Chapter 4: Video 1 - Supplemental slides The Autoregressive Model - - PowerPoint PPT Presentation

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Jan 14, 2024 355 likes •401 views

Chapter 4: Video 1 - Supplemental slides The Autoregressive Model Autoregressive (AR) processes Let 1 , 2 , . . . be White Noise(0, 2 ) innovations, with variance 2 Then, Y 1 , Y 2 , . . . is an AR process if for some constants

SLIDE 1

Chapter 4: Video 1 - Supplemental slides

SLIDE 2

The Autoregressive Model

Autoregressive (AR) processes Let ǫ1, ǫ2, . . . be White Noise(0,σ2

ǫ ) innovations, with variance σ2 ǫ

Then, Y1, Y2, . . . is an AR process if for some constants µ and φ, Yt − µ = φ(Yt−1 − µ) + ǫt

We focus on 1st order case, the simplest AR process

SLIDE 3

AR Processes

Autoregressive (AR) processes Yt − µ = φ(Yt−1 − µ) + ǫt

µ is the mean of the {Yt} process
If φ = 0, then Yt = µ + ǫt, such that Yt is White Noise(µ, σ2

ǫ )

If φ = 0, then observations Yt depend on both ǫt and Yt−1
And the process {Yt} is autocorrelated
If φ = 0, then (Yt−1 − µ) is fed forward into Yt
φ determines the amount of feedback
Larger values of |φ| result in more feedback

SLIDE 4

AR Processes: Properties

If |φ| < 1, then E(Yt) = µ Var(Yt) = σ2

Y =

σ2

ǫ

1 − φ2 Corr(Yt, Yt−h) = ρ(h) = φ|h| for all h

If µ = 0 and φ = 1, then