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# 21 - Introduction to Time Series Analysis Lecture 21 1...

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Introduction to Time Series Analysis. Lecture 21. 1. Review: The periodogram, the smoothed periodogram. 2. Other smoothed spectral estimators. 3. Consistency. 4. Asymptotic distribution. 1

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Review: Periodogram The periodogram is defined as I ( ν )= X 2 c ( ν )+ X 2 s ( ν ) . X c ( ν )= 1 n n summationdisplay t =1 cos(2 πtν ) x t , X s ( ν )= 1 n n summationdisplay t =1 sin(2 πtν j ) x t . Under general conditions, X c ( ν j ) , X s ( ν j ) are asymptotically independent and N (0 , f ( ν j ) / 2) . Thus, E I ν ( n ) ) f ( ν ) , but Var ( I ν ( n ) )) f ( ν ) 2 . 2
Review: smoothed periodogram If f ( ν ) is approximately constant in a band of frequencies [ ν k L/ (2 n ) , ν k + L/ (2 n )] , we can average the periodogram over this band: ˆ f ( ν k )= 1 L ( L - 1) / 2 summationdisplay l = - ( L - 1) / 2 I ( ν k l/n ) = 1 L ( L - 1) / 2 summationdisplay l = - ( L - 1) / 2 ( X 2 c ( ν k l/n )+ X 2 s ( ν k l/n ) ) . 3

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Review: smoothed periodogram Under general conditions, the X c ( ν k l/n ) and X s ( ν k l/n ) are asymptotically independent and N (0 , f ( ν k l/n ) / 2) . Thus, E ˆ f ( ν ( n ) ) f ( ν ) and Var ˆ f ( ν ( n ) ) f 2 ( ν ) /L . Notice the bias-variance trade off : 1. Our assumption that f is approximately constant on [ ν L/ (2 n ) , ν + L/ (2 n )] becomes worse as L increases, so the difference between ˆ f ν ( n ) ) and f ( ν ) (the bias) will increase with L . 2. The variance of our estimate, Var ˆ f ν ( n ) ) decreases with L .
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