ARMA模型案例.doc

ARMA模型的应用举例2.5.1 案例分析的目的本案例拟选取1996年1月到2010年9月我国货币供应量M1的数据来构建ARMA模型，并利用该模型进行外推预测分析。2.5.2 实验数据数据来源于中经网统计数据库。具体数据见表2.3。表2.3 M1月度数据 单位：亿元日期M1 日期M1 日期M1日期M1 1996-01251951999-10422652003-0776152.82007-04127677.81996-0225255.61999-11433702003-08770332007-05130275.81996-03239091999-1245837.242003-0979163.92007-06135847.41996-04241452000-01465702003-1080267.12007-07136237.41996-05244632000-0244679.22003-1180814.92007-08140993.21996-06246002000-0345158.42003-1284118.62007-09142591.61996-07250782000-04463192004-0183805.92007-10144649.31996-0825729.452000-0546490.22004-0283556.42007-11148009.81996-09262302000-0648024.42004-0385815.62007-12152519.21996-1026798.22000-0747803.12004-0485603.62008-01154872.61996-11274222000-08488852004-0586780.42008-02150177.91996-12285152000-0950616.92004-0688627.12008-03150867.51997-01305732000-10499532004-0787982.22008-04151681.41997-02291032000-1150787.52004-0889125.32008-05153344.81997-03290582000-1253147.22004-0990439.12008-06154820.21997-04299912001-0154406.22004-1090782.52008-07154992.41997-05302752001-0251997.72004-1192387.12008-08156889.91997-06310742001-0353033.42004-1295969.72008-091557491997-07311002001-0453261.32005-01970792008-10157194.41997-0831594.992001-05525432005-02928152008-11157826.61997-09322452001-0655187.42005-0394743.22008-12166217.11997-10324222001-0753502.82005-0494593.72009-01165214.31997-11329092001-0855808.92005-05958022009-02166149.61997-1234826.272001-09568242005-0698601.32009-03176541.11998-0135585.62001-1056114.92005-0797674.12009-04178213.61998-02333952001-1156579.62005-0899377.72009-05182025.61998-03331102001-1259871.62005-091009642009-06193138.21998-04333602002-0160576.12005-101017522009-07195889.31998-05335532002-0258702.92005-11104125.82009-08200394.81998-06337762002-0359474.82005-12107278.72009-09201708.11998-07343562002-0460461.32006-01107250.72009-10207545.71998-08350502002-0561284.92006-02104357.12009-11212493.21998-09365012002-06631442006-03106737.12009-12220001.51998-1036786.72002-0763487.82006-04106389.12010-012295891998-11374142002-0864868.82006-05109219.22010-022242871998-1238953.682002-09667972006-06112342.42010-03229397.91999-01390112002-1067100.32006-071126532010-04233909.81999-02387492002-1167992.82006-08114845.72010-05236497.91999-03380542002-1270882.12006-09116814.12010-062405801999-04380532003-0172405.72006-101183602010-07240664.11999-05380042003-0269756.62006-111216452010-08244340.61999-06388222003-0371438.82006-12126035.12010-09243802.41999-07389912003-0471321.22007-01128484.11999-08400952003-0572777.82007-02126258.11999-09419142003-0675923.22007-03127881.32.5.3 ARMA模型的构建1、判断序列的平稳性首先绘制出M1的折线图，结果如下图：图2.1 货币供给量M1曲线图从图中可以看出，M1序列具有较强的非线性趋势性，因此从图形可以初步判断该序列是非平稳的。此外M1在每年同时期出现相同的变动方式，表明M1还存在季节性特征。下面对M1的平稳性和季节性进行进一步检验。