TS Module 19 Seasonal models basics

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Time series practice problems seasonal models

*Question 19.1: Seasonal moving average process

A time series is generated by Y_t = e_t – e_t-12.

What is ñ₁₂?

A.     –1.0

B.     –0.5

C.    0

D.    +0.5

E.     +1.0

Answer 19.1: B

The seasonal correlation is –Θ / (1+ Θ²) = –1 / (1 + 1²) = –0.5

(See Cryer and Chan, page 229: equation 10.1.3)

*Question 19.2: 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

What is ã₀?

A.     0.20

B.     0.80

C.    1.60

D.    3.20

E.     5.80

Answer 19.2: E

(P230: equation 10.2.2: ã₀ = (1 + è²)(1+ Θ²) ó²_e )

(1.16 × 1.25 × 4 = 5.80

*Question 19.3: 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

What is ñ₁₂?

A.     –0.50

B.     –0.40

C.    0

D.    +0.40

E.     +0.50

Answer 19.3: B

(P231: equation 10.2.5: ñ₁₂ = –Θ / (1+ Θ²) = –0.5 / 1.25 = –0.4

*Question 19.4: Seasonal ARMA process

A seasonal ARMA process Y_t = Ö Y_t-12 + e_t – è e_t-1 has Ö = –0.707, è = 1.414, and ó²_e = 4

What is ã₀?

A.     2

B.     4

C.    8

D.    16

E.     24

Answer 19.4: E

See Cryer and Chan, page 232, equation 10.2.11:

ã₀ = ó²_e × (1 + è²) / (1 – Ö²) = 4 × (1 + 2) / (1 – ½) = 24

The variance is the product of the seasonal autoregressive process and the non-seasonal moving average process.

*Question 19.5: Seasonal ARMA process

A seasonal ARMA process Y_t = Ö Y_t-12 + e_t – è e_t-1 has Ö = –0.707, è = 1.414, and ó²_e = 4

What is ñ₁₁ – ñ₁₃ ?

A.     –1.00

B.     –0.50

C.    0

D.    +0.50

E.     +1.00

Answer 19.5: C

(See Cryer and Chan, page 232, equation 10.2.11: ñ_12k-1 = ñ_12k+1 = – Ö^k è / (1 + è²)

For a stationary process, ñ₁ = ñ_–1. We form the autocorrelations for the 12 months seasonal lags, and then add or subtract one month for the non-seasonal lag.

*Question 19.6: Non-stationary seasonal ARIMA process

Suppose Y_t = S_t + e_t, with ó²_e = 2   and      S_t = S_t-s + å_t , with ó²_ε = 1

What is the variance of ∇_s Y_t?

A.     1

B.     2

C.    3

D.    4

E.     5

Answer 19.6: E

∇_s Y_t = Y_t – Y_t-s

➾ ∇_s Y_t = S_t – S_t-s + e_t – e_t-s = å_t + e_t – e_t-s

➾ variance ∇_s Y_t = 1 + 2 + 2 = 5

*Question 19.7: Seasonal AR(1)₁₂ process

A store’s toy sales in constant dollars follow a seasonal 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.

What are projected sales for January 20X3?

A.     $10,000

B.     $11,000

C.    $11,100

D.    $12,000

E.     $12,100

Answer 19.7: E

           The December projection is one period ahead.

           The January projection is two periods ahead.