David A. Dickey - Testing for Unit Roots
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
- Regress Y_{t}-Y_{t-1} on Y_{t-1} WITH INTERCEPT
- Regress Y_{t}-Y_{t-1} on Y_{t-1} , 1, and t (time)
- Higher Order Models
- Y_{t} - μ = α_{1}[Y_{t-1}
- μ] + α_{2} [Y_{t-2
} - μ] +, ..., +
α_{p}[Y_{t-p} - μ ]+e_{t}
- Characteristic polynomial is:
- m^{p}- α_{1} m^{p-1} - α_{2
} m^{p-2} - ... - α_{p}
- If m=1 is a root then 1 - α_{1} - α_{2
}- ... - α_{p}
= 0 (definition of root)
- If m=1 then μ[1 - α_{1} - α_{2
}- ... - 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 - α_{1} - α_{2
}- ... - 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.