It is apparently still not evident to some people that carbon dioxide emissions are a cause of global warming. It is perhaps not surprising that those with a sceptical mind should be more sceptical about global warming. Sceptics are well aware of the propensity for deception and delusion that can affect people's perceptions and thinking. But are concerns over global warming just socially induced panic? Science is usually on the side of sceptics but in this case the apparent preponderance of world scientific opinion on the matter seems to be that global warming is real, human-induced and that the prognosis is dire.

In such a complex issue it is perhaps easy to cite counteracting tendencies as evidence of alternative hypotheses that negate the global warming thesis. Some factors may cause warming, some cooling. Which is of greater significance, if any? It is difficult to find definitive proof of causation that would be convincing to all sceptics. Proponents of the dangers of global warming rely on complex global climate models that are opaque to non-experts.

In view of all this, and as a former teacher of Economic Statistics 101 at Monash University, I have been motivated to get some data together and see what light a multivariate regression analysis might throw on the issue . In this procedure, a simple linear econometric model may be used to provide an explanation of average global temperatures over the last century. Greenhouse gases and solar radiation variation due to sun spots are contended to be warming factors and atmospheric sulphates and particulates are contended to be cooling factors. If atmospheric CO₂ concentrations were found to be a significant explanatory factor in such a model, then this may be convincing to those who suspect that far more complex and less transparent models may provide contrived results.

A simple linear model, therefore, may be as follows:

        T = a₀ + a₁ CO₂ + a₂ Sulphate + a₃ Solar + a₄ Volcano

where
        T is global average temperature, difference from 1951-80 average, (degrees C)
        CO₂ is atmospheric carbon dioxide concentration, (ppm)
        Sulphate is atmospheric sulphate aerosols, (kg/m² x 10^-12)
        Solar is solar radiation, (annual watts/m²)
        Volcano is volcanic aerosol particulate mass, (million tonnes) and
        a₀, a₁, a₂, a₃, and a₄ are parameters to be estimated.

These variables are generally considered to be amongst the most relevant exogenous determinants of climate. The model has been estimated by least squares regression for the period 1880-2007 using data obtained mainly from Meehl et al. For such a linear model, all likely exogenous explanators of global temperature should be included, in order to obtain unbiased results. However due to multicolinearity between greenhouse gases, only CO₂ was included. (Note that water vapour is not an exogenous variable.) The results are as follows:
 

Variable	Estimated Coefficient	Standard  Error	t-statistic	P-value
constant	-110.987	32.9797	-3.36531	[.001]
CO2	.011391	.890797E-03	12.7874	[.000]
Sulphate	-.729352E+09	.166501E+09	-4.38047	[.001]
Solar	.787637E-04	.242126E-04	3.25301	[.000]
Volcano	-.011865	.350217E-02	-3.38799	[.001]

Required = .812143        Durbin-Watson = 1.40151

                    Table 1. Equation estimation results

Applying such an analysis to global aggregates obviously sacrifices the benefits of geographical precision. However the results obtained are satisfactory in that more than 80% of the variation in global temperature is explained by the variables provided. (R² = .81) All the coefficients have the expected sign. The relative numerical magnitude of the estimated coefficients is a function of the units of measurement. What is perhaps more important is whether the estimates are statistically significant.

The results indicate that carbon dioxide is indeed a statistically significant determinant of global temperature. The value of 12.79 means that we can reject the null hypothesis that carbon dioxide has no effect on global temperature, with at least 99% confidence. The probability value indicates that there is less than 1% chance that these results could be obtained if CO₂ had no effect.

The positive sign of the estimated CO₂ coefficient indicates that it does contribute to global warming. The coefficient value of  .01 indicates that every 100 ppm increase in atmospheric CO₂ is associated with a temperature rise of approximately 1 degree C. The results also show that sulphates and volcanoes have a significant negative (cooling) effect, and that variations in solar radiation have a significant positive (warming) effect on temperature. The magnitude of the solar coefficient indicates that peak to trough, the solar cycle contributes about 0.2 degrees C to global temperature, or the equivalent of about ten years of CO₂ emissions.

                    Figure 1. Actual and predicted temperature

The graph shows the temperature data ( blue) and the model's predicted values (red), based on the combined contributions of CO₂ emissions, solar variation, sulphates and volcanic particulates. The green line shows the contribution to temperature of CO₂ emissions alone. The yellow line indicates the contribution made by solar variation. The apparent post 2000 cooling, relative to the CO₂ only projection, (actual and predicted below green line), can be mostly explained by the persistently down-cycle solar variation.

This methodology has its limits, but the results broadly replicate what is reported from the climate models. Do they prove that CO₂ causes global warming? No, because proof is not something that is obtained from statistical inference. Do the results suggest that there is cause for concern, to the extent that we should take significant action? Yes. Prudent risk management means that we should give the planet the benefit of the doubt. If it transpires that our actions mean that we have conserved fossil fuels unnecessarily, so be it. At least our children may survive to thank us for it.

John L Perkins, Senior Economist, National Institute of Economic and Industry Research, Melbourne, Australia.
5 August 2008

Notes:
1.    To all my former Eco Stats students who just could not see the point of learning this stuff :  see, it can be useful!
2.     Temperature data obtained from GISS Surface Temperature Analysis
3. Combinations of Natural and Anthropogenic Forcings in Twentieth-Century Climate, Gerald A. Meehl, Warren M. Washington, Caspar M. Ammann, Julie M. Arblaster, T. M. L. Wigley, and Claudia Tebaldi, National Center for Atmospheric Research, Boulder, Colorado, March 2004.
4.     For a more detailed explanation of the variables see 20C3M experiments.
5.     For full statistical results and input data see TSP output