非平稳时间序列分析.docx
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非平稳时间序列分析
非平稳时间序列分析
1、首先画出时序图如下:
t
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:
difx
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
-10
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:
dif2x
90
80
70
60
50
40
30
20
10
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
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:
Autocorrelations
Lag
Covariance
Correlation
-198765432101234567891
StdError
0
577.333
1.00000|
|********************|
0
1
-209.345
-.36261
|*******|.
|0.071247
2
-52.915660
-.09166
|.**|.
|0.080069
3
9.139195
0.01583|
.|.
|0.080600
4
15.375892
0.02663
.|*.
|0.080615
5
-59.441547
-.10296
.**|.
|0.080660
6
-23.834489
-.04128
|.*|.
|0.081324
7
100.285
0.17370|
.|***
|0.081431
8
-146.329
-.25346
|*****|.
|0.083290
9
52.228658
0.09047
|.|**.
|0.087118
10
21.008575
0.03639
|.|*.
|0.087593
11
134.018
0.23213|
.|*****
|0.087670
12
-181.531
-.31443
|******|.
|0.090736
13
23.268470
0.04030
|.|*.
|0.096108
14
71.112195
0.12317|
.|**.
|0.096194
15
-105.621
-.18295
|****|.
|0.096991
16
37.591996
0.06511
.|*.
|0.098727
17
23.031506
0.03989
|.|*.
|0.098945
18
45.654745
0.07908
|.|**.
|0.099027
19
-101.320
-.17550
|****|.
|0.099347
20
127.607
0.22103
|.|****
|0.100908
21
-61.519663
-.10656
|.**|.
|0.103337
22
35.825317
0.06205
|.|*.
|0.103893
23
-93.627333
-.16217
|.***|.
|0.104081
24
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:
AutocorrelationCheckforWhiteNoise
ToChi-Pr>
LagSquareDFChiSq
Autocorrelations
6
30.70
6
<.0001
-0.363
-0.092
0.016
0.027
-0.103
-0.041
12
84.54
12
<.0001
0.174
-0.253
0.090
0.036
0.232
-0.314
18
97.98
18
<.0001
0.040
0.123
-0.183
0.065
0.040
0.079
24
126.99
24
<.0001
-0.175
0.221
-0.107
0.062
-0.162
0.096
P值都小于0.05,认为不是白噪声。
接下来对模型进行定阶:
MinimumInformationCriterion
Lags
MA0
MA1MA2MA3MA4MA5
AR0
6.356905
6.1418316.149838
6.175552
6.191564
6.203649
AR1
6.236922
6.1681216.15152
6.172674
6.186962
6.193905
AR2
6.193215
6.1808186.177337
6.197407
6.203224
6.207239
AR3
6.19748
6.2030816.202837
6.221083
6.215313
6.188712
AR4
6.220313
6.229496.227445
6.241883
6.162837
6.189358
AR5
6.222131
6.2367396.244025
6.264968
6.185963
6.210425
Errorseriesmodel:
AR(10)
MinimumTableValue:
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:
ConditionalLeastSquaresEstimation
Standard
Approx
Parameter
Estimate
Error
tValue
Pr>|t|Lag
MU
0.40286
0.16900
2.38
0.01810
MA1,1
0.89063
0.03266
27.27
<.00011
Forecastsforvariablex
时间
Forecast
StdError
95%ConfidenceLimits
1997一季度
7759.2061
31.2276
7698.0011
7820.4112
1997二季度
7842.6135
40.3048
7763.6175
7921.6095
1997三季度
7926.4237
48.9444
7830.4945
8022.3530
1997四季度
8010.6368
57.4356
7898.0651
8123.2085
