Summary of the statistical analysis

The Fossil Fuel variable is world consumption of fossil fuels (i.e. coal, petroleum, and natural gas), measured in gigabarrels of oil equivalent per year. These data were pulled together from various sources, principally British Petroleum's Statistical Review of World Energy (2007) for oil and gas, and the History Database of the Environment, a publication of The Netherlands, for coal. Data from the 19th century are unreliable at best. Predictions for 2007 and later are based on my personal Hubbert analyses of coal, natural gas, conventional oil, and extra-heavy oil.

The Solar Activity variable is constructed from a trailing 11-year smooth of the annual sunspot numbers reported by the Royal Observatory of Belgium. The smoothing operation removes almost all of the cyclic activity, leaving only the average intensity of solar activity over the previous 11 years.

The dependent variable in this analysis is HadCrut3v, the global mean temperature "anomaly" as estimated by the Climate Prediction Unit of the University of East Anglia, 1850-2006, measured in degrees Celsius (1°C = 1.8°F). The term "anomaly" in this context means only that these are temperature differences from the 1961-90 global average, not absolute temperatures.

The statistical model used here is an AR(1) linear autoregressive model, which yields an R2 of 86% and an F-statistic of 328 with 3 and 152 degrees of freedom. The model's coefficients and their statistics are summarized in the following table:

Variable
Coefficient
Standard Error
t-Statistic
Significance
Constant
-0.200851
0.0401
-5.01
p < 0.0001
HadCrut3v (lag 1)
0.570599
0.0657
8.69
p < 0.0001
Fossil Fuel (lag 25)
0.00728943
0.0013
5.65
p < 0.0001
Solar Activity
0.0012068
0.0005
2.17
p = 0.0318

For the purposes of simulation, this model can be represented as a stochastic difference equation. Let Xt be temperature, Yt be fossil fuel consumption, and Zt be solar activity, all in year t. Then:

Xt+1 = (-0.200851) + (0.570599)Xt + (0.0012068)Yt-25 + (0.00728943)Zt + Ut ,

where {U} is a series of random i.i.d. variables with mean zero and standard deviation 0.0915.

Finally, here is a table of data for these variables:

