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A SIMPLE ITERATIVE Z-ESTIMATOR FOR SEMIPARAMETRIC MODELS

Published online by Cambridge University Press:  12 April 2018

David T. Frazier
David T. Frazier*
Affiliation:
Monash University
*
*Address correspondence to David Frazier, Department of Econometrics and Business Statistics, Monash University, Melbourne, Australia; e-mail: david.frazier@monash.edu.
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Abstract

We propose a new iterative estimation algorithm for use in semiparametric models where calculation of Z-estimators by conventional means is difficult or impossible. Unlike a Newton–Raphson approach, which makes use of the entire Hessian, this approach only uses curvature information associated with portions of the Hessian that are relatively easy to calculate. Consistency and asymptotic normality of estimators obtained from this algorithm are established under regularity conditions and an information dominance condition. Two specific examples, a quantile regression model with missing covariates and a GARCH-in-mean model with conditional mean of unknown functional form, demonstrate the applicability of the algorithm. This new approach can be interpreted as an extension of the maximization by parts estimation approach to semiparametric models.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 1 , February 2019 , pp. 111 - 141
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

This article has benefited tremendously from the feedback provided by two anonymous referees and a co-editor. Additional thanks are given to Eric Renault, Mervyn Silvapulle, Saraswata Chaudhuri, Yanqin Fan, Dennis Kristensen, and Oliver Linton.

