Humour me please. Let us suppose we have an ARMAX(1,0,0) model with 2 independent variables and a constant.

ie  (1-ΦB)Yt = α +µ1*X1t +µ2*X2t  where B is the lag or backshift operator

Assuming Φ is positive this is equivalent to 

Yt =α + Φ*Yt-1 +µ1*X1t +µ2*X2t 

  which is relatively straightforward to check against Gretl output  using the estimated coefficients and relevant rhs variables .However, the results of doing so do not correspond to the predicted or fitted values  from Gretl.

Estimating the equation via OLS, produces marginally  different coefficient estimates (all the same sign) with the exception of the intercept, and manually checking the equation does correspond to the gretl output fitted values for OLS similarly specified model.  So my question is what is the ARMAX prediction function in this case

 Is it the expansion of the solution for Yt by dividing by  (1-ΦB) ?

Here is the ARMAX estimated equation;

               coef              st.err          z           pval

const       3.81738      0.250722     15.23    2.39e-052 ***
           coef              st.err          z           pval

const    3.81738      0.250722     15.23    2.39e-052   *** 

phi_1     0.492142     0.282172      1.744   0.0811      *
 X1        −0.06129    0.0232348    −2.638   0.0083      ***
  X2        0.183408     0.0832595     2.203   0.0276      **

Adjusted R-squared   0.645  S.D. of innovations  0.230

Here are the OLS equivalents

                  coef              st.err          z           pval

const      1.99713      0.686194      2.910    0.0142     **

Yt-1        0.517002     0.181787      2.844    0.0160    **

  X1       −0.04545      0.0152909    −2.973    0.0127  **

  X2        0.183119     0.0835570     2.192    0.0508    *
 

S.E. of regression   0.2245   Adjusted R-squared   0.719

Log-likelihood       3.447668 Schwarz criterion    3.9368

An explanation would be helpful

Thanks