Lecture 23 – The DickeyFuller Test
We have seen that

the dynamic behavior of I(1) processes is quite different from the behavior of I(0) processes

the way we go about defining and estimating the trend and cyclical components of a time series may depend on whether we assume the series is (trend) stationary or difference stationary.

regressions with difference stationary variables need special care.
For these reasons we might be interested in testing the null hypothesis of a unit root against the stationary or trendstationary alternative.
Consider the following AR(1) model for y_{t}:
y_{t} = ρy_{t1} + ε_{t} , ε_{t }~ iid (0,σ^{2})
1 < ρ < 1
If ρ < 1, y_{t} ~ I(0), mean 0 and var σ^{2}/(1ρ^{2})
If ρ = 1, y_{t} ~ I(1), a random walk

The OLS estimator of ρ is consistent for all ρ; it is superconsistent when ρ = 1.

The OLS tstatistic
where is the OLS s.e. of ρhat,
is asymptotically standard normal when
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