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 ó_ε = 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)

ñ_k = 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 ó_ε = 4.

A.      What is the covariance of Y_t with Y_t-1? (1 period lag)

B.      What is the covariance of Y_t with Y_t-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 ó²_ε 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 Y_t with Y_t-1 (1 period lag) is ã₁.

= 0.6 × 4² / (1 – 0.6²) = 15

Part B: The covariance of Y_t with Y_t-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?