Problems with Statistical Tests of Pre-1958 Atmospheric CO2 Concentration Data

James McCown writes:

A number of climatologists and economists have run statistical tests of the annual time series of greenhouse gas (GHG) concentrations and global average temperatures in order to determine if there is a relation between the variables. This is done in order to confirm or discredit the anthropogenic global warming (AGW) theory that burning of fossil fuels is raising global temperatures and causing severe weather and rises in the sea level. Many economists have become involved in this research because of the use of statistical tests for unit roots and cointegration, that were developed by economists in order to discern relations between macroeconomic variables. The list of economists who have become involved includes James Stock of Harvard, one of the foremost experts at time series statistics.

With a couple of notable exceptions, the conclusions of nearly all the studies are similar to the conclusion reached by Liu and Rodriguez (2005), from their abstract:

Using econometric tools for selecting I(1) and I(2) trends, we found the existence of static long-run steady-state and dynamic long-run steady-state relations between temperature and radiative forcing of solar irradiance and a set of three greenhouse gases series.

Many of the readers of WUWT will be familiar with the issues I raise about the pre-1958 CO₂ data. The purpose of this essay is to explain how the data problems invalidate much of the statistical research that has been done on the relation between the atmospheric CO₂ concentrations and global average temperatures. I suspect that many of the economists involved in this line of research do not fully realize the nature of the data they have been dealing with.

The usual sources of atmospheric CO₂ concentration data, beginning with 1958, are flask measurements from the Scripps Institute of Oceanography and the National Oceanic and Atmospheric Administration, from observatories at Mauna Loa, Antarctica, and elsewhere. These have been sampled on a monthly basis, and sometimes more frequently, and thus provide a good level of temporal accuracy for use in comparing annual average CO₂ concentrations with annual global average temperatures.

Unfortunately, there were only sporadic direct measurements of atmospheric CO₂ concentrations prior to 1958. The late Ernst-Georg Beck collected much of the pre-1958 data and published on his website here: http://www.biomind.de/realCO2/realCO2-1.htm.

Most researchers who have examined pre-1958 relations between GHGs and temperature have used Antarctic ice core data provided by Etheridge et al (1996) (henceforth Etheridge). Etheridge measured the CO₂ concentration of air trapped in the ice on Antarctica at the Law Dome, using three cores that varied from 200 to 1200 meters deep.

There have been several published papers by various groups of researchers that have used the pre-1958 CO₂ concentrations from Etheridge. Recent statistical studies that utilize Etheridge’s data include Liu & Rodriguez (2005), Kaufmann, Kauppi & Stock (2006a), Kaufman, Kauppi, & Stock (2006b), Kaufmann, Kauppi, & Stock (2010), Beenstock, Reingewertz, & Paldor (2012), Kaufmann, Kauppi, Mann, & Stock (2013), and Pretis & Hendry (2013). Every one of these studies treat the Etheridge pre-1958 CO₂ data as though it were annual samples of the atmospheric concentration of CO₂.

Examination of Etheridge’s paper reveals the data comprise only 26 air samples taken at various times during the relevant period from 1850 to 1957. Furthermore, Etheridge state clearly in their paper that the air samples from the ice cores have an age spread of at least 10 – 15 years. They have further widened the temporal spread by fitting a “smoothing spline” with a 20 year window, to the data from two of the cores to compute annual estimates of the atmospheric CO₂. These annual estimates, which form the basis for the 1850 – 1957 data on the GISS website, may have been suitable for whatever purpose Etheridge were using them, but are totally inappropriate for the statistical time series tests performed in the research papers mentioned above. The results from the tests of the pre-1958 data are almost certainly spurious.

