We estimate an econometric model of emission rates motivated by the arguments in section 3, using the panel of data over time by countries described in the last section. We begin by focusing on the bivariate relationship with average income. This will motivate our specification choice for estimating a richer model incorporating inequality within countries.
To capture the issues raised in section 3, we want a specification which allows the third derivative to be non-zero. In Figure 1 we plot the data; there are 738 observations.[10] We also give fitted values from a cubic function of income, estimated as a simple pooled model by OLS, as given in Table 1.[11] In addition to a cubic function of income, the regression includes a time trend and population. The time trend is negative and the population effect is positive.[12]
The bivariate relationship between carbon emissions and average income suggests a decreasing MPE up to relatively medium-high incomes, but a significantly positive third derivative so that the MPE starts to rise above some point. It follows that the trade off between emissions and inequality between countries improves as income grows. Indeed, the relationship becomes convex at sufficiently high incomes, eliminating the trade off.
This result is not, however, robust to allowing for country-level fixed effects; as can also be seen from Table 1, adding country dummy variables renders the cubic term in average income insignificant (Table 1). The evidence of a positive third derivative in Figure 1 is clearly driven by the differences between countries rather than changes over time within countries.
While the cubic model in income levels provides a straightforward test for how the MPE varies with average income, it can be improved upon as a specification for the bivariate relationship. We also tried regressing the log of the emission rate against both a quadratic and cubic function of the log of average income. The cubed term in log income was not needed and even with one less parameter, the quadratic in logs gave a higher R2 than the cubic in levels (0.78, as compared to 0.64 for the cubic model in levels above). However, as Davidson and Mackinnon (1993) point out, it is not strictly valid to compare the R2's of these models. Under the assumption of normally distributed errors, one can compare the values of the log likelihood functions from the two competing models. The loglikelihood function from the linear cubic model is -2978.82 and from the log quadratic model is -2500.69.[13] These values suggest that the log quadratic model fits the data better than the linear model, and is more parsimonious. We thus use the log specification in our subsequent analysis.
As noted in the last section, while the data on carbon emissions and average incomes are fairly complete over time, that on inequality is sparse. In our view, there can be little hope of meaningful results treating inequality as time varying in this context. This entails that rather strong assumptions are needed about omitted variables to identify the effects of inequality on emissions, although (as we will see) effects on the growth elasticity of emissions can be identified more confidently.
We want to introduce inequality within countries into the emission-income relationship in as flexible a way as possible. So we postulate that all parameters in the relationship are a function of measured inequality. Combining these considerations, our econometric model of carbon emissions in country j at date t takes the form:
where the b parameters are assumed to be linear functions of measured income inequality:
Equation (7) also includes a country fixed effect (h) which is a linear function of inequality:
where n is an unobserved country fixed effect.
The income elasticity of carbon emissions implied by equation (7) is
The effect of inequality on emissions is given by:
In defining the elasticity of emissions to population growth one has to be careful about whether one is referring to per capita emissions, and whether or not one is holding total income or income per capita constant. Presumably one is more interested in the effect on total emissions. If total income is held constant then the elasticity is given by:
If instead income per capita is held constant then the elasticity is given by:
We estimate equation (7) under two alternative assumptions. First we assume that n is uncorrelated with the other variables, and estimate (7) as a simple pooled model using a heteroscedasticity-robust OLS method (the mean-deviations of n simply go into the error term, and one adds an intercept and term in I to (7)). The second method recognizes that the unobserved fixed effect could be correlated with the other regressors in the model, creating bias in the OLS estimator of (7). We eliminate the fixed effect by adding country dummy variables.
It is difficult to say on a priori grounds which of these approaches should be preferred. Ignoring measurement error, we would generally expect the income and population elasticities to be better estimated by the fixed effects model, since it purges the estimates of correlated fixed effects. However, with time varying measurement errors it may well be that the signal to noise ratio deteriorates so much in the fixed effects estimator that it may entail an even greater bias than the OLS pooled model (Griliches and Hausman, 1986; Biørn, 1996).