Recently, a number of papers have surfaced that use advanced statistical methods to analyze climate data. The techniques involved have been developed not by climate scientists but by economists and social scientists. These new tools belong to the field of econometrics. The use of statistical break tests and polynomial cointegration to analyze the relationships between time series data for greenhouse gas concentrations, insolation, aerosol levels and temperature have shown that these data are non-stationary. The implication of these findings is that much of the statistical analysis applied by climate scientists is flawed and potentially misleading. So strong is the statistical evidence that a couple of economists are claiming to have refuted the theory of anthropogenic global warming. This, on top of everything else that has recently transpired, may indicate that a climate change paradigm shift is imminent.

The source of most climate predictions are general circulation models (GCM) that attempt to computationally simulate the physical processes found in nature. But GCM are not the only way to model the behavior of Earth's climate system. I have previously commented on the use of the Chow test in time series analysis to test for the presence of a structural break. This statistical model predicts no temperature rise until 2050 and only a slight increase of 0.2°C between 2050 and 2100. Those interested can read “Stat Model Predicts Flat Temperatures Through 2050” or, for the more mathematically adventuresome, the paper by David R.B. Stockwell and Anthony Cox, “Structural break models of climatic regime-shifts: claims and forecasts.”

The world suffers from no shortage of future climate predictions so the addition of one more really did not cause much of a ripple in climate change circles. But now, a new paper that mounts a direct assault on the theory of anthropogenic global warming has emerged. The most controversial aspect of this new work is that its methodology comes, not from climate science or a related field, but from econometrics.

Polynomial Cointegration

Two statisticians have stirred up a hornet's nest on the internet with the release of a draft of an unpublished paper claiming to refute the theory of anthropogenic global warming. In it, Michael Beenstock and Yaniv Reingewertz, both from the Department of Economics at Hebrew University, Israel, have applied econometric statistical methods designed for non-stationary time series to test AGW. Their claim is based on the assertion that, if atmospheric CO₂ levels drive temperature change, time series data for CO₂ and temperature will exhibit statistical correlation linking the two. No correlation, no link between carbon dioxide and climate change.

[T]he methodology of polynomial cointegration is used to test AGW when global temperature and solar irradiance are stationary in 1^st differences, whereas greenhouse gas forcings (CO₂, CH₄ and N₂O) are stationary in 2^nd differences. We show that although greenhouse gas forcings share a common stochastic trend, this trend is empirically independent of the stochastic trend in temperature and solar irradiance. Therefore, greenhouse gas forcings, global temperature and solar irradiance are not polynomially cointegrated, and AGW is refuted. Although we reject AGW, we find that greenhouse gas forcings have a temporary effect on global temperature. Because the greenhouse effect is temporary rather than permanent, predictions of significant global warming in the 21^st century by IPCC are not supported by the data.

To understand why this is such a big deal, statistically speaking, we need to look into the properties of time series data. In statistics, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. In other words, a collection of historical measurements. Examples would be a history of the daily stock market closing price or high temperature readings. Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics from the data.

Time series data showing CO₂ emission by region, source OECD Factbook 2007.

Time series data from complex systems, natural or man-made, often have the properties of random walks. Statistically speaking, properties of such series can have means and variances that vary over time. Time series that show this behavior are called non-stationary series and are by definition unpredictable. A significant correlation between such time series does not necessarily imply a meaningful relationship. The relationship can also be meaningless, i.e. spurious. According to researchers experienced in such analysis, Zhiqiu Lin and Augustine Brannigan, “failure to meet the assumption of stationarity in the process of analyzing time series variables may result in spurious and unreliable statistical inferences.” Plainly stated, if your data are not stationary, normal statistical techniques can lie to you.

That the field of non-stationary time series analysis has attracted a lot of attention from economists should come as no surprise. Econometrics combines economic theory with statistics to analyze and test economic theories and relationships. Because the underpinnings of economic systems are constantly changing, economic statistics gathered over a period of time fall into the category of non-stationary series.

This makes logical sense—the world economy of a decade ago is not the same as the economy today: industries wax and wain, new products are created while old ones fall out of favor, consumer trends change, monetary exchange rates fluctuate, and the balance of trade among nations shifts. Trying to analyze the behavior of any economic system over time is like shooting at a moving target.

