Table 2 gives (under "unrestricted model") our estimate of the pooled regression. The standard errors are high. Yet the joint restriction that all coefficients are zero is comfortably rejected (at probability levels below the 0.01% level). This suggests that multicollinearity may be inflating the standard errors. Two joint restrictions were very easily accepted; the first is the restriction that b21=b41=0 and the second is that b11= b41= 0 (a joint F-test accepted both at the 45% level or better). Standard model selection criteria (adjusted R2, mean square error) do not suggest any non-arbitrary way to choose between these two restricted forms. So we present both in Table 2. There were no other linear parameter restrictions that could be accepted statistically. The standard errors drop substantially in the restricted forms, and most individual coefficients are significantly different from zero at the 1% level or better.
Table 3 gives the fixed effects model. The same two restrictions perform well, and no other restrictions do. Again individual parameters are significantly different from zero in the restricted model, and in most cases this is true at the 1% level or better. There are strong population growth effects, implying a rejection of the commonly imposed homogeneity restriction.
Table 4 gives the implied elasticities at sample mean points. The choice between the two restricted forms matters little to the estimated elasticities. However, the pooled and fixed effects regressions yield quite different results. Those from the pooled model appear to be more believable. Our reasoning is as follows. Holding income per capita constant, total income must grow at the same rate as population. So the population elasticity of emissions when holding average income constant should equal the income elasticity when holding population constant. This prediction is confirmed quite well for the pooled model; both elasticities are close to one. However, for the fixed effects model, the population elasticity is far too high relative to the income elasticity.
Our concerns about the fixed effects model were enhanced when we followed the suggestion of Griliches and Hausman (1986) of comparing it to a difference model (in which we took the first difference over time of equation 7). If there is a substantial time-varying measurement error problem leading to biases in the fixed effects estimator, then its parameter estimates should differ noticeably from those of the difference estimator. That is what we found. Coefficient estimates differed appreciably and many variables which are significant in the fixed effects model were insignificant in the difference model.
So our preferred specification indicates a significant negative impact of higher inequality on the level of emissions and an income elasticity of about one, with total emissions being roughly proportional to total income.
The pooled model in Table 2 indicates that the income elasticity of emissions declines as average income increases. And this effect persists when we allow for country-level fixed effects, as indicated by the results for the fixed effects model in Table 3.
To help see what this implies for the effects of redistribution between countries, we simulated the effect of taking 1% of the income of the richest five countries in our sample (in terms of income per capita) and transferring that sum of money to the poorest five countries; the transfer was assumed to preserve inequality within both the donor and recipient countries (so that the five rich countries lost a fixed percentage of their income, while the poorest five countries gained a fixed percentage). Obviously emissions fall in the five rich countries, and rise in the five poor ones. Taking the 10 countries as a whole, total emissions increased by 0.49%. If instead one transfers 5% of the income of the richest five countries to the poorest five, then emissions increase by 2.40%. Thus the elasticity of emissions to redistribution from the richest tenth of countries to the poorest tenth is about 0.5.
Both the pooled and fixed effects models also indicate that the income elasticity of carbon emissions (equation 9) is an increasing function of the Gini index (b11 and/or b21 positive). The higher the inequality the higher the impact of a given rate of growth on emissions. Figure 2 shows how the income elasticity rises with rising inequality (at the sample mean log income, and using restricted form B of the pooled model; the figure looks very similar for form A). At a relatively low Gini index of 25% the income elasticity is 0.75, while it is 1.98 at a Gini of 60%.
Similarly, the elasticity of emissions to inequality within countries is an increasing function of average income. (By symmetry of second cross-partial derivatives, the derivative of equation (9) w.r.t. I is identical to that of (10) w.r.t. lnY.) We could re-draw Figure 2 with the inequality derivative of emissions on the vertical axis and log mean income on the horizontal axis; with growth in average income the inequality derivative will increase, getting closer to zero. While higher inequality within a country lowers emissions, the impact is smaller at higher average incomes.
One must be skeptical of basing longer-term forecasts on models calibrated to data over only 15 years or so. However, as a means of understanding better the properties of the models we have estimated, it is of interest to ask whether our results suggest that carbon emissions could start to fall in a growing economy with sufficiently low inequality. With lower inequality, the income elasticity falls; with a higher mean income (due to growth) the elasticity falls even further. So could the income elasticity become negative within reasonable bounds? Figure 3 gives forecasts implied by our pooled model for an economy with a growth rate in income per capita of 5% per year. We use the parameters of the unrestricted (pooled) model and all variables are set at the mean points (with income set at the mean initially), except for the Gini index which is set at one standard deviation above and below the mean. (The mean is 35.9 and the standard deviation is 8.6.) The low-inequality trajectory entails an initially higher emission rate, but it also results in a lower elasticity of emissions to growth, with the effect that it eventually achieves a lower emission level than even the high inequality trajectory; the low inequality path overtakes the other two after 27 years, and emissions start to decline after 19 years.