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Robust Nonstationary Regression

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Abstract

This paper provides a robust statistical approach to nonstationary time series regression and inference. Fully modified extensions of traditional robust statistical procedures are developed which allow for endogeneities in the nonstationary regressors and serial dependence in the shocks that drive the regressors and the errors that appear in the equation being estimated. The suggested estimators involve semiparametric corrections to accommodate these possibilities and they belong to the same family as the fully modified least squares (FM-OLS) estimator of Phillips and Hansen (1990). Specific attention is given to fully modified least absolute deviation (FM-LAD) estimation and fully modified M (FM-M)-estimation. The criterion function for LAD and some M-estimators is not always smooth and the paper develops generalized function methods to cope with this difficulty in the asymptotics. The results given here include a strong law of large numbers and some weak convergence theory for partial sums of generalized functions of random variables. The limit distribution theory for FM-LAD and FM-M estimators that is developed includes the case of finite variance errors and the case of heavy-tailed (infinite variance) errors. Some simulations and a brief empirical illustration are reported.

Suggested Citation

  • Peter C.B. Phillips, 1993. "Robust Nonstationary Regression," Cowles Foundation Discussion Papers 1064, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1064
    Note: CFP 912.
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    References listed on IDEAS

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    1. Phillips, P C B, 1991. "Optimal Inference in Cointegrated Systems," Econometrica, Econometric Society, vol. 59(2), pages 283-306, March.
    2. Park, Joon Y. & Phillips, Peter C.B., 1989. "Statistical Inference in Regressions with Integrated Processes: Part 2," Econometric Theory, Cambridge University Press, vol. 5(01), pages 95-131, April.
    3. Phillips, Peter C B, 1995. "Fully Modified Least Squares and Vector Autoregression," Econometrica, Econometric Society, vol. 63(5), pages 1023-1078, September.
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    6. Paulauskas, V. J., 1976. "Some remarks on multivariate stable distributions," Journal of Multivariate Analysis, Elsevier, vol. 6(3), pages 356-368, September.
    7. Park, Joon Y. & Phillips, Peter C.B., 1988. "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, Cambridge University Press, vol. 4(03), pages 468-497, December.
    8. Phillips, P.C.B., 1988. "Weak Convergence of Sample Covariance Matrices to Stochastic Integrals Via Martingale Approximations," Econometric Theory, Cambridge University Press, vol. 4(03), pages 528-533, December.
    9. Knight, Keith, 1991. "Limit Theory for M-Estimates in an Integrated Infinite Variance," Econometric Theory, Cambridge University Press, vol. 7(02), pages 200-212, June.
    10. Hansen, Bruce E., 1992. "Convergence to Stochastic Integrals for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 8(04), pages 489-500, December.
    11. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
    12. Phillips, P.C.B., 1991. "A Shortcut to LAD Estimator Asymptotics," Econometric Theory, Cambridge University Press, vol. 7(04), pages 450-463, December.
    13. repec:cup:etheor:v:7:y:1991:i:4:p:450-63 is not listed on IDEAS
    14. Hunter, John, 1992. "Tests of cointegrating exogeneity for PPP and uncovered interest rate parity in the United Kingdom," Journal of Policy Modeling, Elsevier, vol. 14(4), pages 453-463, August.
    15. Peter C.B. Phillips & Bruce E. Hansen, 1988. "Statistical Inference in Instrumental Variables," Cowles Foundation Discussion Papers 869R, Cowles Foundation for Research in Economics, Yale University, revised Apr 1989.
    16. Pollard, David, 1991. "Asymptotics for Least Absolute Deviation Regression Estimators," Econometric Theory, Cambridge University Press, vol. 7(02), pages 186-199, June.
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    Cited by:

