# Does arimax handle both transfer functions and reg with arima errors?

In The ARIMAX model muddle (I think written by hyndman), the author writes that tsa::arimax fits a transfer function model: $$y_t = \frac{\beta(B)}{v(B)}x_t + \frac{\theta(B)}{\phi(B)}z_t$$ where $$B$$ is the backshift operator, $$\beta, \phi, \theta, v$$ are polynomials in the backshift operator, $$y_t$$ is a dependent time series, and $$x$$ is an independent variable.

However, when looking at the tsa::arimax function, there are options: xreg and xtransf. Also, the function says it is built off of stats::arima, which I think is regression with arima errors.

So, am I right that arimax is really this model: $$y_t = \delta(B)x_t + \frac{\beta(B)}{v(B)}X_t + \frac{\theta(B)}{\phi(B)}z_t$$ Where $$x_t$$ is a regression with arima errors, and $$X_t$$ is a transfer function model? ($$x_t$$ would be set with xreg variable, and $$X_t$$ with the xtrnsf variable)

This would mean, I think, that the $$x_t$$ are differenced if the order of integration is greater than zero, while the $$X_t$$ are just handled like a transfer function with whatever I specify.

Is this correct?

The ARIMAX model muddle: https://robjhyndman.com/hyndsight/arimax/