Abstract
In the case where the lagged dependent variables are included in the regression model, it is known that the ordinary least squares estimates (OLSE) are biased in small sample and that the bias increases as the number of the irrelevant variables increases. In this paper, based on the bootstrap methods, an attempt is made to obtain the unbiased estimates in autoregressive and non-Gaussian cases. We propose the residual-based bootstrap method in this paper. Some simulation studies are performed to examine whether the proposed estimation procedure works well or not. We obtain the results that it is possible to recover the true parameter values and that the proposed procedure gives us the less biased estimators than OLSE.
Similar content being viewed by others
References
Abadir, K.M. (1993) “OLS Bias in A Nonstationary Autoregression,” Econometric Theory, 9 (1), 81–93.
Andrews, D.W.K. (1993) “Exactly Median-Unbiased Estimation of First Order Autoregressive/Unit Root Models,” Econometrica, 61 (1), 139–165.
Andrews, D.W.K. and Chen, H.Y. (1994) “Approximately Median-Unbiased Estimation of Autoregressive Models,” Journal of Business and Economic Statistics, 12 (2), 187–204.
Diebold, F.X. and Rudebusch, G.D. (1991) “Unbiased Estimation of Autoregressive Models,” unpublished manuscript, University of Pennsylvania.
Efron, B. and Tibshirani, R.J. (1993) An Introduction to the Bootstrap.
Enders, W. and Falk, B. (1998) “Threshold-Autoregressive, Median-Unbiased, and Cointegration Tests of Purchasing Power Parity,” International Journal of Forecasting, 14 (2), 171–186.
Greene, W.H. (1993) Econometric Analysis (Second Edition), Prentice Hall.
Grubb, D. and Symons, J. (1987) “Bias in Regressions with a Lagged Dependent Variable”, Econometric Theory, 3, 371–386.
Hurwicz, L. (1950) “Least-Squares Bias in Time Series,” in Statistical Inference in Dynamic Economic Models, ed. T.C. Koopmans, New York, John Wiley, 365–383.
Imhof, J.P. (1961) “Computing the Distribution of Quadratic Forms in Normal Variates,” Biometrika, 48, 419–426.
Kendall, M.G. (1954) “Note on Bias in the Estimation of Autocorrelations,” Biometrika, 41, 403–404.
MacKinnon, J.G., and Smith, A.A., Jr. (1998) “Approximate Bias Correction in Econometrics,” Journal of Econometrics, 85 (2), 205–230.
Maekawa, K. (1983) “An Approximation to the Distribution of the Least Squares Estimator in an Autoregressive Model with Exogenous Variables,” Econometrica, 51 (1), 229–238.
Maekawa, K. (1987) “Finite Sample Properties of Several Predictors from an Autoregressive model,” Econometric Theory, 3, 359–370.
Marriott, F.H.C. and Pope, J.A. (1954) “Bias in the Estimation of Autocorrelations,” Biometrika, 41, 390–402.
Orcutt, G.H. and Winokur, H.S. (1969) “First Order Autoregression: Inference, Estimation, and Prediction,” Econometrica, 37 (1), 1–14.
Peters, T.A. (1989) “The Exact Moments of OLS in Dynamic Regression Models with Non-Normal Errors,” Journal of Econometrics, 40, 279–305.
Quenouille, M.H. (1956) “Notes on Bias in Estimation,” Biometrika, 43, 353–360.
Sawa, T. (1978) “The Exact Moments of the Least Squares Estimator for the Autoregressive Model,” Journal of Econometrics, 8, 159–172.
Shaman, P. and Stine, R.A. (1988) “The Bias of Autoregressive Coefficient Estimators,” Journal of the American Statistical Association, 83, 842–848.
Tanaka, K. (1993) “Asymptotic Expansions Associated with AR(1) Model with Unknown Mean,” Econometrica, 51 (4), 1221–1231.
Tanizaki, H. (1995) “Asymptotically Exact Confidence Intervals of CUSUM and CUSUMSQ Tests: A Numerical Derivation Using Simulation Technique,” Communications in Statistics, Simulation and Computation, 24 (4), 1019–1036.
Tanizaki, H. (2000) “Bias Correction of OLSE in the Regression Model with Lagged Dependent Variables,” Computational Statistics and Data Analysis, 34 (4), 495–511.
Tse, Y.K. (1982) “Edgeworth Approximations in First-Order Stochastic Difference Equations with Exogenous Variables,” Journal of Econometrics, 20, 175–195.
Tsui, A.K. and Ali, M.M. (1994) “Exact Distributions, Density Functions and Moments of the Least Squares Estimator in a First-Order Autoregressive Model,” Computational Statistics and Data Analysis, 17 (4), 433–454.
White, J.S. (1961) “Asymptotic Expansions for the Mean and Variance of the Serial Correlation Coefficient,” Biometrika, 48, 85–94.
Wu, C.F.J. (1986) “Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis,” Annals of Statistics, 14, 1261–1350 (with discussion).
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is a substantial revision of Tanizaki (2000). The normality assumption is adopted in Tanizaki (2000), but it is not required in this paper. The authors are grateful to an anonymous referee for valuable suggestions and comments. This research was partially supported by Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research (C)(2) #14530033, 2002–2005, for H. Tanizaki and Grants-in-Aid for the 21st Century COE Program.
Rights and permissions
About this article
Cite this article
Tanizaki, H., Hamori, S. & Matsubayashi, Y. On least-squares bias in the AR(p) models: Bias correction using the bootstrap methods. Statistical Papers 47, 109–124 (2006). https://doi.org/10.1007/s00362-005-0275-6
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s00362-005-0275-6