# CP3.pdf - STAT 4005 Time Series Chapter 3 ARMA Model ARMA...

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STAT 4005 - Time Series Chapter 3 - ARMA Model ARMA Model (CUHK) Chapter 3 1 / 55

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... Agenda 1 Introduction 2 Moving Average Model (MA) Stationarity Invertibility 3 Autoregressive Model (AR) Stationarity Asymptotic Stationarity Causality Autocovariance of AR( p ) model via Yule-Walker equation 4 ARMA model 5 ARIMA model 6 Seasonal Model 7 Summary ARMA Model (CUHK) Chapter 3 2 / 55
... Introduction ARMA model The most common model for stationary time series Y t - φ 1 Y t - 1 - · · · - φ p Y t - p = Z t - θ 1 Z t - 1 - · · · - θ q Z t - q Autoregressive (AR) Moving Average (MA) Y t is observation, Z t WN (0 , σ 2 ) is white noise. Equivalently, ARMA model can be written as φ ( B ) Y t = θ ( B ) Z t φ ( B ) = I - φ 1 B - φ 2 B 2 - · · · - φ p B p θ ( B ) = I - θ 1 B - θ 2 B 2 - · · · - θ q B q are characteristic polynomials without common roots, i.e., no x s.t. φ ( x ) = θ ( x ) = 0 . ARMA Model (CUHK) Chapter 3 3 / 55

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... Introduction ARIMA model The most common model for non-stationary time series φ ( B ) (1 - B ) d Y t = θ ( B ) Z t Integrated If Y t follows an ARIMA model with integrated parameter d , then d Y t = (1 - B ) d Y t follows an ARMA model. ARMA Model (CUHK) Chapter 3 4 / 55
... No Common Root Consider (1 - ψ B ) Y t = (1 - ψ B ) Z t , i.e., Y t - ψ Y t - 1 = Z t - ψ Z t - 1 , Z t WN (0 , σ 2 ) . (1) Now, φ ( B ) = θ ( B ) = (1 - ψ B ) φ 1 ψ = θ 1 ψ = 0 1 ψ is the common root. Apply (1) repeatedly, we have Y t - Z t = ψ Y t - 1 - ψ Z t - 1 = ψ ( Y t - 1 - Z t - 1 ) = · · · = ψ k ( Y t - k - Z t - k ) for all k 1 . Thus, the only stationary solution for (1) is Y t = Z t for all t 0 . Remark: Y t = Z t + ψ t is a non-stationary solution for (1) ARMA Model (CUHK) Chapter 3 5 / 55

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... Exercises Identify the the following ARIMA models a) Y t - 0 . 5 Y t - 1 + 0 . 06 Y t - 2 = Z t + 0 . 6 Z t - 1 - 0 . 16 Z t - 2 b) Y t - 0 . 8 Y t - 1 - 0 . 2 Y t - 2 = Z t + 0 . 2 Z t - 1 c) Y t - 0 . 8 Y t - 1 - 0 . 2 Y t - 2 = Z t - 0 . 2 Z t - 1 ARMA Model (CUHK) Chapter 3 6 / 55
... Agenda 1 Introduction 2 Moving Average Model (MA) Stationarity Invertibility 3 Autoregressive Model (AR) Stationarity Asymptotic Stationarity Causality Autocovariance of AR( p ) model via Yule-Walker equation 4 ARMA model 5 ARIMA model 6 Seasonal Model 7 Summary ARMA Model (CUHK) Chapter 3 7 / 55

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... Moving Average Model Definition 1 (MA( q ) model) A stochastic process { Y t } t =1 , 2 ,... follows an MA( q ) model if Y t = Z t - θ 1 Z t - 1 - · · · - θ q Z t - q where Z t WN (0 , σ 2 ) . Sometimes, we further assume Z t iid N (0 , σ 2 ) ARMA Model (CUHK) Chapter 3 8 / 55
... Agenda 1 Introduction 2 Moving Average Model (MA) Stationarity Invertibility 3 Autoregressive Model (AR) Stationarity Asymptotic Stationarity Causality Autocovariance of AR( p ) model via Yule-Walker equation 4 ARMA model 5 ARIMA model 6 Seasonal Model 7 Summary ARMA Model (CUHK) Chapter 3 9 / 55

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... Moving Average Model: Stationarity Proposition 2 (Stationarity) For all q and ( θ 1 , . . . , θ q ) R q , the MA( q ) model Y t = Z t - θ 1 Z t - 1 - · · · - θ q Z t - q , Z t WN (0 , σ 2 ) defines a stationary process.
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