[Gretl-users] ARIMA model with exogenous variable

Dear GRTEL users, I have estimated the follwing models 1) ARIMAX of order (2,1,0) with additional constant for the dependent variable GMSL_CW (endogenous) and the independent variable GISS_GSST (exogenous) (arima 2 1 0; GMSL_CW const GISS_GSST) Function evaluations: 24 Evaluations of gradient: 8 Model 4: ARMAX, using observations 1881-2001 (T = 121) Estimated using Kalman filter (exact ML) Dependent variable: (1-L) GMSL_CW Standard errors based on Hessian coefficient std. error t-ratio p-value --------------------------------------------------------- const 1.69314 0.281737 6.010 1.86e-09 *** phi_1 -0.361518 0.0878460 -4.115 3.87e-05 *** phi_2 -0.250534 0.0898182 -2.789 0.0053 *** GSST_GISS 3.72863 1.36854 2.725 0.0064 *** Mean dependent var 1.497521 S.D. dependent var 5.290911 Mean of innovations -0.015051 S.D. of innovations 4.819733 Log-likelihood -362.0990 Akaike criterion 734.1979 Schwarz criterion 748.1769 Hannan-Quinn 739.8753 Real Imaginary Modulus Frequency ----------------------------------------------------------- AR Root 1 -0.7215 -1.8630 1.9979 -0.3088 Root 2 -0.7215 1.8630 1.9979 0.3088 ----------------------------------------------------------- 2) ARIMAX of order (2,0,0) with additional constant for the dependent variable d_GMSL_CW/dt (endogenous) and the independent variable GISS_GSST (exogenous) (arima 2 0 0; d_GMSL_CW const GISS_GSST) Function evaluations: 26 Evaluations of gradient: 8 Model 5: ARMAX, using observations 1881-2001 (T = 121) Estimated using Kalman filter (exact ML) Dependent variable: d_GMSL_CW Standard errors based on Hessian coefficient std. error t-ratio p-value --------------------------------------------------------- const 1.69314 0.281737 6.010 1.86e-09 *** phi_1 -0.361518 0.0878464 -4.115 3.87e-05 *** phi_2 -0.250534 0.0898208 -2.789 0.0053 *** GSST_GISS 3.72863 1.36860 2.724 0.0064 *** Mean dependent var 1.497521 S.D. dependent var 5.290911 Mean of innovations -0.015052 S.D. of innovations 4.819733 Log-likelihood -362.0990 Akaike criterion 734.1979 Schwarz criterion 748.1769 Hannan-Quinn 739.8753 Real Imaginary Modulus Frequency ----------------------------------------------------------- AR Root 1 -0.7215 -1.8630 1.9979 -0.3088 Root 2 -0.7215 1.8630 1.9979 0.3088 ----------------------------------------------------------- As expected the estimated model parameters are identical in both cases as GRETL estimates the differenced model 1) with an additional constant. However, I could not explain how GRETL computes the fitted values and residuals for the levels (not the differences) in model 1). I was expecting the fitted values of model 1) to be the cumulated sums of fitted values of model 2). So, I wonder how GRETL recovers the fitted values from the estimated model parameters (const, phi_1, phi_2, GSST_GISS) in ARIMAX case with differencing? Cheers, Didier