Edgeworth Approximation of a Finite Sample Distribution for an AR(1) Model with Measurement Error
Author(s)
Shuichi Nagata*
ABSTRACT
In this paper, we consider the finite sample property of the ordinary least squares (OLS) estimator for an AR(1) model with measurement error. We present the Edgeworth approximation for a finite distribution of OLS up to O(T1/2). We introduce an instrumental variable estimator that is consistent in the presence of measurement error. Finally, a simulation study is conducted to assess the theoretical results and to compare the finite sample performances of these estimators.
In this paper, we consider the finite sample property of the ordinary least squares (OLS) estimator for an AR(1) model with measurement error. We present the Edgeworth approximation for a finite distribution of OLS up to O(T1/2). We introduce an instrumental variable estimator that is consistent in the presence of measurement error. Finally, a simulation study is conducted to assess the theoretical results and to compare the finite sample performances of these estimators.
Cite this paper
S. Nagata, "Edgeworth Approximation of a Finite Sample Distribution for an AR(1) Model with Measurement Error," Open Journal of Statistics, Vol. 2 No. 4, 2012, pp. 383-388. doi: 10.4236/ojs.2012.24046.
S. Nagata, "Edgeworth Approximation of a Finite Sample Distribution for an AR(1) Model with Measurement Error," Open Journal of Statistics, Vol. 2 No. 4, 2012, pp. 383-388. doi: 10.4236/ojs.2012.24046.
References
[1] F. H. C. Marriott and J. A. Pope, “Bias in the Estimation of Autocorrelations,” Biometrika, Vol. 41, No. 3-4, 1954, pp. 390-402. doi:10.2307/2332719 [2] M. G. Kendall, “Note on the Bias in the Estimation of Autocorrelation,” Biometrika, Vol. 41, No. 3-4, 1954, pp. 403-404. doi:10.2307/2332720 [3] W. A. Fuller, “Measurement Error Models,” John Wiley, New York, 1987. doi:10.1002/9780470316665 [4] J. Staudenmayer and P. Buonaccorsi, “Measurement Error in Linear Autoregressive Model,” Journal of the American Statistical Association,” Vol. 100, No. 471. 2005, pp. 841-851. doi:10.1198/016214504000001871 [5] K. Tanaka, “A Unified Approach to the Measurement Error Problem in Time Series Models,” Econometric Theory, Vol. 18, No. 2, 2002, pp. 278-296. doi: 10.1017/S026646660218203X [6] P. C. B. Phillips, “Approximations to Some Sample Distributions Associated with a First Order Stochastic Difference Equation,” Econometrica, Vol. 45, No. 2, 1977, pp. 463-485. doi.org/10.2307/1911222 [7] K. Tanaka, “Asymptotic Expansions Associated with the AR(1) Model with Unknown Mean,” Econometrica, Vol. 51, No. 4, 1983, pp. 1221-1231. doi:10.2307/1912060 [8] K. Maekawa “An Approximation to the Distribution of the Least Squares Estimator in an Autoregressive Model with Exogenous Variables,” Econometrica, Vol. 51, No.1, 1983, pp. 229-238. doi:10.2307/1912258 [9] J. D. Sargan, “Econometric Estimators and the Edgeworth Expansions,” Econometrica, Vol. 44, No. 3, 1976, pp. 421-448. doi:10.2307/1913972 [10] C. W. J. Granger and M. J. Morris, “Time Series Modeling and Interpretation,” Journal of the Royal Statistical Society A, Vol. 139, No. 2, 1976, pp. 246-257. doi:10.2307/2345178 [11] K. Maekawa “Edgeworth Expansion for the OLS Estimator in a Time Series Regression Model,” Econometric Theory, Vol. 1, No. 2, 1985, pp. 223-239. doi:10.1017/S0266466600011154 [12] K. Hayakawa “A Note on Bias in First-Differenced AR(1) Models,” Economics Bulletin, Vol. 3 No. 27, 2006, pp. 1- 10. [13] P. C. B. Phillips and C. Han, “Gaussian Inference in AR(1) Times Series with or without Unit Root,” Econometric Theory, Vol. 24, No. 3, 2008, pp. 631-650. doi:10.1017/S0266466608080262
[1] F. H. C. Marriott and J. A. Pope, “Bias in the Estimation of Autocorrelations,” Biometrika, Vol. 41, No. 3-4, 1954, pp. 390-402. doi:10.2307/2332719 [2] M. G. Kendall, “Note on the Bias in the Estimation of Autocorrelation,” Biometrika, Vol. 41, No. 3-4, 1954, pp. 403-404. doi:10.2307/2332720 [3] W. A. Fuller, “Measurement Error Models,” John Wiley, New York, 1987. doi:10.1002/9780470316665 [4] J. Staudenmayer and P. Buonaccorsi, “Measurement Error in Linear Autoregressive Model,” Journal of the American Statistical Association,” Vol. 100, No. 471. 2005, pp. 841-851. doi:10.1198/016214504000001871 [5] K. Tanaka, “A Unified Approach to the Measurement Error Problem in Time Series Models,” Econometric Theory, Vol. 18, No. 2, 2002, pp. 278-296. doi: 10.1017/S026646660218203X [6] P. C. B. Phillips, “Approximations to Some Sample Distributions Associated with a First Order Stochastic Difference Equation,” Econometrica, Vol. 45, No. 2, 1977, pp. 463-485. doi.org/10.2307/1911222 [7] K. Tanaka, “Asymptotic Expansions Associated with the AR(1) Model with Unknown Mean,” Econometrica, Vol. 51, No. 4, 1983, pp. 1221-1231. doi:10.2307/1912060 [8] K. Maekawa “An Approximation to the Distribution of the Least Squares Estimator in an Autoregressive Model with Exogenous Variables,” Econometrica, Vol. 51, No.1, 1983, pp. 229-238. doi:10.2307/1912258 [9] J. D. Sargan, “Econometric Estimators and the Edgeworth Expansions,” Econometrica, Vol. 44, No. 3, 1976, pp. 421-448. doi:10.2307/1913972 [10] C. W. J. Granger and M. J. Morris, “Time Series Modeling and Interpretation,” Journal of the Royal Statistical Society A, Vol. 139, No. 2, 1976, pp. 246-257. doi:10.2307/2345178 [11] K. Maekawa “Edgeworth Expansion for the OLS Estimator in a Time Series Regression Model,” Econometric Theory, Vol. 1, No. 2, 1985, pp. 223-239. doi:10.1017/S0266466600011154 [12] K. Hayakawa “A Note on Bias in First-Differenced AR(1) Models,” Economics Bulletin, Vol. 3 No. 27, 2006, pp. 1- 10. [13] P. C. B. Phillips and C. Han, “Gaussian Inference in AR(1) Times Series with or without Unit Root,” Econometric Theory, Vol. 24, No. 3, 2008, pp. 631-650. doi:10.1017/S0266466608080262