A bivariate stochastic Gamma diffusion model: statistical inference and application to the joint modelling of the gross domestic product and CO₂ emissions in Spain

Abstract

This paper examines the use of a bivariate stochastic Gamma diffusion model to represent the co-evolution of the stochastic variables CO₂ emission and gross domestic product (GDP) in Spain. These variables were selected in view of the strong correlation between them. We compare the results obtained to those provided by the Gamma one-dimensional process with exogenous factors, taking CO₂ emission as an endogenous variable and GDP as the exogenous factor. This methodology was applied to a real case, with two dependent variables: firstly, GDP and CO₂ emission from the combustion of fossil fuels (gas, liquid and solid fuels) and cement manufacture in Spain. And secondly, with GDP and CO₂ emission from the consumption and flaring of natural gas in Spain. The joint dynamic evolution of these factors is represented by the proposed model. In addition, a comparison is made with results obtained from fitting the data using the Gamma diffusion process with external factors, in which GDP is the variable containing the external information. This implementation was carried out on the basis of annual observations of the variables over the periods 1986–2008 and 1986–2009, respectively.

Appendix

Correlation function can be calculated for different times. When t ₀ < s ≤ t and i ≠ j, then:

$$ \varrho(x_i(t), x_j(s))=\frac{\left(e^{(s-t_0)b_{ij}}-1\right)}{\left(e^{(t-t_0)b_{ii}}-1\right)^{1/2}\left(e^{(s-t_0)b_{jj}}-1\right)^{1/2}} $$

for t = s:

$$ \varrho(x_i(t), x_j(t))=\frac{\left(e^{(t-t_0)b_{ij}}-1\right)}{\left(e^{(t-t_0)b_{ii}}-1\right)^{1/2}\left(e^{(t-t_0)b_{jj}}-1\right)^{1/2}}. $$

In the above example, we assume that the significant correlation is that which occurs at equal times, because the CO₂ emissions for a given year would logically be related to the GDP for previous years, but are more strongly associated with the GDP for the current year.

Proof

The marginal conditional and non-conditional moments of order r (${r\in {\mathbb{N}}^{*}}$) can be obtained from the function generating the random vector

$$ Z(t)=\log\left[x(t)\mid x(s)=x_s\right], $$

which follows the law ${\mathcal{N}}_2\left( \mu(s, t, x_s); (t-s)B\right),$ and is expressed as follows, for ${\lambda\in {\mathbb{R}}^2}$

$$ {\mathbb{E}}(e^{\lambda'Z(t)})=\exp\left\{ \lambda'\mu(s, t, x_s)+\frac{t-s}{2}\lambda' B\lambda\right\} $$

with

$$ \mu(s, t, x)=\log(x)+ a\log \left(\frac{t}{s}\right) -\left(\beta+\frac{b}{2}\right)(t-s)=\log(x)+\delta (s,t), $$

where

$$ \delta (s,t)=a\log \left(\frac{t}{s}\right) -\left(\beta+\frac{b}{2}\right)(t-s). $$

For particular values of the λ = (0, r)′ or λ = (r, 0)′ (${r\in {\mathbb{N}}^{*}}$), we obtain, for example, the marginal conditional trend functions of order r of the process and which have the following form, for i = 1, 2

$$ {\mathbb{E}}\left( x^r_i(t)\mid x(s)=x_{s}\right)= \exp\left(r\mu_i(s, t, x_{s})+\frac{r^2(t-s)}{2} b_{ii}\right) $$

and for λ = (r ₁, r ₂)′ (${r_1, r_2 \in {\mathbb{N}}^{*}}$), we obtain the joint conditional trend of the process

$$ \begin{aligned} {\mathbb{E}}\left( x^{r_1}_1(t)x^{r_2}_2(t)\mid x(s)=x_{s}\right)=\,&\exp\left(r_1\mu_1(s, t, x_{s})+r_2\mu_2(s, t,x_{s})\right. \\ & \left.+\frac{(t-s)}{2}(r^2_1b_{11}+r^2_2b_{22}+2r_1r_2b_{12})\right). \end{aligned} $$

