In many settings, administrative data record choices only for participants in a public program, while participation outside the public sector (e.g., private provision or non-use) is observed only through coarse aggregates or implicitly ignored. This paper develops a structural framework that jointly models participation in a public program and choice among public providers, using external demographic and epidemiological information to discipline the outside option. We illustrate our approach using hospital choice in the Italian National Health Service. Relative to conventional nested-logit estimates, our method yields stronger estimated substitution patterns, sharper within-nest correlation, and economically meaningful responses on the extensive margin. The framework is applicable to a wide class of public programs characterized by partial observability of participation.
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