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According to wold representation theorem every covariance-stationary time series can be written as a linear combination of lagged values of a white noise process (MA(∞) representation). Now if a MA(∞) representation can be written in terms of an AR(1), why is there a need of using an ARMA model?

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  • $\begingroup$ an MA($\infty$) cannot always be written in terms of an AR(1). Invertible MA models can be written in an AR-way, but it doesn't say anything about the order of the AR model. It's usually greater than $1$. Also, just because we can write it one way or another, usually the more economical way is preferred. $\endgroup$ – Taylor Jul 19 '17 at 20:24
  • $\begingroup$ thanks for replying me! So an ARMA model can be more economical than an AR model. Is there a way to prove it? $\endgroup$ – user169479 Jul 19 '17 at 21:41
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    $\begingroup$ yes. Pick some causal ARMA(1,1) model and try to write it as an AR model. How many parameters does it have? $\endgroup$ – Taylor Jul 19 '17 at 22:41
  • $\begingroup$ Well, the non-null ARMA(1,1) model I just chose corresponds to an AR(0)... but degenerate cases aside, those are great comments. You might want to consider writing them into an answer. $\endgroup$ – Glen_b Jul 19 '17 at 23:31
  • $\begingroup$ Do you have some books to suggest? ( I'm studying system identification for control problems) $\endgroup$ – user169479 Jul 20 '17 at 8:40
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Now if a MA(∞) representation can be written in terms of an AR(1)...

This is not always possible. Consider the following MA($\infty$) model: $$ X_t = \sum_{j=0}^{\infty}\psi_j Z_{t-j}, $$ where $\psi_0 = 1$, $\psi_1 = .7$, and for $k \ge 2$, $\psi_k = .7\psi_{k-2} - .1\psi_{k-3}$. This is equivalent to the causal AR(2) model $$ X_t = .7 X_{t-1} - .1X_{t-2} + Z_t. $$

So an ARMA model can be more economical than an AR model. Is there a way to prove it?

Yes, here is one example. The following is an invertible (and causal but it doesn't matter here) ARMA(1,1) process: $$ X_t - .5X_{t-1} = Z_t + .4Z_{t-1}. $$ If we wanted to write it in terms of only AR components, it would be an AR($\infty$) process. It is equivalent to $$ \sum_{j=0}^{\infty}\pi_j X_{t-j} = Z_t, $$ where $\pi_0 = 1$, $\pi_j = -(.4 + .5)(-.4)^{j-1}$ for $j \ge 1$.

Do you have some books to suggest?

These examples are from Introduction to Time Series and Forecasting by Brockwell and Davis (3rd edition).

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  • $\begingroup$ What do you mean by "causal AR(2) model"? What does "causal" refer to? The context here is forecasting not causal analysis as one might conclude from the "causal" term you use. $\endgroup$ – ColorStatistics Nov 15 '18 at 20:35
  • $\begingroup$ @ColorStatistics "causal" means that the roots of the AR polynomial are outside the unit circle, or that the process' conditional mean can refer back to past values of itself. This is a commonly-used definition in time series. $\endgroup$ – Taylor Nov 16 '18 at 0:59

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