This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Time Series (MATH5/30085) 2004 Exercises 1 1. Properties of covariance. Using the deﬁnition Cov (X, Y ) = E [(X − µX )(Y − µY )] Prove the following: (a) Cov (X, Y ) = Cov (Y, X ) (b) Cov (a + bX, c + dY ) = bdCov (X, Y ) (c) Cov (X, Y ) = E (XY ) − µX µY 2. Find the autocorrelation sequences for the following processes (a) A white noise process with E (Xt ) = µ, V ar(Xt ) = σ 2 ∀t (b) Xt =
t − t−1 t (c) For an MA(1) process, Xt = − θ1 t−1 , show that you cannot identify an MA(1) process uniquely from the autocorrelation by comparing the
− results using θ1 with those if you replaced θ1 by θ1 1 3. Considering a ﬁrst order AR process Xt = φ1 Xt−1 +
t AR(1) – Markov process (a) Find the mean and an expression for the variance (b) Show that for the variance to be ﬁnite, θ1  must be less than one (c) Find the autocorrelation sequence (you may assume stationarity) 4. (a) What is meant by saying that a stochastic process is secondorder stationary? (b) Determine whether the following stochastic process is secondorder stationary, giving full justiﬁcation Xt =
13 Xt−1 4 − 3 Xt−2 + t . 4 1 5. HARDER QUESTION Let {Xt } be the zero mean autoregressive process of order 2 deﬁned by Xt − (g1 + g2 )Xt−1 + g1 g2 Xt−2 = t , where g1 , g2  < 1, and { t } is white noise with mean zero and variance σ 2 . (a) Explain why {Xt } is stationary. (b) Show that {Xt } can be written in the general linear model form Xt = 1 g2 − g 1
∞ k k g2 +1 − g1 +1 k=0 t−k . (c) Hence show that the autocovariance sequence takes the form sτ = g τ +1 (1 − g 2 ) − g τ +1 (1 − g 2 ) σ2 2 1 1 2 2 2 g2 − g1 (1 − g1 )(1 − g2 )(1 − g1 g2 ) τ = 0, ±1, ±2, ... 2 ...
View
Full
Document
This note was uploaded on 10/19/2009 for the course MATH 611 taught by Professor Jsdkasj during the Spring '09 term at Kansas.
 Spring '09
 jsdkasj
 Covariance, Variance

Click to edit the document details