非平稳时间序列实验报告

1、实验报告-时间序列分析08经济统计I60814030王思瑶一、 实验简介针对我国19782002年中国支出法GDP（单位：亿元）进行非平稳性检验、平稳化方法、模型建立及预测，从而掌握对非平稳时间序列的分析。数据如下：19782002年中国支出法GDP（单位：亿元）年份GDP零均值化后GDP19783605.6-32539.9519794073.9-32071.6519804551.3-31594.2519814901.4-31244.1519825489.2-30656.3519836076.3-30069.2519847164.4-28981.1519858792.1-27353.45198

2、610132.8-26012.75198711784-24361.55198814704-21441.55198916466-19679.55199018319.5-17826.05199121280.4-148656-10281.95199334500.6-1644.95199446690.710545522364.95199668330.432184.85199774894.338748.75199879003.342857.75199982673.146527.55200089340.953195.35200198592.962447.

3、352002107897.671752.05二、 非平稳性检验进行非平稳性检验，先用两种方法检验零均值化GDP的平稳性：1、 自相关、偏自相关函数检验法Date: 06/09/11 Time: 22:00Sample: 1978 2002Included observations: 25AutocorrelationPartial CorrelationAC  PAC Q-Stat Prob     . |* |     . |* |10.7270.727

4、14.8770.000     . |* |     . | . |20.5300.00123.1110.000     . |* |     . | . |30.365-0.04427.1990.000     . |*. |     . | . |40.240-0.02229.0550.000&

5、#160;    . |* . |     . | . |50.1780.04830.1290.000     . |* . |     . | . |60.1590.05731.0220.000     . |* . |     . | . |70.1480.02031.8380.000 

6、0;   . |* . |     . | . |80.1360.00632.5720.000     . |* . |     . | . |90.119-0.00133.1650.000     . |* . |     . | . |100.091-0.01433.5400.000  

7、0;  . | . |     . | . |110.057-0.02233.6990.000     . | . |     . | . |120.020-0.03133.7190.001从上图可以看出：自相关函数是拖尾的，偏自相关函数是截尾的，但自相关函数是缓慢衰减的，这说明序列存在一定的非平稳性。2、 单位根检验法在零均值化后数据窗口依次按ViewsUnit Root Text进行单位根检验，如下：Null Hypoth

8、esis: GDP has a unit rootExogenous: ConstantLag Length: 5 (Automatic based on SIC, MAXLAG=5)t-Statistic  Prob.*Augmented Dickey-Fuller test statistic-1.765786 0.3847Test critical values:1% level-3.8315115% level-3.02997010% level-2.655194*MacKinnon (1996) one-sided p-values.Warning: P

9、robabilities and critical values calculated for 20        observations and may not be accurate for a sample size of 19Augmented Dickey-Fuller Test EquationDependent Variable: D(GDP)Method: Least SquaresDate: 06/09/11 Time: 22:07Sample (adjusted): 1984 2002Incl

10、uded observations: 19 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  GDP(-1)-0.0701450.039725-1.7657860.1028D(GDP(-1)1.7803480.2582726.8933010.0000D(GDP(-2)-1.0152820.474855-2.1380880.0538D(GDP(-3)0.5564340.5527061.0067440.3339D(GDP(-4)-0.7314370.537091-1.3618480.1983D(GDP

11、(-5)0.9944990.4185282.3761830.0350C-1668.8691507.690-1.1069050.2900R-squared0.933641    Mean dependent var5359.016Adjusted R-squared0.900461    S.D. dependent var3778.165S.E. of regression1192.002    Akaike info criterion17.28197Sum squared

12、 resid17050421    Schwarz criterion17.62992Log likelihood-157.1787    F-statistic28.13902Durbin-Watson stat2.110286    Prob(F-statistic)0.000002原假设：零均值化后数据有单位根。由结果知假设成立的概率为 0.3847，大于显著性水平0.05，则不能拒绝原假设：序列存在单位根。由于平稳性数据的单位根的绝对值均是小于1的，现在序列

