Identification in Linear Quantile Panel Models

This paper studies identification in linear quantile panel models with unrestricted individual heterogeneity when the number of time periods is fixed and small. We impose quantile strict exogeneity, whereby the conditional quantile restriction holds given the individual’s complete regressor history and latent individual effect, but otherwise allow the disturbances to be arbitrarily dependent over time. We show that the common slope coefficient can be partially identified from the requirement that the residuals in all periods be compatible with a common latent individual effect. We characterize the sharp identified set as the set of coefficients for which there exists a latent coupling satisfying the period-specific quantile restrictions. This characterization yields observable crossing inequalities that provide computationally convenient outer bounds and an observable dual representation that is sharp. When the regressors and outcomes have finite support, the sharp identified set can be computed exactly using finite-dimensional linear programs, without discretizing the latent individual effect; for continuously distributed regressors, we develop nested dual-sieve procedures that converge to the sharp set. We further characterize how identification depends on directional variation in the regressor paths, showing that additional periods need not generate identification unless they provide sufficiently rich variation relative to the support of the composite residual. Under a finite-width composite-residual condition, we obtain explicit bounds on the identified set and give conditions under which even a two-period model is point identified. Finally, we consider a quantile-varying factor-loading extension in which the latent individual effect is common across quantiles, and show how cross-quantile restrictions can identify economically meaningful relative loadings.