4005T4 - STA 4005 Tutorial 4 Stationarity for the...

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STA 4005 Tutorial 4 Stationarity for the Autoregressive Process Def: A time series {Z t } is said to be an AR(p) if Z t = φ 1 Z t-1 + φ 2 Z t-2 +…+ φ t-p Z t-p + a t Or φ (B)Z t = a t where φ (B) = 1- φ 1 B - φ 2 B 2 +…+ φ p B p Fact: An AR(p) is stationary iff the AR characteristic polynomial φ (B) = 1- φ 1 B - φ 2 B 2 +…+ φ p B p has roots all lie outside the unit circle. (i.e. with absolute value > 1) Example: For the AR(1) process, with φ (B) = 1- φ 1 B, is stationary iff the solution of “ φ (B) = 0” >1 in absolute value (i.e. | φ 1 | <1 ) Example: For the AR(2) process with φ (B) = 1- φ 1 B- φ 2 B 2 = (1- r 1 B )(1 - r 2 B ) = 0 is stationary iff | r 1 -1 | > 1 and | r 2 -1 | >1 Note: if r is real, | r | represents its absolute value if r is a complex number (say a + b i ), | r | represents its modulus i.e. (a 2 + b 2 ) 1/2 Example 1: Determine the AR processes, with the following characteristic polynomials, are stationary or not. (a)
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This note was uploaded on 01/20/2012 for the course STA 4005 taught by Professor ? during the Spring '08 term at CUHK.

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4005T4 - STA 4005 Tutorial 4 Stationarity for the...

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