Hostname: page-component-7c8c6479df-xxrs7 Total loading time: 0 Render date: 2024-03-27T19:36:41.467Z Has data issue: false hasContentIssue false

EFFICIENT ESTIMATION USING THE CHARACTERISTIC FUNCTION

Published online by Cambridge University Press:  22 February 2016

Marine Carrasco
Affiliation:
Université de Montréal
Rachidi Kotchoni*
Affiliation:
African School of Economics
*
*Address correspondence to Rachidi Kotchoni, African School of Economics. E-mail: rachidi.kotchoni@africanschoolofeconomics.com.

Abstract

The method of moments procedure proposed by Carrasco and Florens (2000) permits full exploitation of the information contained in the characteristic function and yields an estimator which is asymptotically as efficient as the maximum likelihood estimator. However, this estimation procedure depends on a regularization or tuning parameter α that needs to be selected. The aim of the present paper is to provide a way to optimally choose α by minimizing the approximate mean square error (AMSE) of the estimator. Following an approach similar to that of Donald and Newey (2001), we derive a higher-order expansion of the estimator from which we characterize the finite sample dependence of the AMSE on α. We propose to select the regularization parameter by minimizing an estimate of the AMSE. We show that this procedure delivers a consistent estimator of α. Moreover, the data-driven selection of the regularization parameter preserves the consistency, asymptotic normality, and efficiency of the CGMM estimator. Simulation experiments based on a CIR model show the relevance of the proposed approach.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2016 

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

Ait-Sahalia, Y. & Kimmel, R. (2007) Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics 83, 413452.Google Scholar
Buse, A. (1992) The bias of instrumental variable estimators. Econometrica 60(1), 173180.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, 529573.Google Scholar
Carrasco, M. & Florens, J.P. (2000) Generalization of GMM to a continuum of moment conditions. Econometric Theory 16, 797834.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. 6. pp. 56335751.Google Scholar
Chacko, G. & Viceira, L. (2003) Spectral GMM estimation of continuous-time processes. Journal of Econometrics 116, 259292.CrossRefGoogle Scholar
Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica 53, 385407.Google Scholar
Devroye, L. (1986) Non-Uniform Random Variate Generation. Spinger-Verlag.Google Scholar
Donald, S. & Newey, W. (2001) Choosing the number of instruments. Econometrica 69, 11611191.Google Scholar
Duffie, D. & Singleton, K. (1993) Simulated moments estimation of Markov models of asset prices. Econometrica 61, 929952.Google Scholar
Feller, W. (1951) Two singular diffusion problems. Annals of Mathematics 54, 173182.Google Scholar
Feuerverger, A. & McDunnough, P. (1981a) On efficient inference in symmetry stable laws and processes. In Csorgo, M. (ed.), Statistics and Related Topics, pp. 109122. North Holland.Google Scholar
Feuerverger, A. & McDunnough, P. (1981b) On some Fourier methods for inference. Journal of the American Statistical Association 76, 379387.Google Scholar
Feuerverger, A. & McDunnough, P. (1981c) On the efficiency of empirical characteristic function procedures. Journal of the Royal Statistical Society. Series B (Methodological) 43, 2027.Google Scholar
Feuerverger, A. & Mureika, R. (1977) The empirical characteristic function and its applications. The Annals of Statistics 5(1), 8897.Google Scholar
Gouriéroux, C. & Jasiak, J. (2005) Autoregressive Gamma Processes. Journal of Forecasting 25, 129152.Google Scholar
Gouriéroux, C. & Monfort, A. (1996) Simulation Based Econometric Methods. CORE Lectures. Oxford University Press.Google Scholar
Gouriéroux, C., Monfort, A., & Renault, E. (1993) Indirect inference. Journal of Applied Econometrics 8, S85S118.Google Scholar
Gray, S.F. (1996) Modeling the conditional distribution of interest rates as a regime-switching process. Journal of Financial Economics 42, 2762.Google Scholar
Hansen, L. (1982) Large sample properties of generalized method of moments estimators. Econometrica 50, 10291054.Google Scholar
Heston, S. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies 6(2), 327343.Google Scholar
Jacho-Chavez, D.T. (2010) Optimal bandwidth choice for estimation of inverse conditional–density–weighted expectations. Econometric Theory 26, 94118.Google Scholar
Jiang, G. & Knight, J. (2002) Estimation of continuous time processes via the empirical characteristic function. Journal of Business and Economic Statistics 20, 198212.Google Scholar
Koenker, R., Machado, J.A.F., Skeels, C.L., & Welsh, A.H.I. (1994) Momentary lapses: Moment expansions and the robustness of minimum distance estimators. Econometric Theory 10, 172197.Google Scholar
Koutrouvelis, I.A. (1980) Regression-type estimation of the parameters of stable laws. Journal of the American Statistical Association 75(372), 918928.Google Scholar
Linton, O. (2002) Edgeworth approximation for semiparametric instrumental variable and test statistics. Journal of Econometrics 106, 325368.Google Scholar
Liu, Q. & Pierce, D.A. (1994) A note on Gauss–Hermite quadrature. Biometrika 81(3), 624629.Google Scholar
Madan, D.B. & Senata, E. (1990) The variance gamma model for share market returns. Journal of Business 63(4), 511524.Google Scholar
Nagar, A.L. (1959) The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 573595.Google Scholar
Newey, W.K. & McFadden, D. (1994) Large sample estimation and hypotheses testing. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4, pp. 21112245 Google Scholar
Newey, W.K. & Smith, R.J. (2004) Higher order properties of GMM and generalized empirical likelihood estimators. Econometrica 72(1), 219255.Google Scholar
Nolan, J. P. (2016) Stable Distributions: Models for Heavy Tailed Data. Springer Verlag.Google Scholar
Paulson, A.S., Holcomb, W.E., & Leitch, R.A. (1975) The estimation of the parameters of the stable laws. Biometrika 62(1), 163170.Google Scholar
Phillips, P.C.B. & Moon, H.R. (1999) Linear regression limit theory for nonstationary panel data. Econometrica 67(5), 10571112.Google Scholar
Rilstone, P., Srivastava, V.K., & Ullah, A. (1996) The second-order bias and mean-squared error of nonlinear estimators. Journal of Econometrics 75, 369395.Google Scholar
Rothenberg, T.J. (1983) Asymptotic properties of some estimators in structural models. In Karlin, S., Amemiya, T., & Goodman, L.A. (eds.), Studies in Econometrics, Time Series and Multivariate Statistics. Academic Press.Google Scholar
Rothenberg, T.J. (1984) Approximating the distributions of econometric estimators and test statistics. In Griliches, Z. & Intriligator, M.D. (eds.), Handbook of Econometrics, vol. 2. North-Holland.Google Scholar
Singleton, K.J. (2001) Estimation of affine pricing models using the empirical characteristic function. Journal of Econometrics 102, 111141.Google Scholar
Yu, J. (2004) Empirical characteristic function estimation and its applications. Econometric Reviews 23(2), 93123.Google Scholar
Zhou, H. (2001) Finite sample properties of EMM, GMM, QMLE, and MLE for a square-root interest rate diffusion model. Journal of Computational Finance 2, 89122.Google Scholar