A series follows X_t = ε_t + 0.5 ε_(t−1) + 0.5 ε_(t−2), where ε_t is white noise with variance σ². What is the autocorrelation ρ(2)? Give a fraction in lowest terms.

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Correlation at the last lag

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A series follows $X_{t} = ϵ_{t} + 0.5 ϵ_{t - 1} + 0.5 ϵ_{t - 2}$ , where $ϵ_{t}$ is white noise with variance $σ^{2}$ . What is the autocorrelation $ρ (2)$ ? Give a fraction in lowest terms.

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Hint

Write out $X_{t}$ and $X_{t - 2}$ and find the shocks they share.

Solution

The variance is $(1 + 0.25 + 0.25) σ^{2}$ , which is $1.5 σ^{2}$ .
$X_{t}$ and $X_{t - 2}$ share only $ϵ_{t - 2}$ , with weights $0.5$ and $1$ , so the covariance is $0.5 σ^{2}$ .
Therefore $ρ (2) = \frac{0.5}{1.5} = \frac{1}{3}$ .

Answer: $\frac{1}{3}$

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