Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T18:28:46.339Z Has data issue: false hasContentIssue false
  • English
  • Français

EDGEWORTH AND SADDLEPOINT EXPANSIONS FOR NONLINEAR ESTIMATORS

Published online by Cambridge University Press:  25 February 2013

Gubhinder Kundhi  and
Paul Rilstone
Gubhinder Kundhi*
Affiliation:
Carleton University
Paul Rilstone*
Affiliation:
York University
*
*Address correspondence to Paul Rilstone, Department of Economics, York University, 4700 Keele St., Toronto, Ontario M3J IP3, Canada; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Simple methods are developed for deriving Edgeworth, saddlepoint, and related expansions for the estimators of multivariate and nonlinear models. Illustrations are provided. Simulations are reported indicating the methods work well compared to standard asymptotic and bootstrapped approaches.

Type
MISCELLANEA
Information
Econometric Theory , Volume 29 , Issue 5 , October 2013 , pp. 1057 - 1078
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research funding for Rilstone was provided by the Social Sciences and Humanities Research Council of Canada. The authors gratefully acknowledge the remarks of Lonnie Magee, Barry Smith, Augustine Wong, Peter C.B. Phillips, two anonymous referees, and seminar participants at La Trobe University, the 2005 Canadian Economic Association meetings in Hamilton, the 2009 Australasian Econometric Society meetings in Canberra, the 2009 Canadian Econometric Study Group meetings in Ottawa, and the 2009 International Conference on Computing and Finance in Sydney. Any errors are those of the authors.

References

REFERENCES

Armstrong, M. & Hillier, G. (1999) The density of the maximum likelihood estimator. Econometrica 67(6), 1459–1470.Google Scholar
Bhattacharya, R.N. & Ghosh, J.K. (1978) On the validity of the formal Edgeworth expansion. Annals of Statistics 6(2), 434–451.Google Scholar
Butler, R.W. (2007) Saddlepoint Approximations with Applications. Cambridge University Press.CrossRefGoogle Scholar
Daniels, H.E. (1954) Saddlepoint approximations in statistics. Annals of Mathematical Statistics 25, 631–650.CrossRefGoogle Scholar
Davidson, R. & MacKinnon, J.G. (2000) Bootstrap tests: How many bootstraps? Econometric Reviews 19, 55–68.CrossRefGoogle Scholar
Easton, G.S. & Ronchetti, E. (1986) General saddlepoint approximations with applications to L statistics. Journal of the American Statistical Association 81, 420–430.Google Scholar
Feuerverger, A. (1989) On the empirical saddlepoint approximation. Biometrika 76(3), 457–464.CrossRefGoogle Scholar
Hall, P. (1992) The Bootstrap and the Edgeworth Expansion. Springer-Verlag.CrossRefGoogle Scholar
Hansen, B.E. (2000) Edgeworth Expansions for the Wald and GMM Statistics for Nonlinear Restrictions. Manuscript, University of Wisconsin.Google Scholar
Hengartner, N.W. & Wegkamp, M.H. (2004) Second order asymptotics for M-estimators under non-standard conditions. In Rojo, J. & Prez-Abreu, V. (eds.), IMS Lecture Notes, pp. 107–124.Google Scholar
Holly, A. & Phillips, P.C.B. (1979) A saddlepoint approximation to the distribution of the k-class estimator of a coefficient in a simultaneous system. Econometrica 47(6), 1527–1548.CrossRefGoogle Scholar
Kundhi, G. & Rilstone, P. (2008) The third order bias of nonlinear estimators. Communications in Statistics 37(16), 2617–2633.CrossRefGoogle Scholar
Lahiri, S.N. (1992) Two term Edgeworth expansions for M-estimators of a regression coefficient. Journal of Multivariate Analysis 43, 125–132.CrossRefGoogle Scholar
Lieberman, O. (1994) On the approximation of saddlepoint expansions in statistics. Econometric Theory 10, 900–916.Google Scholar
Lugannani, R. & Rice, S.O. (1980) Saddlepoint approximations for the distributions of the sum of random variables. Advances in Applied Probability 12, 475–490.CrossRefGoogle Scholar
MacRae, E.C. (1974) Matrix derivatives with an application to an adaptive linear decision problem. Annals of Statistics 2, 337–346.CrossRefGoogle Scholar
McCullagh, P. (1987) Tensor Methods in Statistics. Chapman & Hall.Google Scholar
Monti, A.C. & Ronchetti, E. (1993) On the relationship between empirical likelihood and empirical approximations for multivariate M-estimators. Biometrika 80, 329–338.CrossRefGoogle Scholar
Newey, W.K. & Smith, R. (2004) Higher order properties of GMM and generalized empirical likelihood estimators. Econometrica 72(1), 219–255.CrossRefGoogle Scholar
Ohman–Strickland, P. & Casella, G. (2002) Approximate and estimated saddlepoint approximations. Canadian Journal of Statistics 30(1), 97–108.CrossRefGoogle Scholar
Petrov, V.V. (1975) Sums of Independent Random Variables. Springer Verlag.CrossRefGoogle Scholar
Phillips, P.C.B. (1978) Edgeworth and saddlepoint approximations in a first order non-circular autoregression. Biometrika 65(1), 91–98.CrossRefGoogle Scholar
Phillips, P.C.B. & Park, J.Y. (1988) On the formulation of Wald tests of nonlinear restrictions. Econometrica 56(5), 1065–1083.CrossRefGoogle Scholar
Rilstone, P., Srivastava, V.K., & Ullah, A. (1996) The second–order bias and mean squared error of nonlinear estimators. Journal of Econometrics 75(2), 369–395.Google Scholar
Rilstone, P. & Ullah, A. (2005) Corrigendum to: The second–order bias and mean squared error of non-linear estimators. Journal of Econometrics 124(1), 203–204.CrossRefGoogle Scholar
Ronchetti, E. & Welsh, A.H. (1994) Empirical saddlepoint approximations for multivariate M-estimators. Journal of the Royal Statistical Society, Series B 56, 313–326.Google Scholar
Sowell, F. (2007) The Empirical Saddlepoint Approximation for GMM Estimators. MRPA paper no. 3356.Google Scholar
Spady, R.H. (1991) Saddlepoint approximations for regression models. Biometrika 78(4), 879–889.CrossRefGoogle Scholar
Taniguchi, M. & Puri, M.L. (1996) Valid Edgeworth expansions of M-estimators in regression models with weakly dependent residuals. Econometric Theory 12, 331–346.CrossRefGoogle Scholar
Ullah, A. (2004) Finite Sample Econometrics. Oxford University Press.CrossRefGoogle Scholar