×

zbMATH — the first resource for mathematics

Fourth order pseudo maximum likelihood methods. (English) Zbl 1441.62733
Summary: We extend PML theory to account for information on the conditional moments up to order four, but without assuming a parametric model, to avoid a risk of misspecification of the conditional distribution. The key statistical tool is the quartic exponential family, which allows us to generalize the PML2 and QGPML1 methods proposed in [C. Gourieroux et al., Econometrica 52, 681–700 (1984; Zbl 0575.62031)] to PML4 and QGPML2 methods, respectively. An asymptotic theory is developed. The key numerical tool that we use is the Gauss-Freud integration scheme that solves a computational problem that has previously been raised in several fields. Simulation exercises demonstrate the feasibility and robustness of the methods.

MSC:
62P20 Applications of statistics to economics
Software:
nlmdl; OPQ
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agmon, N.; Alhassid, Y.; Levine, R.D., An algorithm for finding the distribution of maximal entropy, Journal of computational physics, 30, 250-259, (1979) · Zbl 0406.65034
[2] Altonji, J.G.; Segal, L.M., Small-sample bias in GMM estimation of covariance structures, Journal of business & economic statistics, 14, 3, 353-366, (1996)
[3] Andersen, T.G.; Sorenson, B., GMM estimation of a stochastic volatility model: a Monte Carlo study, Journal of business & economic statistics, 14, 328-352, (1996)
[4] Arellano-Valle, R.B.; Genton, M.G., On fundamental skew distributions, Journal of multivariate analysis, 96, 93-116, (2005) · Zbl 1073.62049
[5] Barndorff-Nielsen, O.E., Information and exponential families in statistical theory, (1978), Wiley Chichester · Zbl 0387.62011
[6] Barndorff-Nielsen, O.E., Normal inverse Gaussian distributions and stochastic volatility modeling, Scandinavian journal of statistics, 24, 1-13, (1997) · Zbl 0934.62109
[7] Bollerslev, T.; Wooldridge, J.M., Quasi-maximum likelihood estimation and inference in dynamic models with time varying covariances, Econometric reviews, 11, 2, 143-172, (1992) · Zbl 0850.62884
[8] Brown, L.D., Fundamentals of statistical exponential families, (1986), Institute of Mathematical Statistics Hayward, California
[9] Chamberlain, G., Asymptotic efficiency in estimation with conditional moment restrictions, Journal of econometrics, 34, 305-334, (1987) · Zbl 0618.62040
[10] Doran, H.E.; Schmidt, P., GMM estimators with improved finite sample properties using principal components of the weighting matrix, with an application to the dynamic panel data model, Journal of econometrics, 133, 1, 387-409, (2006) · Zbl 1345.62042
[11] Eberlein, E.; Keller, U., Hyperbolic distributions in finance, Bernoulli, 1, 281-299, (1995) · Zbl 0836.62107
[12] Fernandez, C.; Steel, M.F.J., On Bayesian modelling of fat tails and skewness, Journal of the American statistical association, 93, 359-371, (1998) · Zbl 0910.62024
[13] Freud, G., On the greatest zero of an orthogonal polynomial, I, Journal of approximation theory, 46, 16-24, (1986) · Zbl 0603.42022
[14] Gale, D.; Nikaido, H., The Jacobian matrix and global univalence of mappings, Mathematische annalen, 159, 2, 81-93, (1965) · Zbl 0158.04903
[15] Gallant, A.R., Nonlinear statistical models, (1987), John Wiley & Sons New York · Zbl 0611.62071
[16] Gautschi, W., Orthogonal polynomials: computation and approximation, (2004), Oxford Science Publications, Oxford University Press · Zbl 1130.42300
[17] Genton, M.G., Skew-elliptical distributions and their applications: A journey beyond normality, (2004), Chapman & Hall/CRC Boca Raton, Florida. · Zbl 1069.62045
[18] Golan, A.; Judge, G.; Miller, D., Maximum entropy econometrics: robust estimation with limited data, (1996), Wiley Chichester · Zbl 0884.62126
[19] Golub, G.H.; Welch, J.H., Calculation of Gauss quadrature rules, Mathematics of computations, 23, 106, 221-230, (1969) · Zbl 0179.21901
[20] Gourieroux, C.; Monfort, A.; Trognon, A., Pseudo maximum likelihood methods: theory, Econometrica, 52, 1, 681-700, (1984) · Zbl 0575.62031
[21] Gourieroux, C.; Monfort, A., Statistics and econometric models: volume one, (1995), Cambridge University Press Cambridge
[22] Gourieroux, C.; Monfort, A., Statistics and econometric models: volume two, (1995), Cambridge University Press Cambridge
