Arima модель
Построим модель ARIMA для ряда ft
Так как ln ft~I(1), то строим для его первой разности.В командной строке: ls d(ft) c
Dependent Variable: D(FT) |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 20:34 |
|
|
||
Sample (adjusted): 2008M08 2011M10 |
|
|||
Included observations: 39 after adjustments |
|
|||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
C |
1.396410 |
43.62106 |
0.032012 |
0.9746 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.000000 |
Mean dependent var |
1.396410 |
|
Adjusted R-squared |
0.000000 |
S.D. dependent var |
272.4135 |
|
S.E. of regression |
272.4135 |
Akaike info criterion |
14.07783 |
|
Sum squared resid |
2819945. |
Schwarz criterion |
14.12048 |
|
Log likelihood |
-273.5176 |
Hannan-Quinn criter. |
14.09313 |
|
Durbin-Watson stat |
1.653302 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Проверяем на автокорреляцию:
Breusch-Godfrey Serial Correlation LM Test: |
|
|||
|
|
|
|
|
|
|
|
|
|
F-statistic |
0.822032 |
Prob. F(5,33) |
0.5430 |
|
Obs*R-squared |
4.319471 |
Prob. Chi-Square(5) |
0.5044 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Test Equation: |
|
|
|
|
Dependent Variable: RESID |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 20:35 |
|
|
||
Sample: 2008M08 2011M10 |
|
|
||
Included observations: 39 |
|
|
||
Presample missing value lagged residuals set to zero. |
||||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
C |
-1.999542 |
44.32764 |
-0.045108 |
0.9643 |
RESID(-1) |
0.178697 |
0.178487 |
1.001176 |
0.3240 |
RESID(-2) |
-0.169754 |
0.178012 |
-0.953613 |
0.3472 |
RESID(-3) |
0.035472 |
0.184069 |
0.192708 |
0.8484 |
RESID(-4) |
0.246136 |
0.182898 |
1.345750 |
0.1876 |
RESID(-5) |
-0.010348 |
0.186987 |
-0.055340 |
0.9562 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.110756 |
Mean dependent var |
-1.17E-14 |
|
Adjusted R-squared |
-0.023978 |
S.D. dependent var |
272.4135 |
|
S.E. of regression |
275.6601 |
Akaike info criterion |
14.21685 |
|
Sum squared resid |
2507621. |
Schwarz criterion |
14.47279 |
|
Log likelihood |
-271.2286 |
Hannan-Quinn criter. |
14.30868 |
|
F-statistic |
0.822032 |
Durbin-Watson stat |
1.941127 |
|
Prob(F-statistic) |
0.542952 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Автокорреляции нет, так как prob>0,05%
Для улудшения модели попробуем добавить ma(4) (он выступает)
Dependent Variable: D(FT) |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 21:46 |
|
|
||
Sample (adjusted): 2008M08 2011M10 |
|
|||
Included observations: 39 after adjustments |
|
|||
Convergence achieved after 9 iterations |
|
|||
MA Backcast: 2008M04 2008M07 |
|
|
||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
C |
-8.069908 |
64.54687 |
-0.125024 |
0.9012 |
MA(4) |
0.680582 |
0.134092 |
5.075503 |
0.0000 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.223648 |
Mean dependent var |
1.396410 |
|
Adjusted R-squared |
0.202665 |
S.D. dependent var |
272.4135 |
|
S.E. of regression |
243.2478 |
Akaike info criterion |
13.87596 |
|
Sum squared resid |
2189271. |
Schwarz criterion |
13.96127 |
|
Log likelihood |
-268.5812 |
Hannan-Quinn criter. |
13.90657 |
|
F-statistic |
10.65877 |
Durbin-Watson stat |
1.709687 |
|
Prob(F-statistic) |
0.002363 |
|
|
|
Константу убираем, так как она не значима
Dependent Variable: D(FT) |
|
|
||||||
Method: Least Squares |
|
|
||||||
Date: 11/30/11 Time: 21:47 |
|
|
||||||
Sample (adjusted): 2008M08 2011M10 |
|
|||||||
Included observations: 39 after adjustments |
|
|||||||
Convergence achieved after 7 iterations |
|
|||||||
MA Backcast: 2008M04 2008M07 |
|
|
||||||
|
|
|
|
|
||||
|
|
|
|
|
||||
