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Unit‐root testing: on the asymptotic equivalence of Dickey–Fuller with the log–log slope of a fitted autoregressive spectrum

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  • Evangelos E. Ioannidis

Abstract

In this article we consider the problem of testing for the presence of a unit root against autoregressive alternatives. In this context we prove the asymptotic equivalence of the well‐known (augmented) Dickey–Fuller test with a test based on an appropriate parametric modification of the technique of log‐periodogram regression. This modification consists of considering, close to the origin, the slope (in log–log coordinates) of an autoregressively fitted spectral density. This provides a new interpretation of the Dickey–Fuller test and closes the gap between it and log‐periodogram regression. This equivalence is based on monotonicity arguments and holds on the null as well as on the alternative. Finally, a simulation study provides indications of the finite‐sample behaviour of this asymptotic equivalence.

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  • Evangelos E. Ioannidis, 2010. "Unit‐root testing: on the asymptotic equivalence of Dickey–Fuller with the log–log slope of a fitted autoregressive spectrum," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(3), pages 153-166, May.
  • Handle: RePEc:bla:jtsera:v:31:y:2010:i:3:p:153-166
    DOI: 10.1111/j.1467-9892.2010.00652.x
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    1. Perron, P, 1993. "Erratum [The Great Crash, the Oil Price Shock and the Unit Root Hypothesis]," Econometrica, Econometric Society, vol. 61(1), pages 248-249, January.
    2. Perron, Pierre & Vogelsang, Timothy J, 1992. "Testing for a Unit Root in a Time Series with a Changing Mean: Corrections and Extensions," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(4), pages 467-470, October.
    3. Velasco, Carlos, 1999. "Non-stationary log-periodogram regression," Journal of Econometrics, Elsevier, vol. 91(2), pages 325-371, August.
    4. Robinson, Peter M. & Velasco, Carlos, 2000. "Whittle pseudo-maximum likelihood estimation for nonstationary time series," LSE Research Online Documents on Economics 2273, London School of Economics and Political Science, LSE Library.
    5. Peter M Robinson & Carlos Velasco, 2000. "Whittle Pseudo-Maximum Likelihood Estimation for Nonstationary Time Series - (Now published in Journal of the American Statistical Association, 95, (2000), pp.1229-1243.)," STICERD - Econometrics Paper Series 391, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    6. Perron, Pierre, 1990. "Testing for a Unit Root in a Time Series with a Changing Mean," Journal of Business & Economic Statistics, American Statistical Association, vol. 8(2), pages 153-162, April.
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