13、出现了单位根，说明序列不是平稳的。通过上面两种方法说明：序列不是平稳的。三、 对序列进行平稳化处理把不平稳数据变为平稳化数据的方法是进行差分，在命令行中输入genr d1=d(gdp)对序列进行一阶差分，得到差分后的数据d1，然后对d1进行单位根检验，得到如下结果：Null Hypothesis: D1 has a unit rootExogenous: ConstantLag Length: 2 (Automatic based on SIC, MAXLAG=5)t-Statistic  Prob.*Augmented Dickey-Fuller test statist

14、ic-1.385731 0.5693Test critical values:1% level-3.7880305% level-3.01236310% level-2.646119*MacKinnon (1996) one-sided p-values.Augmented Dickey-Fuller Test EquationDependent Variable: D(D1)Method: Least SquaresDate: 06/09/11 Time: 22:28Sample (adjusted): 1982 2002Included observations: 21 afte

15、r adjustmentsVariableCoefficientStd. Errort-StatisticProb.  D1(-1)-0.1183970.085440-1.3857310.1837D(D1(-1)0.9295200.1897724.8980860.0001D(D1(-2)-0.5045280.219858-2.2947910.0347C717.1676455.27911.5752270.1336R-squared0.610638    Mean dependent var426.4095Adjusted R-squar

16、ed0.541927    S.D. dependent var1863.610S.E. of regression1261.312    Akaike info criterion17.28734Sum squared resid27045415    Schwarz criterion17.48629Log likelihood-177.5170    F-statistic8.887063Durbin-Watson stat2.1

17、88072    Prob(F-statistic)0.000913由结果知假设成立的概率为  0.5693，大于显著性水平0.05，则不能拒绝原假设：序列存在单位根。 继续进行差分，在命令行中输入genr d2=d(d1)对序列进行一阶差分，得到差分后的数据d2，然后对d2进行单位根检验，得到如下结果：Null Hypothesis: D2 has a unit rootExogenous: ConstantLag Length: 1 (Automatic based on SIC, MAXLAG=4)t-Statistic

18、0; Prob.*Augmented Dickey-Fuller test statistic-3.969756 0.0067Test critical values:1% level-3.7880305% level-3.01236310% level-2.646119*MacKinnon (1996) one-sided p-values.Augmented Dickey-Fuller Test EquationDependent Variable: D(D2)Method: Least SquaresDate: 06/09/11 Time: 22:31Sample (

19、adjusted): 1982 2002Included observations: 21 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  D2(-1)-0.7001220.176364-3.9697560.0009D(D2(-1)0.6410440.2015123.1811630.0052C222.5020289.71240.7680100.4524R-squared0.498631    Mean dependent var8.571429Adjust

20、ed R-squared0.442923    S.D. dependent var1732.575S.E. of regression1293.152    Akaike info criterion17.29912Sum squared resid30100356    Schwarz criterion17.44833Log likelihood-178.6407    F-statistic8.950848Durbin-Wats

21、on stat2.266213    Prob(F-statistic)0.002002该原假设为：d2有单位根。由结果看：d2有单位根成立的概率为0.0067小于0.05，拒绝原假设，即二阶差分后的数据不再含有单位根，序列平稳。四、 模型建立由于二阶差分后的数据已是平稳的，现可以用之做模型进行预测，下面看序列d2的自相关与偏自相关函数：Date: 06/09/11 Time: 22:37Sample: 1978 2002Included observations: 22AutocorrelationPartial CorrelationAC 

22、 PAC Q-Stat Prob     . |*. |     . |*. |10.2130.2131.13980.286     . *| . |     . *| . |2-0.095-0.1471.37930.502     . *| . |     . | .

23、 |3-0.105-0.0541.68420.640     .*| . |     .*| . |4-0.246-0.2403.46620.483     . *| . |     . | . |5-0.0870.0023.70390.593     . |* . |     . |* . |60

24、.1590.1314.53820.604     . *| . |     .*| . |7-0.114-0.2484.99670.660     . | . |     . |* . |80.0220.0985.01460.756     . | . |     . *| . |90.021-0.

25、0665.03320.831     . | . |     . |* . |100.0010.0815.03320.889     . | . |     . *| . |110.000-0.0925.03320.930     . | . |     . | . |120.000-0.0105.

26、03320.957从上图可以看出：自相关函数大致是拖尾的，偏自相关函数是截尾的，且在二阶处截尾。所以，拟建立AR(2)模型对序列进行拟合。如下：Dependent Variable: D2Method: Least SquaresDate: 06/09/11 Time: 22:39Sample (adjusted): 1982 2002Included observations: 21 after adjustmentsConvergence achieved after 3 iterationsVariableCoefficientStd. Errort-StatisticProb.