[23] Gourieroux, C.; Monfort, A., Pricing with splines, Annales d’economie et de statistique, 82, 4-33, (2006)
[24] Hansen, L.P., Large sample properties of the generalized methods of moments, Econometrica, 50, 1029-1054, (1982) · Zbl 0502.62098
[25] Hansen, B., Autoregressive conditional density estimation, International economic review, 35, 705-730, (1994) · Zbl 0807.62090
[26] Harvey, C.R.; Siddique, A., Autoregressive conditional skewness, Journal of financial and quantitative analysis, 34, 4, 465-487, (1999)
[27] Holly, A., Asymptotic theory for nonlinear econometric models: estimation, ()
[28] Holly, A., 2009. Modeling risk using fourth order pseudo maximum likelihood methods. Mimeo, Institute of Health Economics and Management, University of Lausanne.
[29] Holly, A., Pentsak, Y., 2004. Maximum likelihood estimation of the conditional mean \(E(y \mid x)\) for skewed dependent variables in four-parameter families of distribution. Technical Report. Institute of Health Economics and Management, University of Lausanne.
[30] Hood, W.; Koopmans, T., The estimation of simultaneous linear economic relationships, () · Zbl 0053.28005
[31] Jondeau, E.; Rockinger, M., Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements, Journal of economic dynamics and control, 27, 10, 1699-1737, (2003) · Zbl 1178.91226
[32] Junk, M., Maximum entropy for reduced moment problems, Mathematical models and methods in applied sciences, 10, 7, 1001-1025, (2000) · Zbl 1012.44005
[33] Kitamura, Y.; Stutzer, M., An information-theoretic alternative to generalized method of moments estimation, Econometrica, 65, 4, 861-874, (1997) · Zbl 0894.62011
[34] Levin, E.; Lubinsky, D.S., ()
[35] Maasoumi, E., A compendium to information theory in economics and econometrics, Econometric reviews, 12, 2, 137-181, (1993) · Zbl 0769.62003
[36] Manning, W.; Basu, A.; Mullahy, J., Generalized modeling approaches to risk adjustment of skewed outcomes data, Journal of health economics, 24, 465-488, (2005)
[37] Mead, L.R.; Papanicolaou, N., Maximum entropy in the problem of moments, Journal of mathematical physics, 25, 8, 2404-2417, (1984)
[38] Monfort, A., 1982. Cours de statistique mathmatique. Economica, Paris.
[39] Newey, W.K., Semiparametric efficiency bounds, Journal of applied econometrics, 5, 99-135, (1990) · Zbl 0705.62033
[40] Newey, W.K.; Steigerwald, D.G., Asymptotic bias for quasi-maximum likelihood estimators in conditional heteroskedastic models, Econometrica, 65, 587-599, (1997) · Zbl 0870.62091
[41] Noschese, S.; Pasquini, L., On the nonnegative solution of a freud three-term recurrence, Journal of approximation theory, 99, 54-67, (1999) · Zbl 0955.42013
[42] Ormoneit, D.; White, H., An efficient algorithm to compute maximum entropy densities, Econometric reviews, 18, 2, 127-140, (1999) · Zbl 0932.62006
[43] Sawa, T., Information criteria for discriminating among alternative regression models, Econometrica, 46, 6, 1273-1291, (1978) · Zbl 0393.62025
[44] Stacy, E., A generalization of gamma distribution, Annals of mathematical statistics, 33, 1187-1192, (1962) · Zbl 0121.36802
[45] Stacy, E.; Mihram, G., Parameter estimation for a generalized gamma distribution, Technometrics, 7, 349-358, (1965) · Zbl 0129.11107
[46] Tauchen, G., Statistical properties of generalized method-of-moments estimators of structural parameters obtained from financial market data, Journal of business & economic statistics, 4, 4, 397-416, (1986)
[47] Titterington, D.M.; Smith, A.F.M.; Makov, U.E., Statistical analysis of finite mixture distributions, (1985), John Wiley Chichester · Zbl 0646.62013
[48] White, H., Maximum likelihood of misspecified models, Econometrica, 50, 1, 1-25, (1982) · Zbl 0478.62088
[49] White, H., Estimation, inference, and specification analysis, (1994), Cambridge University Press Cambridge, UK · Zbl 0860.62100
[50] Zellner, A.; Highfield, R.A., Calculation of maximum entropy distributions and approximation of marginal posterior distributions, Journal of econometrics, 37, 2, 195-209, (1988)
[51] Ziliak, J.P., Efficient estimation with panel data when instruments are predetermined: an empirical comparison of moment-condition estimators, Journal of business and economic statistics, 15, 419-431, (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.