Details of the Etheridge et al (1996) Ice Core Data

Etheridge drilled three ice cores at the Law Dome in East Antarctica between 1987 and 1993. The cores were labeled DE08 (drilled to 234 meters deep), DE08-2 (243 meters), and DSS (1200 meters). They then sampled the air bubbles that were trapped in the ice at various depths in order to determine how much CO₂ was in the earth’s atmosphere at various points in the past. They determined the age of the ice and then the air of the air bubbles trapped in the ice. According to Etheridge:

The air enclosed by the ice has an age spread caused by diffusive mixing and gradual bubble closure…The majority of bubble closure occurs at greater densities and depths than those for sealing. Schwander and Stauffer [1984] found about 80% of bubble closure occurs mainly between firn densities of 795 and 830 kg m^-3. Porosity measurements at DE08-2 give the range as 790 to 825 kg m^-3 (J.M. Barnola, unpublished results, 1995), which corresponds to a duration of 8 years for DE08 and DE08-2 and about 21 years for DSS. If there is no air mixing past the sealing depth, the air age spread will originate mainly from diffusion, estimated from the firn diffusion models to be 10-15 years. If there is a small amount of mixing past the sealing depth, then the bubble closure duration would play a greater role in broadening the age spread. It is seen below that a wider air age spread than expected for diffusion alone is required to explain the observed CO₂ differences between the ice cores.

In other words, Etheridge are not sure about the exact timing of the air samples they have retrieved from the bubbles in the ice cores. Gradual bubble closure has caused an air age spread of 8 years for the DE08 and DE08-2 cores, and diffusion has caused a spread of 10 – 15 years. Etheridge’s results for the DE08 and DE08-2 cores are shown below (from their Table 4):

Etheridge Table 4: Core DE08

Core DE08-2

Due to the issues of diffusive mixing and gradual bubble closure, each of these figures give us only an estimate of the average CO₂ concentration over a period that may be 15 years or more. If the distribution of the air age is symmetric about these mean air ages, the estimate of 310.5 ppm from the DE08 core for 1938 could include air from as early as 1930 and as late as 1946.

Etheridge combined the estimates from the DE08 and DE08-2 cores and fit a 20-year smoothing spline to the data, in order to obtain annual estimates of the CO₂ concentrations. These can be seen here: http://cdiac.ornl.gov/ftp/trends/co2/lawdome.smoothed.yr20. These annual estimates, which are actually 20 year or more moving averages, were used by Dr. Makiko Sato, who was then affiliated with NASA-GISS, in order to compile an annual time series of CO₂ concentrations for the period from 1850 to 1957. Dr. Sato used direct measurements of CO₂ from Mauna Loa and elsewhere for 1958 to the present. He references the ice core data from Etheridge on that web page, and adds that it is “Adjusted for Global Mean”. Some of the papers reference the data from the website of NASA’s Goddard Institute for Space Science (GISS) here: http://data.giss.nasa.gov/modelforce/ghgases/Fig1A.ext.txt.

I emailed Dr. Sato (who is now at Columbia University) to ask if he had used the numbers from Etheridge’s 20-year smoothing spline and what exactly he had done to adjust for a global mean. He replied that he could not recall what he had done, but he is now displaying the same pre-1958 data on Columbia’s website here: http://www.columbia.edu/~mhs119/GHG_Forcing/CO2.1850-2013.txt.

I believe Sato’s data are derived from the numbers obtained from Etheridge’s 20-year smoothing spline. For every year from 1850 to 1957, they are less than 1 ppm apart. Because of the wide temporal inaccuracy of the CO₂ estimates of the air trapped in the ice, exacerbated by the use of the 20-year smoothing spline, we have only rough moving average estimates of the CO₂ concentration in the air for each year, not precise annual estimates. The estimate of 311.3 ppm for 1950 that is shown on the GISS and Columbia websites, for example, could include air from as early as 1922 and as late as 1978. Fitting the smoothing spline to the data may have been perfectly acceptable for Etheridge’s purposes, but as we shall see, it is completely inappropriate for use in the time series statistical tests previously mentioned.

Empirical Studies that Utilize Etheridge’s Pre-1958 Ice Core Data

As explained in the introduction, there are a number of statistical studies that attempt to discern a relation between GHGs and global average temperatures. These researchers have included climatologists, economists, and often a mixture of the two groups.

Liu and Rodriguez (2005), Beenstock et al (2012) and Pretis & Hendry (2013) use the annual Etheridge spline fit data for the 1850 – 1957 period, from the GISS website, as adjusted by Sato for the global mean.