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
||||
|
|
|
|
|
||||
|
|
|
|
|
||||
MA(4) |
0.680017 |
0.132099 |
5.147794 |
0.0000 |
||||
|
|
|
|
|
||||
|
|
|
|
|
||||
R-squared |
0.223318 |
Mean dependent var |
1.396410 |
|||||
Adjusted R-squared |
0.223318 |
S.D. dependent var |
272.4135 |
|||||
S.E. of regression |
240.0769 |
Akaike info criterion |
13.82510 |
|||||
Sum squared resid |
2190202. |
Schwarz criterion |
13.86776 |
|||||
Log likelihood |
-268.5895 |
Hannan-Quinn criter. |
13.84041 |
|||||
Durbin-Watson stat |
1.708930 |
|
|
|
||||
|
|
|
|
|
||||
|
|
|
|
|
||||
Inverted MA Roots |
.64-.64i |
.64-.64i |
-.64+.64i |
-.64+.64i |
||||
|
|
|
|
|
||||
|
|
|
|
|
||||
|
|
|
|
|
||||
|
|
|
|
|
||||
Проверяем на автокорреляцию:
Breusch-Godfrey Serial Correlation LM Test: |
|
|||
|
|
|
|
|
|
|
|
|
|
F-statistic |
1.412316 |
Prob. F(5,33) |
0.2455 |
|
Obs*R-squared |
6.872964 |
Prob. Chi-Square(5) |
0.2303 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Test Equation: |
|
|
|
|
Dependent Variable: RESID |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 21:51 |
|
|
||
Sample: 2008M08 2011M10 |
|
|
||
Included observations: 39 |
|
|
||
Presample missing value lagged residuals set to zero. |
||||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
MA(4) |
0.346830 |
0.198788 |
1.744726 |
0.0903 |
RESID(-1) |
0.238410 |
0.177822 |
1.340720 |
0.1892 |
RESID(-2) |
-0.096080 |
0.172519 |
-0.556924 |
0.5813 |
RESID(-3) |
-0.091371 |
0.176740 |
-0.516979 |
0.6086 |
RESID(-4) |
-0.577544 |
0.259315 |
-2.227194 |
0.0329 |
RESID(-5) |
0.110589 |
0.183370 |
0.603093 |
0.5506 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.176230 |
Mean dependent var |
-1.616115 |
|
Adjusted R-squared |
0.051416 |
S.D. dependent var |
240.0713 |
|
S.E. of regression |
233.8181 |
Akaike info criterion |
13.88760 |
|
Sum squared resid |
1804139. |
Schwarz criterion |
14.14353 |
|
Log likelihood |
-264.8082 |
Hannan-Quinn criter. |
13.97943 |
|
Durbin-Watson stat |
1.937309 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Автокорреляции нет, так как pob>0,05
Проверяем остатки на гетероскедастичность:
H0: остатки гомоскедастичны
H1: остатки гетероскедастичны
Heteroskedasticity Test: White |
|
|||
|
|
|
|
|
|
|
|
|
|
F-statistic |
0.051355 |
Prob. F(1,37) |
0.8220 |
|
Obs*R-squared |
0.054056 |
Prob. Chi-Square(1) |
0.8161 |
|
Scaled explained SS |
0.032396 |
Prob. Chi-Square(1) |
0.8572 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Test Equation: |
|
|
|
|
Dependent Variable: RESID^2 |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 21:53 |
|
|
||
Sample: 2008M08 2011M10 |
|
|
||
Included observations: 39 |
|
|
||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
C |
58035.27 |
13267.05 |
4.374391 |
0.0001 |
GRADF_01^2 |
-0.022154 |
0.097759 |
-0.226617 |
0.8220 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.001386 |
Mean dependent var |
56159.03 |
|
Adjusted R-squared |
-0.025604 |
S.D. dependent var |
63926.43 |
|
S.E. of regression |
64739.63 |
Akaike info criterion |
25.04405 |
|
Sum squared resid |
1.55E+11 |
Schwarz criterion |
25.12937 |
|
Log likelihood |
-486.3591 |
Hannan-Quinn criter. |
25.07466 |
|
F-statistic |
0.051355 |
Durbin-Watson stat |
1.452530 |
|
Prob(F-statistic) |
0.821969 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Так как prob>0,05%, то остатки гомоскедастичны
Проверим нормальность остатков:
H0: остатки имеют нормальное распределение
H1: остатки не имею нормальное распределение
Jarque-Bera=1.47
Prob=0,47
Так как prob>0,05 то мы не отвергаем H0 и делаем вывод, что остатки имеют нормальное распределение.
