非平稳时间序列分析1、首先画出时序图如下：从时序图中看出有明显的递增趋势，而该序列是一直递增，不随季节波动，所以认为该序列不存在季节特征。

故对原序列做一阶差分，画出一阶差分后的时序图如下：从中可以看到一阶差分后序列仍然带有明显的增长趋势，再做二阶差分：做完二阶差分可以看到，数据的趋势已经消除，接下来对二阶差分后的序列进行检验：AutocorrelationsLag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error0 577.333 1.00000 | |********************| 01 -209.345 -.36261 | *******| . | 0.0712472 -52.915660 -.09166 | .**| . | 0.0800693 9.139195 0.01583 | . | . | 0.0806004 15.375892 0.02663 | . |* . | 0.0806155 -59.441547 -.10296 | .**| . | 0.0806606 -23.834489 -.04128 | . *| . | 0.0813247 100.285 0.17370 | . |*** | 0.0814318 -146.329 -.25346 | *****| . | 0.0832909 52.228658 0.09047 | . |**. | 0.08711810 21.008575 0.03639 | . |* . | 0.08759311 134.018 0.23213 | . |***** | 0.08767012 -181.531 -.31443 | ******| . | 0.09073613 23.268470 0.04030 | . |* . | 0.09610814 71.112195 0.12317 | . |** . | 0.09619415 -105.621 -.18295 | ****| . | 0.09699116 37.591996 0.06511 | . |* . | 0.09872717 23.031506 0.03989 | . |* . | 0.09894518 45.654745 0.07908 | . |** . | 0.09902719 -101.320 -.17550 | ****| . | 0.09934720 127.607 0.22103 | . |**** | 0.10090821 -61.519663 -.10656 | . **| . | 0.10333722 35.825317 0.06205 | . |* . | 0.10389323 -93.627333 -.16217 | .***| . | 0.10408124 55.451208 0.09605 | . |** . |从其自相关图中可以看出二阶差分后的序列自相关系数很快衰减为零，且都在两倍标准差范围之内，所以认为平稳，白噪声检验结果：Autocorrelation Check for White NoiseTo Chi- Pr >Lag Square DF ChiSq--------------------Autocorrelations--------------------6 30.70 6 <.0001 -0.363 -0.092 0.016 0.027 -0.103 -0.04112 84.54 12 <.0001 0.174 -0.253 0.090 0.036 0.232 -0.31418 97.98 18 <.0001 0.040 0.123 -0.183 0.065 0.040 0.07924 126.99 24 <.0001 -0.175 0.221 -0.107 0.062 -0.162 0.096P值都小于0.05，认为不是白噪声。

接下来对模型进行定阶：Minimum Information CriterionLags MA 0 MA 1 MA 2 MA 3 MA 4 MA 5AR 0 6.356905 6.141831 6.149838 6.175552 6.191564 6.203649AR 1 6.236922 6.168121 6.15152 6.172674 6.186962 6.193905AR 2 6.193215 6.180818 6.177337 6.197407 6.203224 6.207239AR 3 6.19748 6.203081 6.202837 6.221083 6.215313 6.188712AR 4 6.220313 6.22949 6.227445 6.241883 6.162837 6.189358AR 5 6.222131 6.236739 6.244025 6.264968 6.185963 6.210425Error series model: AR(10)Minimum Table Value: BIC(0,1) = 6.141831从sas的定阶结果来看，BIC(0,1)取得最小值，所以选取MA（1）模型，接下来对模型进行拟合：得到模型为：模型检验结果为：Conditional Least Squares EstimationStandard ApproxParameter Estimate Error t Value Pr > |t| LagMU 0.40286 0.16900 2.38 0.0181 0MA1,1 0.89063 0.03266 27.27 <.0001 1检验结果显示都显著。

