TS Module 10 Autocorrelation functions

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Time series practice problems partial autocorrelations

*Question 10.1: partial autocorrelations

A stationary ARMA process has ñ₁ = 0.60 and ñ₂ = 0.52. What is the partial autocorrelation of lag 2 (ö₂₂)?

A.   0.25

B.   0.36

C.   0.52

D.   0.56

E.   0.66

Answer 10.1: A

(Cryer and Chan, P113, equation 6.2.3)

(0.52 – 0.60²) / (1 – 0.60²) = 0.250

Jacob: What is the intuition for this formula?

Rachel: The partial autocorrelation of lag 2 says how much of the observed autocorrelation stems from ö₂, after the effects of ö₁ are considered.

        The observed autocorrelation is ñ₂.

        The effect of ö₁ is ñ₁².

        The difference is attributed to ö₂.

Jacob: What about the denominator of this formula?

Rachel: If ñ₁ is zero, the reasoning above is fine. Now suppose ö₁ is 80%, so this parameter gives ñ₂ of 64%. If the observed ñ₂ is greater, how much of the increase is caused by ö₂?

Illustration: If ñ₂ = 76%, the extra correlation caused by ö₂ is (76% – 64% = 12%), which is one third of the remaining correlation (1 – 64% = 36%).

*Question 10.2: partial autocorrelations

A stationary AR(1) process has ö = 50%.

What is ö₁₁ – ö₂₂, the partial autocorrelation of lag 1 minus the partial autocorrelation of lag 2?

A.   –1.0

B.   –0.5

C.   0

D.   +0.5

E.   +1.0

Answer 10.2: A

(P113, equation 6.2.3): For all AR(1) models:

        ö₁₁ = 1

        ö₂₂ = 0

*Question 10.3: Autocorrelations and partial autocorrelations

Which of the following are true?

1.            For an AR(p) process, the autocorrelations decay exponentially as the lag increases for lags more than p.

2.            For an MA(q) process, the autocorrelations decay exponentially as the lag increases for lags more than q.

3.            For an AR(p) process, the partial autocorrelations decay exponentially as the lag increases for lags more than p.

4.            For an MA(q) process, the partial autocorrelations decay exponentially as the lag increases for lags more than q.