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EXISTENCE AND CHARACTERIZATION OF CONDITIONAL DENSITY PROJECTIONS

Published online by Cambridge University Press:  13 May 2015

Ivana Komunjer*
Affiliation:
University of California
Giuseppe Ragusa
Affiliation:
Luiss University
*
*Address correspondence to Ivana Komunjer, Department of Economics, University of California, San Diego, 9500 Gilman Drive MC 0508, La Jolla, CA 92093-0508; e-mail: komunjer@ucsd.edu
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Abstract

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In this paper we propose primitive conditions under which a projection of a conditional density onto a set defined by conditional moment restrictions exists and is unique. Moreover, we provide an analytic expression of the obtained projection. The range of applications where conditional density projections are used is wide. The derived results are potentially useful in a variety of areas including: semiparametric efficient estimation and optimal testing in (conditional) moment models, Bayesian prior determination and inference in semiparametric models, density forecasting, and simulation-based econometric analysis.

Regarding existence, we propose three different combinations of assumptions that are all sufficient to show that the projection exists and is unique. The proposed conditions exhibit a clear trade off between restrictions put on the divergence between the conditional densities and on the moment function which defines the projection set. Depending on the nature of the application, the researcher can pick and choose which set of conditions to use. Our second set of results characterizes the projection. The expression for the projected density is new though not surprising given the previously obtained results for the unconditional case. The projection is characterized by the dual of the original projection problem. In establishing the strong duality, however, we work with a constraint qualification condition that is weaker than that used by Borwein and Lewis (1991a, 1992a, 1993 in their seminal work concerning the unconditional case.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2015 

Footnotes

We would like to thank the Co-Editor, Yuichi Kitamura, and two anonymous referees for their excellent comments and suggestions. We also thank the seminar participants at Rice University, Iowa State University, University of British Columbia, University of Pennsylvania, USC, University of Texas Austin, Rochester, and Joint Montréal Econometrics seminar for their feedback. Previous versions of this paper were circulated under the title “Existence and Uniqueness of Semiparametric Projections.

