Covariance of a Stationary AR(2) Process

Summary

The document asks for the lagged covariance function of an AR(2) process with a specified mean, two autoregressive coefficients, and Gaussian white-noise innovations. It does not give a covariance formula. The response suggests deriving the variance and standard deviation, finding the correlation between observations at the relevant lag, and combining those quantities to obtain covariance.

This is a general variance-and-correlation route to covariance, rather than a complete derivation tailored to AR(2). The document does not state stationarity conditions, derive the autocorrelation recursion, or provide worked values. Those omissions matter because the usual lag covariance formula assumes a stationary process, and the AR coefficients must support stationarity. It is therefore a limited conceptual pointer, not a self-contained reference for calculating the AR(2) covariance sequence.

Key ideas

• The question concerns lagged covariance in an AR(2) process with Gaussian white-noise innovations.
• Covariance can be obtained from standard deviations and correlation at the same lag.
• The response provides no explicit AR(2) covariance formula or worked derivation.
• A stationary covariance function requires stationarity conditions on the autoregressive coefficients.

Tags

# Covariance AR(2) Process


# Covariance AR(2) Process












I am not sure what the formula is for the covariance of an AR(2) process, described by $X_t - \mu = \phi_1(X_{t-1} - \mu) + \phi_2(X_{t-2} -\mu ) + \epsilon_t$

where $\mu$ denoted the process mean and $\{ \epsilon_t\}$ a Gaussian white noise process with $\epsilon_t \sim 𝑁(0,\sigma^2)$

What is the formula for $Cov(X_𝑡,X_{𝑡−𝑗})$ ?

## Answer by develarist (score -2)

https://quant.stackexchange.com/a/59978

Use variance algebra to derive the variance or standard deviation of $X_t$ and $X_{t-j}$, as well as their correlation, then multiply these three quantities together to get the covariance between the two series

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.