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# Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root

David A. Dickey and Wayne A. Fuller
Econometrica
Vol. 49, No. 4 (Jul., 1981), pp. 1057-1072
DOI: 10.2307/1912517
Stable URL: http://www.jstor.org/stable/1912517
Page Count: 16
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## Abstract

Let the time series Yt satisfy $Y_{t}=\alpha +\rho Y_{t-1}+e_{t}$, where Y1 is fixed and the et are normal independent (0, σ 2) random variables. The likelihood ratio test of the hypothesis that (α, ρ) = (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 α is also discussed. A similar investigation is conducted for the model containing a time trend.

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