预测图:
t
本题代码
dataaa;|
inputx@@;|
difx=dif(x);
dif2x=dif(difx);|
t=intnx('quarter','1jan1947'd,_n_-1);
formattyear4.;|
cards;
227.8231.7236.1246.3252.6259.9266.8268.1
263.0
259.5
261.2
258.9
269.6
279.3
296.9
308.4
323.2
331.1
337.9
373.7
342.3
345.3
345.9
351.7
364.2
371.0
374.5
368.7
417.8
368.4
368.7
373.4
381.9
394.8
403.1
411.4
420.5
444.4
426.0
430.8
439.2
448.1
450.1
457.2
451.7
448.6
461.8
475.0
499.0
512.0
512.5
516.9
530.3
529.2
532.2
527.3
531.8
542.4
553.2
566.3
579.0
586.9
594.1
597.7606.8615.3628.2637.5654.5663.4674.3
679.9
701.2713.9730.4752.6775.6785.2798.6812.5
822.2
828.2844.7861.2886.5910.8926.0943.6966.3
979.9
999.3
1144.4
1008.0
1020.3
1035.7
1053.8
1058.4
1104.2
1124.9
1158.8
1198.5
1231.8
1256.7
1297.0
1347.9
1379.4
1404.4
1449.7
1463.9
1496.8
1526.4
1563.2
1571.3
1608.3
1670.6
1725.3
1783.5
1814.0
1847.9
1899.0
1954.5
2026.4
2088.7
2120.4
2166.8
2293.7
2356.2
2437.0
2491.4
2552.9
2629.7
2687.5
2761.7
2756.1
2818.8
2941.5
3076.6
3105.4
3197.7
3222.8
3221.0
3270.3
3287.8
3323.8
3388.2
3501.0
3596.8
3700.3
3824.4
3911.3
3975.6
4022.7
4100.4
4158.7
4238.8
4306.2
4376.6
4399.4
4455.8
4508.5
4573.1
4655.5
4731.4
4845.2
4914.5
5013.7
5105.3
5217.1
5329.2
5423.9
5501.3
5557.0
5681.4
5767.8
5796.8
5813.6
5849.0
5904.5
5959.4
6016.6
6138.3
6212.2
6281.1
6390.5
6458.4
6512.3
6584.8
6684.5
6773.6
6876.3
6977.6
7062.2
7140.5
7202.4
7293.4
7344.3
7426.6
7537.5
7593.6procgplot;
plotx*tdifx*tdif2x*t;|
symbolc=blacki=joinv=star;run;
procarima;
identifyestimate
var=x(1,1)nlag=8minicp=(0:
5)
q=1;
q=(0:
5);
forecastrun;
lead=5id=tinterval
=quarter
out
=results;|
procgplot
data=results;
plotx*t=1forecast*t=2l95*t=3u95*t=3/overlay
symbol1c=blacki=nonev=star;|
symbol2c=redi=joinv=none;|
symbolrun;
c=greeni=joinv=none丨=32;
2、
首先画出时序图:
x
t
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:
difx
800
700
600
500
400
300
200
100
0
-100
-200
-300
-400
t
可以看到趋势已经消除,但季节性仍存在,对其进行检验:
Autocorrelations
Lag
Covariance
Correlation
-198765432101234567891
StdError
0
16681.747
1.00000|
|********************|
0
1
-3098.631
-.18575|
****|
|0.049568
2
49.867617
0.00299
|.|.
|0.051250
3
-4342.304
-.26030
|*****|
|0.051250
4
-1177.801
-.07060|
.*|.
|0.054402
5
3921.886
0.23510|
|*****
|0.054626
6
-258.497
-.01550
|.|.
|0.057058
7
3392.968
0.20339
||****
|0.057069
8
-1407.632
-.08438
|**|.
|0.058823
9
-4040.701
-.24222
|*****|
|0.059120
10
-1262.123
-.07566
|**|.
|0.061510
11
-1890.805
-.11335
|**|.
|0.061738
12
10239.264
0.61380
||************|
0.062247
13
-2555.185
-.15317
|***|
|0.075671
14
-784.895
-.04705
|.*|.
|0.076429
15
-4767.938
-.28582
|******|
|0.076500
16
-1583.636
-.09493
|.**|.
|0.079080
17
4107.732
0.24624
||*****
|0.079360
18
-931.403
-.05583
|.*|.