References

REFERENCES

Ackerberg, D., Chen, X., & Hahn, J. (2012) A practical asymptotic variance estimator for two-step semiparametric estimators. The Review of Economics and Statistics 94(2), 481–498.CrossRefGoogle Scholar
Ai, C. & Chen, X. (2003) Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71(6), 1795–1843.CrossRefGoogle Scholar
Amado, C. & Terasvirta, T. (2013) Modelling volatility by variance decomposition. Journal of Econometrics 175(2), 142–153.CrossRefGoogle Scholar
Andrews, D.W.K. (1994) Asymptotics for semiparametric econometric models via stochastic equicontinuity. Econometrica 62(1), 43–72.CrossRefGoogle Scholar
Chen, X. (2007) Large sample sieve estimation of semi-nonparametric models. In Heckman, J.J. & Learner, E.E. (eds.), Handbook of Econometrics, vol. 6, Part B, pp. 5549–5632. Elsevier.Google Scholar
Chen, X., Hong, H., & Tarozzi, A. (2008) Semiparametric efficiency in GMM models with auxiliary data. Annals of Statistics 36, 808–843.CrossRefGoogle Scholar
Chen, X., Linton, O., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71(5), 1591–1608.CrossRefGoogle Scholar
Chen, X., Wan, A.T.K., & Zhou, Y. (2015) Efficient quantile regression analysis with missing observations. Journal of the American Statistical Association 110(510), 723–741.CrossRefGoogle Scholar
Christensen, B.J., Dahl, C.M., & Iglesias, E.M. (2012) Semiparametric inference in a garch-in-mean model. Journal of Econometrics 167(2), 458–472.CrossRefGoogle Scholar
Ding, Y. & Nan, B. (2011) A sieve m-theorem for bundled parameters in semiparametric models, with application to the efficient estimation in a linear model for censored data. The Annals of Statistics 39(6), 3032–3061.CrossRefGoogle Scholar
Dominitz, J. & Sherman, R.P. (2005) Some convergence theory for iterative estimation procedures with an application to semiparametric estimation. Econometric Theory 21(4), 838–863.CrossRefGoogle Scholar
Engle, R.F., Lilien, D.M., & Robins, R.P. (1987) Estimating time varying risk premia in the term structure: The arch-m model. Econometrica 55(2), 391–407.CrossRefGoogle Scholar
Escanciano, J.C., Jacho-Chavez, D.T., & Lewbel, A. (2014) Uniform convergence of weighted sums of non and semiparametric residuals for estimation and testing. Journal of Econometrics 178(0), 426–443.CrossRefGoogle Scholar
Fan, Y., Pastorello, S., & Renault, E. (2015) Maximization by parts in extremum estimation. The Econometrics Journal 18(2), 147–171.CrossRefGoogle Scholar
Fiorentini, G., Calzolari, G., & Panattoni, L. (1996) Analytic derivatives and the computation of garch estimates. Journal of Applied Econometrics 11(4), 399–417.3.0.CO;2-R>CrossRefGoogle Scholar
Graham, B.S., Pinto, C.C., & Egel, D. (2012) Inverse probability tilting for moment condition models with missing data. Review of Economic Studies 79, 1053–1079.CrossRefGoogle Scholar
Horvitz, D.G. & Thompson, D.J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association 47(260), 663–685.CrossRefGoogle Scholar
Hu, Z., Wang, N., & Carroll, R.J. (2004) Profile-kernel versus backfitting in the partially linear models for longitudinal/clustered data. Biometrika 91(2), 251–262.CrossRefGoogle Scholar
Huang, J. & Wellner, J. (1997) Interval censored survival data: A review of recent progress. In Lin, D. & Fleming, T. (eds.), Proceedings of the First Seattle Symposium in Biostatistics. Lecture Notes in Statistics, pp. 123–169. Springer.CrossRefGoogle Scholar
Hunter, D.R. & Lange, K. (2000) Quantile regression via an mm algorithm. Journal of Computational and Graphical Statistics 9(1), 60–77.Google Scholar
Ichimura, H. & Lee, S. (2010) Characterization of the asymptotic distribution of semiparametric m-estimators. Journal of Econometrics 159(2), 252–266.CrossRefGoogle Scholar
Lange, K., Hunter, D.R., & Yang, I. (2000) Optimization transfer using surrogate objective functions. Journal of Computational and Graphical Statistics 9(1), 1–20.Google Scholar
Ling, S. (2004) Estimation and testing stationarity for double-autoregressive models. Journal of the Royal Statistical Society Series B (Methodology) 66(1), 63–78.CrossRefGoogle Scholar
Linton, O. & Perron, B. (2003) The shape of the risk premium. Journal of Business & Economic Statistics 21(3), 354–367.CrossRefGoogle Scholar
Mammen, E., Roth, C., & Schienle, M. (2016) Semiparametric estimation with generated covariates. Econometric Theory 32, 1140–1177.CrossRefGoogle Scholar
Nan, B. & Wellner, J.A. (2013) A general semiparametric z-estimator for case-cohort studies. Statistical Sinica 23(23), 1155–1180.Google Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347–370.CrossRefGoogle Scholar
Newey, W.K. (1994) The asymptotic variance of semiparametric estimators. Econometrica 62(6), 1349–1382.CrossRefGoogle Scholar
Noureldin, D., Shephard, N., & Sheppard, K. (2014) Multivariate rotated arch models. Journal of Econometrics 179(1), 16–30.CrossRefGoogle Scholar
Pakes, A. & Pollard, D. (1989) Simulation and the asymptotics of optimization estimators. Econometrica 57(5), 1027–1057.CrossRefGoogle Scholar
Pastorello, S., Patilea, V., & Renault, E. (2003) Iterative and recursive estimation in structural nonadaptive models [with comments, rejoinder]. Journal of Business & Economic Statistics 21(4), 449–482.CrossRefGoogle Scholar
Robins, M., Rotnitzky, A., & Zhao, L. (1994) Estimation of regression coefficients when some regressors are not always observed. Journal of American Statistical Association 427, 846–866.CrossRefGoogle Scholar
Roth, C. & Firpo, S. (2016) Properties of doubly robust estimators when nuisance functions are estimated nonparametrically. Working paper.Google Scholar
Scharfstein, D.O., Rotnitzky, A., & Robins, J.M. (1999) Adjusting for nonignorable drop-out using semiparametric nonresponse models. Journal of the American Statistical Association 94, 1096–1146.CrossRefGoogle Scholar
Song, P.X.K., Fan, Y., & Kalbfleisch, J. (2005) Maximization by parts in likelihood inference [with comments, rejoinder]. Journal of the American Statistical Association 100(472), 1145–1167.CrossRefGoogle Scholar
Song, P., Zhang, P., & Qu, A. (2007) Maximum likelihood inference in robust linear mixed-effects models using multivariate t distributions. Statistica Sinica 17(3), 929–943.Google Scholar
van Keilgom, I. & Carroll, R.J. (2007) Backfitting versus profiling in general criterion functions. Statistica Sinica (17), 797–816.Google Scholar
Wooldridge, J.M. (2007) Inverse probability weighted estimation for general missing data problems. Journal of Econometrics 141(2), 1281–1301.CrossRefGoogle Scholar
Xue, L. (2009) Empirical likelihood confidence intervals for response mean with data missing at random. Scandinavian Journal of Statistics 36(4), 671–685.CrossRefGoogle Scholar
Zhang, R., Czado, C., & Min, A. (2011) Efficient maximum likelihood estimation of copula based meta-distributions. Computational Statistics and Data Analysis 55(3), 1196–1214.CrossRefGoogle Scholar
Zhou, Y., Wan, A.T.K., & Wang, X. (2008) Estimating equations inference with missing data. Journal of the American Statistical Association 103(483), 1187–1199.CrossRefGoogle Scholar