Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T18:44:53.142Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE ASYMPTOTIC EFFICIENCY OF GMM

Published online by Cambridge University Press:  23 October 2013

Marine Carrasco  and
Jean-Pierre Florens
Marine Carrasco*
Affiliation:
Université de Montréal CIREQ and CIRANO
Jean-Pierre Florens
Affiliation:
Toulouse School of Economics
*
*Address correspondence to Marine Carrasco, University of Montreal, Departement de Sciences Economiques, CP 6128, succ Centre Ville, Montreal, QC H3C3J7, Canada; e-mail:[email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

The efficiency of the generalized method of moment (GMM) estimator is addressed by using a characterization of its variance as an inner product in a reproducing kernel Hilbert space. We show that the GMM estimator is asymptotically as efficient as the maximum likelihood estimator if and only if the true score belongs to the closure of the linear space spanned by the moment conditions. This result generalizes former ones to autocorrelated moments and possibly infinite number of moment restrictions. Second, we derive the semiparametric efficiency bound when the observations are known to be Markov and satisfy a conditional moment restriction. We show that it coincides with the asymptotic variance of the optimal GMM estimator, thus extending results by Chamberlain (1987, Journal of Econometrics 34, 305–33) to a dynamic setting. Moreover, this bound is attainable using a continuum of moment conditions.

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 2 , April 2014 , pp. 372 - 406
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.)

