1
$\begingroup$

Suppose we have some set of data $\{x_t\}_{t=1}^T$, which we model as a part of realization of stationary stochastic process $\{X_t|t\in\mathbb{Z}\}$. Now, as I understand, by a virtue of The Wold Decomposition, we can represent this process as $MA(\infty)$- series of elements constituting White Noise. Further, we can approximate this representation by a stationary solution of some $ARMA(p,q)$ equation. I perceive this as a justification of $ARMA$ modelling, where we restrict considered family of stationary stochastic processes to its particular subset: stationary solutions of $ARMA$ equations. Here, my question arises: what we achieve by doing this in a context of prediction? What are advantages of this approach over simply considering a $\textbf{linear projection}$ of a variable of interest on its past values, which minimizes $MSE$ among all linear forecasting functions?

$\endgroup$

1 Answer 1

2
$\begingroup$

Regarding the advantages of ARMA over a linear projection, i.e. an AR model:
In many cases, ARMA(p,q) offers a more parsimonious approximation of MA($\infty$) than an AR(p') could.$^*$ Due to the bias-variance trade-off, parsimonious models tend to be better at prediction.

$^*$I used p' to distinguish from p, as the do not have to coincide.

$\endgroup$
2
  • $\begingroup$ Thank you for your reply. I need one clarification: am I right in saying that due to The Wold Decomposition, we obtain a structure of our process, which we can approximate (e.g. by ARMA or AR models); this in turn enable us to simplify our expressions for optimal forecast procedures like conditional expectation and linear projection? $\endgroup$ Jun 23, 2022 at 0:18
  • $\begingroup$ @Mentossinho, yes, absolutely. $\endgroup$ Jun 23, 2022 at 5:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.