Superior Estimation and Inference Avoiding Heteroscedasticity and Flawed Pivots: R-example of Inflation Unemployment Trade-off

  • H. D. Vinod
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 196)


We use a new solution to heteroscedastic regression problem while avoiding so-called incidental parameters (inconsistency) problem by using recently discovered maps from time domain to numerical values domain and back. This involves a parsimonious fit for sorted logs of squared fitted residuals. Dufour [9] showed that inference based on Fisher’s pivot (dividing by standard errors) can be fundamentally flawed for deep parameters of genuine interest to policy makers. Hence, we use Godambe’s [12] pivot, which is always a sum of T items and asympotically subject to the central limit thory. We provide R functions to implement the ideas using the Phillips curve trade-off between inflation and unemployment for illustration. An Appendix discusses numerical methods to correct for general ARMA errors with an illustration of ARMA(4,3).


KEY WORDS: feasible generalized least squares specification robust smoothness simulation consistency 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alogoskoufis, G., Smith, R.: The phillips curve, the persistence of inflation, and the lucas critique.: Evidence from exchange rate regimes. American Economic Review 81(2), 1254–1275 (1991)Google Scholar
  2. 2.
    Carroll, R.J.: Adapting for heteroscedasticity in linear models. Annals of Statistics 10, 1224–1233 (1982)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Carroll, R.J., Ruppert, D.: A comparison between maximum likelihood and generalized least squares in a heteroscedastic linear model. Journal of the American Statistical Association 77, 878–882 (1982)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cook, D., Weisberg, S.: Diagnostics for heteroscedasticity in regression. Biometrika 70, 1–10 (1983)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cragg, J.G.: More efficient estimation in the presence of heteroscedasticity of unknown form. Econometrica 51(3), 751–763 (1983)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cribari-Neto, F.: Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics & Data Analysis 45, 215–233 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Davidson, R., MacKinnon, J.G.: Econometric Theory and Methods. Oxford Univ. Press, New York (2004)Google Scholar
  8. 8.
    Dixon, S.L., McKean, J.W.: Rank-based analysis of the heteroscedastic linear model. Journal of the American Statistical Association 91, 699–712 (1996)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dufour, J.M.: Some impossibility theorems in econometrics with applications to structural and dynamic models. Econometrica 65, 1365–1387 (1997)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Eicker, F.: Asymptotic normality and consistency of the least squares estimator for families of linear regressions. Annals of Mathematical Statistics 34, 447–456 (1963)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fox, J.: car.: Companion to Applied Regression. R package version 1.2-14 (2009). URL I am grateful to Douglas Bates, David Firth, Michael Friendly, Gregor Gorjanc, Spencer Graves, Richard Heiberger, Georges Monette, Henric Nilsson, Derek Ogle, Brian Ripley, Sanford Weisberg, and Achim Zeileis for various suggestions and contributions
  12. 12.
    Godambe, V.P.: The foundations of finite sample estimation in stochastic processes. Biometrika 72, 419–428 (1985)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Godfrey, L.G.: Tests for regression models with heteroskedasticity of unknown form. Computational Statistics & Data Analysis 50, 2715–2733 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Greene, W.H.: Econometric Analysis, 4 edn. Prentice Hall, New York (2000)Google Scholar
  15. 15.
    Kendall, M.G., Stuart, A.: The Advanced Theory of Statistics, vol. 2. Macmillan Publishing, New York (1979)MATHGoogle Scholar
  16. 16.
    Long, J.S., Ervin, L.H.: Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician 54, 217–224 (2000)CrossRefGoogle Scholar
  17. 17.
    McCullough, B.D., Vinod, H.D.: Verifying the solution from a nonlinear solver: A case study. American Economic Review 93(3), 873–892 (2003)CrossRefGoogle Scholar
  18. 18.
    Newey, W.K.: Efficient estimation of models with conditional moment restrictions. In: G.S. Maddala, C.R. Rao, H.D. Vinod (eds.) Handbook of Statistics: Econometrics, vol. 11, chap. 16. North–Holland, Elsevier Science Publishers, New York (1993)Google Scholar
  19. 19.
    Pregibon, D.: Data analysts captivated by r’s power. The New York Times January 6 (2009). URL\_r=1\&em
  20. 20.
    Priestley, M.B.: Spectral Analysis and Time Series, vol. I and II. Academic Press, London (1981)MATHGoogle Scholar
  21. 21.
    R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2008). URL Cited 14 May 2009
  22. 22.
    R Development Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2009). URL ISBN 3-900051-07-0
  23. 23.
    Rao, P.S.R.S.: Theory of the MINQUE– A Review. Sankhya 39, 201–210 (1977)MATHGoogle Scholar
  24. 24.
    Robinson, P.M.: Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometica 55(4), 875–891 (1987)MATHCrossRefGoogle Scholar
  25. 25.
    Vinod, H.D.: Exact maximum likelihood regression estimation with arma (n, n-l) errors. Economics Letters 17, 355–358 (1985)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Vinod, H.D.: Foundations of statistical inference based on numerical roots of robust pivot functions (fellow’s corner). Journal of Econometrics 86, 387–396 (1998)MATHCrossRefGoogle Scholar
  27. 27.
    Vinod, H.D.: Ranking mutual funds using unconventional utility theory and stochastic dominance. Journal of Empirical Finance 11(3), 353–377 (2004)CrossRefGoogle Scholar
  28. 28.
    Vinod, H.D.: Maximum entropy ensembles for time series inference in economics. Journal of Asian Economics 17(6), 955–978 (2006)CrossRefGoogle Scholar
  29. 29.
    Vinod, H.D.: Hands-on Intermediate Econometrics Using R: Templates for Extending Dozens of Practical Examples. World Scientific Publishers, Hackensack, NJ (2008). URL Cited 14 May 2009
  30. 30.
    Vinod, H. D. and Lopez-de-Lacalle, Javier.: Maximum entropy bootstrap for time series: The meboot R-package. Journal of Statistical Software 29(5), 1–30 (2009). URL Cited 14 May 2009
  31. 31.
    White, H.: A heteroskedasticity-consistent covariance matrix and a direct test for heteroskedasticity. Econometrica 48, 817–838 (1980)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York 2010

Authors and Affiliations

  1. 1.Fordham UniversityBronxUSA

Personalised recommendations