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Partial correlation by hand

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A stationary series has autocorrelations $ρ (1) = 0.6$ and $ρ (2) = 0.5$ . Let $PACF (2)$ be the coefficient on $X_{t - 2}$ in the population regression of $X_{t}$ on $X_{t - 1}$ and $X_{t - 2}$ . What is $PACF (2)$ ? Give a fraction in lowest terms.

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Hint

Compare $ρ (2)$ with what lag $1$ alone would pass on, which is $ρ (1)^{2}$ .

Solution

The formula is $PACF (2) = \frac{ρ ( 2 ) - ρ ( 1 ) ^{2}}{1 - ρ ( 1 ) ^{2}}$ , and $0.6 \times 0.6 = 0.36$ .
The numerator is $0.5 - 0.36 = 0.14$ and the denominator is $1 - 0.36 = 0.64$ .
So $PACF (2) = \frac{0.14}{0.64} = \frac{7}{32}$ .

Answer: $\frac{7}{32}$

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