International Encyclopedia of Statistical Science

2011 Edition
| Editors: Miodrag Lovric

Dickey-Fuller Tests

  • David G. Dickey
Reference work entry

One of the most basic and useful of the time series models is the order 1 (1 lag) autoregressive model, denoted AR(1) and given by Ytμ = ρ(Yt − 1μ) + et where Yt is the observation at time t, μ is the long run mean of the time series and et is an independent sequence of random variables. We use this venerable model to illustrate the Dickey–Fuller test then mention that the results extend to a broader collection of models.

When written as Yt = μ(1 − ρ) + ρYt − 1 + et, or more convincingly as Yt = λ + ρYt − 1 + et, with e independent and identically distributed as N(0, σ2), the AR(1) model looks like a regression with errors satisfying the usual assumptions. Indeed the least squares estimators of the coefficients are asymptotically unbiased and normally distributed under one key condition, namely that the true ρ satisfies | ρ | < 1. It appears that this assumption is quite often violated. Many prominent time series appear to have ρ = 1, in which case Ytμ = ρ(Yt − 1μ) + et...

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References and Further Reading

  1. Billingsley P (1968) Convergence of probability measures. Wiley, New YorkzbMATHGoogle Scholar
  2. Dickey DA, Fuller WA (1979) Distribution of the estimators for autoregressive time series with a unit root. J Am Stat Assoc 74:427–431zbMATHMathSciNetGoogle Scholar
  3. Dickey DA, Fuller WA (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49:1057–1072zbMATHMathSciNetGoogle Scholar
  4. Dickey DA (1984) Power of unit root tests. In: Proceedings of the business and economic statistics section. American Statistical Association, Philadelphia, Washington, DC, pp 489–493Google Scholar
  5. Elliott G, Rothenberg TJ, Stock JH (1996) Efficient tests for an autoregressive unit root. Econometrica 64:813–836zbMATHMathSciNetGoogle Scholar
  6. Fuller WA (1996) Introduction to statistical time series. Wiley, New YorkzbMATHGoogle Scholar
  7. Pantula SG, Gonzalez-Farias G, Fuller WA (1994) A comparison of unit root criteria. J Bus Econ Stat 13:449–459MathSciNetGoogle Scholar
  8. Park HJ, Fuller WA (1995) Alternative estimators and unit root tests for the autoregressive process. J Time Ser Anal 15:415–429MathSciNetGoogle Scholar
  9. Phillips PCB, Perron P (1988) Testing for a unit root in time series regression. Biometrika 75:335–346zbMATHMathSciNetGoogle Scholar
  10. Said SE, Dickey DA (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71:599–607zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • David G. Dickey
    • 1
  1. 1.North Carolina State UniversityRaleighUSA