Econometrica: Mar 1990, Volume 58, Issue 2
The Fractional Unit Root Distribution
Fallaw SowellAsymptotic distributions are derived for the ordinary least squares (OLS) estimate of a first order autoregression when the series is fractionally integrated of order $1 + d$, for $- 1/2 < d < 1/2$. The fractional unit root distribution is introduced to describe the limiting distribution. The unit root distribution $(d = 0)$ is seen to be an atypical member of this family because its density is nonzero over the entire real line. For $- 1/2 < d < 0$ the fractional unit root distribution has nonpositive support, while if $0 < d < 1/2$ the fractional unit root distribution has nonnegative support. Any misspecification of the order of differencing leads to drastically different limiting distributions. Testing for unit roots is further complicated by the result that the $t$ statistic in this model only converges when $d = 0$ Results are proven by means of functional limit theorems.
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