  This paper develops asymptotic theory for estimation of parameters in
regression models for binomial response time series where serial dependence is
present through a latent process. Use of generalized linear model (GLM)
estimating equations leads to asymptotically biased estimates of regression
coefficients for binomial responses. An alternative is to use marginal
likelihood, in which the variance of the latent process but not the serial
dependence is accounted for. In practice this is equivalent to using
generalized linear mixed model estimation procedures treating the observations
as independent with a random effect on the intercept term in the regression
model. We prove this method leads to consistent and asymptotically normal
estimates even if there is an autocorrelated latent process. Simulations
suggest that the use of marginal likelihood can lead to GLM estimates result.
This problem reduces rapidly with increasing number of binomial trials at each
time point but, for binary data, the chance of it can remain over 45% even in
very long time series. We provide a combination of theoretical and heuristic
explanations for this phenomenon in terms of the properties of the regression
component of the model and these can be used to guide application of the method
in practice.
