MCMC conditional maximum likelihood for the two-way fixed-effects logit

Abstract

We propose a Markov Chain Monte Carlo Conditional Maximum Likelihood (MCMC-CML) estimator for the two-way fixed-effects logit model for dyadic data, typically used in network analyses. The proposed MCMC approach, based on a Metropolis algorithm, allows us to overcome the computational issues of evaluating the probability of the outcome conditional on nodes in- and out-degrees, which are sufficient statistics for the incidental parameters. Under mild regularity conditions, the MCMC-CML estimator converges to the exact CML one and is asymptotically normal. Moreover, it is more efficient than the existing pairwise CML estimator. We study the finite sample properties of the proposed approach by means of an extensive simulation study and three empirical applications, where we also show that the MCMC-CML estimator can be applied to logit models for binary panel data with both subject and time-fixed effects. Results confirm the expected theoretical advantage of the proposed approach, especially with small, concentrated, and sparse networks or with rare events in panel data.

Keywords:

• Directed network
• fixed effects
• link formation
• metropolis algorithm
• panel data

JEL Classification::

• C23
• C25
• C63

Acknowledgments

We are grateful to Bryan S. Graham, the Associate Editor and two anonymous Referees for their useful comments and suggestions. This paper also benefited from the reactions of audience at the 27th International Panel Data Conference and the 3rd Italian Workshop of Econometrics and Empirical Economics.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. The procedure in steps 1-2 can be inefficient with large networks and can be performed on a subset of randomly chosen rows and columns in order to speed up computations.

2. ML and BC estimators are computed by the R package alpaca, https://cran.r-project.org/package=alpaca

3. Experiments are run on a computer with two Intel Xeon CPU E5-2640 v4 2.40 GHz, 250 GB of RAM, with Debian GNU/Linux as operating system, running R version 4.2.2.

4. For the sake of brevity, we consider the three most realistic designs proposed in Graham (2017a). Design 1, Design 2, and Design 3 in the present work correspond to the design B.1, B.2, and B.3 presented in Graham (2017b), respectively.

5. Since we rely on the same simulation design, we report the results for the TL estimator presented in the original paper. As a consequence, some of the reported statistics are missing.

6. The proof to Theorem 1 in Geyer (1994) is derived under weaker conditions.

Additional information

Funding

Francesco Valentini would like to acknowledge the financial support by the European Union – Next Generation EU. Project Code: ECS00000041; Project CUP: C43C22000380007; Project Title: Innovation, digitalization and sustainability for the diffused economy in Central Italy – VITALITY.