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Corr(Y₁, Y₁-1) = c C Var(Y₁-k) Corr(Y₁, Y₁-k) = ck Var(Y₂) Hint: Argue that Y₁ is independent of t-1 et. Var(Y₁-1) Var(Y₂) (d) For large t, argue that 0² e and, in general, Then use Cov(Y, Yt-1)= Cov(cY₁-1 + et, Yt -1) - c² C for k> 0 Var(Y₂) ~ and Corr(Y₁, Y₁-k) ck so that {Y} could be called asymptotically stationary. (e) Suppose now that we alter the initial condition and put Y₁ that now {Y} is stationary. = for k > 0 e 1 √1-c² . Show

Corr(Y₁, Y₁-1) = c C Var(Y₁-k) Corr(Y₁, Y₁-k) = ck Var(Y₂) Hint: Argue that Y₁ is independent of t-1 et. Var(Y₁-1) Var(Y₂) (d) For large t, argue that 0² e and, in general, Then use Cov(Y, Yt-1)= Cov(cY₁-1 + et, Yt -1) - c² C for k> 0 Var(Y₂) ~ and Corr(Y₁, Y₁-k) ck so that {Y} could be called asymptotically stationary. (e) Suppose now that we alter the initial condition and put Y₁ that now {Y} is stationary. = for k > 0 e 1 √1-c² . Show

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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0 2.22 Let {e} be a zero-mean white noise process, and let c be a constant with |c| < 1. Define Y recursively by Y₁ = cY₁ - 1 + e₁ with Y₁ = €₁. (a) Show that E(Y₂) = 0. (b) Show that Var(Y) = o² (1 + c² +cª + ··· + c²¹−2). Is {Y₁} stationary? (c) Show that Corr(Y, Y₁₁) = c Yt- Yt-1) Var(Y₁-1) Var(Y₂) Var(Y₁-k) Corr(Y₁, Var(Y₂) t- Hint: Argue that Y₁ - 1 is independent of et. Then use Cov(Y₁, Y₁-1)= Cov(cY₁-1 + e₁, Y₁-1) t- (d) For large t, argue that 0² Y₁ k) = ck Yt-k) ≈ and, in general, for k> 0 Var(Y₂): and Corr(Y₁, Yt_k) ≈ ck 1-c² so that {Y} could be called asymptotically stationary. (e) Suppose now that we alter the initial condition and put Y₁ that now {Y} is stationary. for k > 0 = N e 1 Show
Transcribed Image Text:0 2.22 Let {e} be a zero-mean white noise process, and let c be a constant with |c| < 1. Define Y recursively by Y₁ = cY₁ - 1 + e₁ with Y₁ = €₁. (a) Show that E(Y₂) = 0. (b) Show that Var(Y) = o² (1 + c² +cª + ··· + c²¹−2). Is {Y₁} stationary? (c) Show that Corr(Y, Y₁₁) = c Yt- Yt-1) Var(Y₁-1) Var(Y₂) Var(Y₁-k) Corr(Y₁, Var(Y₂) t- Hint: Argue that Y₁ - 1 is independent of et. Then use Cov(Y₁, Y₁-1)= Cov(cY₁-1 + e₁, Y₁-1) t- (d) For large t, argue that 0² Y₁ k) = ck Yt-k) ≈ and, in general, for k> 0 Var(Y₂): and Corr(Y₁, Yt_k) ≈ ck 1-c² so that {Y} could be called asymptotically stationary. (e) Suppose now that we alter the initial condition and put Y₁ that now {Y} is stationary. for k > 0 = N e 1 Show
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