notes3 - NOTES 3 Trends of a time series: Deterministic...

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NOTES 3 Trends of a time series : Deterministic versus Stochastic Trends Examples : y t = a t - 0 . 5 a t - 1 where a t N (0 2 a ); Purely Stochastic. y t = β 0 + β 1 t + β 2 t 2 ; Purely Deterministic y t = β 0 + β 1 t + β 2 t 2 + a t where a t N (0 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 i =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 0 n [1 + 2 n - 1 X k =1 (1 - k n ) ρ k ] = γ 0 n n - 1 X k = - n +1 (1 - | k | n ) ρ k Note that the first factor γ 0 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 0 n . 1
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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 [ 0 + 2 n - 1 X t =1 n X s = t +1 γ t,s ] = 1 n 2 [ 0 + 2(( n - 1) γ 1 + ( n
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notes3 - NOTES 3 Trends of a time series: Deterministic...

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