12_04_1 - Determine whether the following processes are...

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Determine whether the following processes are stationary: a) Y t + 0.2 Y t – 1 = e t – 0.8 e t – 1 + 0.5 e t – 2 Φ ( B ) = 1 + 0.2 B The root of Φ ( z ) = 0 is z = 5, it is outside the unit circle. This process is stationary. b) Y t – 0.1 Y t – 1 – 0.2 Y t – 2 = e t – 0.6 e t – 1 Φ ( B ) = 1 – 0.1 B – 0.2 B 2 = ( 1 – 0.5 B ) ( 1 + 0.4 B ) The roots of Φ ( z ) = 0 are z 1 = 2 and z 2 = 2.5. All roots of Φ ( z ) = 0 are outside the unit circle. This process is stationary. OR An AR(2) model is stationary if 1 < φ 2 < 1, 2 + 1 < 1, 2 1 < 1. 1 < 0.2 < 1, 0.2 + 0.1 < 1, 0.2 – 0.1 < 1. This process is stationary. c) Y t – 0.7 Y t – 1 – 0.6 Y t – 2 = e t + 0.9 e t – 1 – 0.7 e t – 2 Φ ( B ) = 1 – 0.7 B – 0.6 B 2 = ( 1 – 1.2 B ) ( 1 + 0.5 B ) The roots of Φ ( z ) = 0 are z 1 = 5 / 6 and z 2 = 2. z 1 = 5 / 6 is NOT outside the unit circle, it is inside the unit circle. The roots of Φ ( z ) = 0 must ALL be outside the unit circle for the process to be stationary.
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12_04_1 - Determine whether the following processes are...

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