# Is ARIMAx a transfer function model?

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.

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: $$y_t=\beta x_t+\frac{\theta(B)}{\phi(B)}z_t$$