2、单位根检验为了减少M1的变动趋势及异方差性，先对M1进行对数处理，记为LM1，其曲线图见图2.2。图2.2 LM1曲线图对数后的货币供给量趋势性也较强。下面观察LM1的自相关图，选择滞后期为24，见表2.4。表2.4 LM1的自相关图AutocorrelationPartial CorrelationACPACQ-StatProb.|*|.|*|10.9630.963166.920.000.|*|.|. |20.926-0.017322.180.000.|*|.|. |30.889-0.019466.140.000.|*|.|. |40.853-0.010599.380.000.|* |.|. |50.817-0.014722.390.000.|* |.|. |60.7830.008836.110.000.|* |.|. |70.7510.002941.270.000.|* |.|. |80.720-0.0001038.50.000.|* |.|. |90.6910.0101128.60.000.|* |.|. |100.6640.0091212.30.000.|* |.|. |110.6390.0081290.20.000.|* |.|. |120.6150.0111362.90.000.|* |.|. |130.592-0.0051430.70.000.|* |.|. |140.569-0.0201493.60.000.|* |.|. |150.546-0.0051551.80.000.|* |.|. |160.523-0.0041605.70.000.|* |.|. |170.501-0.0121655.30.000.|* |.|. |180.4800.0091701.20.000.|* |.|. |190.459-0.0041743.50.000.|* |.|. |200.4400.0091782.60.000.|* |.|. |210.4230.0071818.90.000.|* |.|. |220.405-0.0111852.40.000.|* |.|. |230.388-0.0031883.40.000.|* |.|. |240.3730.0131912.10.000从上表可以看出，LM1的PACF只在滞后一期时是显著的，ACF随着滞后阶数增加慢慢衰减至0，因此从偏/自相关系数可以看出该序列表现一定的平稳性。进一步进行单位根检验，打开LM1，单击view/unit root test.，选择存在趋势项的形式，并根据AIC自动选择滞后阶数，单位根检验见过见表2.5。表2.5 LM1的单位根检验结果Null Hypothesis: LM1 has a unit rootExogenous: Constant, Linear TrendLag Length: 13 (Automatic based on SIC, MAXLAG=13)t-StatisticProb.*Augmented Dickey-Fuller test statistic-4.5101880.0020Test critical values:1% level-4.0153415% level-3.43762910% level-3.143037*MacKinnon (1996) one-sided p-values.T统计量的值小于临界值，且相伴概率为0.0020，因此该序列不存在单位根，即该序列是平稳序列。3、季节性分析趋势性往往会掩盖季节性特征，从LM1的图形可以看出，该序列具有较强的趋势性。为了分析季节性，可以对LM1进行差分处理来观察季节性：genr dLM1=lm1-lm1(-1)观察dLM1的自相关图：表2.6 dLM1的自相关图AutocorrelationPartial CorrelationACPACQ-StatProb*|. |*|. |1-0.085-0.0851.28080.258*|. |*|. |2-0.203-0.2128.73240.013.|* |.|* |30.1660.13413.7060.003*|. |*|. |4-0.093-0.11815.2710.004.|. |.|. |5-0.0270.02215.4000.009.|. |*|. |6-0.006-0.07815.4060.017.|. |.|. |7-0.045-0.02115.7810.027*|. |*|. |8-0.081-0.12517.0160.030.|* |.|* |90.0850.07918.3780.031*|. |*|. |10-0.313-0.39036.8210.000.|. |.|* |110.0480.11137.2510.000.|* |.|* |120.5730.466100.040.000*|. |.|. |13-0.1290.063103.220.000*|. |*|. |14-0.179-0.182109.400.000.|. |*|. |150.061-0.149110.130.000*|. |*|. |16-0.081-0.088111.410.000*|. |*|. |17-0.078-0.067112.600.000.|. |*|. |18-0.002-0.063112.600.000*|. |*|. |19-0.083-0.066113.960.000*|. |*|. |20-0.062-0.104114.730.000.|. |.|. |210.0650.064115.580.000*|. |.|. |22-0.2380.035127.100.000.|. |*|. |230.060-0.072127.830.000.|* |.|* |240.4950.162178.260.000*|. |.|* |25-0.0680.087179.210.000*|. |.|. |26-0.154-0.035184.160.000.|* |*|. |270.066-0.093185.070.000*|. |*|. |28-0.086-0.103186.620.000*|. |*|. |29-0.066-0.064187.550.000.|. |.|. |300.011-0.027187.580.000*|. |.|. |31-0.086-0.018189.180.000.|. |.|. |32-0.0100.015189.210.000.|* |.|. |330.0760.053190.460.000*|. |.|. |34-0.2030.055199.560.000.