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Limiting distributions of unconditional maximum likelihood unit root test statistics in seasonal time–series models

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  • Taiyeong Lee
  • David A. Dickey

Abstract

. The likelihood function of a seasonal model, Yt = ρYt−d + et as implemented in computer algorithms under the assumption of stationary initial conditions is a function of ρ which is zero at the point ρ = 1. It is a smooth function for ρ in the above seasonal model with a well‐defined maximum regardless of the data‐generating mechanism. Gonzalez‐Farias (PhD Thesis, North Carolina State University, 1992) proposed tests for unit roots based on maximizing the stationary likelihood function in nonseasonal time series. We extend it to seasonal time series. The limiting distribution of seasonal unit root test statistics based on the unconditional maximum likelihood estimators are shown. Models having a single mean, seasonal means, and a single‐trend variable across the seasons are considered.

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  • Taiyeong Lee & David A. Dickey, 2004. "Limiting distributions of unconditional maximum likelihood unit root test statistics in seasonal time–series models," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(4), pages 551-561, July.
    • Handle: RePEc:bla:jtsera:v:25:y:2004:i:4:p:551-561
    DOI: 10.1111/j.1467-9892.2004.01814.x
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    1. Hylleberg, S. & Engle, R. F. & Granger, C. W. J. & Yoo, B. S., 1990. "Seasonal integration and cointegration," Journal of Econometrics, Elsevier, vol. 44(1-2), pages 215-238.
    2. Richard J. Smith & A. M. Robert Taylor, 1999. "Likelihood Ratio Tests for Seasonal Unit Roots," Journal of Time Series Analysis, Wiley Blackwell, vol. 20(4), pages 453-476, July.
    3. Sook Fwe Yap & Gregory C. Reinsel, 1995. "Results On Estimation And Testing For A Unit Root In The Nonstationary Autoregressive Moving‐Average Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(3), pages 339-353, May.
    4. Dong Wan Shin & Wayne Fuller, 1998. "Unit Root Tests Based on Unconditional Maximum Likelihood Estimation for the Autoregressive Moving Average," Journal of Time Series Analysis, Wiley Blackwell, vol. 19(5), pages 591-599, September.
    5. Pantula, Sastry G & Gonzalez-Farias, Graciela & Fuller, Wayne A, 1994. "A Comparison of Unit-Root Test Criteria," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(4), pages 449-459, October.
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