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The Finite-Sample Effects of VAR Dimensions on OLS Bias, OLS Variance, and Minimum MSE Estimators: Purely Nonstationary Case

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Steve Lawford ()
Michalis P. Stamatogiannis

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Abstract

Vector autoregressions (VARs) are an important tool in time series analysis. However, relatively little is known about the finite-sample behaviour of parameter estimators. We address this issue, by investigating ordinary least squares (OLS) estimators given a data generating process that is a purely nonstationary first-order VAR. Specifically, we use Monte Carlo simulation and numerical optimization to derive response surfaces for OLS bias and variance, in terms of VAR dimensions,given correct and (several types of) over-parameterization of the model:we include a constant, and a constant and trend, and introduce excess lags. We then examine the correction factors required for the least squares estimator to attain minimum mean squared error (MSE). Our results improve and extend one of the main ?nite-sample analytical bias results of Abadir, Hadri and Tzavalis (Econometrica 67 (1999) 163), generalize the univariate variance and MSE ?ndings of Abadir (Econ.Lett. 47 (1995) 263), and complement various asymptotic studies.

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Paper provided by Economics and Finance Section, School of Social Sciences, Brunel University in its series Public Policy Discussion Papers with number 04-05.

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Length: 20 pages
Date of creation: Mar 2004
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Handle: RePEc:bru:bruppp:04-05

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  1. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-72, June. [Downloadable!] (restricted)
  2. Campos, Julia, 1986. "Finite-sample properties of the instrumental-variables estimator for dynamic simultaneous-equation subsystems with ARMA disturbances," Journal of Econometrics, Elsevier, vol. 32(3), pages 333-366, August. [Downloadable!] (restricted)
  3. Abadir, K.M., 1995. "On Efficient Simulations in Dynamic Models," Discussion Papers 95/21, University of Exeter, School of Business and Economics.
  4. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March. [Downloadable!] (restricted)
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  5. MacKinnon, James G & Haug, Alfred A & Michelis, Leo, 1999. "Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 14(5), pages 563-77, Sept.-Oct. [Downloadable!]
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  6. MacKinnon, James G, 1994. "Approximate Asymptotic Distribution Functions for Unit-Root and Cointegration Tests," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(2), pages 167-76, April.
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  7. Peter C.B. Phillips & Joon Y. Park, 1986. "Statistical Inference in Regressions with Integrated Processes: Part 2," Cowles Foundation Discussion Papers 819R, Cowles Foundation, Yale University, revised Feb 1987. [Downloadable!]
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  8. Abadir, Karim M. & Larsson, Rolf, 2001. "The Joint Moment Generating Function Of Quadratic Forms In Multivariate Autoregressive Series," Econometric Theory, Cambridge University Press, vol. 17(01), pages 222-246, February. [Downloadable!]
  9. Karim M. Abadir & Kaddour Hadri & Elias Tzavalis, 1999. "The Influence of VAR Dimensions on Estimator Biases," Econometrica, Econometric Society, vol. 67(1), pages 163-182, January.
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  10. Peter C.B. Phillips, 1985. "Asymptotic Expansions in Nonstationary Vector Autoregressions," Cowles Foundation Discussion Papers 765, Cowles Foundation, Yale University. [Downloadable!]
  11. Cheung, Yin-Wong & Lai, Kon S, 1995. "Lag Order and Critical Values of the Augmented Dickey-Fuller Test," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(3), pages 277-80, July.
  12. Abadir, Karim M & Hadri, Kaddour, 2000. "Is More Information a Good Thing? Bias Nonmonotonicity in Stochastic Difference Equations," Bulletin of Economic Research, Blackwell Publishing, vol. 52(2), pages 91-100, April.
  13. Kiviet, J.F. & Phillips, G.D.A., 1998. "Moment Approximation for Least Squares Estimators in Dynamic Regression Models with a Unit Root," Discussion Papers 99/09, University of Exeter, School of Business and Economics.
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  14. MacKinnon, James G. & Smith Jr., Anthony A., 1998. "Approximate bias correction in econometrics," Journal of Econometrics, Elsevier, vol. 85(2), pages 205-230, August. [Downloadable!] (restricted)
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  15. Neil R. Ericsson & James G. MacKinnon, 2002. "Distributions of error correction tests for cointegration," Econometrics Journal, Royal Economic Society, vol. 5(2), pages 285-318, 06. [Downloadable!] (restricted)
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  16. repec:cup:etheor:v:12:y:1996:i:4:p:682-704 is not listed on IDEAS
  17. Robert F. Engle & David F. Hendry & David Trumble, 1985. "Small-Sample Properties of ARCH Estimators and Tests," Canadian Journal of Economics, Canadian Economics Association, vol. 18(1), pages 66-93, February. [Downloadable!] (restricted)
  18. Abadir, Karim M., 1995. "Unbiased estimation as a solution to testing for random walks," Economics Letters, Elsevier, vol. 47(3-4), pages 263-268, March. [Downloadable!] (restricted)
  19. Karim M. Abadir & Jan R. Magnus, 2002. "Notation in econometrics: a proposal for a standard," Econometrics Journal, Royal Economic Society, vol. 5(1), pages 76-90, June. [Downloadable!] (restricted)
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  20. Andrews, Donald W K, 1993. "Exactly Median-Unbiased Estimation of First Order Autoregressive/Unit Root Models," Econometrica, Econometric Society, vol. 61(1), pages 139-65, January. [Downloadable!] (restricted)
  21. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-38, May. [Downloadable!] (restricted)
  22. Tanizaki, Hisashi, 2000. "Bias correction of OLSE in the regression model with lagged dependent variables," Computational Statistics & Data Analysis, Elsevier, vol. 34(4), pages 495-511, October. [Downloadable!] (restricted)
  23. repec:cup:etheor:v:12:y:1996:i:3:p:481-99 is not listed on IDEAS
  24. MacKinnon, James G, 1996. "Numerical Distribution Functions for Unit Root and Cointegration Tests," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(6), pages 601-18, Nov.-Dec.. [Downloadable!] (restricted)
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  25. Evans, G B A & Savin, N E, 1981. "Testing for Unit Roots: 1," Econometrica, Econometric Society, vol. 49(3), pages 753-79, May. [Downloadable!] (restricted)
  26. Ericsson, Neil R, 1991. "Monte Carlo Methodology and the Finite Sample Properties of Instrumental Variables Statistics for Testing Nested and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 59(5), pages 1249-77, September. [Downloadable!] (restricted)
  27. 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 482-93, October.
  28. repec:cup:etheor:v:9:y:1993:i:1:p:81-93 is not listed on IDEAS
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