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Mean and Autocovariance Function Estimation Near the Boundary of Stationarity




We analyze the applicability of standard normal asymptotic theory for linear process models near the boundary of stationarity. The concept of stationarity is refined, allowing for sample size dependence in the array and paying special attention to the rate at which the boundary unit root case is approached using a localizing coefficient around unity. The primary focus of the present paper is on estimation of the the mean, autocovariance and autocorrelation functions within the broad region of stationarity that includes near boundary cases which vary with the sample size. The rate of consistency and the validity of the normal asymptotic approximation for the corresponding estimators is determined both by the sample size n and a parameter measuring the proximity of the model to the unit root boundary. An asymptotic result on the estimation of the localizing coefficient is also presented. To assist in the development of the limit theory in the present case, a suitable asymptotic theory for the behavior of quadratic forms in the vicinity of the boundary of stationarity is provided.

Suggested Citation

  • Liudas Giraitis & Peter C. B. Phillips, 2009. "Mean and Autocovariance Function Estimation Near the Boundary of Stationarity," Cowles Foundation Discussion Papers 1690, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1690
    Note: CFP 1363

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    References listed on IDEAS

    1. Liudas Giraitis & Peter C. B. Phillips, 2006. "Uniform Limit Theory for Stationary Autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(1), pages 51-60, January.
    2. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
    3. Hosking, Jonathan R. M., 1996. "Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series," Journal of Econometrics, Elsevier, vol. 73(1), pages 261-284, July.
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    Cited by:

    1. Yabe, Ryota, 2017. "Asymptotic distribution of the conditional-sum-of-squares estimator under moderate deviation from a unit root in MA(1)," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 220-226.
    2. repec:eee:stapro:v:135:y:2018:i:c:p:54-59 is not listed on IDEAS
    3. Bailey, N. & Giraitis, L., 2013. "Weak convergence in the near unit root setting," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1411-1415.
    4. Stauskas, Ovidijus, 2019. "On the Limit Theory of Mixed to Unity VARs: Panel Setting With Weakly Dependent Errors," Working Papers 2019:2, Lund University, Department of Economics.

    More about this item


    Asymptotic normality; Integrated periodogram; Linear process; Local to unity; Localizing coefficient; Moderate deviation; Unit root;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes


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