[Gretl-users] positive and negative signs of the coefficients in the ARIMA equations

Dear all: I have a question about positive and negative signs of the coefficients of ARIMA when it is expressed in the equation form. There is the output of ARIMA(1,1,1)(1,1,1) below: Function evaluations: 132 Evaluations of gradient: 45 Model 1: ARIMA, using observations 1977:03-1986:12 (T = 118) Estimated using BHHH method (conditional ML) Dependent variable: (1-L)(1-Ls) y coefficient std. error z p-value --------------------------------------------------------- phi_1 0.0767983 1.75029 0.04388 0.9650 Phi_1 -0.133120 0.159001 -0.8372 0.4025 theta_1 -0.0385106 1.77542 -0.02169 0.9827 Theta_1 -0.662710 0.131924 -5.023 5. 08e-07 *** The equation should be written in(1-0.0767983B)(1+0.133120B^12)(1-L)(1-L_12) y_t=(1-0.0385106B)(1-0.662710L^12)a_t , (1-0.0767983B)(1+0.133120B^12)(1-L)(1-L_12) y_t=(1+0.0385106B)(1+0.662710L^12)a_t or (1+0.0767983B)(1-0.133120B^12) (1-L)(1-L_12) y_t=(1-0.0385106L)(1-0.662710L^12)a_t ?B is backshit oprator which equation is correct? I don’t know the correct positive and negative signs of the coefficients in the ARIMA equation. Another sample is ARIMAX. I add two dummy variables: dm1 is for January, dm2 is for Febuary. There is the output of ARIMAX(1,1,1) below: Model 4: ARMAX, 使用中之子樣本範圍: 1997:03-2006:12 (T = 118) Estimated using BHHH method (最大概似法 conditional ML) 應變數 (Dependent variable): (1-L) new_zwaland coefficient std. error z p-value -------------------------------------------------------------- const 6971.19 4629.76 1.506 0.1321 phi_1 -0.0428580 0.174013 -0.2463 0.8055 theta_1 0.719310 0.107846 6.670 2.56e-011 *** dm1 -78779.9 14357.2 -5.487 4.08e-08 *** dm2 9786.05 16383.1 0.5973 0.5503 The ARIMAX equation should be written in (1+0.0.428580B) (1-L)y_t= 6971.19+(1+0.719310B) a_t - 78779.9dm1 + 9786.05dm2, (1-0.0.428580B) (1-L)y_t= 6971.19+(1+0.719310B) a_t - 78779.9dm1 + 9786.05dm2or other forms?B is backshift oprator Thanks a lot