Assuming the initial condition P(x(t ₀) = x _{t_0}) = 1, the marginal non conditional trend of the process is expressed as

$$ {\mathbb{E}}\left( x^r_i(t)\right)=\exp\left(r\mu_i(t_0, t, x_{t_0})+\frac{r^2(t-t_0)}{2} b_{ii}\right) $$

and the joint non conditional trend of the process is expressed as

$$ \begin{aligned} {\mathbb{E}}\left( x^{r_1}_1(t)x^{r_2}_2(t)\right)=\,&\exp\left(r_1\mu_1(t_0, t, x_{t_0})+r_2\mu_2(t_0, t,x_{t_0})\right. \\ &\left. +\frac{(t-t_0)}{2}\left(r^2_1b_{11}+r^2_2b_{22}+2r_1r_2b_{12}\right)\right). \end{aligned}$$

Expressed in terms of δ(s, t) this is

$$ \begin{aligned} {\mathbb{E}}\left( x^r_i(t)\mid x(s)=x_{s}\right)=& \exp\left(r\log(x_{s,i})+r\delta_i (s,t)+\frac{r^2(t-s)}{2} b_{ii}\right) \\ =& x_{s,i}^r\exp\left(r \delta_i (s,t)+\frac{r^2(t-s)}{2} b_{ii}\right) \end{aligned} $$

and

$$ \begin{aligned} {\mathbb{E}}\left( x^{r_1}_1(t)x^{r_2}_2(t)\mid x(s)=x_{s}\right)=& x_{s,1}^{r_1}x_{s,2}^{r_2}\exp\left(r_1\delta_1 (s,t)+r_2\delta_2 (s,t)\right. \\ &\left. +\frac{(t-s)}{2}\left(r^2_1b_{11}+r^2_2b_{22}+2r_1r_2b_{12}\right)\right). \end{aligned} $$

For t ₀ < s ≤ t the covariance function is

$$ \hbox{Cov}\left( x_1(t), x_2(s)\right)={\mathbb{E}}\left(x_1(t)x_2(s)\right)-{\mathbb{E}}\left(x_1(t)\right){\mathbb{E}}\left(x_2(s)\right) $$

$$ \begin{aligned} {\mathbb{E}}\left(x_1(t)x_2(s)\right)=& {\mathbb{E}}\left[{\mathbb{E}}\left(x_1(t)x_2(s)/x(s)\right)\right] =\, {\mathbb{E}}\left[x_2(s){\mathbb{E}}\left(x_1(t)/x(s)\right)\right] \\ =\,&{\mathbb{E}}\left[x_2(s)x_1(s)\right]\exp\left(\delta_1 (s,t)+\frac{(t-s)}{2}b_{11}\right) \\ =\,& x_{t_0,1}x_{t_0,2}\exp\left(\delta_1 (t_0,s)+\delta_2 (t_0,s)+\frac{(s-t_0)}{2}\left(b_{11}+b_{22}+2b_{12}\right)\right)\exp\left(\delta_1 (s,t)+\frac{(t-s)}{2} b_{11}\right) \\ =\,&x_{t_0,1}x_{t_0,2}\exp\left(\delta_1 (t_0,s)+\delta_1 (s,t)+\delta_2 (t_0,s)+\frac{(s-t_0)}{2}b_{11}+\frac{(t-s)}{2}b_{11}+\frac{(s-t_0)}{2}b_{22}+(s-t_0)b_{12}\right). \\ \end{aligned} $$

Using the fact that

$$ \begin{aligned} &\delta_1 (t_0,s)+\delta_1 (s,t)=\delta_1 (t_0,t) \\ &\frac{(t-s)}{2}b_{11}+\frac{(s-t_0)}{2}b_{11}=\frac{t-t_0}{2}b_{11}. \end{aligned} $$

we obtain

$$ {\mathbb{E}}\left(x_1(t)x_2(s)\right)= x_{t_0,1}x_{t_0,2}\exp\left(\delta_1 (t_0,t)+ \frac{t-t_0}{2}b_{11}+\,\delta_2 (t_0,s)+\frac{(s-t_0)}{2}b_{22}+ (s-t_0)b_{12}\right). $$

