  A common approach to analyzing categorical correlated time series data is to
fit a generalized linear model (GLM) with past data as covariate inputs. There
remain challenges to conducting inference for short time series length. By
treating the historical data as covariate inputs, standard errors of estimates
of GLM parameters computed using the empirical Fisher information do not fully
account the auto-correlation in the data. To overcome this serious limitation,
we derive the exact conditional Fisher information matrix of a general logistic
autoregressive model with endogenous covariates for any series length $T$.
Moreover, we also develop an iterative computational formula that allows for
relatively easy implementation of the proposed estimator. Our simulation
studies show that confidence intervals derived using the exact Fisher
information matrix tend to be narrower than those utilizing the empirical
Fisher information matrix while maintaining type I error rates at or below
nominal levels. Further, we establish that the exact Fisher information matrix
approaches, as T tends to infinity, the asymptotic Fisher information matrix
previously derived for binary time series data. The developed exact conditional
Fisher information matrix is applied to time-series data on respiratory rate
among a cohort of expectant mothers where it is found to provide narrower
confidence intervals for functionals of scientific interest and lead to greater
statistical power when compared to the empirical Fisher information matrix.