References

REFERENCES

Ali, S.M. & Silvey, S.D. (1966) A general class of coefficients of divergence of one distribution from another. Journal of the Royal Statistical Society Ser. B, 28, 131142.Google Scholar
Aliprantis, C.D. & Border, K.C. (2007) Infinite Dimensional Analysis, Berlin: Springer-Verlag, 3rd ed.Google Scholar
Bernardo, J. (1979) Reference Posterior Distributions for Bayesian Inference. Journal of the Royal Statistical Society. Series B (Methodological), 41, 113147.CrossRefGoogle Scholar
Bernardo, J. (2005) Reference analysis, Handbook of Statistics, 25, 1790.CrossRefGoogle Scholar
Borwein, J.M. & Lewis, A.S. (1991a) Duality Relationships for Entropy-Like Minimization Problems. SIAM Journal on Control and Optimization, 29, 325338.CrossRefGoogle Scholar
Borwein, J.M. & Lewis, A.S. (1991b) On the Convergence of Moment Problems. Transactions of the American Mathematical Society, 325, pp. 249271.CrossRefGoogle Scholar
Borwein, J.M. & Lewis, A.S. (1992a) Partially finite convex programming, part I: quasi relative interiors and duality theory, interiors and duality theory. Math. Program., 57, 1548.CrossRefGoogle Scholar
Borwein and Lewis (1992b) Partially finite convex programming, part II: explicit lattice models, Math. Program., 57, 4983.CrossRefGoogle Scholar
Borwein, J.M. & Lewis, A.S. (1993) Partially-finite Programming in L 1 and the Existence of Maximum Entropy Estimates. SIAM Journal on Optimization, 3, 248267.CrossRefGoogle Scholar
Buck, B. & Macaulay, V. (1991) Maximum entropy in action: a collection of expository essays, Oxford University Press.Google Scholar
Burg, J. (1967) Maximum Entropy Spectrum Analysis, Paper presented at 37th Annual Mtg. of Exploration Geophyslcrists, Oklahoma City.Google Scholar
Chor-Yiu, S. & White, H. (1996) Information criteria for selecting possibly misspecified parametric models. Journal of Econometrics, 71, 207225.Google Scholar
Cressie, N. & Read, T. (1984) Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society. Series B (Methodological), 440464.CrossRefGoogle Scholar
Csiszár, I. (1967) Information-type measures of the difference of probability distributions and indirect observations, Studia Scientiarum Mathematicarum Hungarica, 299318.Google Scholar
Csiszár, I. (1975) I-Divergence Geometry of Probability Distributions and Minimization Problems. Annals of Probability, 3, 146–58.CrossRefGoogle Scholar
Csiszár, I. (1995) Generalized projections for non-negative functions, Acta Mathematica Hungarica, 68, 161186.CrossRefGoogle Scholar
Ekeland, I. & Témam, R. (1987) Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics.Google Scholar
Giacomini, R. & Ragusa, G. (2013) Theory-coherent forecasting. Journal of Econometrics (in press).Google Scholar
Gourieroux, C. & Monfort, A. (1997) Simulation-Based Econometric Methods, Oxford University Press.CrossRefGoogle Scholar
Gowda, M. & Teboulle, M. (1990) A Comparison of Constraint Qualifications in Infinite-Dimensional Convex Programming. SIAM Journal on Control and Optimization, 28, 925935.CrossRefGoogle Scholar
Hiriart-Urruty, J.B. & Lemarechal, C. (1993) Convex Analysis and Minimization Algorithms I: Fundamentals (Grundlehren Der Mathematischen Wissenschaften), Springer.CrossRefGoogle Scholar
Holmes, R.D. (1974) Geometric Functional Analysis and Its Applications. Springer-Verlag.Google Scholar
Kim, J. (2002) Limited information likelihood and Bayesian analysis. Journal of Econometrics, 107, 175193.CrossRefGoogle Scholar
Kitamura, Y. (2001) Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions. Econometrica, 69, 16611672.CrossRefGoogle Scholar
Kitamura, Y., Otsu, T., & Evdokimov, K. (2009) Robustness, Infinitesimal Neighborhoods, and Moment Restrictions, Cowles Foundation Discussion Papers.CrossRefGoogle Scholar
Kitamura, Y. & Stutzer, M. (1997) An Information-Theoretic Alternative to Generalized Method of Moments Estimation. Econometrica, 65, 861874.CrossRefGoogle Scholar
Kitamura, Y., Tripathi, G., & Ahn, H. (2004) Empirical Likelihood-Based Inference in Conditional Moment Restriction Models. Econometrica, 72, 16671714.CrossRefGoogle Scholar
Komunjer, I. & Vuong, Q. (2009) Semiparametric Efficiency Bound in Time-Series Models for Conditional Quantiles, Econometric Theory, forthcoming.Google Scholar
Krasnosel’skii, M. & Rutickii, Y. (1961) Convex Functions and Orlicz Spaces, P.Noordhoff Ltd.Google Scholar
Kullback, S. & Khairat, M.A. (1966) A Note on Minimum Discrimination Information. The Annals of Mathematical Statistics, 37, 279280.CrossRefGoogle Scholar
Liese, F. (1975) On the existence of f – projections, Colloquia of Mathematical Society Janos Bolyai, 16, 431446, budapest.Google Scholar
Liese, F. & Vajda, I. (1987) Convex Statistical distances. Leipzig: Teubner.Google Scholar
Newey, W. & Smith, R.J. (2004) Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. Econometrica, 72, 219256.CrossRefGoogle Scholar
Otsu, T., Seo, M.H., & Whang, Y.-J. (2008) Testing for Non-Nested Conditional Moment Restrictions using Unconditional Empirical Likelihood, Cowles Foundation Discussion Paper No. 1660.Google Scholar
Ragusa, G. (2011) Minimum divergence, generalized empirical likelihoods, and higher order expansions. Econometric Reviews, 30, 406456.CrossRefGoogle Scholar
Rockafellar, R.T. (1968) Integrals which are convex functionals. Pacific Journal of Mathematics, 24, 525539.CrossRefGoogle Scholar
Rockafellar, R.T. (1970) Convex Analysis. Princeton, New Jersey: Princeton University Press.CrossRefGoogle Scholar
Rockafellar, R.T. (1971) Integrals which are convex functionals. II. Pacific Journal of Mathematics, 39, 439469.CrossRefGoogle Scholar
Rockafellar, R.T. (1974) Conjugate Duality and Optimization, volume 16 of Regional Conferences Series in Applied Mathematics, SIAM, Philadelphia.Google Scholar
Sawa, T. (1978) Information Criteria for Discriminating Among Alternative Regression Models. Econometrica, 46, 12731291.CrossRefGoogle Scholar
Schennach, S.M. (2014) Entropic Latent Variable Integration via Simulation. Econometrica, 82, 345385.Google Scholar
Shi, X. (2014) A Non-Degenerate Vuong Test, Manuscript.Google Scholar
Stein, C. (1956) Efficient Nonparametric Testing and Estimation, In Proceedings of the Third Berkeley Symposium in Mathematical Statistics and Probability, Berkeley: University of California Press, vol. 1, 187–196.Google Scholar
Stinchcombe, M.B. & White, H. (1998) Consistent Specification Testing with Nuisance Parameters Present Only under the Alternative. Econometric Theory, 14, pp. 295325.CrossRefGoogle Scholar
Tang, Y. & Ghosal, S. (2007) Posterior Consistency of Dirichlet Mixtures for Estimating a Transition Density. Journal of Statistical Planning and Inference, 137, 17111726.CrossRefGoogle Scholar
Ullah, A. (1996) Entropy, divergence and distance measures with econometric applications. Journal of Statistical Planning and Inference, 49, 137162.CrossRefGoogle Scholar
Vuong, Q. (1989) Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses. Econometrica, 57, 307333.CrossRefGoogle Scholar
White, H. (1982) Maximum Likelihood Estimation of Misspecified Models. Econometrica, 50, 125.CrossRefGoogle Scholar
White, H. (1994) Estimation, Inference and Specification Analysis. Cambridge University Press.CrossRefGoogle Scholar
Zellner, A. (1996): Models, prior information, and Bayesian analysis. Journal of Econometrics, 75, 5168.CrossRefGoogle Scholar
Zellner, A. (2002) Information processing and Bayesian analysis, Journal of Econometrics, 107, 4150.CrossRefGoogle Scholar
Zellner, A. (2003) Some Recent Developments in Econometric Inference. Econometric Reviews, 22, 203215.CrossRefGoogle Scholar
Zellner, A. & Tobias, J. (2001) Further Results on Bayesian Method of Moments Analysis of the Multiple Regression Model. International Economic Review, 42, 121139.CrossRefGoogle Scholar
Zălinescu, C. (1999) A comparison of constraint qualifications in infinite-dimensional convex programming revisited, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 40, 353378.CrossRefGoogle Scholar