The Fractional Unit Root Distribution

Asymptotic 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 ths model only converges when
d = 0. Results are proven by means of functional limit theorems.
KEYWORDS: Unit root distribution, fractional differencing, functional limit theorem.
1. INTRODUCTION
THENEED TO TEST economic theories which imply random walks has stimulated a
large literature involving the unit root distribution (see Dickey and Fuller (1979,
1981), Evans and Savin (1981, 1984), Sargan and Bhargava (1983), Phillips
(1987)). One facet of the unit root literature has concerned weakening the
assumption of IID errors. In particular Phillips (1987) shows that the unit root
distribution can be used to test for a random walk if the errors satisfy a strong
mixing condition. Unfortunately, this condition may not be justified for some
economic time series. For example, dependency greater than allowed in Phillips
(1987), is permitted by fractionally integrated models which extend the
ARIMA(p, d, q) model to real values of d. Furthermore, studies that have
looked for fractional integration (Granger and Joyeux (1981), Geweke and
Porter-Hudak (1983)) have concluded that some economic time series possess
fractional unit roots.