Посмотрим, являются ли остатки белым шумом:
Остатки похожи на белый шум
Модель ARIMA (0,1,4)
Строим ARIMA для ряда micex
Dependent Variable: D(MICEX) |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 22:08 |
|
|
||
Sample (adjusted): 2008M08 2011M10 |
|
|||
Included observations: 39 after adjustments |
|
|||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
C |
-1.625897 |
18.51941 |
-0.087794 |
0.9305 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.000000 |
Mean dependent var |
-1.625897 |
|
Adjusted R-squared |
0.000000 |
S.D. dependent var |
115.6537 |
|
S.E. of regression |
115.6537 |
Akaike info criterion |
12.36438 |
|
Sum squared resid |
508279.2 |
Schwarz criterion |
12.40704 |
|
Log likelihood |
-240.1055 |
Hannan-Quinn criter. |
12.37969 |
|
Durbin-Watson stat |
1.072483 |
|
|
|
|
|
|
|
|
|
|
|
|
|
П |
|
|
|
|
Проверяем на автокорреляцию:
Breusch-Godfrey Serial Correlation LM Test: |
|
|||
|
|
|
|
|
|
|
|
|
|
F-statistic |
1.964763 |
Prob. F(5,33) |
0.1101 |
|
Obs*R-squared |
8.946627 |
Prob. Chi-Square(5) |
0.1112 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Test Equation: |
|
|
|
|
Dependent Variable: RESID |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 22:13 |
|
|
||
Sample: 2008M08 2011M10 |
|
|
||
Included observations: 39 |
|
|
||
Presample missing value lagged residuals set to zero. |
||||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
C |
1.147096 |
17.50553 |
0.065528 |
0.9481 |
RESID(-1) |
0.452290 |
0.175260 |
2.580685 |
0.0145 |
RESID(-2) |
-0.043807 |
0.197519 |
-0.221784 |
0.8258 |
RESID(-3) |
0.064469 |
0.201525 |
0.319904 |
0.7511 |
RESID(-4) |
0.100233 |
0.204677 |
0.489714 |
0.6276 |
RESID(-5) |
-0.219869 |
0.186028 |
-1.181913 |
0.2457 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.229401 |
Mean dependent var |
-8.38E-15 |
|
Adjusted R-squared |
0.112643 |
S.D. dependent var |
115.6537 |
|
S.E. of regression |
108.9453 |
Akaike info criterion |
12.36021 |
|
Sum squared resid |
391679.6 |
Schwarz criterion |
12.61614 |
|
Log likelihood |
-235.0240 |
Hannan-Quinn criter. |
12.45203 |
|
F-statistic |
1.964763 |
Durbin-Watson stat |
1.962771 |
|
Prob(F-statistic) |
0.110059 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Так как prob>0,05 то автокорреляции нет
Смотрим:
Для улудшения модели добавляем ar(1)
Dependent Variable: D(MICEX) |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 22:11 |
|
|
||
Sample (adjusted): 2008M09 2011M10 |
|
|||
Included observations: 38 after adjustments |
|
|||
Convergence achieved after 3 iterations |
|
|||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
C |
6.621267 |
30.35542 |
0.218125 |
0.8286 |
AR(1) |
0.443241 |
0.146247 |
3.030778 |
0.0045 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.203286 |
Mean dependent var |
2.184211 |
|
Adjusted R-squared |
0.181155 |
S.D. dependent var |
114.6988 |
|
S.E. of regression |
