SARIMAX(1,0,0)(0,0,0,0): Regression with ARMA errors or ARMAX? I'm trying to unravel the influences of various exogenous regressors on a time series dataset. I'm getting good results with a sarimax(1,0,0)(0,0,0,0) specification, but I'm confused about the mathematical specification behind this model. With this, I'm also questioning the interpretability of the coefficients. I'm using the statsmodels.tsa.statespace.sarimax.SARIMAX package in python.
Literature gives me two options (see here):


*

*The model is specified as a regression model with ARMA errors:


*The model is specified as an ARMAX model:

Which model am I actually using?
 A: The interpretation of $SARIMAX(p,d,q)(P,D,Q)_m$ is the following:
p: your auto-regressive order.
d: your differencing order.
q: your moving average order. 
P: your seasonal auto-regressive order.
D: your seasonal differencing order.
Q: your seasonal moving average order.   
m: your seasonality (e.g. 24, 7, 52, 12, etc...) 
for your model everything is equal to 0 except for p, which means your process is a $AR(1)$ auto-regressive process: 
$y_t = \phi_1 y_{t-1} + \mu + \epsilon_t$ 
(with $\mu$ the mean of the process and $\epsilon_t$ the error) 
A: Answer:
If you use Statsmodels' SARIMAX, then you are using the first specification you listed: regression with ARMA (or more generally SARIMA) errors.

Details:
Although it may not be so clear because of the generality of the mathematical notation that is used, this is what is described in the statsmodels documentation (https://www.statsmodels.org/dev/generated/statsmodels.tsa.statespace.sarimax.SARIMAX.html):
"In terms of this model, regression with SARIMA errors can be represented easily as:
$$
\begin{split}
y_t & = \beta_t x_t + u_t \\
\phi_p(L) \tilde \phi_p (L^s) \Delta^d \Delta_s^D u_t & = A(t) + \theta_q(L) \tilde \theta_q(L^s) \zeta_t
\end{split}
$$
"
in your case, we have the following restrictions:


*

*$u_t \equiv n_t$ (just a notation difference)

*$\tilde \phi_p(L^s) = 1$, $\Delta^d = 1$, $\Delta_s^D = 1,$ and $\tilde \theta_q(L^s) = 1$, since you don't have seasonal or differencing components

*$A(t) = 0$ since you don't have any trend components

*$\beta_t = \beta$ since the regression coefficients are not time-varying

*$\phi_p(L) = (1 - \phi_1 L - \dots - \phi_p L^p)$ is the autoregressive lag polynomial

*$\theta_q(L) = (1 + \theta_1 L + \dots + \theta_q L^q)$ is the moving average lag polynomial

A: Please read this article by PennState College- 
https://onlinecourses.science.psu.edu/stat510/node/67/
Arima is nothing but a regression. All external will be added as regressors. -
http://machinelearningstories.blogspot.com/2016/08/time-series-and-fitting-regression-on.html
