Question

4.6 A first-order autoregressive model is generated from the white noise series wt using the generating equations xt = ?xt-1 + wt, where ?, for |?| < 1, is a parameter and the wt are independent random variables with mean zero and variance ?w^2. (a) Show that the power spectrum of xt is given by fx(?) = ?w^2 / (1 + ?^2 - 2? cos(2??)). (b) Verify the autocovariance function of this process is ?x(h) = (?w^2 ?^|h|) / (1 - ?^2), h = 0, ±1, ±2, . . . , by showing that the inverse transform of ?x(h) is the spectrum derived in part (a).

          4.6 A first-order autoregressive model is generated from the white noise series wt using the generating equations
xt = ?xt-1 + wt,
where ?, for |?| < 1, is a parameter and the wt are independent random variables with mean zero and variance ?w^2.
(a) Show that the power spectrum of xt is given by
fx(?) = ?w^2 / (1 + ?^2 - 2? cos(2??)).
(b) Verify the autocovariance function of this process is
?x(h) = (?w^2 ?^|h|) / (1 - ?^2),
h = 0, ±1, ±2, . . . , by showing that the inverse transform of ?x(h) is the spectrum derived in part (a).

Show more…

4.6 A first-order autoregressive model is generated from the white noise series wt using the generating equations
xt = ?xt-1 + wt,
where ?, for |?| < 1, is a parameter and the wt are independent random variables with mean zero and variance ?w^2.
(a) Show that the power spectrum of xt is given by
fx(?) = ?w^2 / (1 + ?^2 - 2? cos(2??)).
(b) Verify the autocovariance function of this process is
?x(h) = (?w^2 ?^|h|) / (1 - ?^2),
h = 0, ±1, ±2, . . . , by showing that the inverse transform of ?x(h) is the spectrum derived in part (a).

Added by Kathleen C.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Supreeta N

05/26/2023

Step 1

The model given is a first-order autoregressive (AR(1)) model, defined by the equation \(X_t = \phi X_{t-1} + W_t\), where \(|\phi| < 1\) and \(W_t\) is white noise with mean zero and variance \(\sigma^2\). We need to show that the power spectrum of \(X_t\) is Show more…

Show all steps

lock

Thanks for your feedback!

4.6 A first-order autoregressive model is generated from the white noise series wt using the generating equations xt = ϕxt-1 + wt , where ϕ, for |ϕ| < 1, is a parameter and the wt are independent random variables with mean zero and variance σw^2. (a) Show that the power spectrum of xt is given by fx(ω) = σw^2 / (1 + ϕ^2 - 2ϕ cos(2πω)). (b) Verify the autocovariance function of this process is γx(h) = (σw^2 ϕ^|h|) / (1 - ϕ^2), h = 0, ±1, ±2, . . . , by showing that the inverse transform of γx(h) is the spectrum derived in part (a).

Play audio

Feedback

Powered by NumerAI

Supreeta N and 59 other subject Algebra educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Recommended Textbooks

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later