Econometrica: Jul, 1981, Volume 49, Issue 4

Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root

https://doi.org/0012-9682(198107)49:4<1057:LRSFAT>2.0.CO;2-4
p. 1057-1072

David A. Dickey, Wayne A. Fuller

Let the time series Y"t satisfy Y"t = @a + pY"t-1 + e"t, where Y"1 is fixed and the e"t are normal independent (0,@s^2) random variables. The likelihood ratio test of the hypothesis that (@a, p) = (0, 1) is investigated and a limit representation for the test statistic is presented. Percentage points for the limiting distribution and for finite sample distributions are estimated. The distribution of the least squares estimator of @a is also discussed. A similar investigation is conducted for the model containing a time trend.