# Why does Statsmodels' ARMAX function fit a linear regression model

I want to fit an AR(MA)X model to some data. The application is to describe the temperature of a single room. As disturbances/exogenous inputs to the room temperature is solar radiation, outdoor air temperature, and water-based radiator. I want to use statsmodels in Python which offers functions to fit ARMA models with exogenous input.

According to the documentation of the ARMAX implementation, the function fits a "regression with ARIMA errors" to the data. That is a model on the form

$$\phi(L) ( y_t - X_t \beta) = \theta(L) \epsilon_t$$,

$$\implies y_t = \frac{\theta(L)}{\phi(L)} \epsilon_t + X_t \beta$$

where $$\theta$$, $$\phi$$ are polynomials in the lag operator $$L$$. But this would not be the intuitive (for me at least) to formulate such a system since there is no transfer function from the exogenous inputs - only the errors. I would have appreciated a form like

$$y_t = \frac{\theta(L)}{\phi(L)} \epsilon_t + \frac{\alpha(L)}{\phi(L)} X_t$$,

where $$\alpha$$ is again a polynomial in the lag operator $$L$$.

So my question is: There must be a good reason for formulating the model in this way, but I don't see why? Hope you can help me! Cheers!