solutions4 - Matlab Solutions Time Series (2DD23) Exercise...

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

View Full Document Right Arrow Icon
Matlab Solutions Time Series (2DD23) Week 4 Exercise 3.4 X t - μ = 0 . 7 ( X t - 1 - μ) + Z t = 0 . 7 ( 0 . 7 ( X t - 2 - μ) + Z t - 1 ) + Z t = 0 . 7 ( 0 . 7 ( 0 . 7 ( X t - 3 - μ) + Z t - 2 ) + Z t - 1 ) + Z t = 0 . 7 3 ( X t - 3 - μ) + 0 . 7 2 Z t - 2 + 0 . 7 Z t - 1 + Z t = X i = 0 0 . 7 i Z t - i Now we can compute the expectation and the variance of X t - μ : E ( X t - μ) = 0 Var ( X t - μ) = σ 2 Z X i = 0 ( 0 . 7 2 ) i = 1 1 - 0 . 7 2 σ 2 Z The autocovariance function: γ ( k ) = Cov ( X t , X t + k ) = Cov ( X t - μ, X t + k - μ) = Cov ( X i = 0 0 . 7 i Z t - i , X i = 0 0 . 7 i Z t + k - i ) = Cov ( X i = 0 0 . 7 i Z t - i , X j =- k 0 . 7 j + k Z t - j ) = σ 2 Z X i = 0 0 . 7 i 0 . 7 i + k = σ 2 Z 0 . 7 k X i = 0 ( 0 . 7 2 ) i = σ 2 Z 0 . 7 k 1 - 0 . 7 2 The autocorrelation function: ρ( k ) = γ ( k ) γ ( 0 ) = 0 . 7 k 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
Exercise 3.7 X t = 1 12 X t - 1 + 1 12 X t - 2 + Z t X t - 1 12 X t - 1 - 1 12 X t - 2 = Z t ( 1 - 1 12 B - 1 12 B 2 ) X t = Z t X t = Z t 1 - 1 12 B - 1 12 B 2 The auxiliary equation becomes: y 2 - 1 12 y - 1 12 = 0 ( y - 1 24 ) 2 = 1 12 + 1 24 2 = ( 7 24 ) 2 y - 1 24 = 7 24 y - 1 24 = - 7 24 y = 1 3 y = - 1 4 The roots lie within the unit circle, so X t is stationary. Now we will compute the autocorrelation- function: ρ( k ) = A 1 π | k | 1 + A 2 π | k | 2 , where π 1 = 1
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/23/2009 for the course MATH 200 taught by Professor Gur during the Spring '09 term at Accreditation Commission for Acupuncture and Oriental Medicine.

Page1 / 5

solutions4 - Matlab Solutions Time Series (2DD23) Exercise...

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