Definition of a Unit Root Process and How to Test for Unit Roots

• AR(1) models
  • Model: Y_t - μ = ρ ( Y_t-1 - μ ) + e_t
  • Y_t = observation at time t
  • e_t = 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 ( Y_t-1 - μ ) from both sides of model
  • Reparameterized Model: Y_t - Y_t-1 = (ρ -1)( Y_t-1 - μ ) + e_t
  • Compute **First Difference** D_t = Y_t-Y_t-1
  • Regress D_t on 1, Y_t-1
  • Test coefficient of Y_t-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 Y_t-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
  • Regress Y_t-Y_t-1 on Y_t-1 NO INTERCEPT
    • Coefficient ρ - 1 and n[ ρ - 1 ] distribution
    • Student t test that we call τ distribution
  • Regress Y_t-Y_t-1 on Y_t-1 WITH INTERCEPT
    • Get NEW n[ ρ - 1 ] distribution and mean adjusted τ distribution - (τ_μ )
  • Regress Y_t-Y_t-1 on Y_t-1 , 1, and t (time)
    • Get NEW n[ ρ -1 ] distribution and trend adjusted τ distribution - (τ_t )

• Higher Order Models
• Y_t - μ = α₁[Y_t-1 - μ] + α₂ [Y_t-2 - μ] +, ..., + α_p[Y_t-p - μ ]+e_t
  • Characteristic polynomial is:
  • m^p- α₁ m^p-1 - α₂ m^p-2 - ... - α_p
    • If m=1 is a root then 1 - α₁ - α₂ - ... - α_p = 0 (definition of root)
    • If m=1 then μ[1 - α₁ - α₂ - ... - α_p] 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 D_t = Y_t -Y_t-1 and its lagged values.
  • Regress D_t= Y_t -Y_t-1 on 1, Y_t-1, D_t-1 ,...,D_t-p-1
    • True coefficient on Y_t-1 is -[1 - α₁ - α₂ - ... - α_p] (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 Y_t-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.