TS Module 10 Autocorrelation functions
(The attached PDF file has better formatting.)
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:
*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.
A. 1 and 2 only
B. 3 and 4 only
C. 1 and 3 only
D. 2 and 4 only
E. 1 and 4 only
Answer 10.3: E
We use sample autocorrelations and partial autocorrelations to select the type of model.
(Cryer and Chan, P113-114)