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# 3 - Introduction to Time Series Analysis Lecture 3 Peter...

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Introduction to Time Series Analysis. Lecture 3. Peter Bartlett 1. Review: Autocovariance, linear processes 2. Sample autocorrelation function 3. ACF and prediction 4. Properties of the ACF 1

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Mean, Autocovariance, Stationarity A time series { X t } has mean function μ t = E [ X t ] and autocovariance function γ X ( t + h,t ) = Cov ( X t + h ,X t ) = E [( X t + h - μ t + h )( X t - μ t )] . It is stationary if both are independent of t . Then we write γ X ( h ) = γ X ( h, 0) . The autocorrelation function (ACF) is ρ X ( h ) = γ X ( h ) γ X (0) = Corr ( X t + h ,X t ) . 2
Linear Processes An important class of stationary time series: X t = μ + summationdisplay j = −∞ ψ j W t j where { W t } ∼ WN (0 2 w ) and μ,ψ j are parameters satisfying summationdisplay j = −∞ | ψ j | < . 3

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Linear Processes X t = μ + summationdisplay j = −∞ ψ j W t j Examples: White noise: ψ 0 = 1 . MA(1): ψ 0 = 1 , ψ 1 = θ . AR(1): ψ 0 = 1 , ψ 1 = φ , ψ 2 = φ 2 , ... 4
Estimating the ACF: Sample ACF Recall: Suppose that { X t } is a stationary time series. Its mean is μ = E [ X t ] . Its autocovariance function is γ ( h ) = Cov ( X t + h ,X t ) = E [( X t + h - μ )( X t - μ )] . Its autocorrelation function is ρ ( h ) = γ ( h ) γ (0) . 5

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Estimating the ACF: Sample ACF For observations x 1 ,...,x n of a time series, the sample mean is ¯ x = 1 n n summationdisplay t =1 x t . The sample autocovariance function is ˆ γ ( h ) = 1 n n −| h | summationdisplay t =1 ( x t + | h | - ¯ x )( x t - ¯ x ) , for - n<h<n . The sample autocorrelation function is ˆ ρ ( h ) = ˆ γ ( h ) ˆ γ (0) . 6
Estimating the ACF: Sample ACF Sample autocovariance function: ˆ γ ( h ) = 1 n n −| h | summationdisplay t =1 ( x t + | h | - ¯ x )( x t - ¯ x ) . the sample covariance of ( x 1 ,x h +1 ) ,..., ( x n h ,x n ) , except that we normalize by n instead of n - h , and we subtract the full sample mean.

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