Problem 1. If temperature is adequately represented by a deterministic trend due to increasing GHGs, why be concerned with the presence of a unit root?

Rather than bloviate over the implications of a unit root (integrative behavior) in the global temperature series, a more productive approach is to formulate an hypothesis, and test it.

A deterministic model of global temperature (y) and anthropogenic forcing (g) with random errors e is:

yt=a+b.gt

An autoregressive model of changes in temperature Δyt uses a difference equation with a deterministic trend b.gt-1 and the previous value of y or yt-1:

Δyt =b.gt-1+c.yt-1

Written this way, the presence of the unit root in an AR1 series y is equivalent to the coefficient c equaling zero (see http://en.wikipedia.org/wiki/Dickey%E2%80%93Fuller_test).

I suspect the controversy can be reduced to two simple hypotheses:

H0: The size of the coefficient b is not significantly different from zero.
Ha: The size of the coefficient b is significantly different from zero.

The size of the coefficient should be indicative of the contribution of the deterministic trend (in this case anthropogenic warming) to the global temperature.

We transform the global temperature by differencing (an autoregressive or AR coordinate system), and then fit a model just as we would with any model.

In the deterministic coordinate system, b is highly significant with a strong contribution from AGW. For the AGW forcing I use the sum of the anthropogenic forcings in the RadF.txt file W-M_GHGs, O3, StratH2O, LandUse, and AIE.

fig1


Call: lm(formula = y ~ g)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.34054 0.01521 -22.39 <2e-16 ***
g 0.31573 0.01802 17.52 <2e-16 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1251 on 121 degrees of freedom
Multiple R-squared: 0.7172, Adjusted R-squared: 0.7149
F-statistic: 306.9 on 1 and 121 DF, p-value: < 2.2e-16

The result is very different in the AR coordinate system. The coefficient of y is not significantly greater than zero (at 95%) and neither is b.

fig2


Call: lm(formula = d ~ y + g + 0)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
y -0.06261 0.03234 -1.936 0.0552 .
g 0.01439 0.01088 1.322 0.1887

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.101 on 121 degrees of freedom
Multiple R-squared: 0.0389, Adjusted R-squared: 0.02302
F-statistic: 2.449 on 2 and 121 DF, p-value: 0.09066

Perhaps the main contribution of AGW is since 1960, so we restrict the data to this period and examine the effect. The deterministic trend in AGW is greater, but still not significant.

fig3


Prob1(window(CRU,start=1960),GHG)
Call: lm(formula = d ~ y + g + 0)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
y -0.24378 0.10652 -2.289 0.0273 *
g 0.03050 0.01512 2.017 0.0503 .

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1149 on 41 degrees of freedom
Multiple R-squared: 0.1284, Adjusted R-squared: 0.08591
F-statistic: 3.021 on 2 and 41 DF, p-value: 0.05974

But what happens when we use another data set. Below is the result using GISS. The coefficients are significant but the effect is still small.

fig4


> Prob1(GISS,GHG)
Call: lm(formula = d ~ y + g + 0)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
y -0.27142 0.06334 -4.285 3.69e-05 ***
g 0.06403 0.01895 3.379 0.00098 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1405 on 121 degrees of freedom
Multiple R-squared: 0.1375, Adjusted R-squared: 0.1232
F-statistic: 9.645 on 2 and 121 DF, p-value: 0.0001298

So why be concerned with the presence of a unit root? It has been argued that while the presence of a unit indicates that using OLS regression is wrong, this does not contradict AGW because the effect of greenhouse gas forcings can still be incorporated as deterministic trends.

I am not 100% sure of this, as the differencing removes most of the deterministic trend that could be potentially explained by g.

If the above is true, there is a problem. When the analysis respects the unit root on real data, the deterministic trend due to increasing GHGs is so small that the null hypothesis is not rejected, i.e. the large contribution of anthropogenic global warming suggested by a simple OLS regression model is a spurious result.

Here is my code. Orient is a functions that matches two time series to the same start and end date.


Prob1<-function(y,g) {
v<-orient(list(y,g))
d<-diff(v[,1]);y<-v[1:(dim(v)[1]-1),1];
g<-v[1:(dim(v)[1]-1),2]
l<-lm(d~y+g+0)
print(summary(l))
plot(y,type="l")
lines((g*l$coef[2]+y[1]),col="blue")
}

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