SimpleFractionalDickeyFullerTest Bensalma,Ahmed MunichPersonalRePEcArchive

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Simple Fractional Dickey Fuller Test

Bensalma, Ahmed

Ecole National Superieure de la Statistique et de l’Economie Appliquée (ENSSEA)

24 July 2013

Online at https://mpra.ub.uni-muenchen.de/50315/

MPRA Paper No. 50315, posted 01 Oct 2013 05:31 UTC

XXIX-th European Meeting of Statisticians, 2013, Budapest

Simple Fractional Dickey-Fuller Test

Ahmed BENSALMA

Ecole National de la Statistique et de l’Economie Applique (ENSSEA), Algiers, Algeria email: bensalma.ahmed@gmail.com; bensalma.ahmed@enssea.dz

This paper proposes a new testing procedure for the degree of fractional integration of a time series inspired on the unit root test of Dickey-Fuller (1979). The composite null hypothesis is that ofd≥d0

against d < d0. The test statistics is the same as in Dickey-Fuller test using as output ∆^d0y_t instead of ∆yt and as input ∆^−1+d0y_t−1 instead ofy_t−1, exploiting the fact that ify_t isI(d) then ∆^−1+d0y_t is I(1) under the nulld=d0. Ifd≥d0, using the generalization of Sowell’s results (1990), we propose a test based on the least favorable case d= d0, to control type I error and when d < d0 we show that the usual tests statistics diverges to−∞, providing consistency. By noting that d−d0 can always be decomposed as d−d0=m+δ, wherem∈Nand δ∈]−0.5,0.5], the asymptotic null and alternative of the Dickey-Fuller, normalized bias statistic nρb_n and the Dickey-Fuller t-statistic t_bρn are provided by the theorem 1.

Theorem 1. Let {y_t} be generated according DGP ∆^dy_t = ε_t. If regression model ∆^d0y_t = b

ρ_n∆^−1+d0y_t−1+bϵ_t is fitted to a sample of size nthen, as n↑ ∞, nρb_n and t_ρbn verifies that b

ρ_n=O_p(log⁻¹n) and (logn)ρb_n→ −∞,^p ifd−d0 =−0.5, (1) b

ρ_n=O_p(n^−1−2δ) and nρb_n→ −∞,^p if −0.5< d−d0<0, (2) b

ρ_n=O_p(n⁻¹) and nρb_n⇒

1 2

{w²(1)−1}

∫1

0 w²(r)dr , ifd−d0 = 0, (3) b

ρ_n=O_p(n⁻¹) and nρb_n⇒

1 2w²

δ,m+1(1)

∫¹

0 w²

δ,m+1(r)dr, ifd−d0>0. (4) t_bρn =O_p(n^−0.5log^−0.5n) and t_ρbn

→ −∞,p ifd−d0 =−0.5, (5)

t_ρbn =O_p(n^−δ) andt_ρbn

→ −∞,p if −1

2 < d−d0<0, (6)

t_ρbn =O_p(1) andt_ρbn ⇒

1 2

{w²(1)−1} [∫1

0 w²(r)dr]¹/2, ifd−d0= 0, (7) t_bρn =O_p(n^δ) and t_bρn

→p +∞, if 0< d−d0<0.5, (8) t_bρn =O_p(n^0.5) and t_bρn

→p +∞, ifd−d0≥0.5. (9)

where w_δ,m(r) is (m−1)−fold integral of w_δ(r) recursively defined as w_δ,m(r) =∫r

0 w_δ,m−1(s)ds,with w_δ,1(r) =w_δ(r) and w(r) is the standard Brownian motion.

These properties and distributions are the generalization of those established by Sowell (1990) for the cases−¹₂ < d−1<0,d−1 = 0 and 0< d−1< ¹₂.

References

[A. BENSALMA. (2012)] BENSALMA, A. 2012: Unified theoretical framework for unit root and fractional unit root, arXiv:1209.1031 (September 2012).

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