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Unformatted text preview: NOTES 3 Trends of a time series : Deterministic versus Stochastic Trends Examples : y t = a t . 5 a t 1 where a t ∼ N (0 ,σ 2 a ); Purely Stochastic. y t = β + β 1 t + β 2 t 2 ; Purely Deterministic y t = β + β 1 t + β 2 t 2 + a t where a t ∼ N (0 ,σ 2 a ); Stochastic + Deterministic Estimation of a constant mean Model: Z t = μ + X t , E ( X t ) = 0 for all t. First, we wish to estimate μ with observed time series Z 1 ,Z 2 ,...,Z n . The most common estimate of μ is the sample mean ¯ Z = 1 n n X t =1 Z t . Since E ( ¯ Z ) = μ , ¯ Z is an unbiased estimate of μ . To investigate the precision of ¯ Z as an estimate of μ , we need to make further assump tions concerning X t . Theorem Suppose { X t } is a stationary time series, then V ar ( ¯ Z ) = γ n [1 + 2 n 1 X k =1 (1 k n ) ρ k ] = γ n n 1 X k = n +1 (1  k  n ) ρ k Note that the first factor γ n is the population variance assuming the observations are independent. If the { X t } series is in fact just white noise, then ρ k = 0 for k ≥ 1 and V ar ( ¯ Z ) simply reduce to γ n . 1 Proof: V ar ( ¯ Z ) = V ar ( 1 n n X t =1 Z t ) = 1 n 2 V ar ( n X t =1 Z t ) = 1 n 2 V ar ( n X t =1 X t ) = 1 n 2 [ n X t =1 V ar ( X t ) + 2 n 1 X t =1 n X s = t +1 Cov ( X t ,X s )] = 1 n 2 [ nγ + 2 n 1 X t =1 n X s = t +1 γ t,s ] = 1 n 2 [ nγ + 2(( n 1) γ 1 + ( n 2) γ 2 + ( n 3) γ 3 + ... + γ n 1 )] = 1 n 2 [ nγ + 2 n 1 X k =1 ( n k ) γ k ] = γ n [1 + 2 n 1 X k =1 (1 k n ) ρ k ] Since ρ k = ρ k , V ar ( ¯ Z ) = γ n [1 + n 1 X k =1 (1 k n ) ρ k + 1 X k = ( n 1) (1  k  n ) ρ k ] = γ n n 1 X k = n +1 (1  k  n ) ρ k 2 Example: Z t = a t 1 2 a t 1 , ρ 1 = . 4 , ρ k = 0 , k > 1 . V ar ( ¯ Z ) = γ n [1 + 2(1 1 n )( . 4)] = γ n [1 . 8 n 1 n ] When n is large, V ar ( ¯ Z ) ≈ . 2 γ n ....
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This note was uploaded on 11/02/2011 for the course STAT 4005 taught by Professor Wu,kaho during the Spring '08 term at CUHK.
 Spring '08
 WU,KaHo

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