Second order longitudinal dynamic models with covariates: estimation and forecasting

In this paper, we propose an extension to the first-order branching process with immigration in the presence of fixed covariates and unobservable random effects. The extension permits the possibility that individuals from the second generation of the process may contribute to the total number of offsprings at time <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$t$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>t</mi> </math> </EquationSource> </InlineEquation> by producing offsprings of their own. We will study the basic properties of the second order process and discuss a generalized quasilikelihood (GQL) estimation of the mean and variance parameters and the generalized method of moments estimation of the correlation parameters. We will discuss the asymptotic distribution of the GQL estimator by first deriving the influence curve of the estimator. For the fixed effects model we shall derive a forecasting function and the variance of the forecast error. The performance of the proposed estimators and forecasts will be examined through a simulation study. Copyright Springer-Verlag Berlin Heidelberg 2014