TS module 20 seasonal models practice problems

(The attached PDF file has better formatting.)

** Exercise 20.1: Seasonal moving average process

A time series is generated by Y_t = e_t – Θ₁ e_t-12 – Θ₂ e_t-24, with ó²_ε = ó²)

A.    What is ã₀?

B.    What is ã₁₂ = ã_{1 × 12}?

C.    What is ñ₁₂ = ñ_{1 × 12}?

Part A: The variance of Y_t = variance (e_t – Θ₁ e_t-12 – Θ₂ e_t-24) = ó² + Θ₁² × ó² + Θ₂² × ó².

Part B: The covariance of (Y_t, Y_t-12) = covariance (e_t – Θ₁ e_t-12 – Θ₂ e_t-24, e_t-12 – Θ₁ e_t-24 – Θ₂ e_t-36) =

– Θ₁ × ó² + Θ₁ × Θ₂ × ó²

Part C: The autocorrelation for one seasonal lag = ñ₁₂ = ñ_{1 × 12} =

(– Θ₁ × ó² + Θ₁ × Θ₂ × ó² ) / (ó² + Θ₁² × ó² + Θ₂² × ó²) = (– Θ₁ + Θ₁ × Θ₂) / (1 + Θ₁² + Θ₂²)

** Exercise 20.2: Seasonal moving average process

A seasonal moving average process has a characteristic polynomial of (1 – èx)(1 – Θx¹²).

For simplicity, assume ó²_e = 1, so we can ignore the ó²_ε term in the covariances.

A.    What is ã₀?

B.    What is ã₁?

C.    What is ñ₁?

D.    What is ã₁₂?

E.    What is ñ₁₂?

F.    What is ã₁₁?

G.    What is ñ₁₁?

H.    What is ã₁₃?

I.      What is ñ₁₃?

Part A: This time series can be written as Y_t = e_t – è e_t-1 – Θ e_t-12 + è Θ e_t-13 (see equation 10.1.2 on page 230). 

ã₀ = 1 + è² + Θ² + è² × Θ² = (1 + è²) × (1 + Θ²).

Part B: ã₁ = covariance (e_t – è e_t-1 – Θ e_t-12 + è Θ e_t-13, e_t-1 – è e_t-2 – Θ e_t-13 + è Θ e_t-14 ) =

– è – è Θ² = – è × (1 + Θ²).

Part C: ñ₁ = ã₁ / ã₀ = – è × (1 + Θ²) / (1 + è²) × (1 + Θ²) = – è / (1 + è²).

Part D: ã₁₂ = covariance (e_t – è e_t-1 – Θ e_t-12 + è Θ e_t-13, e_t-12 – è e_t-13 – Θ e_t-24 + è Θ e_t-25 ) =

– Θ – è² Θ = – Θ × (1 + è²).

Part E: ñ₁₂ = ã₁₂ / ã₀ = – Θ × (1 + è²) / (1 + è²) × (1 + Θ²) = – Θ / (1 + Θ²) .

Part F: ã₁₁ = covariance (e_t – è e_t-1 – Θ e_t-12 + è Θ e_t-13, e_t-11 – è e_t-12 – Θ e_t-23 + è Θ e_t-24 ) = + Θ × è.

Part G: ñ₁₁ = ã₁₁ / ã₀ = Θ × è / (1 + è²) × (1 + Θ²).

Part H: ã₁₃ = covariance (e_t – è e_t-1 – Θ e_t-12 + è Θ e_t-13, e_t-13 – è e_t-14 – Θ e_t-25 + è Θ e_t-26 ) = + Θ × è.

Part I: ñ₁₃ = ã₁₃ / ã₀ = Θ × è / (1 + è²) × (1 + Θ²).

Jacob: Do exam problems give the characteristic polynomial for seasonal models?

Rachel: Yes. A characteristic polynomial of (1 – èx)(1 – Θx¹²) means Y_t = e_t – è e_t-1 – Θ e_t-12 + è Θ e_t-13

** Exercise 20.3: Seasonal moving average process

A seasonal moving average process has a characteristic polynomial of (1 – èx)(1 – Θx¹²), with è = 0.4, Θ = 0.5, and ó²_e = 4.

A.    What is ã₀?

B.    What is ã₁?

C.    What is ñ₁?

D.    What is ã₁₂?

E.    What is ñ₁₂?

F.    What is ã₁₁?

G.    What is ñ₁₁?

H.    What is ã₁₃?

I.      What is ñ₁₃?

Part A: ã₀ = (1 + è²) × (1 + Θ²) × ó²_ε = (1 + 0.16) × (1 + 0.25) × 4 = 5.800.

Part B: ã₁ = – è × (1 + Θ²) × ó²_ε = – 0.4 × 1.25 × 4 = -2.000.

Part C: ñ₁ – è / (1 + è²) = –0.4 / 1.16 = -0.345.

