Beenstock’s radical theory needs to be tested. As discussed here, he proposed that CHANGE in greenhouse gases (delta GHGs or dGHGs) not absolute values produces global warming. A simple test is to develop linear regression models predicting temperature, with and without GHG and dGHG. If Beenstock’s theory is correct, then models containing dGHG should be more accurate.

The protocol was to develop and test linear regression models on all the temperature data from 1900 to 2004 (internal test), and two external tests on held back data. That is, the data is divided in half, the model is developed in one half and tested on the other. This gives two external tests.

The index of fit is the Nash-Sutcliffe coefficient of model power. The NSE compares the skill of a prediction to a mean value. The NSE is positive if the prediction has more skill, zero if skill is the same as the mean, and negative if less than the mean.

I chose the following variables based on previous models. I decided to include an ocean oscillation term as I have seen a 60 year cycle in the residuals (eg here), indicating the presence of an unexplained periodic. Here are the variables:

TEMP — temperature

OO — The sum of a standardized AMO an PDO indices

GHG — The sum of all anthropogenic columns in RadF.txt, mostly the radiant effect of CO2.

dGHG — The first difference of the above

V — Stratospheric aerosols (a proxy for volcanic eruptions)

SS — Sun spot count, a proxy for solar isolation

1) Incredibly, on the first test on all the variables, GHG is not even significant, being entirely screened by dGHG.

TEMP ~ -0.49(***)+0.06*OO(***) + 0.72*GHG() -11.1*dGHG(***) + 4.0*V() -0.09*SS() R-squared: 0.8709

The NSE coefficients that follow are: model on 1900-1950 and testing on 1950-2000, model on 1950-2000 and testing on 1900-1950, and finally model development and testing on 1900-2000.

[1] -4.83 0.625 0.871

The NSE indicates the model has some difficulty predicting temperature post 1950 from a model developed on data prior to 1950.

I then ran the model again with only GHG and not dGHG. The predictions are shown on the graph, where blue is prediction from a model developed on pre 1950 data, green the prediction from a model developed on post 1950 data, and observed global temperature is black.

You can see the difficulty in predicting post 1950 temperatures, reflected in the NSE values as well:

[1] -14.809 0.639 0.820

Finally, I ran it again with dGHG not GHG. The result when using dGHG was improved over GHG, also reflected in the NSE values. The prediction of post 1950 temperatures was still slightly worse than a mean value.

[1] -1.046 0.709 0.878

This test shows two sets of evidence in support of Beenstock’s theory:

1. dGHG is highly significant and GHG is not significant when regressed together.

2. dGHG is a more powerful predictor of global temperature than the absolute GHG in independent tests.

In other words, the empirical evidence from 1900 supports Beenstock’s theory (developed in a cointegration analysis) of a transitory global warming effect from increases in greenhouse gasses, but no long-term harmful effect on global temperature.

Call: lm(formula = TEMP ~ OO + dGHG + V + SS)

Coefficients: Estimate Std. Error t value Pr(>|t|)

(Intercept) -0.679982 0.084692 -8.029 2.31e-10 ***

s$OO 0.068453 0.008863 7.723 6.61e-10 ***

s$dGHG -11.548800 2.599261 -4.443 5.37e-05 ***

s$V 12.015335 2.380680 5.047 7.17e-06 ***

s$SS 0.135458 0.243381 0.557 0.58

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Signif. codes: 0 â€˜***â€™ 0.001 â€˜**â€™ 0.01 â€˜*â€™ 0.05 â€˜.â€™ 0.1 â€˜ â€™ 1Multiple R-squared: 0.8652