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Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power

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  • Serena Ng

    ()
    (Boston College)

  • Pierre Perron

    (Universite de Montreal)

Abstract

It is widely known that when there are negative moving average errors, a high order augmented autoregression is necessary for unit root tests to have good size, but that information criteria such as the AIC and BIC tend to select a truncation lag that is very small. Furthermore, size distortions increase with the number of deterministic terms in the regression. We trace these problems to the fact that information criteria omit important biases induced by a low order augmented autoregression. We consider a class of Modified Information Criteria (MIC) which account for the fact that the bias in the sum of the autoregressive coefficients is highly dependent on the lag order k. Using a local asymptotic framework in which the root of an MA(1) process is local to -1, we show that the MIC allows for added dependence between k and the number of deterministic terms in the regression. Most importantly, the k selected by the recommended MAIC is such that both its level and rate of increase with the sample size are desirable for unit root tests in the local asymptotic framework, whereas the AIC, MBIC and especially the BIC are less attractive in at least one dimension. In monte-carlo experiments, the MAIC is found to yield huge size improvements to the DF(GLS) and the feasible point optimal P(t) test developed in Elliot, Rothenberg and Stock (1996). We also extend the M tests developed in Perron and Ng (1996) to allow for GLS detrending of the data. The M(GLS) tests are shown to have power functions that lie very close to the power envelope. In addition, we recommend using GLS detrended data to estimate the required autoregressive spectral density at frequency zero. This provides more efficient estimates on the one hand, and ensures that the estimate of the spectral density is invariant to the parameters of the deterministic trend function, a property not respected by the estimation procedure currently employed by several studies. The MAIC along with GLS detrended data yield a set of Mbar(GLS) tests with desirable size and power properties.

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Bibliographic Info

Paper provided by Boston College Department of Economics in its series Boston College Working Papers in Economics with number 369.

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Length: 43 pages
Date of creation: 01 May 1997
Date of revision: 01 Sep 2000
Handle: RePEc:boc:bocoec:369

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Keywords: unit root test; truncation lag; GLS detrending; information criteria;

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References

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  1. Perron, P. & Ng, S., 1994. "Useful Modifications to Some Unit Root Tests with Dependent Errors and Their Local Asymptotic Properties," Cahiers de recherche 9427, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  2. Lopez, J. Humberto, 1997. "The power of the ADF test," Economics Letters, Elsevier, vol. 57(1), pages 5-10, November.
  3. Serena Ng & Pierre Perron, 2005. "A Note on the Selection of Time Series Models," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 67(1), pages 115-134, 02.
  4. Nabeya, S. & Perron, P., 1991. "Local Asymtotic Distributions Related to the AR(1) MOdel with Dependent Errors," Papers 362, Princeton, Department of Economics - Econometric Research Program.
  5. Graham Elliott & Thomas J. Rothenberg & James H. Stock, 1992. "Efficient Tests for an Autoregressive Unit Root," NBER Technical Working Papers 0130, National Bureau of Economic Research, Inc.
  6. Perron, Pierre & Ng, Serena, 1998. "An Autoregressive Spectral Density Estimator At Frequency Zero For Nonstationarity Tests," Econometric Theory, Cambridge University Press, vol. 14(05), pages 560-603, October.
  7. Ng, S. & Perron, P., 1994. "Unit Root Tests ARMA Models with Data Dependent Methods for the Selection of the Truncation Lag," Cahiers de recherche 9423, Universite de Montreal, Departement de sciences economiques.
  8. Schwert, G William, 1989. "Tests for Unit Roots: A Monte Carlo Investigation," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(2), pages 147-59, April.
  9. Phillips, P.C.B., 1986. "Testing for a Unit Root in Time Series Regression," Cahiers de recherche 8633, Universite de Montreal, Departement de sciences economiques.
  10. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  11. Perron, Pierre & Rodriguez, Gabriel, 2003. "GLS detrending, efficient unit root tests and structural change," Journal of Econometrics, Elsevier, vol. 115(1), pages 1-27, July.
  12. Dufour, J-M. & King, M.L., 1989. "Optimal Invariant Tests For The Autocorrelation Coefficient In Linear Regressions With Stationary And Nonstationary Ar(1) Errors," Cahiers de recherche 8921, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  13. DeJong, David N. & Nankervis, John C. & Savin, N. E. & Whiteman, Charles H., 1992. "The power problems of unit root test in time series with autoregressive errors," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 323-343.
  14. Franses, Philip Hans & Haldrup, Niels, 1994. "The Effects of Additive Outliers on Tests for Unit Roots and Cointegration," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(4), pages 471-78, October.
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