ARIMA模型预测

1、5ARIMA模型预测5J模型选取u前，学术界较为成熟的预测方法很多，各种不同的预测方法有其所面向的特定对象，不存在一种普遍最好”的预测方法。GM(1,1)模型预测是以灰色系统理论为基础，通过原始数据的分析处理和建立灰色模型，对系统未来状态作出科学的定量预测的一种方法。我们采用GM(1,1)模型是基于以下两方面的考虑：第一，GM(1,1)模型对数据要求较低，而其他多数预测方法以数理统计为基础，对样本量有较高要求。我们用来做预测的数据时序只有14年，预测使用GM(1,1)模型较好；第二，GM(1,1)模型的计算量相对较小，计算方法相对简单，适用性较好。5.2模型假设前提1、假设未来望庆地区经济发展

2、基本态势不变：2、假设未来中央政府对重庆实施的政策方向基本不变；3、假设未来不会出现战争、瘟疫及其它不可抗拒的自然或社会因素。预测数据来源预测样本为1997-2008年的重庆市农资价格指数、化学肥料价格指数、饲料价格指数。具体预测样本数据如下：表5.119972008年重庆部分农资价格指数单位：%19971998199920002001200220032004200520062007农资105.990.493.296.8100.5103.5102.5109.8106.9101.2110.0化肥93.687.496.597.39&999.7101.0106.3108.399.9102.2

3、饲料96.691.091.995.810L1106.6105.6115.7105.5101.1109.2为提高数据预测的科学性，我们以1996年(直辖前)的农资价格为基期，假设1996年农资产品价格为100元，则以后第，年的农资产品价格讣算公式如下:G.=p100xZ99?xxZf经此换算，得到1997-2008年的预测样本。其中，NZJG表示换算后的农资，HXFL表示换算后的化肥，SL表示换算后的饲料。具体见下表：表5.219972008年转换后的预测样本单位：元19971998199920002001200220032004200520062007NZJG105.995.789.286.4

4、86.889.892.1101.1108.1109.4120.3HXFL93.681.87&976.876.075.776.581.388.188.089.9SL96.687.980.877.47&283.48&1101.9107.5108.7118.754GM(1J)模型建立与检验541序列的建立iailfn个原始数据组成的原始序列为X(k)二X(9)，x(9)，,x数(n)。那么可以得到四个样本原始序列：NZJGX(6)(k)二105.9,95.7,，120.3)；HXFLX(6)(k)二93.6,81.&89.9)；SLxe(k)96.6,87.9,，11

5、&7。542级比检验级别检验是GM(1,1)建模的数据检验，经讣算可得：NZJG级比序列二0.904,0.932，1.198)；HXFL地区序列二0.874,0.965,，L200；SL地区序列二0,910,0.919,-,LI70)：都落在界区(0.7515,1.3307)内。这表明，以上三个样本序列均可以进行GM(1,1)模型建模。543模型的方程通过一次累加生成新序列：X（k）二仗，X，则GMd.1）模型相应的微分方程为：其中，&称为发展灰度，为内生控制灰度，它们是方程中重要的参数。通过求解微分方程，即可得到预测模型。山于GM（1,1）预测模型种类较多，我们选取其中较常用

6、的一种如下：通过测算，我们得到三组、值和相应的四个拟合预测方程如下表表5.3拟合预测方程和.值a拟合预测方程NZJG-0.04691572.875525Vd)-(1呼9_72.875525)1(-o.(M69丈JL72.875525-0.046915-0.046915HXFL-0.02734869.157548¥山一(936-69.157548).0.02734®*j_69.157548g”-0.027348-0.027348SL-0.05742164.105061v(h-(966-64.105061)C-(.o.o5742i)a十64.105061g”-0.057421-

7、0,057421544后验差检验后验差检验包括残差的方差比（C）和小误差概率（P）。首先算得知残差平均值为1x1xr1Q-Z£（6）=-L占（Q胛6）.«i-l«77历史数据方差为町二（1/八）卜伙）7其中历史数据平均值为1»乍叫）残差方差为后验差比值为C吕小误差概率为尸=/V&（Q-f<0.6745*1经计算，我们同样得到四组方差比（C）和小误差概率（P）如下:表54后验差检验结果表CPNZJG0.2739161HXFL0.406991SL0.17281根据下列后验差检验结果判别表，我们认为四个模型均通过了后验差检验，有较好的预测精度。表

