  Binomial time series in which the logit of the probability of success is
modelled as a linear function of observed regressors and a stationary latent
Gaussian process are considered. Score tests are developed to first test for
the existence of a latent process and, subsequent to that, evidence of serial
dependence in that latent process. The test for the existence of a latent
process is important because, if one is present, standard logistic regression
methods will produce inconsistent estimates of the regression parameters.
However the score test is non-standard and any serial dependence in the latent
process will require consideration of nuisance parameters which cannot be
estimated under the null hypothesis of no latent process. The paper describes
how a supremum-type test can be applied. If a latent process is detected,
consistent estimation of its variance and the regression parameters can be done
using marginal estimation which is easily implemented using generalised linear
mixed model methods. The test for serial dependence in a latent process does
not involve nuisance parameters and is based on the covariances between
residuals centered at functions of the latent process conditional on the
observations. This requires numerical integration in order to compute the test
statistic. Relevant asymptotic results are derived and confirmed using
simulation evidence. Application to binary and binomial time series is made.
For binary series in particular, a complication is that the variance of the
latent process, even if present, can be estimated to be zero with a high
probability.