103.7910 |
Akaike info criterion |
12.17383 |
|
Sum squared resid |
387812.5 |
Schwarz criterion |
12.26002 |
|
Log likelihood |
-229.3028 |
Hannan-Quinn criter. |
12.20450 |
|
F-statistic |
9.185614 |
Durbin-Watson stat |
1.956559 |
|
Prob(F-statistic) |
0.004499 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Inverted AR Roots |
.44 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Так как констатнта незначима (prob>0,05), то убираем ее:
Dependent Variable: D(MICEX) |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 22:11 |
|
|
||
Sample (adjusted): 2008M09 2011M10 |
|
|||
Included observations: 38 after adjustments |
|
|||
Convergence achieved after 2 iterations |
|
|||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
AR(1) |
0.442299 |
0.144290 |
3.065345 |
0.0040 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.202226 |
Mean dependent var |
2.184211 |
|
Adjusted R-squared |
0.202226 |
S.D. dependent var |
114.6988 |
|
S.E. of regression |
102.4469 |
Akaike info criterion |
12.12253 |
|
Sum squared resid |
388328.4 |
Schwarz criterion |
12.16562 |
|
Log likelihood |
-229.3281 |
Hannan-Quinn criter. |
12.13786 |
|
Durbin-Watson stat |
1.952102 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Inverted AR Roots |
.44 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Проверяем на гетероскедастичность:
Heteroskedasticity Test: White |
|
|||
|
|
|
|
|
|
|
|
|
|
F-statistic |
3.634241 |
Prob. F(1,36) |
0.0646 |
|
Obs*R-squared |
3.484390 |
Prob. Chi-Square(1) |
0.0620 |
|
Scaled explained SS |
3.401638 |
Prob. Chi-Square(1) |
0.0651 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Test Equation: |
|
|
|
|
Dependent Variable: RESID^2 |
|
|
||
Method: Least Squares |
|
|
||
Date: 11/30/11 Time: 22:15 |
|
|
||
Sample: 2008M09 2011M10 |
|
|
||
Included observations: 38 |
|
|
||
|
|
|
|
|
|
|
|
|
|
Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
|
|
|
|
|
|
|
|
|
|
C |
7499.461 |
2731.622 |
2.745424 |
0.0094 |
GRADF_01^2 |
0.205014 |
0.107541 |
1.906369 |
0.0646 |
|
|
|
|
|
|
|
|
|
|
R-squared |
0.091694 |
Mean dependent var |
10219.17 |
|
Adjusted R-squared |
0.066464 |
S.D. dependent var |
14862.23 |
|
S.E. of regression |
14359.84 |
Akaike info criterion |
22.03345 |
|
Sum squared resid |
7.42E+09 |
Schwarz criterion |
22.11964 |
|
Log likelihood |
-416.6356 |
Hannan-Quinn criter. |
22.06412 |
|
F-statistic |
3.634241 |
Durbin-Watson stat |
1.383997 |
|
Prob(F-statistic) |
0.064610 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Так как prob>0,05%, то остатки гомоскедастичны
Проверим нормальность остатков:
H0: остатки имеют нормальное распределение
H1: остатки не имею нормальное распределение
Jarque-Bera=3,8981
Prob=-0,142409
Так как prob>0,05 то мы не отвергаем H0 и делаем вывод, что остатки имеют нормальное распределение.
Посмотрим, являются ли остатки белым шумом:
Остатки похожи на белый шум
ARIMA (1,1,0)