We study a fixed-$T$ panel data logit model for ordered outcomes that accommodates fixed effects and state dependence. We provide identification results for the autoregressive parameter, regression coefficients, and the threshold parameters in this model. Our results require only four observations on the outcome variable. We provide conditions under which a composite conditional maximum likelihood estimator is consistent and asymptotically normal. We use our estimator to explore the determinants of self-reported health in a panel of European countries over the period 2003-2016. We find that: (i) the autoregressive parameter is positive and analogous to a linear AR(1) coefficient of about 0.25, indicating persistence in health status
(ii) the association between income and health becomes insignificant once we control for unobserved heterogeneity and persistence.