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Estimation for Autoregressive Time Series with a Root Near 1

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  • Roy, Anindya
  • Fuller, Wayne A

Abstract

Estimators for the parameters of autoregressive time series are compared, emphasizing processes with a unit root or a root close to 1. The approximate bias of the sum of the autoregressive coefficients is expressed as a function of the test for a unit root. This expression is used to construct an estimator that is nearly unbiased for the parameter of the first-order scalar process. The estimator for the first-order process has a mean squared error that is about 40% of that of ordinary least squares for the process with a unit root and a constant mean, and the mean squared error is smaller than that of ordinary least squares for about half of the parameter space. The maximum loss of efficiency is 6n[superscript -1] in the remainder of the parameter space. The estimation procedure is extended to higher-order processes by modifying the estimator of the sum of the autoregressive coefficients. Limiting results are derived for the autoregressive process with a mean that is a linear trend.

Suggested Citation

  • Roy, Anindya & Fuller, Wayne A, 2001. "Estimation for Autoregressive Time Series with a Root Near 1," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(4), pages 482-493, October.
  • Handle: RePEc:bes:jnlbes:v:19:y:2001:i:4:p:482-93
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    3. Giuseppe Cavaliere & Dimitris N. Politis & Anders Rahbek & Michael Wolf & Dan Wunderli, 2015. "Recent developments in bootstrap methods for dependent data," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(3), pages 352-376, May.
    4. Alonso Fernández, Andrés Modesto & Bastos, Guadalupe & García-Martos, Carolina, 2017. "BIAS correction for dynamic factor models," DES - Working Papers. Statistics and Econometrics. WS 24029, Universidad Carlos III de Madrid. Departamento de Estadística.
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    6. Gao, Jie & Xu, Zhen-yuan & Zhang, Li-ting, 2009. "Approximating long-memory DNA sequences by short-memory process," Physica A: Statistical Mechanics and its Applications, Elsevier, pages 3475-3485.
    7. Han, Chirok & Phillips, Peter C. B. & Sul, Donggyu, 2011. "Uniform Asymptotic Normality In Stationary And Unit Root Autoregression," Econometric Theory, Cambridge University Press, pages 1117-1151.
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    9. Mohitosh Kejriwal & Claude Lopez, 2013. "Unit Roots, Level Shifts, and Trend Breaks in Per Capita Output: A Robust Evaluation," Econometric Reviews, Taylor & Francis Journals, pages 892-927.
    10. Han, Chirok & Phillips, Peter C. B. & Sul, Donggyu, 2011. "Uniform Asymptotic Normality In Stationary And Unit Root Autoregression," Econometric Theory, Cambridge University Press, pages 1117-1151.
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    12. Jarociński, Marek & Marcet, Albert, 2010. "Autoregressions in small samples, priors about observables and initial conditions," Working Paper Series 1263, European Central Bank.
    13. Carlos A. Medel & Pablo M. Pincheira, 2016. "The out-of-sample performance of an exact median-unbiased estimator for the near-unity AR(1) model," Applied Economics Letters, Taylor & Francis Journals, pages 126-131.
    14. Falk, Barry & Roy, Anindya, 2005. "Forecasting using the trend model with autoregressive errors," International Journal of Forecasting, Elsevier, pages 291-302.
    15. Lawford, Steve & Stamatogiannis, Michalis P., 2009. "The finite-sample effects of VAR dimensions on OLS bias, OLS variance, and minimum MSE estimators," Journal of Econometrics, Elsevier, pages 124-130.
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    17. Gonçalves Mazzeu, Joao Henrique & Ruiz, Esther & Veiga, Helena, 2015. "Model uncertainty and the forecast accuracy of ARMA models: A survey," DES - Working Papers. Statistics and Econometrics. WS ws1508, Universidad Carlos III de Madrid. Departamento de Estadística.
    18. Kim, Jae H. & Silvapulle, Param & Hyndman, Rob J., 2007. "Half-life estimation based on the bias-corrected bootstrap: A highest density region approach," Computational Statistics & Data Analysis, Elsevier, pages 3418-3432.
    19. Ghoshray, Atanu & Ordóñez, Javier & Sala, Hector, 2016. "Euro, crisis and unemployment: Youth patterns, youth policies?," Economic Modelling, Elsevier, pages 442-453.
    20. Sun, Jingwei & Shi, Wendong, 2015. "Breaks, trends, and unit roots in spot prices for crude oil and petroleum products," Energy Economics, Elsevier, pages 169-177.
    21. Hendrik Kaufmannz & Robinson Kruse, 2013. "Bias-corrected estimation in potentially mildly explosive autoregressive models," CREATES Research Papers 2013-10, Department of Economics and Business Economics, Aarhus University.

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