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Does anyone know, considering an ARIMAX model that fitting a stationary process Y, then do the exogenous components for the model need to be (weakly) stationary?
I think exogenous components can be any process, even non-deterministic ones, am I right?
Look at the simplest form of ARIMAX(0,1,0) or IX(1): $$\Delta y_t=c+x_t+\varepsilon_t$$ where $x_t$ - exogenous variables. Take an expectation: $$E[\Delta y_t]=c+E[x_t]$$ If you think that your $\Delta y_t$ is stationary, then $x_t$ must be statrionary too. The same with ARX(1): $$y_t=\phi_1 y_{t-1}+c+x_t+\varepsilon_t$$ and expectation: $$E[y_t]=\phi_1 E[y_{t-1}]+c+E[x_t]$$ $$E[y_t]=\frac{c+E[x_t]}{1-\phi_1}$$
This is known as transfer function model:
A(L)y(t)=B(L)e(t)+C(L)x(t)
y(t)=inv(A(L))B(L)e(t)+inv(A(L))C(L)x(t)
For stabity and invertibility you have to have some restrictions on the characteristic polynomial of this representation. Which means processes jointly has to satisfy these conditions.
You can have seasonal event dummies, intervention dummies or other deterministic components here without causing too much trouble. Problem is to identify them on the first place!
You can think ARIMAX model as a stochastic difference equation which can have Dirac delta function type of events with their own constant coefficients.
