A Statistical Analysis of Global Warming

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A Statistical Analysis of Global

Warming

Gaetan Lion. July 2006

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Global Warming basics

• The anthropogenic emission of CO2 is increasing CO2 concentration in the atmosphere.

• CO2 increasing level is causing Global temperature to rise.

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Independent Variables to test

1. CO2 emission

2. CO2 concentration

We test each of these independent variables separately against the dependent variable:

Land Air temperature.

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Data SourcesCarbon Emission

1751-2000 from G. Marland et al., "Global, Regional, and National Fossil Fuel CO2 Emissions," in Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, U.S. Department of Energy, Trends: A Compendium of Data on Global Change (Oak Ridge, TN).

2001-2003 calculated by Worldwatch with data from BP, Statistical Review of World Energy 2004 (London, 2004). Updated 3 February 2005.

2004 estimated by Worldwatch with data from press reports and International Energy Agency, Oil Market Report, 18 January 2005, and press reports.

Carbon Concentration

Atmospheric CO2 concentrations (ppmv) derived from in situ

Air samples collected at Mauna Loa Observatory, Hawaii

Source: C.D. Keeling, T.P. Whorf, and the Carbon Dioxide Research Group

Scripps Institution of Oceanography (SIO)

University of California La Jolla, California USA 92093-0444

Land Air temperature

NASA, Goddard Institute of Space Studies, "Global Temperature Anomalies in .01 C, base period 1951-1980" (January-December), at

www.giss.nasa.gov/data/update/gistemp/GLB.Ts.txt

Time series:

Carbon Emission and Air temperature: 1867 – 2004.

Carbon Concentration: 1958 – 2004.

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Testing CO2 Emission

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The Cause?

CO2 emission from fossil fuels (in mm tons)

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1,000

2,000

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1867

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2002

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The Effect?

Land temperature (Celsius)

13.40

13.50

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13.90

14.00

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1867

1873

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1891

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1927

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A Perfect Granger Causality set upA Granger causes B

Base case Model Test ModelAutoregressive Multivariate

X1 = Lag B X1 = Lag BY = B X2 = Lag A

Y = B

Square Residuals Square Residuals

Hypothesis testing

F or t TestDo the 2 samples ofresiduals come from same population?

Linear Regression

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Models’ preliminary results

The tested independent variable is CO2 emission level in mm of tons a year ago. The dependent variable is avg. global temperature in Celsius in current year.

CO2 emission vs Air temperature

Regression Statistics

Base TestModel Model

Multiple R 0.801 0.842R Square 0.642 0.710Adjusted R Square 0.639 0.705Standard Error 0.149 0.135Observations 137 137

Over Land area

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Checking for residual serial correlation

Test modelDurbin Watson 2.05D-U Max 2.42Residual serial correl. 0.00

A Durbin Watson score close to 2.00 indicates there is no residual serial correlation. We confirmed this by also calculating the actual residual serial correlation that was indeed clause to zero.

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Checking for Heteroskedasticity Land. Residual in Celsius.

Residual Land Air temp (in Celsius). Test Model.

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1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

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Residual Land Air temp (in Celsius). Base Model

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1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

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Heteroskedasticity looks like this.

The two larger graphs above indicate that the residuals are not heteroskedastic.

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How Should we test the Residuals?

The Jarque-Berra test calculates the probability a sample (square residuals) comes from a normally distributed population. The probability is close to zero. Thus, we should weigh more on nonparametric test (Mann Whitney).

Over Land areaBase Test

Model Modeln 137 137Skewness 2.92 2.12Kurtosis 11.00 4.25JB 885 205DF 2 2p-value 0.0% 0.0%

Jarque-Berra test

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Granger Causality output

We observe a large difference in P values between the t test and Mann-Whitney test. Given the Jarque Berra test result, we should rely more on the Mann-Whitney test P values. At end of presentation, we’ll see a way to reconcile between the t test and Mann-Whitney.

Two tail P valuesUnpaired t test 24.2%Mann-Whitney 53.0%

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Modify the variables

• For the tested independent variable, we will change it from CO2 emission level to % change in CO2 emission.

• For the dependent variable, instead of looking at temperature level, we’ll take the change in temperature.

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The tested independent variable is CO2 emission % change a year ago. The dependent variable is avg. global temperature change in Celsius in current year.

