A bivariate stochastic Gompertz diffusion model: statistical aspects and application to the joint modeling of the Gross Domestic Product and CO2 emissions in Spain

In this paper we propose a bivariate stochastic Gompertz diffusion model as the solution for a system of two Itô stochastic differential equations (SDE) that are similar as regards the drift and diffusion coefficients to those considered in the univariate Gompertz diffusion model, which has been the object of much study in recent years. We establish the probabilistic characteristics of this model, such as the bivariate transition density, the bidimensional moment functions, the conditioned trend functions and in particular, the correlation function between each of the components of the model. We then go on to study the maximum likelihood estimation of the bidimensional drift and the diffusion matrix of the diffusion in question, proposing a computational statistical methodology for this purpose based on discrete observations over time, for both components of the model. By these means we are able to achieve the maximum likelihood estimation of the trend and correlation functions and thus establish a method for trend analysis, which we apply to the real case of two dependent variables, Gross Domestic Product (GDP) and CO2 emission in Spain, the joint dynamic evolution of which is modeled by the proposed Gompertz bidimensional model. This implementation is carried out on the basis of annual observations of the variables over the period 1986–2003. The application is a new methodology in environmental and climate change studies, and provides an alternative to other approaches of a more econometric nature, or those corresponding to the methodology of secular trends in Time Series. Copyright © 2008 John Wiley & Sons, Ltd.