Part D: ã₁₂ = – Θ × (1 + è²) × ó²_ε = -2.320.

Part E: ñ₁₂ = – Θ / (1 + Θ²) = -0.5 / 1.25 = -0.400.

Part F: ã₁₁ = + Θ × è × ó²_ε = 0.4 × 0.5 × 4 = 0.800.

Part G: ñ₁₁ = ã₁₁ / ã₀ = 0.13793.

Part H: ã₁₃ = + Θ × è × ó²_ε = 0.4 × 0.5 × 4 = 0.800.

Part I: ñ₁₃ = ã₁₃ / ã₀ = 0.13793.

** Exercise 20.4: Seasonal ARMA process

A seasonal ARMA process Y_t = Ö Y_t-12 + e_t – è e_t-1 has a variance of the error term ó²_e.

This is a multiplicative seasonal ARMA(p,q) × (P,Q) process, with p = 0, q = 1, P = 1, Q = 0.

Let k be an integer (1, 2, 3, …).

A.    What is ã₀?

B.    What is ñ₁₂?

C.    What is ñ₁₃?

D.    What is ñ₁₁?

E.    What is ñ_12k?

F.    What is ñ_12k+1?

G.    What is ñ_12k–1?

Part A: The variance of the multiplicative ARMA process is the product of the autoregressive and moving average parts.

        An MA(1) process has ã₀ = ó²_å (1 + è²). 

        An AR(1) process has ã₀ = ó²_å / (1 – ö²). 

        A seasonal AR(1) process has ã₀ = ó²_å / (1 – Ö²). 

        A multiplicative seasonal ARMA(p,q) × (P,Q) process has ã₀ = ó²_e × (1 + è²) / (1 – Ö²).

Part B: The moving average part of this process has a one-period effect. The 12 month autocorrelation stems from the autoregressive process: ñ₁₂ = Ö

Part C: The 13 month autocorrelation is the product of the one-month moving average autocorrelation and the 12 month autoregressive correlation: ñ₁₃ = – Ö è / (1 + è²)

Part D: The 11 month autocorrelation is the product of the moving average autocorrelation for a lag of negative one month and the 12 month autoregressive correlation: ñ₁₃ = – Ö è / (1 + è²)

Jacob: How did we get the moving average autocorrelation for a lag of negative one month?

Rachel: A stationary ARMA process is symmetric: ñ_k = ñ_–k

Part E: The autocorrelation for a lag of 12k months is Ö^k.

Part F: The autocorrelation for a lag of 12k+1 months is – Ö^k è / (1 + è²).

Part G: The autocorrelation for a lag of 12k-1 months is – Ö^k è / (1 + è²).

Jacob: For the multiplicative seasonal moving average process, we wrote the process as a combination of error terms to compute the covariance at each lag. Can we do the same with the multiplicative seasonal ARMA process?

Rachel: Y_t = e_t + Ö Y_t-12 can be written as Y_t = e_t + Ö e_t-12 + Ö² e_t-24 + Ö³ e_t-36 + … Combine this infinite series with the moving average piece to get a single expression of error terms. Compute the covariances to get the formulas in the textbook. Each covariance is a infinite series of Ö terms, which is rewritten as Ö^k / (1 – Ö²).

** Exercise 20.5: Seasonal ARMA process

A seasonal ARMA process Y_t = Ö Y_t-12 + e_t – è e_t-1 has Ö = 0.4, è = 0.5, and ó²_e = 4

A.    What is ã₀?

B.    What is ñ₁₂?

C.    What is ñ₁₃?

D.    What is ñ₁₁?

E.    What is ñ₂₄?

F.    What is ñ₂₃??

Part A: ã₀ = ó²_e × (1 + è²) / (1 – Ö²) = 4 × (1 + 0.5²) / (1 – 0.4²) = 6

Part B: ñ₁₂ = Ö = 0.4

Part C: ñ₁₃ = – Ö è / (1 + è²) = – (0.4) × 0.5 / (1 + 0.5²) = -0.160.

Part D: ñ₁₁ = – Ö è / (1 + è²) = – (0.4) × 0.5 / (1 + 0.5²) = -0.160.

Part E: ñ₂₄ = Ö² = (0.4)² = 0.160.

Part F: ñ₂₃ = – Ö² è / (1 + è²) = – (0.4)² × 0.5 / (1 + 0.5²) = -0.064.

** Exercise 20.6: Seasonal non-stationary AR(1)₁₂ process

A store’s toy sales in constant dollars follow a seasonal non-stationary AR(1)₁₂ process: Y_t = Ö Y_t-12 + e_t.

        Sales are $10,000 in January 20X1 and $100,000 in December 20X1.

        Projected sales are $110,000 in December 20X2.

A.    What is Ö?

B.    What are projected sales for January 20X2?

C.    What are projected sales for January 20X3?