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Verifying the existence of maximum likelihood estimates for generalized linear models

Sergio Correia, Paulo Guimarães, Thomas Zylkin

arXiv 5 Mar 2019 · Econometrics · publishedEconometric Reviews (2026) · 39 citations (OpenAlex)

arXiv:1903.01633 · PDF · DOI · OpenAlex · Extracted main text

Abstract

A fundamental problem with nonlinear models is that maximum likelihood estimates are not guaranteed to exist. Though nonexistence is a well known problem in the binary choice literature, it presents significant challenges for other models as well and is not as well understood in more general settings. These challenges are only magnified for models that feature many fixed effects and other high-dimensional parameters. We address the current ambiguity surrounding this topic by studying the conditions that govern the existence of estimates for (pseudo-)maximum likelihood estimators used to estimate a wide class of generalized linear models (GLMs). We show that some, but not all, of these GLM estimators can still deliver consistent estimates of at least some of the linear parameters when these conditions fail to hold. We also demonstrate how to verify these conditions in models with high-dimensional parameters, such as panel data models with multiple levels of fixed effects.

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Most heavily cited references

The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.

ReferenceIntensityMentionsSectionsMain text
1Verbeek, A (1989) The Compactification of Generalized Linear Models, in1.000124100%
2Geyer, C. J (2009) Likelihood Inference in Exponential Families and Directions of Recession1.00063100%
3Santos Silva, J. M. C. and S. Tenreyro (2010) On the Existence of the Maximum Likelihood Estimates in Poisson Regression0.97413592%
4Albert, A. and J. A. Anderson (1984) On the Existence of Maximum Likelihood Estimates in Logistic Regression Models0.9619489%
5Gourieroux, A. C., A. Monfort, and A. Trognon (1984) Pseudo Maximum Likelihood Methods: Theory0.9416483%
6Clarkson, D. B. and R. I. Jennrich (1991) Computing Extended Maximum Likelihood Estimates for Linear Parameter Models0.92815580%
7Eck, D. J. and C. J. Geyer (2021) Computationally efficient likelihood inference in exponential families when the maximum likelihood estimator does not exist0.87472100%
8Manning, W. G. and J. Mullahy (2001) Estimating Log Models: To Transform or Not to Transform?0.87462100%
9Aickin, M (1979) Existence of MLEs for discrete linear exponential models0.87452100%
10Geyer, C. J (1990) Likelihood and Exponential Families, Ph.D0.87452100%

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