Lecture 23 The Dickey-Fuller Test We have seen that

ρ < 1; it has a non-standard asymptotic

distribution when ρ = 1, being skewed to

the left.
Therefore we cannot apply the standard t-test to test the unit root null. This test will reject the null too often; the actual test size will be greater than the nominal test size.
White (1956) showed that under H₀:ρ=1, τ does have a stable limiting distribution.
Dickey (1976) and Dickey and Fuller (1979) tabulated the percentiles of the this distribution. (How?) The percentiles of the Dickey-Fuller distribution are available in many time series textbooks (including the Enders and Hamilton texts).
See Table

Notes –

• 
  the unit root test is a one-sided test. Reject H₀ if τ is “too negative”, i.e., if ρ-hat is too much less than one.
  
• 
  the median of the DF distribution is about
  

• 
  the 0.025 percentile of the standard normal is -1.96. The 0.025 percentile of the DF distribution is -2.23. The 0.05 percentile of the standard normal is -1.65. The 0.05 of the DF is -1.95
  
• 
  The asymptotic distribution seems to be appropriate even if T is as small as 25.
  
• 
  an equivalent test: Regress Δy_t on y_t and use as the test statistic.
  
• 
  Under H₀ the test-statistic
  

also converges in distribution. The

percentiles of this distribution were also

tabulated by Dickey and Fuller. Unit

root tests based on this test statistic

seem to be less powerful and, so, less

widely used.

Suppose we modify the model and null hypothesis as follows:

H_A: ρ < 1

Under the null hypothesis, the t-statistic

from the regression of y_t on 1, y_t-1, or, equivalently, the t-statistic