I have a question about the presample states of an ARMA model. Let assume that we have an AR(2) model $$x_t=\alpha_0 + \alpha_1x_{t-1} +\alpha_2x_{t-2}. $$
If $t=1, t=2$, then $x_1,x_2$ states are = ??.
I am using Matlab and the program has a function which says the following:
If your model is stable:
The model is AR-stable and without a regression component, so compute the unconditional (i.e., long-run) mean of the y(t) process directly from the parameters of the model and use it to initialize any required presample observations.
If your model is not stable:
The model is not AR-stable, and so a long-run mean of the y(t) process cannot be calculated from the model. The following simply assumes zeros for any required presample observations for y(t).
Could somenone explain this to me.
Thank you.
You just use that $x_t=0\,\forall\, t<1$. So for $t=1$ you get:
$$ x_1=\alpha_0 $$
and for $t=2$
$$ x_2=\alpha_0+\alpha_1x_1=\alpha_0(1+\alpha_1). $$