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

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 (1-0.0767983L)(1+0.133120L^12) ▽^1▽_12^1 y_t=(1+0.0385106L)(1+0.662710L^12)a_t or (1+0.0767983L)(1-0.133120L^12) ▽^1▽_12^1 y_t=(1-0.0385106L)(1-0.662710L^12)a_t ? Maybe the two equations are both wrong. What is the correct equation? 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: Function evaluations: 41 Evaluations of gradient: 14 Model 3: ARMAX, using observations 1975:03-1986:12 (T = 142) Estimated using BHHH method (conditional ML) Dependent variable: (1-L) y coefficient std. error z p-value -------------------------------------------------------------- phi_1 0.675418 0.0430006 15.71 1.35e-055 *** theta_1 -0.999999 0.0168669 -59.29 0.0000 *** dm1 -52980.0 14985.0 -3.536 0.0004 *** dm2 60226.2 14925.8 4.035 5.46e-05 *** The ARIMAX equation should be written in (1-0.675418b) ▽▽^1 y_t=(1+0.999999b) a_t+52980dm1-60226.2dm2 、 (1+0.675418b) ▽▽^1 y_t=(1-0.999999b) a_t-52980dm1+60226.2dm2 or other forms? Thanks a lot