接下来利用此模型对1997年的四个季度进行预测：Forecasts for variable x时间Forecast Std Error 95% Confidence Limits1997一季度7759.2061 31.2276 7698.0011 7820.41121997二季度7842.6135 40.3048 7763.6175 7921.60951997三季度7926.4237 48.9444 7830.4945 8022.35301997四季度8010.6368 57.4356 7898.0651 8123.2085预测图：本题代码data aa;input x@@;difx=dif(x);dif2x=dif(difx);t=intnx('quarter','1jan1947'd,_n_-1);format t year4.;cards;227.8 231.7 236.1 246.3 252.6 259.9 266.8 268.1 263.0259.5 261.2 258.9 269.6 279.3 296.9 308.4 323.2 331.1337.9 342.3 345.3 345.9 351.7 364.2 371.0 374.5 373.7368.7 368.4 368.7 373.4 381.9 394.8 403.1 411.4 417.8420.5 426.0 430.8 439.2 448.1 450.1 457.2 451.7 444.4448.6 461.8 475.0 499.0 512.0 512.5 516.9 530.3 529.2532.2 527.3 531.8 542.4 553.2 566.3 579.0 586.9 594.1597.7 606.8 615.3 628.2 637.5 654.5 663.4 674.3679.9701.2 713.9 730.4 752.6 775.6 785.2 798.6 812.5 822.2828.2 844.7 861.2 886.5 910.8 926.0 943.6 966.3 979.9999.3 1008.0 1020.3 1035.7 1053.8 1058.4 1104.2 1124.9 1144.41158.8 1198.5 1231.8 1256.7 1297.0 1347.9 1379.4 1404.4 1449.71463.9 1496.8 1526.4 1563.2 1571.3 1608.3 1670.6 1725.3 1783.51814.0 1847.9 1899.0 1954.5 2026.4 2088.7 2120.4 2166.8 2293.72356.2 2437.0 2491.4 2552.9 2629.7 2687.5 2761.7 2756.1 2818.82941.5 3076.6 3105.4 3197.7 3222.8 3221.0 3270.3 3287.8 3323.83388.2 3501.0 3596.8 3700.3 3824.4 3911.3 3975.6 4022.7 4100.44158.7 4238.8 4306.2 4376.6 4399.4 4455.8 4508.5 4573.1 4655.54731.4 4845.2 4914.5 5013.7 5105.3 5217.1 5329.2 5423.9 5501.35557.0 5681.4 5767.8 5796.8 5813.6 5849.0 5904.5 5959.4 6016.66138.3 6212.2 6281.1 6390.5 6458.4 6512.3 6584.8 6684.5 6773.66876.3 6977.6 7062.2 7140.5 7202.4 7293.4 7344.3 7426.6 7537.57593.6;proc gplot;plot x*t difx*t dif2x*t;symbol c=black i=join v=star;run;proc arima;identify var=x(1,1) nlag=8minic p=(0:5) q=(0:5);estimate q=1;forecast lead=5id=t interval=quarter out=results;run;proc gplot data=results;plot x*t=1 forecast*t=2 l95*t=3 u95*t=3/overlay;symbol1c=black i=none v=star;symbol2c=red i=join v=none;symbol c=green i=join v=none l=32;run;2、首先画出时序图：从时序图中可以看出序列存在递增趋势，而且存在季节性特征，接下来对序列进行一阶差分，画出差分后的时序图：可以看到趋势已经消除，但季节性仍存在，对其进行检验：AutocorrelationsLag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error0 16681.747 1.00000 | |********************| 01 -3098.631 -.18575 | ****| . | 0.0495682 49.867617 0.00299 | . | . | 0.0512503 -4342.304 -.26030 | *****| . | 0.0512504 -1177.801 -.07060 | .*| . | 0.0544025 3921.886 0.23510 | . |***** | 0.0546266 -258.497 -.01550 | . | . | 0.0570587 3392.968 0.20339 | . |**** | 0.0570698 -1407.632 -.08438 | **| . | 0.0588239 -4040.701 -.24222 | *****| . | 0.05912010 -1262.123 -.07566 | **| . | 0.06151011 -1890.805 -.11335 | **| . | 0.06173812 10239.264 0.61380 | . |************ | 0.06224713 -2555.185 -.15317 | ***| . | 0.07567114 -784.895 -.04705 | . *| . | 0.07642915 -4767.938 -.28582 | ******| . | 0.07650016 -1583.636 -.09493 | .**| . | 0.07908017 4107.732 0.24624 | . |***** | 0.07936018 -931.403 -.05583 | . *| . | 0.08121520 -2035.458 -.12202 | .**| . | 0.08291221 -3762.045 -.22552 | *****| . | 0.08335222 -868.587 -.05207 | . *| . | 0.08483823 -1587.006 -.09513 | .**| . | 0.08491624 9517.308 0.57052 | . |*********** | 0.085178从自相关系数图中可以看到，在其延迟12阶时，相关系数变大，说明序列存在明显季节性特征，对序列进行12步差分，时序图如下：检验结果为：Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error0 12557.531 1.00000 | |********************| 01 -1734.813 -.13815 | ***| . | 0.0503152 2375.783 0.18919 | . |**** | 0.0512673 279.359 0.02225 | . | . | 0.0530054 759.808 0.06051 | . |*. | 0.0530285 138.150 0.01100 | . | . | 0.0532036 655.488 0.05220 | . |*. | 0.0532097 -1028.202 -.08188 | **| . | 0.0533388 533.734 0.04250 | . |*. | 0.0536559 -158.605 -.01263 | . | . | 0.05374110 -1606.237 -.12791 | ***| . | 0.05374812 -5698.457 -.45379 | *********| . | 0.05473013 521.120 0.04150 | . |* . | 0.06354514 -509.219 -.04055 | . *| . | 0.06361415 -1020.660 -.08128 | .**| . | 0.06367916 -730.212 -.05815 | . *| . | 0.06394117 429.071 0.03417 | . |* . | 0.06407518 -825.235 -.06572 | . *| . | 0.06412119 592.947 0.04722 | . |* . | 0.06429220 -565.282 -.04502 | . *| . | 0.06437921 206.681 0.01646 | . | . | 0.06445922 -117.966 -.00939 | . | . | 0.06447023 774.691 0.06169 | . |* . | 0.06447324 -929.421 -.07401 | . *| . | 0.064622平稳性检验显示该序列相关系数迅速衰减为0，且在两倍标准差之内，序列已经平稳，接下来进行白噪声检验：Autocorrelation Check for White NoiseTo Chi- Pr >Lag Square DF ChiSq--------------------Autocorrelations--------------------6 24.69 6 0.0004 -0.138 0.189 0.022 0.061 0.011 0.05212 121.08 12 <.0001 -0.082 0.043 -0.013 -0.128 0.068 -0.45418 128.87 18 <.0001 0.041 -0.041 -0.081 -0.058 0.034 -0.06624 134.72 24 <.0001 0.047 -0.045 0.016 -0.009 0.062 -0.074p值均小于0.05，该序列不是白噪声。