A Fractional Dickey-Fuller Test for Unit Roots

This paper presents a new test for fractionally integrated (FI) processes. In particular, we propose a testing procedure in the time domain that extends the well-known Dickey-Fuller approach, originally designed for the I(1) versus I(0) case, to the more general setup of FI(d0) versus FI(d1), with $...

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Bibliographic Details
Published in:	Econometrica 2002-09, Vol.70 (5), p.1963-2006
Main Authors:	Dolado, Juan J., Gonzalo, Jesús, Mayoral, Laura
Format:	Article
Language:	eng
Subjects:	Applications Applied sciences ARFIMA Critical values Dickey-Fuller test Econometrics Economic models Economics Estimate reliability Estimators Exact sciences and technology fractional processes Hypotheses Hypothesis testing Inference from stochastic processes time series analysis Insurance, economics, finance long memory Macroeconomics Mathematical models Mathematics Memory Monte Carlo simulation Null hypothesis Operational research and scientific management Operational research. Management science Polynomials Probability and statistics Random walk Regression analysis Root test Sciences and techniques of general use Statistical theories Statistics Studies Time series Trends unit roots White noise
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Summary:	This paper presents a new test for fractionally integrated (FI) processes. In particular, we propose a testing procedure in the time domain that extends the well-known Dickey-Fuller approach, originally designed for the I(1) versus I(0) case, to the more general setup of FI(d0) versus FI(d1), with $d_1 < d_0$. When d0 = 1, the proposed test statistics are based on the OLS estimator, or its t-ratio, of the coefficient on Δd1yt-1 in a regression of Δ yt on Δd1yt-1 and, possibly, some lags of Δ yt. When d1 is not taken to be known a priori, a pre-estimation of d1 is needed to implement the test. We show that the choice of any T1/2-consistent estimator of d1 ∈ [0,1) suffices to make the test feasible, while achieving asymptotic normality. Monte-Carlo simulations support the analytical results derived in the paper and show that proposed tests fare very well, both in terms of power and size, when compared with others available in the literature. The paper ends with two empirical applications.
ISSN:	0012-9682 1468-0262