“Small Data”: Inference with Occasionally Observed States
Alexandros Gilch, Andreas Lanz, Philipp Müller, Gregor Reich, Ole Wilms
Management Science · 2026 · Volume 72 · Issue 6
Abstract
We study the estimation of dynamic economic models for which some of the state variables are observed only occasionally by the econometrician—a common problem in many fields, ranging from marketing to finance to industrial organization. If those occasional state observations are serially correlated, the likelihood function of the model becomes a high-dimensional integral over a nonstandard domain. We generalize the recursive likelihood function integration procedure to incorporate the occasional observations, enabling likelihood-based inference in such estimation problems. In extensive Monte Carlo studies, we demonstrate the favorable properties of the proposed method for identifying all model parameters and compare it to alternative methods. This paper was accepted by Raphael Thomadsen, marketing. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Collaborative Research Center Transregio 224]. P. Müller and G. Reich gratefully acknowledge the financial support of Kenneth Judd, senior fellow at the Hoover Institution. Supplemental Material: The online appendices and data files are available at https://doi.org/10.1287/mnsc.2022.00246 .
Keywords
maximum likelihood estimation, occasional state observations, recursive likelihood function integration, interpolation, numerical quadrature, Markov models, dynamic discrete choice models, long-run risk models
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