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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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    2. Marques, André M. & Lima, Gilberto Tadeu & Troster, Victor, 2017. "Unemployment persistence in OECD countries after the Great Recession," Economic Modelling, Elsevier, vol. 64(C), pages 105-116.
    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.
    5. Ghoshray, Atanu & Ordóñez, Javier & Sala, Hector, 2016. "Euro, crisis and unemployment: Youth patterns, youth policies?," Economic Modelling, Elsevier, vol. 58(C), pages 442-453.
    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, vol. 388(17), 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, vol. 27(06), pages 1117-1151, December.
    8. Josep Lluís Carrion-i-Silvestre & María Dolores Gadea & Antonio Montañés, 2017. "“Unbiased estimation of autoregressive models forbounded stochastic processes," AQR Working Papers 201710, University of Barcelona, Regional Quantitative Analysis Group, revised Dec 2017.
    9. Michael Wolf & Dan Wunderli, 2012. "Bootstrap joint prediction regions," ECON - Working Papers 064, Department of Economics - University of Zurich, revised May 2013.
    10. 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, vol. 148(2), pages 124-130, February.
    11. Mohitosh Kejriwal & Claude Lopez, 2013. "Unit Roots, Level Shifts, and Trend Breaks in Per Capita Output: A Robust Evaluation," Econometric Reviews, Taylor & Francis Journals, vol. 32(8), pages 892-927, November.
    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, vol. 23(2), pages 126-131, February.
    14. Falk, Barry & Roy, Anindya, 2005. "Forecasting using the trend model with autoregressive errors," International Journal of Forecasting, Elsevier, vol. 21(2), pages 291-302.
    15. J. Huston McCulloch, 2005. "The Kalman Foundations of Adaptive Least Squares: Applications to Unemployment and Inflation," Computing in Economics and Finance 2005 239, Society for Computational Economics.
    16. 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.
    17. 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, vol. 51(7), pages 3418-3432, April.
    18. Pierre Perron & Mototsugu Shintani & Tomoyoshi Yabu, 2017. "Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 79(5), pages 822-850, October.
    19. Sun, Jingwei & Shi, Wendong, 2015. "Breaks, trends, and unit roots in spot prices for crude oil and petroleum products," Energy Economics, Elsevier, vol. 50(C), pages 169-177.
    20. 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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