|* |.|. |350.074-0.029200.770.000.|* |.|. |360.4270.033241.540.000dLM1在滞后期为12、24、36.等处的自相关系数均显著异于0，因此该序列以周期12呈现季节性，而且季节自相关系数并没有衰减至0，因此为了考虑这种季节性，进行季节性差分：genr sdLM1=dLM1- dLM1(-12)表2.7 sdLM1的自相关图再做关于sdLM1的自相关图： AutocorrelationPartial CorrelationACPACQ-StatProb*|. |*|. |1-0.100-0.1001.66790.197.|* |.|* |20.0800.0712.74360.254.|* |.|* |30.2060.2249.91600.019.|. |.|. |4-0.0020.0389.91670.042.|* |.|. |50.0750.04510.8840.054.|. |.|. |6-0.005-0.04410.8890.092.|. |.|. |70.0240.00010.9870.139.|. |.|. |80.015-0.00411.0250.200.|. |.|. |90.0100.01711.0410.273*|. |*|. |10-0.174-0.19316.3710.089.|* |.|. |110.1010.06418.1750.078*|. |*|. |12-0.274-0.26231.6410.002.|. |.|. |13-0.036-0.02131.8740.003.|. |.|. |140.0400.04232.1680.004.|. |.|* |15-0.0250.14532.2840.006.|. |.|. |16-0.008-0.00232.2970.009.|. |.|. |17-0.036-0.00532.5310.013.|. |*|. |18-0.015-0.07332.5750.019.|. |.|. |19-0.056-0.04433.1670.023.|. |*|. |20-0.052-0.10033.6860.028*|. |.|. |21-0.058-0.01234.3270.033.|. |*|. |22-0.015-0.11234.3700.045.|. |.|. |23-0.0150.05334.4110.059*|. |*|. |24-0.062-0.11435.1480.066sdLM1在滞后期24之后的季节ACF和PACF已经衰减至0，下面对sdLM1建立SARMA模型。4、滞后阶数的初步决定观察sdLM1的自相关-偏自相关图（表2.7），ACF和PACF在滞后期3和12异于0，其余均与0无差异。因此SARMA(p,q)(k,m)S中p和q均不超过3，k和m均不超过1。考虑到高阶移动平均模型估计较为困难，而且自回归模型可以表示无穷阶的移动平均过程，因此q尽可能取较小的取值。本例拟选择SARMA(1,0)(1,0)12、SARMA(1,0)(1,1)12、SARMA(1,1)(1,0)12、SARMA(1,1)(1,1)12、SARMA(2,0)(1,0)12、SARMA(2,0)(1,1)12、SARMA(3,0)(1,0)12、SARMA(3,0)(1,1)12八个模型来拟合sdLM1。5、ARMA模型的参数估计以SARMA(1,0)(1,0)12模型为例，分析该模型的估计及残差的检验。其他模型类似考虑。在主窗口中点击quick/estimate equation，或者在命令行中输入：ls sdLM1 c ar(1) sar(12) 回归结果为：表2.8 SARMA(1,0)(1,0)12模型的估计结果Dependent Variable: SDLM1Method: Least SquaresDate: 05/17/11 Time: 09:56Sample (adjusted): 1998M03 2010M09Included observations: 151 after adjustmentsConvergence achieved after 7 iterationsVariableCoefficientStd. Errort-StatisticProb.C0.0004740.0006610.7169940.4745AR(1)-0.1499570.078911-1.9003250.0593SAR(12)-0.5541590.069940-7.9233180.0000R-squared0.297412Mean dependent var0.000344Adjusted R-squared0.287917S.D. dependent var0.017200S.E. of regression0.014514Akaike info criterion-5.607696Sum squared resid0.031178Schwarz criterion-5.547750Log likelihood426.3811F-statistic31.32483Durbin-Watson stat1.918914Prob(F-statistic)0.000000Inverted AR Roots.92-.25i.92+.25i.67-.67i.67+.67i.25-.92i.25+.92i-.15-.25+.92i-.25-.92i-.67-.67i-.67-.67i-.92-.25i-.92+.25i表2.9 SARMA(1,0)(1,0)12模型的残差检验结果回归结果的最后一部分表示该模型滞后多项式的反特征根，显然各根的模均小于1，因此该模型是平稳的。下面对残差进行检验。观察残差的自相关图：AutocorrelationPartial CorrelationACPACQ-StatProb.|. |.|. |10.0370.0370.2053.|* |.|* |20.1190.1172.3885.|* |.|* |30.2070.2029.10520.003.|. |.|. |40.0540.0349.56760.008.|* |.|. |50.0730.02710.4200.015.|. |*|. |6-0.049-0.10710.8060.029.|* |.|. |70.0700.04511.6010.041.|. |.|. |8-0.037-0.04711.8220.066.|. |.|. |9-0.022-