12:54 p.m.
Dear all:
I make my question clearer. ARIMA and X-12-ARIMA have almost the same outcomes under most
combinations of AR and MA. For example, Using the same sample, the output of
ARIMA(1,1,1)(1,1,0 ):
Function evaluations: 22
Evaluations of gradient: 8
Model 5: ARIMA, using observations 1982:03-1989:12 (T = 94)
Estimated using BHHH method (conditional ML)
Dependent variable: (1-L)(1-Ls) z
coefficient std. error z p-value
---------------------------------------------------------
phi_1 0.0386387 0.490287 0.07881 0.9372
Phi_1 -0.547450 0.103980 -5.265 1.40e-07 ***
theta_1 0.134454 0.505469 0.2660 0.7902
Mean dependent var -595.9894 S.D. dependent var 35113.05
Mean of innovations -657.4065 S.D. of innovations 29171.20
Log-likelihood -1099.788 Akaike criterion 2207.577
Schwarz criterion 2217.750 Hannan-Quinn 2211.686
Real Imaginary Modulus Frequency
-----------------------------------------------------------
AR
Root 1 25.8808 0.0000 25.8808 0.0000
AR (seasonal)
Root 1 -1.8266 0.0000 1.8266 0.5000
MA
Root 1 -7.4375 0.0000 7.4375 0.5000
-----------------------------------------------------------
the output of X-12-ARIMA(1,1,1)(1,1,0 ):
Model 6: ARIMA, using observations 1982:03-1989:12 (T = 94)
Estimated using X-12-ARIMA (conditional ML)
Dependent variable: (1-L)(1-Ls) z
coefficient std. error z p-value
---------------------------------------------------------
phi_1 0.0383739 0.602274 0.06371 0.9492
Phi_1 -0.547423 0.0911210 -6.008 1.88e-09 ***
theta_1 0.134554 0.597619 0.2252 0.8219
Mean dependent var -595.9894 S.D. dependent var 35113.05
Mean of innovations -657.4774 S.D. of innovations 29171.20
Log-likelihood -1099.788 Akaike criterion 2207.577
Schwarz criterion 2217.750 Hannan-Quinn 2211.686
Real Imaginary Modulus Frequency
-----------------------------------------------------------
AR
Root 1 26.0594 0.0000 26.0594 0.0000
AR (seasonal)
Root 1 -1.8267 0.0000 1.8267 0.5000
MA
Root 1 -7.4320 0.0000 7.4320 0.5000
-----------------------------------------------------------
The outcomes of ARIMA(1,1,1)(1,1,0 ) and X-12-ARIMA(1,1,1)(1,1,0 ) are almost the same.
But there are a few exceptions. For example, under the same sample, the output of
ARIMA(1,1,2)(2,1,0 ):
Model 7: ARIMA, using observations 1983:03-1989:12 (T = 82)
Estimated using BHHH method (conditional ML)
Dependent variable: (1-L)(1-Ls) z
coefficient std. error z p-value
-------------------------------------------------------
phi_1 -0.590308 0.200862 -2.939 0.0033 ***
Phi_1 -0.683313 0.134247 -5.090 3.58e-07 ***
Phi_2 -0.240713 0.113586 -2.119 0.0341 **
theta_1 0.873512 0.207170 4.216 2.48e-05 ***
theta_2 0.361254 0.0966288 3.739 0.0002 ***
Mean dependent var -1074.305 S.D. dependent var 36698.54
Mean of innovations -1019.087 S.D. of innovations 28580.42
Log-likelihood -957.7121 Akaike criterion 1927.424
Schwarz criterion 1941.864 Hannan-Quinn 1933.222
Real Imaginary Modulus Frequency
-----------------------------------------------------------
AR
Root 1 -1.6940 0.0000 1.6940 0.5000
AR (seasonal)
Root 1 -1.4194 -1.4628 2.0382 -0.3726
Root 2 -1.4194 1.4628 2.0382 0.3726
MA
Root 1 -1.2090 -1.1430 1.6638 -0.3795
Root 2 -1.2090 1.1430 1.6638 0.3795
-----------------------------------------------------------
the output of X-12-ARIMA(1,1,2)(2,1,0 ):
Model 8: ARIMA, using observations 1983:03-1989:12 (T = 82)
Estimated using X-12-ARIMA (conditional ML)
Dependent variable: (1-L)(1-Ls) z
coefficient std. error z p-value
-------------------------------------------------------
phi_1 0.653709 0.209156 3.125 0.0018 ***
Phi_1 -0.675406 0.113095 -5.972 2.34e-09 ***
Phi_2 -0.244173 0.113191 -2.157 0.0310 **
theta_1 -0.566737 0.220105 -2.575 0.0100 **
theta_2 -0.222901 0.115118 -1.936 0.0528 *
Mean dependent var -1074.305 S.D. dependent var 36698.54
Mean of innovations -2724.431 S.D. of innovations 29295.00
Log-likelihood -959.7371 Akaike criterion 1931.474
Schwarz criterion 1945.914 Hannan-Quinn 1937.272
Real Imaginary Modulus Frequency
-----------------------------------------------------------
AR
Root 1 1.5297 0.0000 1.5297 0.0000
AR (seasonal)
Root 1 -1.3830 1.4774 2.0237 0.3698
Root 2 -1.3830 -1.4774 2.0237 -0.3698
MA
Root 1 1.1990 0.0000 1.1990 0.0000
Root 2 -3.7416 0.0000 3.7416 0.5000
-----------------------------------------------------------
The outcomes of ARIMA(1,1,2)(2,1,0 ) and X-12-ARIMA(1,1,2)(2,1,0 ) are hugely different.
The question above puzzles me.
I also want to know When I choose the options Model/Time series/ARIMA/Using X-12-ARIMA to
run the X-12-ARIMA model. Is the set of equation of X-12-ARIMA in gretl the same as
RegARIMA(X-12-ARIMA – Reference Manual, Version 0.3. U.S. Census Bureau):
φ(B)Φ(B)▽^d ▽_s^D[y-Σβ_i x_it]= θ(B)Θ(B)a_t
I can not see the outcome of any seasonality adjusting regression variables(the part of
y-Σβ_i x_it, such as length-of-month、Trend constant、Trading day、level shift at t_0 and so
on).
Thanks a lot