An alternative approach to approximating the moments of least squares estimators

Liu-Evans, Gareth (2010): An alternative approach to approximating the moments of least squares estimators.

Abstract

A new methodology is presented for approximating the moments of least squares coefficient estimators in situations where endogeneity and dynamics are present. The OLS estimator is the focus here, but the method, which is valid under a simple set of smoothness and moment conditions, can be applied to related estimators. An O(T−1) approximation is presented for the bias in OLS estimation of a general ARX(p) model.

• Item Type: MPRA Paper
• Language: English
• Keywords: moment approximation; bias; finite sample
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

Bao, Y. (2007). The approximate moments of the least squares estimator for the stationary autoregressive model under a general error distribution. Econometric Theory 23(05), 1013–1021.

Bao, Y. & Ullah, A. (2002). The second-order bias and mean-squared error of nonlinear estimators in time-series. Manuscript, University of California, Riverside.

Bao, Y. & Ullah, A. (2007). The second-order bias and mean-squared error of estimators in time-series models. Journal of Econometrics 140(2), 650–669.

Bhansali, R. (1981). Effects of not knowing the order of an autoregressive process on the mean squared error of prediction-i. Journal of the Americal Statistical Association 76, 588–597.

Kadane, J. (1971). Comparison of k-class estimators when the disturbances are small. Econometrica 39, 723–737.

Kendall, M. (1954). Note on bias in the estimation of autocorrelation. Biometrika 61, 403–404.

Kiviet, J. & Phillips, G. (1993). Alternative bias approximations in regressions with a lagged dependent variable. Econometric Theory 9, 62– 80.

Kiviet, J. & Phillips, G. (1994). Bias assessment and reduction in linear error-correction models. Journal of Econometrics 63, 215–243.

Kiviet, J. & Phillips, G. (1996). The bias of the ordinary least squares estimator in simultaneous equation models. Economics Letters 53, 161– 167.

Kiviet, J. & Phillips, G. (2005). Moment approximation for least-squares estimators in dynamic regression models with a unit root. Econometrics Journal 8, 1–28.

Kiviet, J. & Phillips, G. (2010). Higher-order asymptotic expansions of the least-squares estimation bias in first-order dynamic regression models. Forthcoming at Computational Statistics and Data Analysis, 6th Special Issue on Computational Econometrics.

Kunitomo, N. & Yamamoto, T. (1985). Properties of predictors in misspecified autoregressive time series models. Journal of the American Statistical Association 80, 941–950.

Maekawa, K. (1983). An approximation to the distribution of the least squares estimator in an autoregressive model with exogenous variables. Econometrica 51, 229–238.

Magnus, J. & Neudecker, H. (1979). The commutation matrix: some properties and applications. The Annals of Statistics 7(2), 318–394.

Magnus, J. & Neudecker, H. (1988). Matrix differential calculus with applications in statistics and econometrics (Revised 2002 edition). Wiley, 2nd ed.

Marriott, F. & Pope, J. (1954). Bias in the estimation of autocorrelations. Biometrika 61, 393–403.

Nagar, A. (1959). The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 575–595.

Phillips, G. (2000). An alternative approach to obtaining nagar-type moment approximations in simultaneous equation models. Journal of Econometrics 97(2), 345–364.

Phillips, G. (2007). Nagar-type moment approximations in simultaneous equation models: some further results. In The Refinement of Econometric Estimation and Test Procedures: Finite Sample and Asymptotic Analysis,

G. Phillips, ed. Cambridge University Press, pp. 60–99. Phillips, G. & Liu-Evans, G. (2010). The robustness of the 2SLS moment approximations to non-normal disturbances. An earlier version was presented at the Econometric Society European Meeting 2008.

Rilestone, P., Srivastava, S. & Ullah, A. (1996). The second-order bias and mean squared error of nonlinear estimators. Journal of Econometrics 75, 369–395.

Sargan, J. (1974). The validity of nagar’s expansion for the moments of econometric estimators. Econometrica 42, 169–176.

Shaman, P. & Stine, R. (1988). The bias of autoregressive coefficient estimators. Journal of The American Statistical Association 83, 842–848.

Shao, J. (1988). On resampling methods for variance and bias estimation in linear models. Annals of Statistics 16, 986–1008.

Shao, J. & Tu, D. (1995). The Jackknife and Bootstrap. 175th Avenue, New York: Springer-Verlag, 1st ed.

Tanaka, K. (1984). Asymptotic expansions associated with the AR(1) model with unknown mean. Econometrica 51, 1221–1231.

Tjostheim, D. & Paulsen, J. (1983). Bias of some commonly-used time series estimates. Biometrika 70, 389–399.

Ullah, A. (2005). Finite Sample Econometrics. Oxford University Press.

Yamamoto, T. & Kunitomo, N. (1984). Asymptotic bias of the least squares estimator for multivariate autoregressive models. Annals of the Institute of Statistical Mathematics 36, 419–430.