### WUWT and how NOT to test the relationship between CO2 and temperature

WUWT published a piece by Danle Wolfe which purports to measure the correlation between CO2 and global temperature.  As you can probably predict, Wolfe's conclusion is that there is no relationship.
"Focusing on the most recent hiatus below, both visually and in a 1st order linear regression analysis there clearly is effectively zero correlation between CO2 levels and global mean temperature."
Unfortunately for Wolfe, all he's produced is a fine example of mathturbation as well as an example of forming a conclusion first then warping the evidence to fit.

What Wolfe did was cross-correlate GISS land temperature data and Mauna Loa CO2 records, with two vertical lines dividing the plot into three sections.  The first section is marked "~18 years", the middle is marked "~21 years", and the last section is marked "~17 years".

 Figure 1.  Danle Wolfe's plot from WUWT
Why land temperatures rather than land + ocean temperatures?  We don't know as he failed to justify his choice.  There's one other curiosity about his plot.  We know his plot starts in 1959 as he gave that information.  That would make the first section from 1959 to 1977, the middle section from 1977 to 1998, and the last from 1997 to 2014, which means there's an overlap of 1 year between his middle and last sections.  The problem?  The maximum CO2 levels in 1997 (367 ppmv) does not match the vertical line on his graph.

 Figure 2.  Temperatures vs CO2 with loess trend line.
His second line looks to be around 372, a level first crossed in 2001, not 1997.  That makes his last section at most 13 years long rather than the 17 years he claimed.  Furthermore, a loess regression reveals that his lines do not divide the graph into "no correlation" and "correlation" sections as he implied.  His "no correlation" sections are nowhere near as long as he claimed them to be.

The next deception in his graph?  He failed to remove the annual cycle from both the temperature record and the CO2 record before cross-correlating them.

 Figure 3.  Seasonal cycles in both CO2 records and GISS temperatures.
Time series decomposition shows that both GISS temperatures and CO2 records have 12-month cycles—and also shows that the cycles are out-of-phase.  This makes sense as the CO2 annual cycle is tied in with the Northern Hemisphere growing season and therefore only indirectly tied to global average temperatures.  Accordingly, the cycles must be removed to get the true relationship.  Just compare the cross-correlation graph without removing the annual cycles with one with the annual cycles removed via a 12-month moving average:

 Figure 4.  Scatterplots of CO2 versus temperatures, both with and without seasonal cycles removed.
Just removing the annual cycles via a 12-month moving average removed much of the noise Wolfe depended upon to make it look like there was no correlation.  Even when he tried a moving average to remove the cycle, he failed.  Simply put, a 10-month moving average does not eliminate a 12-month cycle.  You can see that in his graph, especially the CO2 line.  If you want to remove an annual cycle, you must use a 12-month moving average, not a 10-month moving average.

 Figure 5.  Ten- versus twelve-month moving averages.  Note that the seasonal cycle is still apparent in the 10-month moving average whereas it is fully removed in the 12-month moving average.
Last, Wolfe failed to account for ENSO, aerosols, solar output, or any of the other non-CO2-related influences on global temperature.  His viewpoint that CO2 must be the only thing that influences global temperature is dead wrong.  There have been several studies over the past decade quantifying and then removing non-CO2 influences on global temperatures via multiple regression (e.g. Lean and Rind 2008, Foster and Rahmstorf 2011, Rahmstorf et al. 2012).  Yet it appears that Wolfe is either ignorant of that work or deliberately ignoring it.

What difference does factoring out the seasonal cycle and non-CO2 influences like El NiĆ±o/Southern Oscillation, sulfur aerosols, and solar output make on the correlation between CO2 and global temperatures?  Quite a bit.

 Figure 6.  Adjusted GISS temperatures versus CO2 with annual cycles removed.
I added a loess regression line to highlight the trend.  Compare that to Wolfe's graph in figure 1 and my graph in figure 2.  Note the differences?  Once seasonal cycles and non-CO2-climate factors are removed, the correlation between global temperatures and CO2 is clear.

And just for Wolfe: Beyond fudging your second vertical line and  "forgetting" to account for seasonal cycles and climate influences like ENSO, solar output, and sulfur aerosols, you also forgot to account for autocorrelation when you did your regression since 1999.  Hint: There's a world of difference between a white noise model and an ARMA(2,1) model, especially after you take out the seasonal cycle, ENSO, aerosols, and changes in the solar cycle.  In "statistician speak," you only got the results you did because of your sloppy, invalid "analysis."
 Figure 7.  Adjusted GISS vs CO2 since 1999 after seasonal cycles are removed
Generalized least squares fit by REML
Model: GISS ~ CO2
Data: monthly
Subset: Time >= 1999
AIC       BIC   logLik
-1311.377 -1292.253 661.6886

Correlation Structure: ARMA(2,1)
Formula: ~1
Parameter estimate(s):
Phi1               Phi2           Theta1
1.3709034   -0.4152224    0.9999937

Coefficients:
Value             Std.Error      t-value         p-value
(Intercept) -1.9335047   0.6987245   -2.767192    0.0062
CO2          0.0058194     0.0018289    3.181950    0.0017

Correlation:
(Intr)
CO2  -1

Standardized residuals:
Min                   Q1                Med                  Q3               Max
-1.54739247   -0.69113533    -0.06509602    0.78913695    1.69592575

Residual standard error: 0.05121744
Degrees of freedom: 181 total; 179 residual