This article 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 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 that 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 generalized linear model 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.Condition 4. The latent process, ¹˛t º, is strictly stationary, Gaussian and strongly mixing with the mixing coefficients satisfying P 1 hD0 .h/ =.2C / < 1 for some > 0.Conditions for a unique asymptotic limit of the marginal likelihood estimators are also required. Denote the marginal probability of j successes in m t trials at time t as 1 x nt defines a sequence of non-random vectors. Let U nt D .U 1;nt ; U 2;nt / be the joint vector at time t ; then each dimension of ¹U nt º is uniformly bounded, strongly mixing and E.U nt / D 0. We need to prove that a 1 U 1;n C a T 2 U 2;n has normal distribution for arbitrary

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.

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Discrete time series data is seen in a wide variety of disciplines including biology, medicine, psychology, criminology and economics. However, traditional methods of detecting serial correlation in time series are not specifically designed for detecting serial dependence in discrete-valued time series. Thus new methods are needed to provide informative and implementable testing approaches.This thesis is concerned with detection and estimation of serial dependence for a variety of observation-driven and parameter-driven models for regression analysis in binary and binomial time series. Generalised linear models (GLMs) are widely used for modelling discrete-valued data but do not allow for serial dependence and, as a result, inferences about regression effects may be invalid for time series application. Two classes of extended GLM have arisen to deal with this issue: observation-driven models and parameter-driven models, in which the serial dependence of the former relies on previous observations and residuals, and the serial dependence of the latter derives from an unobserved latent process. This thesis is structured in two parts corresponding to these two model classes. Chapter 1 provides a review of these models and existing methods for detecting and estimating serial dependence in them. Chapters 2 to 4 focus on observation-driven models and Chapters 5 to 7 focus on parameterdriven models. Chapter 8 distils the main results and conclusions from the thesis and suggests future research opportunities. The thesis proposes the use of score tests because they can be implemented using standard GLM fitting software or, for the parameter-driven models, software for fitting generalised linear mixed models, which is also readily available in most advanced statistical packages.