8、5.5后验差检验结果判别表CP<0.35>0.95好<0.50>0.80合格<0.65>0.70勉强合格>0.65W0.70不合格545相对误差检验经过对模型进行拟合测算，得到以下四个相对误差表：表5.6NZJGGM(1,1)模型拟合相对误差表时期实际值拟合值相对误差()199895.779.70-16.72199989.283.53-6.362008144.2127.41-11.64平均相对误二6.56%表5.7HXFLGM(1,1)模型拟合相对误差表时期实际值拟合值相对误差()199881.872.71-11.12199978.974.72-5.2

9、9表5.8SLGM(1,1)模型拟合相对误差表时期实际值拟合值相对误差()199896.671.69-18.44199987.975.93-6.032008138.8127.30-8.28平均相对误差二6.26%2008107.995. 58-11. 42平均相对误差二5. 41%综上分析，我们发现：以上四个模型的相对误差还是在合理的范ffl内，模型可靠，拟合精度较高，可以进行预测。5.5模型预测利用模型预测2009-2020年XZJG、HXFL.SL的具体数值,结果见表5.9。表5.9GM(1,1)模型预测结果单位：元年份NZJGHXFLSL2009133.598.2134.82010139

10、.9100.9142.82011146.7103.7151.22012153.7106.6160.22013161.1109.6169.62014168.8112.6179.72015176.9115.7190.32016185.411&9201.52017194.3122.2213.42018203.7125.6226.12019213.5129.1239.42020223.7132.7253.6经过换算，我们可以得到2009-2020年重庆市农资价格指数、化学肥料价格指数、饲料价格指数的预测值如下：表5.10GM(1,1)模型预测换算结果单位：%年份农资价格指数化肥价格指数饲料价格

11、指数200992.691.097.12010104.8102.8105.92011104.8102.8105.92012104.8102.8105.92013104.8102.8105.92014104.8102.8105.92015104.8102.8105.92016104.8102.8105.92017104.8102.8105.92018104.8102.8105.92019104.8102.8105.92020104.8102.8105.9图5.1GM11模型预测结果图农资价格指数一一化肥价格指数一饲料价格指数附录CM(1,1)模型模拟结果NZJGHXFL期数实际值模拟值残差期数实际

12、值模拟值残差199879.70-16.00-16.721998199872.71-9.09199983.53-5.67-6.361999199974.72-4.18200087.54L14L322000200076.79-0.01200191.744.945.70200120017&922.92200296.156.357.07200220028L115.412003100.778.679.412003200383.366.862004105.614.514.462004200485.674.372005110.682.582.39200520058&05-0.05200611

13、6.006.606.032006200690.492.492007121.571.27L062007200793.003.102008127.41-16.79-11.642008200895.58-12.32SL期数实际值模拟值残差199871.69-16.21-18.44199975.93-4.87-6.03200080.423.023.90200185.176.97&91200290.206.80&16200395.537.43&442004101.18-0.72-0.712005107.16-0.34-0.322006113.49-1.794.412007120.

14、201.50L262008127.30-1L50-8.28NZJGDNZJGDDNZJGNullHypothesis:NZJGhasaunitrootExogenous:NoneLagLength:1(AutomaticbasedonSIC,MAXLAG=11)t-SlatisticProb?AugmentedDickey-Fullerteststatistic04843870.5029Testcriticalvalues:1%level2.5953405%level-1.94508110%level-1.614017MacKinnon(1996)one-sidedp-values.Augme

15、ntedDickey-FullerTestEquationDependentVariable:D(NZJG)Method:LeastSquaresDale:08/31/09Time;20:34Sample(adjusted);2003M032009M07Includedobservations:77afteradjustmentsVariableCoefficientStd.Errort-StatisticProb,XZJG(-I)-0.0006700,001384-0.4843870.6295D(NZJG(-1)0.6556820,0873467.5066840.0000R-squaredA

16、djusted R-squaredS E of regression Sum squared residLog likelihood Durbin-Watson stat0429121Mean dep endent var0.421509S D dependentvar1.311403Akaike info criterion128.9833Schwarz criterion-129.11951. 936552Hannan-Quinn enter.0.10519517242013.4057023.4665803. 430053NullHypothesis:D(NZJG)hasaunitroot

