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/
tsa::arimax: https://www.rdocumentation.org/packages/TSA/versions/1.2/topics/arimax
