chap6 - Non-Stationary and Seasonal Time Series (Chap 6)...

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1 1 Non-Stationary and Seasonal Time Series (Chap 6) 6.1 ARIMA Models DEFINITION: { X t } is an ARIMA(p,d,q) process if Y t := (1 −Β) d X t is a causal ARMA(p,q) process. Remarks : 1. { X t } satisfies the difference equation φ * (Β) X t = (1 d φ(Β) X t = θ(Β) Z t 2 Ex 6.1.1. (An ARIMA(1,1,0) process). (1 −φ B) (1 X t = Z t , { Z t } ~ WN(0, σ 2 ) . Write, X t = X 0 + Y j , where Y t = (1 X t = ψ j Z t-j . where φ * ( z ) has a zero of order d at z=1. 2. If d > 0 , a polynomial of degree d can be added to { X t } without violating the difference equation φ(Β) (1 d X t = θ(Β) Z t . j t = 0 j = 0
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2 3 0 50 100 150 200 0 2 04 06 08 0 Figure 6.1 (200 observations from ARIMA(1,1,0) (1 −.8 B) (1 −Β) X t = Z t , { Z t } ~ WN(0, 1) with X t = 0. ) 4 Lag ACF 0 1 02 03 0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Figs 6.2 & 6.3 (Sample ACF and PACF of data in Fig 6.1) Lag PACF 0
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3 5 MLE AR(2) model : (1 −1.808Β−.801Β 2 ) X t = Z t , (1 − .825 B) (1 − .983Β) X t = Z t , { Z t } ~ WN(0, .970). ( Fitted model is nearly non-stationary with a root of the AR polynomial near the unit circle.) MLE AR(1) Model for Y t = (1 −Β) X t : (1 −.808 B) (1 X t = Z t , { Z t } ~ WN(0, .978), ( Fitted model is close to true: φ=.8, σ 2 =1 .) 6 6.2 Identification Techniques (a) Preliminary Transformations Goal : Transform the data (if necessary) to achieve a more plausible realization of a stationary ts. Box-Cox Transformation. Useful for transforming skewed data to symmetric data. stabilizing the variance. f λ ( U t ) = λ −1 ( U t λ − 1), U t > 0, λ > 0, ln ( U t ), U t > 0, λ= 0,
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4 7 (thousands) 1980 1982 1984 1986 1988 1990 1992 0.5 1.0 1.5 2.0 2.5 3.0 Figs 1.1 & 1.17 (Red wine data & ln (red wine).) 1980 1982 1984 1986 1988 1990 1992 6.5 7.0 7.5 8.0 8 Elimination of trend and seasonality .
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chap6 - Non-Stationary and Seasonal Time Series (Chap 6)...

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