Kaufmann, Kauppi, & Stock (2006a), (2006b), and (2010), and Kaufmann, Kauppi, Mann, & Stock (2013) also use the pre-1958 Etheridge (1996) data, and their own interpolation method. Their data source for CO₂ is described in the appendix to Stern & Kaufmann (2000):

Prior to 1958, we used data from the Law Dome DE08 and DE08-2 ice cores (Etheridge et al., 1996). We interpolated the missing years using a natural cubic spline and two years of the Mauna Loa data (Keeling and Whorf, 1994) to provide the endpoint.

The research of Liu and Rodriguez (2005), Beenstock et al (2012), Pretis & Hendry (2013), and the four Kaufmann et al papers use a pair of common statistical techniques developed by economists. Their first step is to test the time series of the GHGs, including CO₂, for stationarity. This is also called testing for a unit root, and there are a number of tests devised for this purpose. The mathematical expression for a time series with a unit root is, from Kaufmann et al (2006a):

(1)

Where ɛ is a random error term that represents shocks or innovations to the variable Y. The parameter λ is equal to one if the time series has a unit root. In such a case, any shock to Y will remain in perpetuity, and Y will have a nonstationary distribution. If λ is less than one, the ɛ shocks will eventually die out and Y will have a stationary distribution that reverts to a given mean, variance, and other moments. The statistical test used by Kaufmann et al (2006a) is the augmented Dickey-Fuller (ADF) test devised by Dickey and Fuller (1979) in which they run the following regression of the annual time series data of CO₂, other GHGs, and temperatures:

(2)

Where ∆ is the first difference operator, t is a linear time trend, ɛ is a random error term, and

γ = λ – 1. The ADF test is for the null hypothesis that γ = 0, therefore λ = 1 and Y is a nonstationary variable with a unit root, also referred to as I(1).

There are several other tests for unit roots used by the various researchers, including Phillips & Perron (1988), Kwiatkowski, Phillips, Schmidt, & Shin (1992) , and Elliott, Rothenberg, & Stock (1996). The one thing they have in common is some form of regression of the time series variable on lagged values of itself as in equation (2).

Conducting a regression such as (2) can only be conducted properly on non-overlapping data. As explained previously, the pre-1958 Etheridge data from the ice cores may include air from 20 or more years before or after the given date. This problem is further complicated by the fact that Etheridge are not certain of the amount of diffusion, nor do we know the distribution of how much air from each year is in each sample. Thus, instead of regressing annual CO₂ concentrations on past values (such as 1935 on 1934, 1934 on 1933, etc), these researchers are regressing some average of 1915 to 1955 on an average from 1914 to 1954, and then 1914 to 1954 on 1913 to 1953, and so forth. This can only lead to spurious results, because the test mostly consists of regressing the CO₂ data for some period on itself.

The second statistical method used by the researchers is to test for cointegration of the GHGs (converted to radiative forcing) and the temperature data. This is done in order to determine if there is an equilibrium relation between the GHGs and temperature. The concept of cointegration was first introduced by Engle & Granger (1987), in order to combat the problem of discerning a relation between nonstationary variables. Traditional ordinary least squares regressions of nonstationary time series variables often lead to spurious results. Cointegration tests were first applied to macroeconomic time series data such as gross domestic product, money supply, and interest rates.

In most of the papers the radiative forcings from the various GHGs are added up and combined with estimates of solar irradiance. Aerosols and sulfur are also considered in some of the papers. Then a test is run of these measures to determine if they are cointegrated with annual temperature data (usually utilizing the annual averages of the GISS temperature series). The cointegration test involves finding a linear vector such that a combination of the nonstationary variables using that vector is itself stationary.

A cointegration test can only be valid if the data series have a high degree of temporal accuracy and are matched up properly. The temperature data likely have good temporal accuracy but the pre- 1958 Etheridge CO₂ concentration data, from which part of the radiative forcing data are derived, are 20 year or greater moving averages, of unknown length and distribution. They cannot be properly tested for cointegration with annual temperature data without achieving spurious results. For example, instead of comparing the CO₂ concentration for 1935 with the temperature of 1935, the cointegration test would be comparing some average of CO₂ concentration for 1915 to 1955 with the temperature for 1935.

