STK 320 / WST 321time series analysis exercse 2 memo

1


EXERCISE 2 SUGGESTED SOLUTION
1.(a) Z t = 0.2 − 0.2 Z t −1 + at

O (1 + 0.2 x ) = 0
∴ x = −5
∴ the root is greater than one in absolute value

OR
O φ = − 0 .2
∴ −1 < φ < 1
∴ the process is STATIONARY

1.(b) Z t = at − Z t −1

O (1 + x ) = 0
∴ x = −1
∴ the root is not greater than one in absolute value

OR
O φ = −1
∴ −1 φ < 1
∴ the process is NON-STATIONARY

1.(c) Z t = 100 + 12 Z t −1 + 12 Z t − 2 + at

O (1 − 1
2
)
x − 12 x 2 = 0

∴ (1 − x ) (1 + 12 x ) = 0

∴ x = 1 or x = −2
∴ one of the roots is not greater than one in absolute value

OR
O φ1 = φ 2 = 12
∴φ1 + φ 2 = 1 1
∴ the process is NON-STATIONARY




WST321

, 2


1.(d) Z t = Z t −1 − 0.09 Z t − 2 + at

O (1 − x + 0.09 x ) = 0
2


∴ (1 − 0.1x )(1 − 0.9 x ) = 0
∴ x = 10 or x = 109

∴ both roots are greater than one in absolute value

OR
O φ1 = 1 and φ 2 = −0.09
∴φ2 − φ1 = −1.09 < 1 , φ1 + φ 2 = 0.91 < 1 and − 1 < φ 2 = −0.09 < 1
∴ the process is STATIONARY

1.(e) (1 − L)(1 + 0.25 L)(1 − 0.5 L ) Z t = at

O (1 − x)(1 + 0.25 x)(1 − 0.5 x) = 0
∴ x = 1 , x = −4 or x = 2
∴ one of the roots is not greater than one in absolute value
∴ the process is NON-STATIONARY




WST321