BA3202-L11 - BA3202 Actuarial Statistics Lecture 10 Summary...

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BA3203 L11 BA3202 Actuarial Statistics Lecture 10 Summary: - Time Series (1)
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BA3203 L11 Stationarity Weakly stationary requirements: Mean of the process remains constant 𝜇 𝑡 = 𝐸 𝑋 𝑡 ≡ 𝜇 for all 𝑡 The covariance of 𝑋 𝑠 and 𝑋 𝑡 depends only on the time difference 𝑡 − 𝑠 Autocovariance function: 𝛾 𝑘 ≡ 𝐶𝐶𝐶 𝑋 𝑡 , 𝑋 𝑡+𝑘 = 𝐸 𝑋 𝑡 𝑋 𝑡+𝑘 − 𝐸 𝑋 𝑡 𝐸 𝑋 𝑡+𝑘 Does not depend on 𝑡 𝛾 0 = 𝐶𝐶𝐶 𝑋 𝑡 , 𝑋 𝑡 = 𝐶𝑣𝑣 𝑋 𝑡 is constant 𝛾 𝑘 = 𝐶𝐶𝐶 𝑋 𝑡−𝑘 , 𝑋 𝑡−𝑘+𝑘 = 𝐶𝐶𝐶 𝑋 𝑡−𝑘 , 𝑋 𝑡 = 𝛾 −𝑘 Autocorrelation function (ACF): 𝜌 𝑘 = 𝐶𝐶𝑣𝑣 𝑋 𝑡 , 𝑋 𝑡+𝑘 = 𝛾 𝑘 / 𝛾 0 𝜌 0 = 1 , 𝜌 𝑘 = 𝜌 −𝑘 For a purely indeterministic process, 𝜌 𝑘 0 as 𝑘 → ∞ (i.e. the longer the time lag, the historical observations become less useful for predicting future) PACF Actuarial Science NTU Jade Nie [email protected] 2
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BA3203 L11 Main linear models of time series An autoregressive process 𝑋 𝑡 of order 𝑝 , AR(p), is defined by: 𝑋 𝑡 = 𝜇 + 𝛼 1 𝑋 𝑡−1 − 𝜇 + 𝛼 2 𝑋 𝑡−2 − 𝜇 + + 𝛼 𝑝 𝑋 𝑡−𝑝 − 𝜇 + 𝑒 𝑡 A moving average process 𝑋 𝑡 of order 𝑞 , MA(q), is defined by: 𝑋 𝑡 = 𝜇 + 𝑒 𝑡 + 𝛽 1 𝑒 𝑡−1 + 𝛽 2 𝑒 𝑡−2 + + 𝛽 𝑞 𝑒 𝑡−𝑞 ARMA(p,q): 𝑋 𝑡 = 𝜇 + 𝛼 1 𝑋 𝑡−1 − 𝜇 + 𝛼 2 𝑋 𝑡−2 − 𝜇 + + 𝛼 𝑝 𝑋 𝑡−𝑝 − 𝜇 + 𝑒 𝑡 + 𝛽 1 𝑒 𝑡−1 + 𝛽 2 𝑒 𝑡−2 + + 𝛽 𝑞 𝑒 𝑡−𝑞 Non-stationary 𝐼 ( 𝑑 ) process- 𝑋 𝑡 is an ARIMA(p,d,q) process if 𝑋 𝑡 needs to be differenced at least 𝑑 times to reduce it to stationary and if the 𝑌 𝑡 = 𝛻 𝑑 𝑋 𝑡 is an ARMA(p, q) process 1 − 𝛼 1 𝐵 − 𝛼 2 𝐵 2 − ⋯ − 𝛼 𝑝 𝐵 𝑝 1 − 𝐵 𝑑 𝑋 𝑡 − 𝜇 = 1 + 𝛽𝐵 + 𝛽 2 𝐵 2 + + 𝛽 𝑞 𝐵 𝑞 𝑒 𝑡 Actuarial Science NTU Jade Nie [email protected] 3
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BA3203 L11 AR(1) Vs MA(1) AR(1) MA(1) Definition 𝑋 𝑡 = 𝜇 + 𝛼 ( 𝑋 𝑡−1 − 𝜇 ) + 𝑒 𝑡 𝑋 𝑡 = 𝜇 + 𝑒 𝑡 + 𝛽𝑒 𝑡−1 𝜇 𝑡 = 𝜇 𝐶𝑣𝑣 𝑋 𝑡 = 𝜎 2 1 − 𝛼 2 𝜇 𝑡 = 𝜇 𝐶𝑣𝑣 𝑋 𝑡 = 1 + 𝛽 2 𝜎 2 Stationarity 𝜇 0 = 𝜇 𝛼 < 1 𝐶𝑣𝑣 𝑋 0 = 𝜎 2 1−𝛼 2 Naturally stationary Invertibility N/A 𝛽 < 1 ACF 𝜌 𝑘 = 𝛾 𝑘 𝛾 0 = 𝛼 𝑘 𝜌 𝑘 = 1 , 𝑘 = 0 𝛽 1 + 𝛽 2 , 𝑘 = 1 0 , 𝑘 > 1 PACF 𝜙 𝑘 = 𝛼 , 𝑘 = 1 0, 𝑘 > 1 𝜙 𝑘 = 1 𝑘+1 1 − 𝛽 2 𝛽 𝑘 1 − 𝛽 2 ( 𝑘+1 ) Actuarial Science NTU Jade Nie [email protected] 4
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BA3203 L11 AR(p) model AR(p) model: 𝑋 𝑡 = 𝜇 + 𝛼 1 𝑋 𝑡−1 − 𝜇 + 𝛼 2 𝑋 𝑡−2 − 𝜇 + + 𝛼 𝑝 𝑋 𝑡−𝑝 − 𝜇 + 𝑒 𝑡 We can write the definition in terms of backward shift operator ( 𝐵 ): 1 − 𝛼 1 𝐵 − 𝛼 2 𝐵 2
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