Fourth Order Pseudo Maximum Likelihood Methods

The objective of this paper is to extend the results on Pseudo Maximum Likelihood(PML) theory derived in Gourieroux, Monfort, and Trognon (GMT)(1984) to a situation where the rst four conditional moments are specied.Such an extension is relevant in light of pervasive evidence that conditional distributionsare non-Gaussian in many economic situations. The key statisticaltool here is the quartic exponential family, which allows us to generalize thePML2 and QGPML1 methods proposed in GMT(1984) to PML4 and QGPML2methods, respectively.[...]

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Year of publication:	2008-08-01
Authors:	Holly, Alberto ; Monfort, Alain ; Rockinger, Michael
Institutions:	Finrisk
Subject:	Kurtosis \| Schiefe

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1. Introduction
2. Preliminaries
2.1. Exponential Families.
2.2. Quartic Exponential Family.
2.3. M-estimators and Quasi-Generalized M-estimators.
3. Properties of the Exponential Quartic Family
4. PML 4 Method
4.1. Denition.
4.2. Asymptotic Properties.
5. QGPML2 Method
5.1. Alternative Parametrization.
5.2. Efficient Semiparametric Estimator of Conditional Means andVariances.
5.3. QGPML2 Method.
6. Numerical Implementation
6.1. Canonical Quartic Family.
6.2. Implementation of the PML4 Method
6.3. Computation of the Functions (s; k); (s; k) Using the Gauss-Freud Method.
6.4. Implementation of the QGPML2 Method.
7. Numerical Examples
7.1. Computation of the Quartic Exponential.
7.2. A First Experiment.
7.3. A Second Experiment.
7.4. A Third Experiment.
7.5. A Fourth Experiment: QGPML2.
8. Conclusion
Appendix
References

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