27、0; AR(1)0.9642700.1899045.0776700.0001AR(2)-0.6354540.199195-3.1901010.0048R-squared0.552457    Mean dependent var426.4095Adjusted R-squared0.528902    S.D. dependent var1863.610S.E. of regression1279.118    Akaike info criterion17.236

28、12Sum squared resid31086710    Schwarz criterion17.33560Log likelihood-178.9793    Durbin-Watson stat2.235500Inverted AR Roots .48-.63i     .48+.63i从拟合结果看：AR(1)的伴随概率为0.0001，AR(2)的伴随概率为0.0048，说明通过了显著性检验。再看AR(3)模型拟合情况：Dependent Vari

29、able: D2Method: Least SquaresDate: 06/09/11 Time: 22:52Sample (adjusted): 1983 2002Included observations: 20 after adjustmentsConvergence achieved after 3 iterationsVariableCoefficientStd. Errort-StatisticProb.  AR(1)0.8429110.2376233.5472590.0025AR(2)-0.4330570.303566-1.4265660.1718AR(3)-

30、0.2332120.256790-0.9081850.3765R-squared0.574780    Mean dependent var435.8450Adjusted R-squared0.524754    S.D. dependent var1911.509S.E. of regression1317.758    Akaike info criterion17.34273Sum squared resid29520276   

31、0;Schwarz criterion17.49209Log likelihood-170.4273    Durbin-Watson stat2.097523Inverted AR Roots .57+.67i     .57-.67i       -.30从回归结果看：AR(3)模型的残差平方和为29520276，大于AR(2)模型的残差平方和31086710，说明拟合效果变好；AR(3)模型的AIC为17.34273，大

32、于AR2)模型的AIC值17.23612，说明拟合效果变差。又因为AR(2)的伴随概率为0.1718，AR(3)的伴随概率为0.3765，说明通不过显著性检验。所以，AR(2)模型比AR(3)模型更适合。对d1取对数，在命令行中输入genr d3=dlog(d1)，对d3进行单位根检验，得到如下结果：Null Hypothesis: D3 has a unit rootExogenous: ConstantLag Length: 0 (Automatic based on SIC, MAXLAG=4)t-Statistic  Prob.*Augmented Dickey-F

33、uller test statistic-3.737676 0.0107Test critical values:1% level-3.7695975% level-3.00486110% level-2.642242*MacKinnon (1996) one-sided p-values.Augmented Dickey-Fuller Test EquationDependent Variable: D(D3)Method: Least SquaresDate: 06/09/11 Time: 23:12Sample (adjusted): 1981 2002Included obs

34、ervations: 22 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  D3(-1)-0.8230550.220205-3.7376760.0013C0.1110000.0865151.2830200.2142R-squared0.411249    Mean dependent var-0.000617Adjusted R-squared0.381812    S.D. dependent var0.48438

35、8S.E. of regression0.380850    Akaike info criterion0.993687Sum squared resid2.900938    Schwarz criterion1.092873Log likelihood-8.930555    F-statistic13.97022Durbin-Watson stat1.910194    Prob(F-statistic)0.001298由结果看：

36、d3有单位根成立的概率为 0.0107小于0.05，拒绝原假设，即取对数后的数据不再含有单位根，序列平稳。由于d3已是平稳的，现可以用之做模型进行预测，下面看序列d3的自相关与偏自相关函数：AutocorrelationPartial CorrelationAC  PAC Q-Stat Prob     . |* . |     . |* . |10.1840.1840.88840.346     

37、;. *| . |     . *| . |2-0.074-0.1121.03890.595     . | . |     . |* . |30.0480.0881.10420.776     *| . |     *| . |4-0.321-0.3784.22970.376     . *| .

38、 |     . | . |5-0.1670.0035.12550.401     . |* . |     . |* . |60.1570.1285.95450.428     . *| . |     .*| . |7-0.135-0.1976.61300.470     . |* . |     . |* . |80.1240.1867.20250.515     . |* . |     . *| . |90.147-0.0778.08840.525     . | . |     . |* . |10-0.0070.1628.09050.620     . | . |