    1. Serttas, Fatma Ozgu, 2010. "Essays on infinite-variance stable errors and robust estimation procedures," ISU General Staff Papers 201001010800002742, Iowa State University, Department of Economics.
    2. Lima Luiz Renato & Xiao Zhijie, 2010. "Testing Unit Root Based on Partially Adaptive Estimation," Journal of Time Series Econometrics, De Gruyter, vol. 2(1), pages 1-34, June.
    3. Phillips, Peter C. B. & McFarland, James W., 1997. "Forward exchange market unbiasedness: the case of the Australian dollar since 1984," Journal of International Money and Finance, Elsevier, vol. 16(6), pages 885-907, December.
    4. Michael T. K. Horvath & Mark W. Watson, 1994. "Testing for Cointegration When Some of the Contributing Vectors are Known," NBER Technical Working Papers 0171, National Bureau of Economic Research, Inc.
    5. Phillips, Peter C B & McFarland, James W & McMahon, Patrick C, 1996. "Robust Tests of Forward Exchange Market Efficiency with Empirical Evidence from the 1920s," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(1), pages 1-22, Jan.-Feb..
    6. Hodgson, Douglas J., 1998. "Adaptive estimation of cointegrating regressions with ARMA errors," Journal of Econometrics, Elsevier, vol. 85(2), pages 231-267, August.
    7. Xiao, Zhijie, 2004. "Estimating average economic growth in time series data with persistency," Journal of Macroeconomics, Elsevier, vol. 26(4), pages 699-724, December.
    8. Douglas J. Hodgson & Oliver Linton & Keith Vorkink, 2004. "Testing forward exchange rate unbiasedness efficiently: a semiparametric approach," Journal of Applied Economics, Universidad del CEMA, vol. 7, pages 325-353, November.
    9. Xiao, Zhijie, 2012. "Robust inference in nonstationary time series models," Journal of Econometrics, Elsevier, vol. 169(2), pages 211-223.
    10. Peter C.B. Phillips, 1994. "Nonstationary Time Series and Cointegration: Recent Books and Themes for the Future," Cowles Foundation Discussion Papers 1081, Cowles Foundation for Research in Economics, Yale University.
    11. John W. Galbraith & Victoria Zinde-Walsh, 2000. "Properties of Estimates of Daily GARCH Parameters Based on Intra-Day Observations," Econometric Society World Congress 2000 Contributed Papers 1800, Econometric Society.
    12. Ted Juhl & Zhijie Xiao, 2000. "N-Consistent Semiparametric Regression: Partially Linear Models with Unit Roots," Econometric Society World Congress 2000 Contributed Papers 1532, Econometric Society.
    13. Zinde-Walsh, Victoria, 2008. "Kernel Estimation When Density May Not Exist," Econometric Theory, Cambridge University Press, vol. 24(03), pages 696-725, June.
    14. Rothenberg, Thomas J. & Stock, James H., 1997. "Inference in a nearly integrated autoregressive model with nonnormal innovations," Journal of Econometrics, Elsevier, vol. 80(2), pages 269-286, October.
    15. Theologos Pantelidis & Nikitas Pittis, 2009. "Estimation and forecasting in first-order vector autoregressions with near to unit roots and conditional heteroscedasticity," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 28(7), pages 612-630.
    16. Caner, Mehmet, 1998. "Tests for cointegration with infinite variance errors," Journal of Econometrics, Elsevier, vol. 86(1), pages 155-175, June.
    17. Delgado, Miguel A. & Velasco, Carlos, 2005. "Sign tests for long-memory time series," Journal of Econometrics, Elsevier, vol. 128(2), pages 215-251, October.
    18. Shin, Dong Wan & Oh, Man-Suk, 2004. "Fully modified semiparametric GLS estimation for regressions with nonstationary seasonal regressors," Journal of Econometrics, Elsevier, vol. 122(2), pages 247-280, October.
    19. Alessio Moneta & Doris Entner & Patrik O. Hoyer & Alex Coad, 2013. "Causal Inference by Independent Component Analysis: Theory and Applications," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 75(5), pages 705-730, October.
    20. Bruno Bosco & Lucia Parisio & Matteo Pelagatti & Fabio Baldi, 2010. "Long-run relations in european electricity prices," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 25(5), pages 805-832.
    21. Serguei Zernov & Victoria Zindle-Walsh & John Galbraith, 2006. "Asymptotics For Estimation Of Truncated Infinite-Dimensional Quantile Regressions," Departmental Working Papers 2006-16, McGill University, Department of Economics.
    22. ZINDE-WALSH, Victoria, 2007. "Errors-in-Variables Models : A Generalized Functions Approach," Cahiers de recherche 14-2007, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    23. Zernov, Serguei & Zinde-Walsh, Victoria & Galbraith, John W., 2009. "Asymptotics for estimation of quantile regressions with truncated infinite-dimensional processes," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 497-508, March.
    24. Chen, Yi-Ting, 2012. "A simple approach to standardized-residuals-based higher-moment tests," Journal of Empirical Finance, Elsevier, vol. 19(4), pages 427-453.

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