David A. Dickey - Testing for Unit Roots Definition of a Unit Root Process and How to Test for Unit Roots

• AR(1) models
• Model: Yt - μ = ρ ( Yt-1 - μ ) + et
• Yt = observation at time t
• et = error or "shock" at time t (assumed iid normal)
• μ = series mean (assumed constant over time)
• ρ = Autoregressive coefficient

• Test of Ho:ρ =1
• If ρ =1 then mean μ drops out of model.
• If ρ =1 forecast does NOT revert to mean.
• Stock price example: if ρ =1 then can't make money by buying low and selling high.

• Test construction
• subtract ( Yt-1 - μ ) from both sides of model
• Reparameterized Model: Yt - Yt-1 = (ρ -1)( Yt-1 - μ ) + et
• Compute **First Difference** Dt = Yt-Yt-1
• Regress Dt on 1, Yt-1
• Test coefficient of Yt-1
• Distribution is nonstandard. n( ρ _hat - 1) is Op(1).
• Tables in Fuller, Introduction to Statistical Time Series
• Distribution (of coefficient) does NOT hold in higher order models (more lags)
• t-test on coefficient of Yt-1
• Distribution is nonstandard. t test is Op(1).
• Tables in Fuller, Introduction to Statistical Time Series
• Distribution of t DOES hold (is same asymptotically) in higher order models.

• Detrending

• Higher Order Models
• Yt - μ = α1[Yt-1 - μ] + α2 [Yt-2 - μ] +, ..., + αp[Yt-p - μ ]+et

• Characteristic polynomial is:
• mp- α1 mp-1 - α2 mp-2 - ... - αp
• If m=1 is a root then 1 - α1 - α2 - ... - αp = 0 (definition of root)
• If m=1 then μ[1 - α1 - α2 - ... - αp] is 0 regardless of μ
• If m=1 then no mean μ in model ( μ is not identifiable)
• No mean reversion
• Unit root is generalization of ρ =1 to higher order models.
• Compute first difference Dt = Yt -Yt-1 and its lagged values.
• Regress Dt= Yt -Yt-1 on 1, Yt-1, Dt-1 ,...,Dt-p-1
• True coefficient on Yt-1 is -[1 - α1 - α2 - ... - αp] (do the algebra!).
• True coefficient is negative of characteristic polynomial evaluated at 1.
• True coefficent value is 0 <-> a root is 1.
• Test coefficient of Yt-1 against t (τ ) tables in Fuller.
• Test available in SAS (and other packages)
• Test discussed in Box and Jenkins (newest edition), Fuller, and many other texts.