pkk偏自相關(guān)系數(shù)ekkkkAR(p)拖尾p階截尾MA(q)q階截尾拖尾ARMA(p,q)拖尾拖尾■模型優(yōu)化指標(biāo)當(dāng)一個(gè)擬合模型在指定的置信水平a下通過(guò)了檢驗(yàn)，說(shuō)明了在這個(gè)置信水平a下該擬合模型能有效地?cái)M合時(shí)間序列觀察值的波動(dòng)。但是這種有效的擬合模型并不是惟一的。如果同一個(gè)時(shí)間序列可以構(gòu)造兩個(gè)擬合模型，且兩個(gè)模型都顯著有效，那么應(yīng)該選擇哪個(gè)擬合模型用于統(tǒng)計(jì)推斷呢？通常采用AIC和SBC信息準(zhǔn)則來(lái)進(jìn)行模型優(yōu)化。1.AIC準(zhǔn)則AIC準(zhǔn)則是由日本統(tǒng)計(jì)學(xué)家赤池弘次(Akaike)于1973年提出，AIC全稱是最小信息量準(zhǔn)則(aninformationcriterion)。AIC準(zhǔn)則是一種考評(píng)綜合最優(yōu)配置的指標(biāo)，它是擬合精度和參數(shù)未知個(gè)數(shù)的加權(quán)函數(shù)：AIC=—2ln(模型中極大似然函數(shù)值)+2(模型中未知參數(shù)個(gè)數(shù))(6.68)使AIC函數(shù)達(dá)到最小值的模型被認(rèn)為是最優(yōu)模型。2.BIC準(zhǔn)則AIC準(zhǔn)則也有不足之處：如果時(shí)間序列很長(zhǎng)，相關(guān)信息就越分散，需要多自變量復(fù)雜擬合模型才能使擬合精度比較高。在AIC準(zhǔn)則中擬合誤差等于nln(b2)，即擬合誤差隨樣本容量n￡放大。但是模型參數(shù)個(gè)數(shù)的懲罰因子卻與n無(wú)關(guān)，權(quán)重始終為常數(shù)2。因此在樣本容量n趨于無(wú)窮大時(shí)，由AIC準(zhǔn)則選擇的擬合模型不收斂于真實(shí)模型，它通常比真實(shí)模型所含的未知參數(shù)個(gè)數(shù)要多。為了彌補(bǔ)AIC準(zhǔn)則的不足，Akaike于1976年提出BIC準(zhǔn)則。而Schwartz在1978年根據(jù)Bays理論也得出同樣的判別準(zhǔn)則，稱為SBC準(zhǔn)則。SBC準(zhǔn)則定義為：SBC=—2ln(模型中極大似然函數(shù)值)+ln(n)(模型中未知參數(shù)個(gè)數(shù))(6.69)它對(duì)AIC的改進(jìn)就是將未知參數(shù)個(gè)數(shù)的懲罰權(quán)重由常數(shù)2變成了樣本容量n的對(duì)數(shù)ln(n)。在所有通過(guò)檢驗(yàn)的模型中使得AIC或SBC函數(shù)達(dá)到最小的模型為相對(duì)最優(yōu)模型。之所以稱為相對(duì)最優(yōu)模型是因?yàn)椴豢赡鼙容^所有模型。表6.2河南省歷年國(guó)民生產(chǎn)總值數(shù)據(jù)年份(Year)生產(chǎn)總值(億元)(GDP)人均生產(chǎn)總值(元)(PGDP)年份(Year)生產(chǎn)總值(億元)(GDP)人均生產(chǎn)總值(元)(PGDP)1978162.92232.319921279.751452.31979190.09266.719931662.761867.41980229.16316.719942224.432475.21981249.69340.119953002.743312.81982263.3035319963661.184007.41983327.95432.919974079.264430.11984370.04481.619984356.604695.11985451.74579.719994576.104893.71986502.91635.320005137.6654441987609.60755.820015640.115923.61988749.09909.920026168.736436.51989850.711012.320037048.597570.21990934.651090.620048815.099469.919911045.731201.2■付：完整代碼datagdp;infiledatalines;inputyeargdppgdp;formatgdpBEST12.2pgdpBEST12.2;datalines;162.92232.3190.09266.7229.16316.7249.69340.1263.30353327.95432.9370.04481.6451.74579.7502.91635.3609.60755.8749.09909.9850.711012.3934.651090.61045.731201.21279.751452.31662.761867.42224.432475.23002.743312.83661.184007.44079.264430.14356.604695.14576.104893.75137.6654445640.115923.66168.736436.57048.597570.28815.099469.9run;/**原始數(shù)據(jù)散點(diǎn)圖**/procgplotdata=gdp;plotgdp*year=1;symbolc=redi=joinv=star;run;/*注symbol常用參數(shù)C—圖形顏色，red_紅色，black_黑色，green_綠色，blue_藍(lán)色，pink_洋紅等*//*V—觀測(cè)值的圖形，star_*,dot_.,cicle_圓圈，diamond.菱形，none_不標(biāo)*//*I—觀察值的鏈接方式，join_線連，spline_光滑連接，needle_作觀察值到橫軸懸垂線，none_不連*/procarimadata二gdp;identifyvar二gdpstationarity