The literature on global warming presumes that greenhouse effect of water vapor does not have to be considered explicitly because the level of water vapor will be a function of temperature and and the level of temperature will be determined by the concentration of the other greenhouse gases, notably CO₂. If the level of water vapor in the atmosphere were only a function of temperature then this might be valid but the relationship is more in the nature of a stochastic relationship; i.e.,

W = f(T) + u
 

where W is the water vapor concentration, T is the temperature and u is the effect of the the unmeasured variables affecting W. The variable u may be considered a random variable. The magnitude of the greenhouse effect from water vapor is so large compared to the other greenhouse gases that the fluctuations in u may be quantitatively more important than the fluctuations in the concentrations of the other greenhouse gases.

The complete model is of the nature

T = g(Z+W)
W = f(T) + u
 

where Z is the concentration of the other greenhouse gases besides water vapor. These should be weighted by their radiative efficiencies relative to CO₂. Likewise W should be measured in terms of equivalent amount of CO₂. Climate Change 2001, the report of the Intergovernmental Panel on Climate Change (IGCC) does not give the radiative efficiency of water vapor or of that CO₂ so this is not a feasible computation at this time, but in principle it could be done.

The incremental changes about an average would be approximately

ΔT = g'(Z+W)[ΔZ + f'(T)ΔT + u]
and solving for ΔT
ΔT = [1/(1 − g'(Z+W)f'(T)]g'(Z+W))[ΔZ + u]
 

The factor 1/(1 − g'(Z+W)f'(T)] is approximately two, according to the IGCC 2001. This means that g'(Z+W)f'(T) is approximately one half.

The correlation coefficient between ΔT and ΔZ would be the same as that between ΔT and u. The relative importance of variation in ΔZ and variation in u on the variation in ΔT depends upon the relative variances of ΔZ and u.

A regression of the global temperature on the CO₂ level (Mauna Loa data) for the years 1959 to 2004 gives a coefficient of determination (R²) of 0.775, meaning that 77.5 percent of the variation in global temperature over that period is explained by the variation in the CO₂ level over that period. This means that the variance of u is about 29 percent of the variance of ΔZ.

An R² of 0.775 corresponds to a correlation of 0.88 between ΔT and ΔZ. This seems to be a reasonably high level, but it also means that 22.5 percent of the variation is not attributable to the variation in CO₂. Furthermore, a significant amount of the correlation stems from the fact that both variables have a trend. If there is a causal relationship between the two variables then the common trend is important, but suppose there is no causal relationship. The existence of a trend in any two variables would result in a possibly spurious correlation between. A method for checking to see how much of the correlation stems from the trends is to determine the correlation between the deviations of the variables from their trends. This done by regressing both variables on time and computing the deviations from the trend lines. When this is carried out the coefficient of determination is is 0.2967, which corresponds to a correlation coefficient of 0.54. This correlation could be more impressive than the 0.88 correlation coefficient between the two undetrended variables. If there were no causal relationship between the two variables the expected value of the correlation between the two detrended variables would zero. The question is whether the value of 0.54 is statistically significantly different from zero and this is a matter of what is the standard deviation of correlation coefficients computed between two random variables.

(To be continued.)