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I once heard some statements regarding the relationship between ARMA and AR process, such as

An average of severl lags of an autoregression forms an ARMA process


A weighted mixture of lags of an AR(P) model is ARMA

I am not very clear about these two statements, whether they are right? If they are right, how to get them? Thanks.

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up vote 3 down vote accepted

As I'm sure you are aware, ARMA is an acronym for AutoRegressive Moving Average (Stochastic Process). More fully, we use ARMA(p, q) where p is the order of the autoregressive portion and q the order of the moving average portion.

Both of the statements are saying the same thing: if you form a linear combination of AR processes an ARMA process of some order results. This is a consequence of an inversion theorem in Time Series analysis which states that all stationary ARMA processes have a representation as a (possibly infinite) stationary AR process.

The ARMA model expresses $X_t$ as the sum of polynomials in the lagged values of X_t and the lagged values of $\epsilon_t$ (the i.i.d. noise terms). So an AR(1) model looks like

$X_t = \theta X_{t-1} + \epsilon_t$.

The single lagged value of the series is

$X_{t-1} = \theta X_{t-2} + \epsilon_{t-1}$.

Now, let $0 < \phi < 1$ and consider $Y_t = \phi X_t + (1-\phi) X_{t-1}$. $Y_t$ is a weighted average of lags of an AR(1) model. Let's define $\xi = \phi \theta$

$Y_t = \phi\theta X_{t-1} + (1-\phi)\theta X_{t-2} + \phi \epsilon_t + (1-\phi) \epsilon_{t-2}$.

Some mild algebra (basically arranging things to make a $Y_{t-1}$ appear on the right hand side) puts this in the form

$Y_t - \epsilon_t = (\xi_1 Y_{t-1} + \xi_2 Y_{t-2}) + (\phi \epsilon_{t-1} + \phi^2 \epsilon_{t-2})$

Note that this is the sum of a lagged polynomial in $Y_t$ and a lagged polynomial in the noise vector, $\epsilon$. This is an ARMA process.

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