To deal with such unpredictability, economists have developed sophisticated computer models based on their best estimates of economic factors and interrelationships. Cointegration and error-correction models have been developed over the past decade as remedies to the problems of “spurious regression” arising from non-stationary series. It should also not be surprising that another field that requires a lot of time series data analysis and relies on large complex computer models for future predictions would also involve non-stationary data. That other field is climate science.

Cointegration is used when two time series are both non-stationary, but there is a stationary linear combination of the variables (or their nth differences). In other words, the difference between them has a constant distribution. This sort of thing happens if there is some sort of error-correction or feedback mechanism linking the two time series. Michal P. Murray published a humorous paper in The American Statistican back in 1994, entitled “A Drunk and Her Dog: An Illustration of Cointegration and Error Correction.” In it Murray states his analogy this way:

The drunk is not the only creature whose behavior follows a random walk. Puppies, too, sander aimlessly when unleashed. Each new scent that crosses the puppy's nose dictates a direction for the pup's next step, with the last scent forgotten as soon as the new one arrives. Thus, the meandering, x_i and y_i, of both drunks and dogs along the real line can be modeled by the random walk...

But what if the dog belongs to the drunk? The drunk sets out from the bar, about to wander aimlessly in a random-walk fashion. But periodically she intones "Oliver, where are you?", and Oliver interrupts his aimless wandering to bark. He hears her; she hears him. He thinks, "Oh, I can't let her get too far off; she'll lock me out." She thinks, "Oh, I can't let him get too far off; he'll wake me up in the middle of the night with his barking." Each assesses how far away the other is and moves to partially close that gap.

What Murray is describing is a situation where there are two sets of nonstationary time series data, but there is a meaningful relationship between the two. As the drunk and her dog wander their mutual calls serve as an error-correction mechanism that keeps them from drifting apart. This feedback keeps the distribution of the distance between them statistically stationary. Under these circumstances, it is possible to test for the existence of such a relationship.

Significantly, despite the nonstationarity of the paths, one might still say, "If you find her, the dog is unlikely to be very far away." If this is right, then the distance between the two paths is stationary, and the walks of the woman and her dog are said to be cointegrated of order zero...

Notice that cointegration is a probabilistic concept. The dog is not on a leash, which would enforce a fixed distance between the drunk and the dog. The distance between the drunk and the dog is instead a random variable, but a stationary one despite the nonstationarity of the two paths.

The Murray article goes on to define integrated series and what the term integrated of order n means. Those seeking further details should read the article. For our purposes it is sufficient to state that for a set of series to be cointegrated, each member of the set must be integrated of the same order, n. This is the test that Beenstock and Reingewertz applied to climate data in their paper. The crux of their argument is that for CO₂ to control global temperature there must exist a statisitcal correlation between their respective time series, for a relationship to exist between series they must be cointegrated, and to be cointegrated they must be integrated of the same order.

According to their investigation, CO₂ follows the statistical pattern you’d get if you took a stationary time series and integrated it twice—order I(2). Temperature, however, matches a stationary series integrated only once—order I(1). Note that the quote from Beenstock and Reingewertz above talks about 1^st and 2^nd differences. This is because, when dealing with discreet data, difference equations are used, not the integral or differential equations found in continuous mathematics.

Bottom line, what this means for CO₂ and temperature is that they exhibit two completely different sorts of behavior, behavior that can’t remain correlated for long. The further implication is that any correlations found between CO₂ and temperature must be spurious, the result of inappropriate statistics. But their analysis did not stop there, they performed additional tests for more subtle and complex relationships:

Normally, this difference would be sufficient to reject the hypothesis that global temperature is related to the radiative forcing of greenhouse gases, since I(1) and I(2) variables are asymptotically independent. An exception, however, arises when greenhouse gases, global temperature and solar radiation turn out to be polynomially cointegrated. In polynomial cointegration the greenhouse gases that are stationary in second differences must share a common stochastic trend, henceforth the "greenhouse trend", that is stationary in first differences. If this "greenhouse trend" exists and if it is cointegrated with global temperature and solar irradiance, we may conclude that greenhouse gases are polynomially cointegrated with global temperature and solar irradiance.

The result of this further analysis was that “although greenhouse gases share a common stochastic trend, this "greenhouse trend" is not cointegrated with global temperature and solar irradiance.” They go on to perform other tests, including decomposing the causes of global warming using data from NASA GISS. From the decomposition they calculate the contributions of various forcings to global temperature change during the 20th century. Results are reported in the Table 4 from the paper shown below.

