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{title2=$\frac{\sigma^2}{2}[\Theta(1)^2+\Theta(-1)^2]$}

For coefficients $\theta_0=1,\theta_1,\ldots,\theta_q$, the <long-run variance of a stationary process> sampled every second time is $\sigma^2[(\sum_{j\text{ even}}\theta_j)^2+(\sum_{j\text{ odd}}\theta_j)^2]$. It equals $\sigma^2[\Theta(1)^2+\Theta(-1)^2]/2$.

Odd-subsample long-run variance of a moving average

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 37 / 3 / e / Solution

Odd-subsample long-run variance of a moving average ( $\frac{σ ^{2}}{2} [Θ (1)^{2} + Θ (- 1)^{2}]$ )

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Odd-subsample long-run variance of a moving average (2σ2​[Θ(1)2+Θ(−1)2])

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Odd-subsample long-run variance of a moving average ( $\frac{σ ^{2}}{2} [Θ (1)^{2} + Θ (- 1)^{2}]$ )

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