% change in CO2 causes change inAir temperature

Regression StatisticsOver Land area


Multiple R 0.285 0.297R Square 0.081 0.088Adjusted R Square 0.074 0.074Standard Error 0.149 0.149Observations 136 136

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The probability is very close to zero that these two samples would come from a normally distributed population. Thus, we should rely more on nonparametric test (Mann Whitney) test.


n 137 137Skewness 2.9 3.0Kurtosis 10.4 10.8JB 816 869DF 2 2p-value 0.0% 0.0%

Over Land area

Jarque-Bera Test

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Here the P values from the t test and the Mann-Whitney test are really close. They both tell us that % change in CO2 does not Granger cause change in average global temperature.

Two tail P valuesUnpaired t test 96.4%Mann-Whitney 91.4%

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Scatter Plot1

Yearly change in CO2 emission vs change in land based air temperature the same year

R2 = 0.0247

(0.60)

(0.50)

(0.40)

(0.30)

(0.20)

(0.10)

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0.10

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0.50

-20.0% -15.0% -10.0% -5.0% 0.0% 5.0% 10.0% 15.0% 20.0%

Change in CO2 emission in %

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Testing CO2 Concentration

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CO2 Concentration history

Carbon Concentration (ppmv) 58 - 04 in July of each Year

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The tested independent variable is CO2 concentration level a year ago. The dependent variable is avg. global temperature in Celsius in current year.

Regression StatisticsBase Test

Model ModelMultiple R 0.79 0.87R Square 0.62 0.76Adj. R Square 0.62 0.75Standard Error 0.15 0.12Observations 46 46

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Checking for residual serial correlation

Test modelDurbin Watson score 2.03D-U Max 2.38Residual serial correlation -0.03

Per Durbin Watson score and actual serial correlation calculation, residual serial correlation is close to zero.

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The probability is very close to zero that samples come from a normally distributed population. Thus, we should weigh much more on nonparametric test (Mann Whitney).

Base Testn 46 46Skewness 1.60 2.01Kurtosis 2.13 4.21JB 28 65DF 2 2p-value 0.0% 0.0%

Jarque-Bera test

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The difference in P value is huge. We will shortly reconcile the difference between the two. Given the result from the Jarque-Berra test, we should definitely weight the result of the Mann-Whitney test more.

Two tail P valueUnpaired t test 9.5%Mann-Whitney test 69.9%

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Using different variables

• For the tested independent variable, we will change it from CO2 concentration level to change in CO2 concentration level (% change over previous year).

• For the dependent variable, instead of looking at temperature level, we’ll take the change in temperature (in Celsius) over previous year.

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The tested independent variable is change in CO2 concentration (% change a year ago). The dependent variable is avg. global temperature change in Celsius in current year.

Regression StatisticsBase Test

Model ModelMultiple R 0.369 0.374R Square 0.136 0.140Adj. R Square 0.116 0.099Standard Error 0.150 0.152Observations 45 45

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Probability is very close to zero. Thus, we should weigh much more on nonparametric test (Mann Whitney) in our hypothesis testing.


n 45 45Skewness 1.79 1.85Kurtosis 2.43 2.65JB 35 39DF 2 2p-value 0.0% 0.0%

Jarque-Bera test

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Here the P values from the t test and the Mann-Whitney test are closer. They both tell us that % change in CO2 concentration does not appear to Granger cause change in average global temperature.

Two tail P valueUnpaired t test 98.7%Mann-Whitney test 79.6%

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Scatter Plot2Change in CO2 concentration vs change in land air

temperature

R2 = 0.1223

(0.40)

(0.30)

(0.20)

(0.10)

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0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

Change in CO2 concentration in ppmv

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(C

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Granger Causality Summary

Mann-t test Whitney

Temperature levelCO2 emission level 24.2% 53.0%CO2 concentration level 9.5% 69.9%

Change in temperature levelChange in CO2 emission 96.4% 91.4%Change in CO2 concentration 98.7% 79.6%

Two tail P value

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T test vs Mann-Whitney reconciliation

Usingmedian Mann-

t test t test WhitneyTemperature levelCO2 emission level 24.2% 99.0% 53.0%CO2 concentration level 9.5% 46.3% 69.9%

Change in temperature levelChange in CO2 emission 96.4% 94.4% 91.4%Change in CO2 concentration 98.7% 89.6% 79.6%

Two tail P value

If we recalculate the unpaired t test using Medians instead of Averages, the resulting P values get a lot closer to the ones generated by the Mann-Whitney test.

Education

A Statistical Analysis of Global Warming