Conditional and Unconditional Moments of an ARMA(1,1) Process

Summary

The document poses a question about deriving conditional and unconditional variance, autocovariance, and autocorrelation for an ARMA(1,1) process. It identifies the process form, including an autoregressive term, a moving-average term, and an innovation, and asks specifically about the first two lags.

No derivation, worked solution, or empirical evidence is included. The document therefore serves as a focused prompt for studying ARMA moment calculations rather than as a complete explanation. Any answer would depend on assumptions about the innovation process, such as its variance and whether it is independent across time; these assumptions are not specified here.

Key ideas

• The question concerns conditional and unconditional variance for an ARMA(1,1) process.
• It also asks for autocovariance and autocorrelation at the first two lags.
• The document does not provide a solution or specify assumptions about the innovations.

Tags

# Conditional and unconditional variance, autocovariance and autocorrelation of an ARMA process


# Conditional and unconditional variance, autocovariance and autocorrelation of an ARMA process












Given an ARMA(1,1) process $x_t = a + bx_{t-1} + \varepsilon_t + \theta\varepsilon_{t-1}$, how can we

- find the conditional variance, i.e. $Var_{t-1}(x_t)$,

- find the unconditional variance, i.e. $Var(x_t)$,

- find the autocovariance and autocorrelation for the lags 1 and 2?

I would appreciate a detailed answer.

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