References

REFERENCES

Bates, C. & White, H. (1993) Determination of estimators with minimum asymptotic covariance matrices. Econometric Theory 9, 633–648.Google Scholar
Berlinet, A. & Thomas-Agnan, C. (2004) Reproducing Kernel Hilbert Spaces in Probability & Statistics. Kluwer Academic Publishers.Google Scholar
Bickel, P., Klaassen, C., Ritov, Y., & Wellner, J. (1993) Efficient and Adaptive Estimation for Semiparametric Models. Springer-Verlag.Google Scholar
Bierens, H. (1990) A consistent conditional moment test of functional form. Econometrica 58, 1443–1458.CrossRefGoogle Scholar
Billingsley, P. (1995) Probability and Measure. Wiley.Google Scholar
Böttcher, A., Dijksma, A., Langer, H., Dritschel, M., Rovnyak, J., & Kaashoek, M. (1996) Lectures on Operator Theory and Its Applications. Fields Institute Monographs. American Mathematical Society.Google Scholar
Brezis, H. (2002) Analyse Fonctionnelle: Theorie et Applications. Dunod.Google Scholar
Calzolari, G., Fiorentini, G., & Sentana, E. (2004) Constrained indirect estimation. Review of Economic Studies 71, 945–973.Google Scholar
Carrasco, M. (2012) A regularization approach to the many instruments problem. Journal of Econometrics 170, 383–398.Google Scholar
Carrasco, M., Chernov, M., Florens, J.P., & Ghysels, E. (2007) Efficient estimation of general dynamic models with a continuum of moment conditions. Journal of Econometrics 140, 529–573.CrossRefGoogle Scholar
Carrasco, M. & Florens, J.P. (2002) Simulation based method of moments & efficiency. Journal of Business & Economic Statistics 20, 482–492.Google Scholar
Carrasco, M., Florens, J.P., & Renault, E. (2007) Linear inverse problems in structural econometrics: Estimation based on spectral decomposition and regularization. In Heckman, J.J. & Leamer, E.E. (eds.), Handbook of Econometrics, vol. 6B, pp. 5633–575. Elsevier.Google Scholar
Chamberlain, G. (1987) Asymptotic efficiency in estimation with conditional moment restrictions. Journal of Econometrics 34, 305–334.Google Scholar
Chen, X. (2007) Large sample sieve estimation of semi-nonparametric models. In Heckman, J.J. & Leamer, E.E. (eds.), Handbook of Econometrics, vol. 6B, pp. 5549–5632. Elsevier.Google Scholar
Chen, X. & White, H. (1998) Central limit and functional central limit theorems for Hilbert space-valued dependent processes. Econometric Theory 14, 260–284.CrossRefGoogle Scholar
Davidson, R. (2000) Efficiency and robustness in a geometrical perspective. In Marriott, P. &Salmon, M. (eds.), Applications of Differential Geometry to Econometrics, pp. 151–183. Cambridge University Press.Google Scholar
Debnath, L. & Mikusinski, P. (2005) Introduction to Hilbert Spaces with Applications. Academic Press.Google Scholar
Donald, S. & Newey, W. (2001) Choosing the number of instruments. Econometrica 69, 1161–1191.Google Scholar
Durbin, J. (1960) Estimation of parameters in time-series regression models. Journal of the Royal Statistical Society, Series B 22, 139–153.Google Scholar
Engl, H., Hanke, M., & Neubauer, A. (2000) Regularization of Inverse Problems. Kluwer Academic Publishers.Google Scholar
Feuerverger, A. (1990) An efficiency result for the empirical characteristic function in stationary time-series models. Canadian Journal of Statistics 18, 155–161.CrossRefGoogle Scholar
Fisher, R.A. (1922) On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London, Series A 222, 309–368.Google Scholar
Gallant, A.R. & Long, J.R. (1997) Estimating stochastic differential equations efficiently by minimum chi-squared. Biometrika 84, 125–141.Google Scholar
Gallant, A.R. & Nychka, D. (1987) Semi-nonparametric maximum likelihood estimation, Econometrica 55, 363–390.Google Scholar
Gallant, A.R. & Tauchen, G. (1996) Which moments to match? Econometric Theory 12, 657–681.CrossRefGoogle Scholar
Godambe, V.P. (1960) An optimum property of regular maximum likelihood estimation. Annals of Mathematical Statistics 31, 1208–1212.Google Scholar
Godambe, V.P. (1985) The foundations of finite sample estimation in stochastic processes. Biometrika 72, 419–428.CrossRefGoogle Scholar
Gourieroux, C., Monfort, A., & Renault, E. (1993) Indirect inference. Journal of Applied Econometrics 8, S85–S118.Google Scholar
Greenwood, P., Müller, U., & Wefelmeyer, W. (2004) An introduction to efficient estimation for semiparametric time series. In Mikulin, M.S., Balakrishnan, N., Mesbak, M., & Limnios, N. (eds.), Parametric & Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life, pp. 253–272. Birkhauser.Google Scholar
Hajek, J. (1970) A characterization of limiting distributions of regular estimates. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 14, 323–330.Google Scholar
Hall, A., Inoue, A., Jana, K., & Shin, C. (2007) Information in generalized method of moments estimation and entropy-based moment selection. Journal of Econometrics 138, 488–512.Google Scholar
Hansen, L. (1982) Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–1054.Google Scholar
Hansen, L. (1985) A method for calculating bounds on the asymptotic covariance matrices of generalized method of moments estimators. Journal of Econometrics 30, 203–238.Google Scholar
Hansen, L. (1993) Semiparametric efficiency bounds for linear time-series models. In Phillips, Peter C.B. (ed.), Models, Methods, and Applications of Econometrics: Essays in Honor of A.R. Bergstrom, pp. 253–271. Blackwell.Google Scholar
Hansen, L., Heaton, J. & Ogaki, M. (1988) Efficiency bounds implied by multiperiod conditional moment restrictions. Journal of the American Statistical Association 83, 863–871.Google Scholar
Hansen, L., Heaton, J., & Yaron, A. (1996) Finite-sample properties of some alternative GMM estimators. Journal of Business & Economic Statistics 14, 262–280.Google Scholar
Imbens, G.W. (1997) One-step estimators for over-identifed generalized method of moment models. Review of Economic Studies 64, 359–383.Google Scholar
Kannan, D. & Bharucha-Reid, A.T. (1970) Note on covariance operators of probability measures on a Hilbert space. Proceedings of the Japan Academy 46, 124–129.Google Scholar
Kitamura, Y. & Stutzer, M. (1997) An information-theoretic alternative to generalized method of moments estimation. Econometrica 65, 861–874.Google Scholar
Komunjer, I. & Vuong, Q. (2010) Semiparametric efficiency bound in time-series models forconditional quantiles. Econometric Theory 26, 383–405.Google Scholar
Kress, R. (1999) Linear Integral Equations. Springer-Verlag.Google Scholar
Kuersteiner, G. (2001) Optimal instrumental variables estimation for ARMA models. Journal of Econometrics 104, 359–405.Google Scholar
Kuo, H.-H. (1975) Gaussian Measures in Banach Spaces. Springer-Verlag.Google Scholar
McNeney, B. & Wellner, J. (2000) Application of convolution theorems in semiparametric models with non-i.i.d. data. Journal of Statistical Planning and Inference 91, 441–480.Google Scholar
Newey, W. (1993) Efficient estimation of models with conditional moment restrictions. In Maddala, G.S., Rao, C.R., & Vinod, H.D. (eds.), Handbook of Statistics, vol. 11, pp. 419–454.Elsevier.Google Scholar
Newey, W. & McFadden, D. (1994) Large sample estimation & hypothesis testing. In Engle, R.F. and McFadden, D. (eds.), Handbook of Econometrics, vol. IV, pp. 2111–2245. Elsevier.Google Scholar
Parzen, E. (1959) Statistical Inference on time series by Hilbert space methods, I. Technical report 23, Applied Mathematics & Statistics Laboratory, Stanford. Reprinted in Time Series Analysis Papers (1967) Holden-Day.Google Scholar
Parzen, E. (1970) Statistical inference on time series by RKHS methods. In: Pyke, R. (ed.) 12th Biennial Seminar Canadian Mathematical Congress Proceedings, pp. 1–37. Canadian Mathematical Society.Google Scholar
Pearson, K. (1894) Contribution to the mathematical theory of evolution. Philosohical Transactions of the Royal Society of London, Series A 185, 71–110.Google Scholar
Roussas, G.G. (1965) Asymptotic inference in markov processes. Annals of Mathematical Statistics 36, 987–992.Google Scholar
Saitoh, S. (1997) Integral Transforms, Reproducing Kernels and Their Applications. Longman.Google Scholar
Singleton, K. (2001) Estimation of affine pricing models using the empirical characteristic function. Journal of Econometrics 102, 111–141.Google Scholar
Stinchcombe, M. & White, H. (1998) Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory 14, 295–325.Google Scholar
Stout, W. (1974) Almost Sure Convergence. Academic Press.Google Scholar
Tauchen, G. (1997) New Minimum Chi-Square Methods in Empirical Finance. In: Kreps, D. & Wallis, K., (eds.) Advances in Econometrics, Seventh World Congress, pp. 279–317. Cambridge University Press.Google Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.Google Scholar
Wefelmeyer, W. (1996) Quasi-likelihood models and optimal inference. Annals of Statistics 24, 405–422.Google Scholar