TS module 7 autocovariance and autocorrelations practice problems
(The attached PDF file has better formatting.)
For AR(1), AR(2), MA(1), MA(2), and ARMA(1,1) processes, know how to calculate ã0, ã1, ã2, ñ1, and ñ2 from ö1, ö2, è1, and è2.
** Exercise 7.1: MA(2) process
An MA(2) process has è1 = 0.7, è2 = 0.5, and óε = 2.
A. What is ã0?
B. What is ã1?
C. What is ã2?
D. What is ñ1?
E. What is ñ2?
For an MA(2) process:
ã0 =(1 + è12 + è22) × ó2
ã1 = (–è1 + è1 × è2) × ó2
ã2 = (–è2) × ó2
ñ1 = (–è1 + è1 × è2) / (1 + è12 + è22)
ñ2 = (–è2) / (1 + è12 + è22)
ñk = 0 for k = 3, 4, …
See Cryer and Chan, page 63 (equation 4.2.3)
Part A: ã0 = (1 + è12 + è22) × ó2 = (1 + 0.49 + 0.25 ) × 22 = 6.960
Part B: ã1 = (–è1 + è1 × è2) × ó2 = (–0.7 + 0.7 × 0.5) × 22 = -1.400
Part C: ã2 = (–è2) × ó2 = –0.5 × 22 = -2.000
Part D: ñ1 = (–è1 + è1 × è2) / (1 + è12 + è22) = -0.201
Part E: ñ2 = (–è2) / (1 + è12 + è22) = -0.287
** Exercise 7.2: Covariance of AR(1) process
An AR(1) process has an autoregressive parameter ö = 0.6 and óε = 4.
A. What is the covariance of Yt with Yt-1? (1 period lag)
B. What is the covariance of Yt with Yt-2? (2 period lag)
Part A: See Cryer and Chan, page 66, equation 4.3.3:
ã0 is the variance of the time series elements. It is greater than ó2ε because the time series observations are autocorrelated. If an observation is high (low) one period, it is similarly high (low) the next period, instead of reverting to the mean. An autoregressive parameter ö closer to one causes higher autocorrelation and a higher variance for the time series.
The autocovariance (and autocorrelation) decline by exponential decay; see page 67, equation 4.3.5:
The covariance of Yt with Yt-1 (1 period lag) is ã1.
= 0.6 × 42 / (1 – 0.62) = 15
Part B: The covariance of Yt with Yt-2 (2 period lag) is ã2.
= 0.62 × 42 / (1 – 0.62) = 9
** Question 7.3: Standard deviation
If ó (the standard deviation of the time series) doubles, what is the effect on ã1 and ñ1?
A. ã1 doubles, ñ1 doesn’t change
B. ã1 quadruples, ñ1 doesn’t change
C. ã1 doubles, ñ1 doubles
D. ã1 quadruples, ñ1 doubles
E. ã1 doesn’t change, ñ1 doesn’t change
Answer 7.3: B
See Cryer and Chan, page 66, equation 4.3.3:
and page 67, equation 4.3.5:
If ó doubles, all the ã terms quadruple. The autocorrelations (the ñ terms) are ãk / ã0, so they do not change.
Equation 4.3.5 gives the effect of ö and ó on the ã terms. Know this equation.