|0.081215
19
3860.394
0.23141|
|*****
|0.081309
20
-2035.458
-.12202|
.**|.
|0.082912
21
-3762.045
-.22552|
*****|
|0.083352
22
-868.587
-.05207|
.*|.
|0.084838
23
-1587.006
-.09513|
.**|.
|0.084916
24
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:
dif12x
500
-400
检验结果为:
Lag
Covariance
Correlation
-198765432101234567891
StdError
0
12557.531
1.00000|
|********************|
0
1
-1734.813
-.13815|
***|
|0.050315
2
2375.783
0.18919|
|****
|0.051267
3
279.359
0.02225
|.|.
|0.053005
4
759.808
0.06051|
.|*.
|0.053028
5
138.150
0.01100|
.|.
|0.053203
6
655.488
0.05220
|.|*.
|0.053209
7
-1028.202
-.08188
|**|.
|0.053338
8
533.734
0.04250
|.|*.
|0.053655
9
-158.605
-.01263
|.|.
|0.053741
10
-1606.237
-.12791
|***|
|0.053748
11
858.206
0.06834
|
.|*.
|
0.054513
12
-5698.457
-.45379
|
*********|.
|
0.054730
13
521.120
0.04150
|
.|*.
|
0.063545
14
-509.219
-.04055
|
.*|.
|
0.063614
15
-1020.660
-.08128
|
.**|.
|
0.063679
16
-730.212
-.05815
|
.*|.
|
0.063941
17
429.071
0.03417
|
.|*.
|
0.064075
18
-825.235
-.06572
|
.*|.
|
0.064121
19
592.947
0.04722
|
.|*.
|
0.064292
20
-565.282
-.04502
|
.*|.
|
0.064379
21
206.681
0.01646
|
.|.
|
0.064459
22
-117.966
-.00939
|
.|.
|
0.064470
23
774.691
0.06169
|
.|*.
|
0.064473
24
-929.421
-.07401
|
.*|.
|
、/■.、r
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
:
AutocorrelationCheckforWhiteNoise
To
Chi-
Pr>
Lag
Square
DFChiSq
Autocorrelations
6
24.69
6
0.0004
-0.138
0.189
0.022
0.061
0.011
0.052
12
121.08
12
<.0001
-0.082
0.043
-0.013
-0.128
0.068
-0.454
18
128.87
18
<.0001
0.041
-0.041
-0.081
-0.058
0.034
-0.066
24
134.72
24
<.0001
0.047
-0.045
0.016
-0.009
0.062
-0.074
p值均小于0.05,该序列不是白噪声。
接下来对模型进行定阶:
TheARIMAProcedure
MinimumInformationCriterion
LagsMA0MA1MA2MA3MA4MA5
AR0
9.421702
9.42178
9.416333
9.430297
9.436893
9.438915
AR1
9.415292
9.425825
9.428108
9.436029
9.440215
9.440263
AR2
9.394416
9.409291
9.378723
9.365584
9.354303
9.359478
AR3
9.403315
9.416418
9.344995
9.352347
9.354318
9.337039
AR4
9.415716
9.426209
9.326993
9.34213
9.350261
9.302074
AR5
9.417148
9.426017
9.318538
9.321328
9.290061
9.305156
Errorseriesmodel:
AR(10)
MinimumTableValue:
BIC(5,4)=9.290061
从结果中可以看出,BIC(5,4)最小,所以取ARMA(5,4)模型进行拟合,得到
模型,模型检验结果:
ConditionalLeastSquaresEstimation
Standard
Approx
Parameter
Estimate
Error
tValue
Pr>|t|
Lag
MU
0.27979
5.67672
0.05
0.9607
0
MA1,1
1.03722
0.02745
37.78
<.0001
1
MA1,2
-1.96578
0.02662
-73.86
<.0001
2
MA1,3
1.01221
0.02816
35.94
<.0001
3
MA1,4
-0.93338
0.02327
-40.11
<.0001
4
AR1,1
0.88707
0.05783
15.34
<.0001
1
AR1,2
-1.56109
0.07112
-21.95
<.0001
2
AR1,3