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Robust Nonstationary Regression

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  • Phillips, Peter C.B.

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 that 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, Review of Economic Studies 57,99–125). 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 this 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 heavytailed (infinite variance) errors. Some simulations and a brief empirical illustration are reported.

Suggested Citation

  • Phillips, Peter C.B., 1995. "Robust Nonstationary Regression," Econometric Theory, Cambridge University Press, vol. 11(5), pages 912-951, October.
    • Handle: RePEc:cup:etheor:v:11:y:1995:i:05:p:912-951_00
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    References listed on IDEAS

    as
    1. Phillips, P C B, 1991. "Optimal Inference in Cointegrated Systems," Econometrica, Econometric Society, vol. 59(2), pages 283-306, March.
    2. Peter C. B. Phillips & Bruce E. Hansen, 1990. "Statistical Inference in Instrumental Variables Regression with I(1) Processes," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 57(1), pages 99-125.
    3. Park, Joon Y. & Phillips, Peter C.B., 1989. "Statistical Inference in Regressions with Integrated Processes: Part 2," Econometric Theory, Cambridge University Press, vol. 5(1), pages 95-131, April.
    4. Phillips, Peter C B, 1995. "Fully Modified Least Squares and Vector Autoregression," Econometrica, Econometric Society, vol. 63(5), pages 1023-1078, September.
    5. Resnick, Sidney & Greenwood, Priscilla, 1979. "A bivariate stable characterization and domains of attraction," Journal of Multivariate Analysis, Elsevier, vol. 9(2), pages 206-221, June.
    6. repec:cup:etheor:v:8:y:1992:i:4:p:489-500 is not listed on IDEAS
    7. Paulauskas, V. J., 1976. "Some remarks on multivariate stable distributions," Journal of Multivariate Analysis, Elsevier, vol. 6(3), pages 356-368, September.
    8. Park, Joon Y. & Phillips, Peter C.B., 1988. "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, Cambridge University Press, vol. 4(3), pages 468-497, December.
    9. Phillips, P.C.B., 1988. "Weak Convergence of Sample Covariance Matrices to Stochastic Integrals Via Martingale Approximations," Econometric Theory, Cambridge University Press, vol. 4(3), pages 528-533, December.
    10. Knight, Keith, 1991. "Limit Theory for M-Estimates in an Integrated Infinite Variance," Econometric Theory, Cambridge University Press, vol. 7(2), pages 200-212, June.
    11. Hansen, Bruce E., 1992. "Convergence to Stochastic Integrals for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 8(4), pages 489-500, December.
    12. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
    13. Phillips, P.C.B., 1991. "A Shortcut to LAD Estimator Asymptotics," Econometric Theory, Cambridge University Press, vol. 7(4), pages 450-463, December.
    14. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
    15. repec:cup:etheor:v:7:y:1991:i:4:p:450-63 is not listed on IDEAS
    16. 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.
    17. 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.
    18. Pollard, David, 1991. "Asymptotics for Least Absolute Deviation Regression Estimators," Econometric Theory, Cambridge University Press, vol. 7(2), pages 186-199, June.
    Full references (including those not matched with items on IDEAS)

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