17、Exogenous:Nonet-StatisticProb?AugmentedDickey-Fullerteststatistic3.9485590.0001Testcriticalvalues:1%level2.595340LagLength:0(AutomaticbasedonSIC,MAXLAG=11)5% level10% level1.945081-1.614017*MacKinnon(1996)one-sidedp-values.AugmentedDickey-FullerTestEquationDependentVariable:D(NZJG,2)Method:LeastSqua

18、resDate:08/31/09Time;20:35Sample(adjusted);2003M032009M07Includedobservations;77afteradjustmentsVariablecStd.Errort-Coefficientr.StatisticProb,D(NZJG(-1)03429800,08686239485590.0002R-SQuared0J69859Meandependentvar-0.029870AdjustedR-squared0J69859SDdependentvar1432064SEofregression1.304783Akaikeinfoc

19、riterion3.382852Sumsquaredresid129.3868Schwarzcriterion3.413291LoglikelihoodDurbin-Watsonstat-129.23981.934426Hannan-Quinnenter.3.395027NullHypothesis:D(NZJG,2)hasaunitrootExogenous:NoneLagLength:1(AutomaticbasedonSIC,MAXLAG=11)t-StatisticProb?AugmentedDickey-Fullerteststatistic9.5992680.0000Testcri

20、ticalvalues:1%level2.5961605%level1.94519910%level1.613948*MacKinnon(1996)one-sidedp-values.AugmentedDickey-FullerTestEquationDependentVariable:D(NZJG,3)Method:LeastSquaresDale:08/31/09Time;20:36Sample(adjusted);2003M052009M07Includedobservations:75afteradjustmentsVariableCoefficientStd.Errort-Stati

21、sticProb,D(NZJG(-1),2)1.5468980,161147*9.5992680.0000D(NZJG(-1),3)0.3704290,1064843,4787210.0009R-SQuared0.630862Meandependentvar0.041333AdjustedR-squared0.625805SDdependentvar2J47223SE.ofregression1.313488Akaikeinfocriterion3.409554Sumsquaredresid125.9432Schwarzcriterion3.471353Loglikelihood125.858

22、3Hannan-Quinnenter.3.434229Durbin-Watsonstat2.099426Dale:08/31/09Time;20:51Sample;2003M012009M07Includedobservations:78AutocorrelationPartialCorrelationACPAGQ-SlatProb171.Igl10.6490.64934.1530.000n1-1-120.3890,05646.5920.000广1r130.3700.24157.9650,000i+1-1-140.3680.07169.3540.000r1-1-150,2990.00676.9

23、760.000r1r.160.2730.08183.4230,000.r1-1-170,2180,06387.5990,0001111.180.1290.06689.0830.000-1.1190.0510.08089.3220,000-1.1-1-1100.0240,02989.3720,0001.1L1110147031291.3890.00011-11203310243101730.000“L1.r11302100.221105.960.000.1.1J.1140160-0J14108.460.000“L1-1-11502470,020114,480.000“L1-1-11603000.

24、021123.520,000“L1r.11702290.124128.900,000L1-1-11802180.006133,860.000“L1-1-11902430.032140,130,000“I.1J.12002580,073147.300,000“L11.1210268-0J06155J40.000“L1J.122-0,295-0.071164.830.000“I.1.*i12302440,150171,590,0001.1-1-12401710054174.970.0001.1r.12501200.182176.650.000T.1-1-12601120050178-160.000

25、T.1J.127-0J110082179.680.000.1.1-1-12800810.056180.500.0001-1-129-0,0900.033181.530,0001-1-13000830.048182.420,000-1.1-1-13100190.006182.460.000-1.1-1-1320.031-0.019182.590.000偏相关1阶白相关4阶拖尾ARMA(1,4)DNZJGCMA(l)MAMA(3)MAAR(1)DependentVariable:DNZJGMethod:LeastSquaresDate:08/31/09Time:21:15Sample(adjusted);2003M032009M07Includedobservations:77afteradjustmentsConvergenceachievedafter16iterationsMABackcasl:2002M112003M02VariableCoefficientStd.Errort-StatisticProb.C02713300-675130-0.4018