# Covariance function iif Moving Average process

1. Feb 4, 2009

### Pere Callahan

Hi,

While teaching myself Time Series Analysis and ARMA processes in particular, I came across the question, whether two ARMA(p,q) processes
$$\varphi(B)X_t=\theta(B)Z_t \qquad\qquad \tilde \varphi(B)\tilde X_t=\tilde \theta(B)\tilde Z_t\$$
with different autoregressive and/or moving average polynomials would necessarily have different covariance functions.

I know that the covariance function is given by
$$\gamma_X(n)=\sum_{j\geq 0}{\psi_j\psi_{j+|n|}}$$
where
$$\sum_{j\geq 0}{\psi_j z^j}=\psi(z)=\frac{\theta(z)}{\varphi(z)}$$

Equating the covariance of $X_t$ and $\tilde X_t$ at lags n=0,1,... gives and infinite number of relations between $\psi_j$ and $\tilde \psi_j$. I was trying to use these relations to show that these coefficients actually coincide but since they are not linear there seems to be no easy inversion scheme available.
Any help would be greatly appreciated.

Pere

2. Aug 23, 2009

### MeChaState

Hi Pere,
I was trying to understand the covariance function I came across your posting. I have been working in Fiber Optics Sensing. Could you please give me and practical example of "ARMA processes"? It looks to me very abstract.

Tnx David

3. Aug 24, 2009

### John Creighto

I find the notation a little hard to follow and am not completly sure what is being asked but I would think that if two ARMA processes had the same covariance function then they probably only differ in phase and therefore the impulse response of one ARMA process should be a delayed version of the other.

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