[Gretl-users] ARIMA model with exogenous variable

Didier.Monselesan at csiro.au Didier.Monselesan at csiro.au
Wed Jul 1 01:10:26 EDT 2009


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



More information about the Gretl-users mailing list