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I would like to know if the ARIMAx model is considered a transfer function model. If the answer is no, further explanation on what are differences would be appreciated.

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Yes, Hyndman has a brief explanation of the subject here. ARIMAX is a transfer model $$y_t=\frac {\beta(B)} {v(B)} x_t+\frac{\theta(B)}{\phi(B)}z_t$$ where $x_t$ is contemporaneous exogenous variables and $\beta(B)=\beta$ is a simple coefficient matrix. The transfer function approach would have $\beta(B)$ with a set of possibly lagged exogenous variables.

A thing to be aware of is that sometimes by ARIMAX people, including Hyndman himself, call a different special case of transfer function model that is referred to as regARIMA in MATLAB or regression with ARIMA errors. A case in point is SARIMAX in Python statsmodels. The regression with arima errors model is: $$y_t=\beta x_t+\frac{\theta(B)}{\phi(B)}z_t$$

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  • $\begingroup$ today i posted a question on a similar topic (i.e. regression with GARCH error term), I provide the link here do you know the answer? Thanks in advance $\endgroup$
    – Fr1
    Commented Aug 5, 2019 at 19:52
  • $\begingroup$ Thank you for your great answer @Aksakal. $\endgroup$ Commented Aug 5, 2019 at 21:45
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    $\begingroup$ I have a blog post detailing all the naming confusion about ARIMAX, transfer function models (in its narrower sense and broader sense), and why different people are using different terms. If you want more details: ruqinren.wordpress.com/2020/02/21/… $\endgroup$ Commented Apr 2, 2020 at 12:44

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