So now the fun starts. We have established the integration order of the variables in the RadF file, we impose the rule that only variables of the same order can be combined, and in particular that they cannot be cointegrated with temperature which is I(1). In this case all the anthropogenic variables in RadF are I(2) — W-M_GHGs, O3, StratH2O, LandUse, SnowAlb, BC, ReflAer, AIE — while Solar and StratAer are I(1) or I(0).
Adding these AGW variables together would be one way, as they are all I(2) and are in the same units of Watts/m2. However, the result is still I(2), as it takes two differencings before the ADF test rejects.
d Root ADF Padf
[1,] 0 0.9662216 -1.088566 0.9208762
[2,] 1 0.5309850 -2.192266 0.4966766
[3,] 2 -0.4976567 -4.875666 0.0100000
What we can do is see if the variables cointegrate. To do this, we fit a linear regression model to the variables and test the residuals for integration order. If the residual is a lower order I(1) then the variables are all related and can be treated as a common trend. It turns out the residuals are I(1).
d Root ADF Padf
[1,] 0 0.9988194 -3.087874 0.1245392
[2,] 1 0.4176963 -4.468787 0.0100000
[3,] 2 -0.4492167 -6.912500 0.0100000
Some of the variables are not significant however, so we restrict the cointegration vector to the following:
W-M_GHGs = -4.51*O3 + 54.56*StratH2O + 11.21*LandUse – 0.35*AIE, R2=0.9984
The plot of the residuals of the full (black) and restricted (red) model is below. Interestingly, htere is a clear 20 year cycle not apparent in the original data.
Finally we have the information we need to develop our full model. We have Temperature and Solar as I(1), and we will leave out StratAer (volcanics) as it turns out to be non-significant. The sum of the I(2) variables is AGW=I(2) so we use the first difference (or rate of change) deltaAGW as a variable which will be I(1). We also use the residuals of the cointegration g=I(1).
The resulting linear model turns out to have marginal cointegration on the ADF test (as also found by Beenstock) but the PP test clearly rejects, so the result is I(0) and the variables cointegrate.
d Root ADF Padf
[1,] 0 0.7316112 -3.132276 0.1060901
[2,] 1 -0.1910926 -6.674352 0.0100000
[3,] 2 -0.9669920 -8.286269 0.0100000
Phillips-Perron Unit Root Test
Dickey-Fuller Z(alpha) = -33.0459, Truncation lag parameter = 4, p-value = 0.01
T = -0.36007 + Solar*1.22403 + deltaAGW*5.31922 + g*1.12901, R2=0.629
Below I plot the temperature (black), and the prediction from the model (red).
The plot also explores the impact of different levels of CO2. The upper (green) is what would have happened to temperature if CO2 had increased at twice the rate. The lower (blue) is the result if CO2 had been zero throughout.
So far the reproduction of the Beenstock analysis on an alternate data set resembles his analysis, suggesting some robustness to the result. The message of this analysis is that higher emissions of CO2 make no difference to the eventual temperature levels, although they change faster, arriving at the equilibrium levels sooner.