ARMAX vs ARIMAX models: numerically different estimates

I am getting surprising results when using the "estimate" function from the arima class in Matlab. Although I have found resources regarding either ARIMAX or ARMAX models, I could not find anything mentioning the kind of discrepancies I am getting; I would expect and arima(1,1,1) fitted on a series Y to provide the same estimates as an arima(1,0,1) on the first-difference of the series (i.e. fitted on "diff(Y)"). Writing the equations for both models shows that both are equivalent when using a change of variable Yarmax = diff( Yarimax ).

However, below is some test code and results. As you can see, there are two aspects to these inconsistencies: 1. When starting with the simulated differenced series (ARMAX) and then compounding that series to create the ARIMAX series, there are slight differences which are bugging me (more for reproducibility than for estimation since the differences are small). These differences could be due to the fact the ARIMAX series needs to discard one more observation of X's, still I would appreciate being provided the correct parameters to provide to the Matlab functions in order to retrieve the same results. 2. When starting with the integrated series (ARIMAX) and then first-differencing it to estimate the ARMAX model, the ARIMAX estimation is "all over the place" in terms of its AR and MA components...

    % parameters
    cst = 0;
    phi1 = 0.4;
    theta1 = -0.95;
    sigmaE2 = 1;
    beta = [3 -2 5];
    T = 1000;
    X = rand(T, 3)+[0,10,2];
    Y0 = 20;
    E0 = 0;

    % simulate and estimate ARMAX
    mdlArmaxUnknown = arima( 1,0,1 );
    mdlArmax = arima('AR',phi1,'D',0,'MA',theta1,'Constant',cst,'Beta',beta,'Variance',sigmaE2);
    armaxMaxPQ = max( [ mdlArmaxUnknown.P, mdlArmaxUnknown.Q ] ) + 1;
    rng(1); Yarmax = mdlArmax.simulate(T,'Y0',0,'X',X,'V0',sigmaE2,'E0',E0);
    armaxEst = mdlArmaxUnknown.estimate( Yarmax( armaxMaxPQ:end ), 'Y0', 0, 'X', X, 'display', 'off' );
    armaxSummary = armaxEst.summarize();

    % take the simulated ARMAX model and compound it, then estimate it as
    % an ARIMAX model
    mdlArimaxUnknown = arima( 1,1,1 );
    arimaxMaxPQ = max( [ mdlArimaxUnknown.P, mdlArimaxUnknown.Q ] ) + 1;
    Yarimax = cumsum( [ Y0; Yarmax ] );
    arimaxEst = mdlArimaxUnknown.estimate( Yarimax( arimaxMaxPQ:end ), 'Y0', Y0([1;1]), 'X', X, 'display', 'off' );
    arimaxSummary = arimaxEst.summarize();

    % compare ARMAX and ARIMAX
    results = table( armaxSummary.Table.Value, arimaxSummary.Table.Value, 'RowNames', arimaxSummary.Table.Properties.RowNames, 'VariableNames', {'armax','arimax'} );
    disp(results);

    % Now do the reverse: start with ARIMAX series, differentiate to get
    % ARMAX, and compare estimations
    mdlArimax = arima('AR',phi1,'D',1,'MA',theta1,'Constant',cst,'Beta',beta,'Variance',sigmaE2);
    rng(1); Yarimax = mdlArimax.simulate(T,'Y0',Y0([1;1]),'X',X,'V0',sigmaE2,'E0',E0);
    arimaxEst = mdlArmaxUnknown.estimate( Yarimax( arimaxMaxPQ:end ), 'Y0', Y0([1;1]), 'X', X, 'display', 'off' );
    arimaxSummary = arimaxEst.summarize();
    %ARMAX
    Yarmax = diff( Yarimax );
    armaxEst = mdlArmaxUnknown.estimate( Yarmax( armaxMaxPQ:end ), 'Y0', 0, 'X', X, 'display', 'off' );
    armaxSummary = armaxEst.summarize();
    results = table( armaxSummary.Table.Value, arimaxSummary.Table.Value, 'RowNames', arimaxSummary.Table.Properties.RowNames, 'VariableNames', {'armax','arimax'} );
    disp(results);
    

Decent but imperfect results for the first test:

                   armax      arimax 
                ________    ________

    Constant    -0.21152     0.10106
    AR{1}        0.37322     0.35694
    MA{1}       -0.88944    -0.72669
    Beta(1)         3.12      3.1908
    Beta(2)      -2.0455     -2.0918
    Beta(3)       5.1264      5.1049
    Variance      1.0618      1.3021

"Wild" results for the second test:

                   armax      arimax 
                ________    ________

    Constant    -0.22134     -4.3607
    AR{1}        0.36867     0.99999
    MA{1}       -0.84813    -0.12774
    Beta(1)       3.1323      3.2114
    Beta(2)      -2.0536     -2.0788
    Beta(3)       5.1406      5.1568
    Variance      1.1138      2.2961

The question therefore encompasses both these cases: I am looking for the correct use of parameters and Matlab function (i.e. those from the "econ" toolbox) which enable me to use whichever series first (differenced or integrated) and get correct and numerically equivalent estimations either way thereafter. Thank you!

[Redho redho]

hi bro, could you help me with give this codes because i need this code to toward a bachelor degree

thanks before