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Let {w t = 0,1,...} be a white noise process with variance σ2 and let | < 1 be a constant. Consider the process x0 = wo, and xt = xt−1 + wt, t = 1, 2, ... We might use this method to simulate an AR(1) process from simulated white noise. (a) [6 points] Show that x₁ = Σ o³ w₁_; for any t = 0,1,…….. (b) [3 points] Find the E(x+). j=0 (c) [6 points] Show that for t = 0, 1, · · · Var(x+) = (1 − ø²(t+1)). (1 (d) [5 points] Show that, for h≥ 0, Cov(xt+h, xt) == Var(x+).

Let {w t = 0,1,...} be a white noise process with variance σ2 and let | < 1 be a constant. Consider the process x0 = wo, and xt = xt−1 + wt, t = 1, 2, ... We might use this method to simulate an AR(1) process from simulated white noise. (a) [6 points] Show that x₁ = Σ o³ w₁_; for any t = 0,1,…….. (b) [3 points] Find the E(x+). j=0 (c) [6 points] Show that for t = 0, 1, · · · Var(x+) = (1 − ø²(t+1)). (1 (d) [5 points] Show that, for h≥ 0, Cov(xt+h, xt) == Var(x+).

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Let {w t = 0,1,...} be a white noise process with variance σ2 and let | < 1 be a constant. Consider the process x0 = wo, and xt = xt−1 + wt, t = 1, 2, ... We might use this method to simulate an AR(1) process from simulated white noise. (a) [6 points] Show that x₁ = Σ o³ w₁_; for any t = 0,1,…….. (b) [3 points] Find the E(x+). j=0 (c) [6 points] Show that for t = 0, 1, · · · Var(x+) = (1 − ø²(t+1)). (1 (d) [5 points] Show that, for h≥ 0, Cov(xt+h, xt) == Var(x+).
Transcribed Image Text:Let {w t = 0,1,...} be a white noise process with variance σ2 and let | < 1 be a constant. Consider the process x0 = wo, and xt = xt−1 + wt, t = 1, 2, ... We might use this method to simulate an AR(1) process from simulated white noise. (a) [6 points] Show that x₁ = Σ o³ w₁_; for any t = 0,1,…….. (b) [3 points] Find the E(x+). j=0 (c) [6 points] Show that for t = 0, 1, · · · Var(x+) = (1 − ø²(t+1)). (1 (d) [5 points] Show that, for h≥ 0, Cov(xt+h, xt) == Var(x+).
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