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I have difficulties finding the implementation of ARIMAX, ARMAX model where the exogenous variable would be also included with the time lags:

$X_{t}=\varepsilon_{t}+\sum_{i=1}^{p} \varphi_{i} X_{t-i}+\sum_{i=1}^{q} \theta_{i} \varepsilon_{t-i}-\sum_{i=1}^{p} \beta_{i} u_{t-i}$

Could you please advise on which software uses this approach for ARMAX/ARIMAX? Statmodels from python return only one coefficient for an exogenous variable called sigma, while the model here would ideally return the coefficients for all lagged exogenous values. I suppose the order of the lagged exogenous variables would be the same as the AR order, but I am not sure about this either. Is this what is called transfer function model rather than ARIMAX?

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    $\begingroup$ ARMAX models with lagged values of exogenous variables are called ARDL (AR Distributed Lag) models. You can search the implementation of ARDL for whichever statistical software that you are using. $\endgroup$
    – Dayne
    Commented Jun 29, 2020 at 4:01
  • $\begingroup$ Thank you for the clarification! I suppose that it doesn't have Moving average component. $\endgroup$
    – Sanja
    Commented Jun 29, 2020 at 12:29
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    $\begingroup$ Good point. I guess there is surely a version of ARDL models that allow for MA also. Frankly I have never implemented such a model but I am sure some library in Python/R will have it. $\endgroup$
    – Dayne
    Commented Jun 29, 2020 at 18:04

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