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, and we find evidence for state dependence in self-reported health.