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# 16 - Introduction to Time Series Analysis Lecture 16 1...

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Introduction to Time Series Analysis. Lecture 16. 1. Review: Spectral density 2. Examples 3. Spectral distribution function. 4. Autocovariance generating function and spectral density. 1

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Review: Spectral density If a time series { X t } has autocovariance γ satisfying h = -∞ | γ ( h ) | < , then we define its spectral density as f ( ν )= summationdisplay h = -∞ γ ( h ) e - 2 πiνh for −∞ <ν< . 2
Review: Spectral density 1. f ( ν ) is real. 2. f ( ν ) 0 . 3. f is periodic, with period 1 . So we restrict the domain of f to 1 / 2 ν 1 / 2 . 4. f is even (that is, f ( ν )= f ( ν ) ). 5. γ ( h )= integraldisplay 1 / 2 - 1 / 2 e 2 πiνh f ( ν ) . 3

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Examples White noise: { W t } , γ (0)= σ 2 w and γ ( h )=0 for h negationslash =0 . f ( ν )= γ (0)= σ 2 w . AR(1): X t = φ 1 X t - 1 + W t , γ ( h )= σ 2 w φ | h | 1 / (1 φ 2 1 ) . f ( ν )= σ 2 w 1 - 2 φ 1 cos(2 πν )+ φ 2 1 . If φ 1 > 0 (positive autocorrelation), spectrum is dominated by low frequency components—smooth in the time domain. If φ 1 < 0 (negative autocorrelation), spectrum is dominated by high frequency components—rough in the time domain. 4
Example: AR(1) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 10 20 30 40 50 60 70 80 90 100 ν f( ν ) Spectral density of AR(1): X t = +0.9 X t-1 + W t 5

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Example: AR(1) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 10 20 30 40 50 60 70 80 90 100 ν f( ν ) Spectral density of AR(1): X t = -0.9 X t-1 + W t 6
Example: MA(1) X t = W t + θ 1 W t - 1 . γ ( h )= σ 2 w (1+ θ 2 1 ) if h =0 , σ 2 w θ 1 if | h | =1 , 0 otherwise. f ( ν )= 1 summationdisplay h = - 1 γ ( h ) e - 2 πiνh = γ (0)+2 γ (1)cos(2 πν ) = σ 2 w ( 1+ θ 2 1 +2 θ 1 cos(2 πν ) ) .

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16 - Introduction to Time Series Analysis Lecture 16 1...

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