 Logistic-normal regression with autoregressive random effects

A logistic-normal count model takes $Y_i\mid {\theta }_i\sim \mathrm{Bin}(n_i,{\mathrm{logit}}^{-1}{\theta }_i)$ and ${\theta }_i=x_i^T\beta +\lambda Z_i+{\eta }_i$, where $Z$ is a stationary unit-variance autoregressive process of order one with coefficient $a$ and the independent ${\eta }_i$ have normal distribution with variance $v$. The marginal latent variance is ${\lambda }^2+v$, and its off-diagonal covariance is ${\lambda }^2{a}^{|i-j|}$. Random success probabilities produce overdispersion relative to the binomial distribution.