Year
Temperature (HadCrut3v)
Fossil Fuel (gboe)
Solar Activity (smoothed)
1825
0.10
20.8
1826
0.11
20.9
1827
0.11
21.3
1828
0.12
23.4
1829
0.12
26.7
1830
0.13
31.0
1831
0.14
33.9
1832
0.14
35.8
1833
0.15
36.2
1834
0.16
37.3
1835
0.17
41.7
1836
0.18
51.2
1837
0.19
60.5
1838
0.20
65.4
1839
0.21
67.3
1840
0.22
67.1
1841
0.23
64.0
1842
0.24
61.8
1843
0.25
60.3
1844
0.27
60.9
1845
0.28
63.4
1846
0.30
63.8
1847
0.31
61.7
1848
0.33
60.4
1849
0.35
59.8
1850
-0.439
0.36
58.1
1851
-0.306
0.39
58.1
1852
-0.309
0.42
59.7
1853
-0.342
0.44
61.0
1854
-0.299
0.47
61.9
1855
-0.341
0.50
61.1
1856
-0.409
0.52
57.9
1857
-0.482
0.55
54.4
1858
-0.490
0.57
50.4
1859
-0.360
0.60
47.6
1860
-0.392
0.63
47.5
1861
-0.409
0.66
48.5
1862
-0.521
0.70
48.0
1863
-0.315
0.74
47.1
1864
-0.503
0.78
47.8
1865
-0.308
0.82
48.7
1866
-0.314
0.86
49.6
1867
-0.338
0.90
49.9
1868
-0.297
0.93
51.2
1869
-0.307
0.97
53.0
1870
-0.303
1.01
57.1
1871
-0.349
1.07
58.5
1872
-0.275
1.13
60.7
1873
-0.336
1.19
61.3
1874
-0.376
1.25
61.4
1875
-0.409
1.31
58.7
1876
-0.375
1.37
56.9
1877
-0.129
1.43
56.6
1878
-0.018
1.49
56.2
1879
-0.260
1.55
53.3
1880
-0.249
1.61
49.6
1881
-0.250
1.70
41.9
1882
-0.255
1.79
37.2
1883
-0.306
1.88
33.7
1884
-0.370
1.97
33.5
1885
-0.362
2.06
34.2
1886
-0.289
2.15
34.9
1887
-0.375
2.24
35.1
1888
-0.340
2.33
34.6
1889
-0.223
2.42
34.8
1890
-0.422
2.51
34.9
1891
-0.383
2.63
35.2
1892
-0.478
2.76
36.9
1893
-0.502
2.88
39.3
1894
-0.435
3.00
40.6
1895
-0.416
3.13
40.6
1896
-0.238
3.25
39.7
1897
-0.258
3.38
39.7
1898
-0.399
3.50
41.0
1899
-0.319
3.62
41.4
1900
-0.253
3.75
41.7
1901
-0.316
3.95
41.3
1902
-0.427
4.15
38.6
1903
-0.494
4.35
34.1
1904
-0.537
4.55
30.2
1905
-0.424
4.75
28.9
1906
-0.349
4.95
28.0
1907
-0.518
5.15
29.8
1908
-0.554
5.36
31.8
1909
-0.559
5.56
33.4
1910
-0.544
5.76
34.0
1911
-0.573
5.84
33.6
1912
-0.496
5.92
33.7
1913
-0.485
6.00
33.4
1914
-0.318
6.08
32.1
1915
-0.246
6.16
32.5
1916
-0.432
6.25
32.0
1917
-0.493
6.33
36.5
1918
-0.387
6.41
38.2
1919
-0.333
6.49
39.6
1920
-0.327
6.57
39.0
1921
-0.268
6.69
39.7
1922
-0.376
6.80
40.5
1923
-0.345
6.91
40.7
1924
-0.356
7.03
42.1
1925
-0.274
7.14
45.2
1926
-0.179
7.26
46.7
1927
-0.257
7.37
47.8
1928
-0.255
7.49
45.4
1929
-0.358
7.60
44.0
1930
-0.171
7.71
41.5
1931
-0.139
7.90
40.0
1932
-0.161
8.10
38.6
1933
-0.282
8.29
37.8
1934
-0.161
8.48
38.1
1935
-0.185
8.67
39.9
1936
-0.149
8.86
43.1
1937
-0.041
9.05
47.7
1938
0.002
9.24
51.4
1939
-0.003
9.43
52.4
1940
0.008
9.62
52.6
1941
0.062
9.89
53.7
1942
-0.020
10.15
54.5
1943
-0.018
10.41
55.0
1944
0.099
10.67
55.4
1945
-0.024
10.93
57.6
1946
-0.190
11.19
62.7
1947
-0.193
11.46
69.3
1948
-0.195
11.72
71.3
1949
-0.206
11.98
73.5
1950
-0.293
12.24
73.1
1951
-0.168
13.06
73.2
1952
-0.094
13.88
71.8
1953
-0.045
14.70
70.3
1954
-0.245
15.52
69.2
1955
-0.268
16.34
71.8
1956
-0.336
17.17
81.6
1957
-0.083
17.99
90.5
1958
-0.019
18.81
93.5
1959
-0.073
19.63
95.6
1960
-0.118
20.45
93.6
1961
-0.031
21.29
90.8
1962
-0.034
23.09
87.9
1963
-0.009
25.06
87.6
1964
-0.277
26.77
87.3
1965
-0.210
26.73
88.2
1966
-0.150
28.17
89.1
1967
-0.147
28.92
84.7
1968
-0.159
30.95
77.0
1969
-0.025
32.92
69.8
1970
-0.073
34.67
64.9
1971
-0.181
35.93
60.7
1972
-0.066
37.57
62.1
1973
0.058
39.50
62.1
1974
-0.206
39.45
62.7
1975
-0.161
39.33
63.2
1976
-0.240
41.58
63.0
1977
0.006
42.84
61.2
1978
-0.059
44.10
61.1
1979
0.046
45.67
65.6
1980
0.071
44.57
70.1
1981
0.110
44.02
73.3
1982
0.016
43.62
77.8
1983
0.171
43.93
77.6
1984
-0.019
45.87
78.3
1985
-0.037
46.75
76.8
1986
0.034
47.81
76.6
1987
0.178
49.33
78.1
1988
0.174
50.99
84.7
1989
0.109
52.05
90.7
1990
0.247
52.57
89.5
1991
0.203
52.63
88.7
1992
0.070
52.86
84.5
1993
0.104
53.02
78.9
1994
0.169
53.57
75.6
1995
0.270
54.77
73.0
1996
0.138
56.59
72.1
1997
0.347
57.05
72.9
1998
0.526
57.02
76.1
1999
0.302
57.73
75.4
2000
0.277
59.30
72.0
2001
0.406
59.75
69.1
2002
0.455
60.90
65.3
2003
0.465
63.23
62.5
2004
0.444
66.14
61.3
2005
0.476
67.74
61.2
2006
0.422
68.19
61.0
2007
69.07
2008
69.90
2009
70.71
2010
71.47
2011
72.18
2012
72.85
2013
73.48
2014
74.06
2015
74.59
2016
75.07
2017
75.49
2018
75.87
2019
76.19
2020
76.46
2021
76.67
2022
76.83
2023
76.94
2024
76.98
2025
76.98
2026
76.92
2027
76.81
2028
76.64
2029
76.42
2030
76.15
2031
75.83
2032
75.47
2033
75.05
2034
74.59
2035
74.08
2036
73.53
2037
72.94
2038
72.31
2039
71.64
2040
70.94
2041
70.20
2042
69.43
2043
68.64
2044
67.81
2045
66.96
2046
66.09
2047
65.20
2048
64.28
2049
63.35
2050
62.41
2051
61.45
2052
60.48
2053
59.50
2054
58.51
2055
57.52
2056
56.52
2057
55.52
2058
54.52
2059
53.52
2060
52.52
2061
51.52
2062
50.53
2063
49.54
2064
48.56
2065
47.59
2066
46.62
2067
45.66
2068
44.72
2069
43.78
2070
42.85
2071
41.93
2072
41.03
2073
40.14
2074
39.26
2075
38.39
2076
37.54
2077
36.70
2078
35.87
2079
35.06
2080
34.27
2081
33.48
2082
32.71
2083
31.96
2084
31.21
2085
30.49
2086
29.77
2087
29.07
2088
28.39
2089
27.72
2090
27.06
2091
26.42
2092
25.78
2093
25.17
2094
24.56
2095
23.97
2096
23.39
2097
22.82
2098
22.27
2099
21.73
2100
21.20