Thanks John C. Frain & Dr RJF Hudson. It is kind of you to answer my question. But there are still some questions. The equations of seasonal ARIMA and X-12-ARIMA are different, so their outcomes should be different. But I get almost the same outcomes when I run seasonal ARIMA and X-12-ARIMA using the same AR and MA.(ex: ARIMA(1,1,1)(1,1,0) and X-12-ARIMA(1,1,1)(1,1,0)) There are a few excepios. The outcomes of ARIMA(1,1,1)(1,1,0) and X-12-ARIMA(1,1,1)(1,1,0) above may be almost the same. But when I run ARIMA(1,1,2)(2,1,0) and X-12-ARIMA(1,1,2)(2,1,0) under the same sample. Their outcomes are hugely different. Most of them are almost the same. A few exceptions of them are different. In a word, why are the most outcomes of seasonal ARIMA and X-12-ARIMA almost the same(ex:ARIMA(1,1,1)(1,1,0) and X-12-ARIMA(1,1,1)(1,1,0), ARIMA(2,1,1)(0,1,2) and X-12-ARIMA(2,1,1)(0,1,2)). Their general equations are different. Shouldn't their outcomes be different? I don't know it is a question about statistics, or I run X-12-ARIMA incorrectly in gretl. If I choose the options of Model/Time series/ARIMA/Using X-12-ARIMA to run the X-12-ARIMA model, it has already included original seasonality adjusting variables of regARIMA or I have to choose these variables by myself?(I just want to include the variables defined by X-12-ARIMA – Reference Manual, Version 0.3. U.S. Census Bureau) Thanks a lot The examples are below: 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 ----------------------------------------------------------- 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 ----------------------------------------------------------- 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 ----------------------------------------------------------- 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