The non conditional marginal trend expressions for x ₁(t) and x ₂(s) are:

$$ \begin{aligned} {\mathbb{E}}\left(x_1(t)\right)=\,& x_{t_0,1}\exp\left(\delta_1 (t_0,t)+ \frac{t-t_0}{2}b_{11}\right)\\ {\mathbb{E}}\left(x_2(s)\right)=\,& x_{t_0,2}\exp\left(\delta_2 (t_0,t)+ \frac{s-t_0}{2}b_{22}\right) \end{aligned} $$

it is then deduced that

$$ \begin{aligned} \hbox{Cov}\left( x_1(t), x_2(s)\right)=\,& x_{t_0,1}x_{t_0,2}\exp\left(\delta_1 (t_0,t)+ \delta_2 (t_0,s)+\frac{(t-t_0)}{2}b_{11}+\frac{(s-t_0)}{2}b_{22}+(s-t_0)b_{12}\right) \\ &-x_{t_0,1}x_{t_0,2}\exp\left(\delta_1 (t_0,t)+ \delta_2 (t_0,s)+\frac{(t-t_0)}{2}b_{11}+ \frac{(s-t_0)}{2}b_{22}\right) \\ =\,&x_{t_0,1}x_{t_0,2}\exp\left(\delta_1 (t_0,t)+ \delta_2 (t_0,s)+\frac{(t-t_0)}{2}b_{11}+ \frac{(s-t_0)}{2}b_{22}\right)\left(e^{(s-t_0)b_{12}}-1\right). \cr \end{aligned} $$

The marginal variance function of the process is:

$$ \begin{aligned} \hbox{Var}\left(x_1(t)\right)=\,& {\mathbb{E}}\left(x_1(t)^2\right)-{\mathbb{E}}\left(x_1(t)\right)^2=x_{t_0,1}^2\exp\left\{2\delta_1 (t_0,t)+ 2(t-t_0)b_{11} \right\}- x_{t_0,1}^2\exp\left\{2\delta_1 (t_0,t)+ (t-t_0)b_{11}\right\} \\ =\,&x_{t_0,1}^2\exp\left\{2\delta_1 (t_0,t)+ (t-t_0)b_{11} \right\}\left(e^{(t-t_0)b_{11}}-1\right). \end{aligned} $$

In the same way

$$ \hbox{Var}\left(x_2(s)\right)=x_{t_0,2}^2\exp\left\{2\delta_2 (t_0,s)+\, (s-t_0)b_{22} \right\}\left(e^{(s-t_0)b_{22}}-1\right). $$

Therefore

$$ \begin{aligned} \varrho(x_1(t), x_2(s))=&\frac{\hbox{Cov}\left( x_1(t), x_2(s)\right)}{\sqrt{\hbox{Var}\left(x_1(t)\right)}\sqrt{\hbox{Var}\left(x_2(s)\right)}} \\ =&\frac{x_{t_0,1}x_{t_0,2}\exp\left\{\delta_1 (t_0,t)+ \delta_2 (t_0,s)+\frac{(t-t_0)}{2}b_{11}+ \frac{(s-t_0)}{2}b_{22}\right\}\left(e^{(s-t_0)b_{12}}-1\right)}{x_{t_0,1}x_{t_0,2}\exp\left\{\delta_1 (t_0,t)+\frac{t-t_0}{2}b_{11}+\delta_2 (t_0,s)+\frac{s-t_0}{2}b_{22} \right\}\left(e^{(t-t_0)b_{11}}-1\right)^{1/2}\left(e^{(s-t_0)b_{22}}-1\right)^{1/2}} \\ =&\frac{\left(e^{(s-t_0)b_{12}}-1\right)}{\left(e^{(t-t_0)b_{11}}-1\right)^{1/2}\left(e^{(s-t_0)b_{22}}-1\right)^{1/2}}. \end{aligned} $$

In the same way

$$ \varrho(x_1(s), x_2(t))=\frac{\left(e^{(s-t_0)b_{12}}-1\right)}{\left(e^{(s-t_0)b_{11}}-1\right)^{1/2}\left(e^{(t-t_0)b_{22}}-1\right)^{1/2}} $$

which can be expressed as

$$ \varrho(x_i(t), x_j(s))=\frac{\left(e^{(s-t_0)b_{ij}}-1\right)}{\left(e^{(t-t_0)b_{ii}}-1\right)^{1/2}\left(e^{(s-t_0)b_{jj}}-1\right)^{1/2}} $$

for i ≠ j and t ₀ < s ≤ t.