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

Abstract

SUMMARYIn 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 CO 2 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][1987][1988][1989][1990][1991][1992][1993][1994][1995][1996][1997][1998][1999][2000][2001][2002][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.

Detection, modelling and estimation of non-linear trends by using a non-homogeneous Vasicek stochastic diffusion. Application to CO2 emissions in Morocco

“…The exogenous variables for the application under analysis were selected on the basis of the relevant information concerning the role played by GDP and TEC in the evolution of CO 2 emissions in countries such as Morocco. The influence of GDP on CO 2 emissions in countries with a sharply increasing level of GDP, as is the case of Morocco, has been compared with previously studied cases, for example that of Spain (see, for example, Gutiérrez et al 2008a). With respect to the exogenous factor TEC, note that, in Morocco, a high percentage of the electrical energy consumed is generated in thermal power stations, and that these are known to be an important source of CO 2 emissions; for example, in 2002, 91.2% of the electricity consumed in Morocco was derived from this source.…”

The Stochastic Modified Lundqvist-Korf Diffusion Process: Statistical and Computational Aspects and Application to Modeling of the CO2 Emission in Morocco

“…Note that the non-homogeneity in this process is obtained by adding exogenous factors such as external variables that affect the drift of the homogeneous Vasicek model. Moreover, Gutiérrez et al [10] proposed the bivariate stochastic Gompertz diffusion model as the solution for a system of two Itô's stochastic differential equations (SDEs). The drift and diffusion coefficients are similar to those considered in the univariate stochastic Gompertz diffusion process.…”

The effect of nested grid sampling on the parameter estimation of a spatial Gompertz diffusion

“…In energy studies, Gutiérrez et al (2005a, b) used them for explaining the growth of the natural gas consumption in Spain in comparison with other models. Recently, in cell growth studies, Albano and Giorno (2006) considered a Gompertz model for modeling the tumor growth and, in environmental sciences, Gutiérrez et al (2008a) used a bivariate stochastic Gompertz diffusion for modeling the gross domestic product and CO 2 emissions in Spain.…”