3 - NOTES 3 Trends of a time series Deterministic versus...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

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.

Page1 / 8

3 - NOTES 3 Trends of a time series Deterministic versus...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online