In defense of Beenstock et al (2012), the primary purpose of their paper was to show that the CO₂ data, which they and the other researchers found to be I(2) (two unit roots), cannot be cointegrated with the I(1) temperature data unless it is polynomially cointegrated. They do not claim to find a relation between the pre-1958 CO₂ data and the temperature series.

The conclusion of Kaufmann, Kauppi, & Stock (2006a), from their abstract:

Regression results provide direct evidence for a statistically meaningful relation

between radiative forcing and global surface temperature. A simple model based on these results indicates that greenhouse gases and anthropogenic sulfur emissions are largely responsible for the change in temperature over the last 130 years.

The other papers cited in this essay, except Beenstock et al (2012), come to similar conclusions. Due to the low level of temporal accuracy of the CO₂ data pre-1958, their results for that period cannot be valid. The only proper way to use such data would be if an upper limit to the time spread caused by the length of bubble closure and diffusion of gases through the ice could be determined. For example, if an upper limit of 20 years could be established, then the researchers could then determine an average CO₂ concentration for non-overlapping 20 year periods, and then perform the unit root and cointegration tests. Unfortunately, for the period from 1850 to 1957 that would include only five complete 20 year periods. Such a small sample is not useful. Unless and until a source of pre-1958 CO₂ concentration data is found that has better temporal accuracy, there is no point in conducting cointegration tests with temperature data for that period.

References

Beenstock, M., Y. Reingewertz, and N. Paldor (2012). Polynomial cointegration tests of anthropogenic impact on global warming. Earth Syst. Dynam., 3, 173–188.

Dickey, D. A. and Fuller, W. A.: 1979, ‘Distribution of the estimators for autoregressive time series with a unit root’, J. Am. Stat. Assoc. 74, 427–431.

Elliott, G., Rothenberg, T. J., and Stock, J. H.: Efficient tests for an autoregressive unit root, Econometrica, 64, 813–836, 1996.

Engle, R. F. and Granger, C. W. J.: Co-integration and error correction: representation, estimation and testing, Econometrica, 55, 251–276, 1987.

Etheridge, D. M., Steele, L. P., Langenfelds, L. P., and Francey, R. J.: 1996, ‘Natural and anthropogenic changes in atmospheric CO2 over the last 1000 years from air in Antarctic ice and firn’, J. Geophys. Res. 101, 4115–4128.

Kaufmann, R. K., Kauppi, H., Mann, M. L., and Stock, J. H.: Does temperature contain a stochastic trend: linking statistical results to physical mechanisms, Climatic Change, 118, 729–743, 2013.

Kaufmann, R., Kauppi, H., and Stock, J. H.: Emissions, concentrations and temperature: a time series analysis, Climatic Change, 77, 248–278, 2006a.

Kaufmann, R., Kauppi, H., and Stock J. H.: The relationship between radiative forcing and temperature: what do statistical analyses of the instrumental temperature record measure?, Climatic Change, 77, 279–289, 2006b.

Kaufmann, R., H. Kauppi, and J. H. Stock, (2010) Does temperature contain a

stochastic trend? Evaluating Conflicting Statistical Results, Climatic Change, 101, 395-405.

Kwiatkowski, D., Phillips, P. C. B., Schmidt, P., and Shin, Y.: Testing the null hypothesis of stationarity against the alternative of a unit root, J. Economet., 54, 159–178, 1992.

Liu, H. and G. Rodriguez (2006), Human activities and global warming: a cointegration analysis. Environmental Modelling & Software 20: 761 – 773.

Phillips, P. C. B. and Perron, P.: Testing for a unit root in time series regression, Biometrika, 75, 335–346, 1988.

Pretis, F. and D. F. Hendry (2013). Comment on “Polynomial cointegration tests of anthropogenic impact on global warming” by Beenstock et al. (2012) – some hazards in econometric modelling of climate change. Earth Syst. Dynam., 4, 375–384.

Stern, D. I., and R. K. Kaufmann, Detecting a global warming signal in hemispheric temperature series: A structural time series analysis, Clim. Change, 47, 411 –438, 2000.