  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.
