Can an ARIMAX estimate this model?

So I have this AR data series, AR1, mostly (for the argument). I also have an exogeneous regressor, only the DGP I suspect is something in the line of: $$y_t=y_{t-1}+\alpha x_ty_{t-1}+\varepsilon_t$$

Can an ARIMAX estimate this? What's the specification? If not, what?

• Your equation does not have an error term, so $y_t$ is a deterministic function of $y_{t-1}$ and $x_t$. Is that right? Regardless of that, this is not within the class of ARIMAX. May 10, 2022 at 5:58
• Meant the stochastic version. Will use AR(1). Thank you Richard May 11, 2022 at 7:39

1 Answer

A simple ARIMAX(1,0,0) should be of the form $$y_t = α_1y_{t-1} + α_2x_t + e_t$$. From your question it seems that is what you wanted to write, isn't it?

As @Richard pointed out, the equation you wrote is not an ARIMAX, but solving it would give $$log(y_t/y_{t-1}) ≈ αx_t$$, which a different but well known type of model. Please, clarify.

• Yeah, I mean the stochastic version. Thank you very much :) May 11, 2022 at 7:38