Chapter 4: Video 1 - Supplemental slides

The Autoregressive Model

Autoregressive (AR) processes

Let ε1, ε2, . . . be White Noise(0,σ2ε ) innovations, with variance σ2

ε

Then, Y1, Y2, . . . is an AR process if for some constants µ and φ,

Yt − µ = φ(Yt−1 − µ) + εt

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

AR Processes

Autoregressive (AR) processes

Yt − µ = φ(Yt−1 − µ) + εt

• µ is the mean of the {Yt} process

• If φ = 0, then Yt = µ+ εt, such that Yt is White Noise(µ, σ2ε )

• If φ 6= 0, then observations Yt depend on both εt and Yt−1

• And the process {Yt} is autocorrelated

• If φ 6= 0, then (Yt−1 − µ) is fed forward into Yt

• φ determines the amount of feedback

• Larger values of |φ| result in more feedback

AR Processes: Properties

If |φ| < 1, then

E(Yt) = µ

Var(Yt) = σ2Y = σ2

ε

1− φ2

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

• If µ = 0 and φ = 1, then

Yt = Yt−1 + εt

which is a random walk process, and {Yt} is NOT stationary