# Covariance function iif Moving Average process

• Pere Callahan
In summary, the conversation discusses the question of whether two ARMA processes with different autoregressive and/or moving average polynomials would necessarily have different covariance functions. It is noted that the covariance function is given by a sum of coefficients, and it is mentioned that there may be an infinite number of relations between these coefficients. A practical example of ARMA processes is requested, and it is suggested that if two ARMA processes have the same covariance function, their impulse responses should only differ in phase.
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

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

I find the notation a little hard to follow and am not completely 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.

## 1. What is a covariance function in a Moving Average (MA) process?

A covariance function in a Moving Average (MA) process is a mathematical function that describes the relationship between two random variables within the process. It measures the extent to which changes in one variable are associated with changes in another variable. In simple terms, it shows how two variables move together.

## 2. How is the covariance function calculated for a Moving Average process?

The covariance function for a Moving Average process is calculated by taking the product of the lagged values of the two random variables and then averaging them over time. This can be represented mathematically as Cov(Xt, Yt) = E[(Xt - μx)(Yt - μy)], where μx and μy are the means of X and Y, respectively.

## 3. What does the covariance function tell us about a Moving Average process?

The covariance function tells us about the strength and direction of the relationship between two variables in a Moving Average process. A positive covariance indicates a positive relationship, meaning that increases in one variable tend to be associated with increases in the other variable. A negative covariance indicates a negative relationship, meaning that increases in one variable tend to be associated with decreases in the other variable. A covariance of zero indicates no relationship between the variables.

## 4. How does a covariance function differ from a correlation function in a Moving Average process?

A covariance function and a correlation function both measure the relationship between two variables in a Moving Average process, but they differ in their interpretation. A covariance function shows the strength and direction of the relationship, while a correlation function also takes into account the scale of the variables, making it easier to compare relationships between different variables.

## 5. Can the covariance function be used to predict future values in a Moving Average process?

No, the covariance function cannot be used to predict future values in a Moving Average process. It only describes the relationship between two variables at a specific point in time. To make predictions, a separate forecasting method, such as regression or time series analysis, would need to be used.

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