Chambers, Marcus J. and Kyriacou, Maria (2012): Jackknife bias reduction in autoregressive models with a unit root.
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This paper is concerned with the application of jackknife methods as a means of bias reduction in the estimation of autoregressive models with a unit root. It is shown that the usual jackknife estimator based on non-overlapping sub-samples does not remove fully the first-order bias as intended, but that an ‘optimal’ jackknife estimator can be de- fined that is capable of removing this bias. The results are based on a demonstration that the sub-sample estimators converge to different limiting distributions, and the joint moment generating function of the numerator and denominator of these distributions (which are func- tionals of a Wiener process over a sub-interval of [0,1]) is derived and utilised to extract the optimal weights. Simulations demonstrate the ability of the jackknife estimator to produce substantial bias reductions in the parameter of interest. It is also shown that incorporating an intercept in the regressions allows the standard jackknife estimator to be used and it is able also to produce substantial bias reduction despite the fact that the distributions of the full-sample and sub-sample estimators have greater bias in this case. Of interest, too, is the fact that the jackknife estimators can also reduce the overall root mean squared error compared to the ordinary least squares estimator, this requiring a larger (though still small) number of sub-samples compared to the value that produces maximum bias reduction (which is typically equal to two).
|Item Type:||MPRA Paper|
|Original Title:||Jackknife bias reduction in autoregressive models with a unit root|
|Keywords:||Jackknife; bias reduction; unit root; moment generating function|
|Subjects:||C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General
C - Mathematical and Quantitative Methods > C2 - Single Equation Models; Single Variables > C22 - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models
C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics
|Depositing User:||Maria Kyriacou|
|Date Deposited:||30. Apr 2012 15:17|
|Last Modified:||13. Feb 2013 11:39|
Abadir, K.M. (1993), “The Limiting Distribution of the Autocorrelation Coefficient Under a Unit Root,” Annals of Statistics, 21, 1058–1070.
Andrews, D.W.K. (1993), “Exactly Median Unbiased Estimation of First Order Autoregres- sive/Unit Root Models,” Econometrica, 61, 139–165.
Gonzalo, J., and Pitarakis, J.-Y. (1998), “On the Exact Moments of Nonstandard Asymptotic Distributions in an Unstable AR(1) with Dependent Errors,” International Economic Review, 39, 71–88.
Kendall, M.G. (1954), “Note on Bias in the Estimation of Autocorrelation,” Biometrika, 41, 403–404.
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, 1–14.
Perron, P. (1991), “A Continuous Time Approximation to the Unstable First-Order Autore- gressive Process: The Case without an Intercept,” Econometrica, 59, 211-236.
Phillips, P. C. B. (1987), “Time Series Regression with a Unit Root,” Econometrica, 55, 277–301.
Phillips, P. C. B., and Yu, J. (2005), “Jackknifing Bond Option Prices,” Review of Financial Studies, 18, 707–742.
Quenouille, M.H. (1956), “Notes on Bias in Estimation,” Biometrika, 43, 353–360.
Shaman, P., and Stine, R. (1988), “The Bias of Autoregressive Coefficient Estimators,” Journal of the American Statistical Association, 83, 842–848. 26
Shenton, L.R., and Johnson, W.I. (1965), “Moments of a Serial Correlation Coefficient,” Journal of the Royal Statistical Society Series B, 27, 308–320.
Shin, D.W., and So, B.S. (2001), “Recursive Mean Adjustment for Unit Root Tests,” Journal of Time Series Analysis, 22, 595–612.
Stine, R. (1987), “Estimating Properties of Autoregressive Forecasts,” Journal of the Amer- ican Statistical Association, 82, 1072–1078.
Tanaka, K. (1996), Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, New York: Wiley.
Tukey, J.W. (1958), “Bias and Confidence in not-quite Large Samples,” Annals of Mathe- matical Statistics, 29, 614.
White, J.S. (1958), “The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case,” Annals of Mathematical Statistics, 29, 1188–1197.