From these data the authors argue that, even though greenhouse gas forcing has made a larger contribution to global warming since 1940 the effect is only temporary. This argument is rather subtle and based on changes in the level of warming, not the level of warming itself. Here is the explanation by Beenstock and Reingewertz:

Between 1880 and 2000 global temperature rose by 0.54 degrees Celsius of which 0.48 occurred since 1940. Equation (2) attributes 0.4 of this to solar irradiance and the balance to greenhouse gas forcings. Table 4 shows, however, that since 1940 greenhouse gas forcing have made a larger contribution to global warming than before, and solar irradiance a smaller one. This creates the misleading impression that the level of greenhouse gas forcings have been the main cause of global warming in the 20^th century. However, our results clearly indicate that it is not the level of greenhouse gas forcings that matters, but the change in the level. During 1880-1940 the level of greenhouse gas forcing increased, but the change in the level decreased. This is why Table 4 implies that greenhouse gas forcings reduced global temperature before 1940.

During the second half of the 20th century greenhouse gas forcings accelerated due in particular to increased carbon emissions. Our model predicts that this effect will be temporary unless these forcing continue to accelerate. Since carbon emissions depend on the level of global economic activity, this continued acceleration would unreasonably imply faster economic growth in the 21^st century than in the 20^th. Our results also imply that cutting carbon emissions will only induce a short-term reduction in global temperature, leaving no long run effect.

In other words, it is not the magnitude of the forcing by CO₂ that causes temperature to rise, it is the change in that magnitude that causes temperature to rise. Adding more CO₂ to the atmosphere will cause a positive change in the associated forcing, which will cause a temporary increase in temperature. This is a result of climatic feedback that acts as an error-correction to the system.

Based on these results and several other statistical tests reported in the paper, the authors claim that AGW stands refuted. They conclude that “there is no evidence relating global warming in the 20^th century to the level of greenhouse gases in the long run.” As I interpret these results, this behavior of the Earth system is only based on data from 1880-2000 and significant change in conditions from those present during that time period could alter the underlying system and its responses. But even with that caveat, this is a stunning conclusion.

Climate Science Revolution

As the ancient Greek philosopher Heraclitus observed, you can not step into the same river twice. By this he meant that a river is constantly changing. The same can be said of both the world economy and Earth's climate system. The way climate regulation works is not a static thing. Can anyone doubt that Earth's climate system during the Cretaceous was different from that in place at the beginning of the Paleocene, or that a different climate balance existed during the last glacial period compared to today? Like Heraclitus' ever changing river, climate is constantly adjusting, finding new balancing points and adapting to change. As temperatures rise and fall new mechanisms and factors come into play: ice cover waxes and wains, precipitation patterns shift, ground cover changes and ocean currents are rearranged. This changing nature implies that climate data are non-stationary and that more sophisticated statistical methods need to be applied to their analysis.

So is it valid to compare economics with climate science, econometrics with climate data analysis? Economic data are generally observational, rather than being derived from controlled experiments. Climate data also mostly falls into the observational category—performing controlled experiments on an entire planet is difficult at best. The individual units in an economy interact with each other in complex ways, as do the myriad of factors involved in Earth's climate system. In either field, observed data tend to reflect complex equilibrium conditions rather than simple behavioral relationships. These similarities suggest that sophisticated techniques developed for analyzing data in one field might be profitably applied to the other.

Of course, hubris stands in the way of easy adoption of skills or techniques from outside of one's own discipline. Plus, the drubbing that climate science has been taking in the media recently has caused the already insular climate science community to become even more distrustful of outsiders. There is also a very wide terminology gap. Both economics and climate science are rife with insider jargon and technical terms, often borrowed from other disciplines though given different meanings.

Consider that, in climate science speak, any temperature variation from some baseline is an “anomaly” while any factor that might exert an influence on the climate is a “forcing.” In economics, “sunk costs” are costs that have been incurred and cannot be reversed, while the “reservation wage” is the lowest wage for which a person will work. The Beenstock and Reingewertz paper on cointegration has been greeted with jeers and charged with being “pure crap” at least partially because of the unfamiliar terminology. One commenter on the WattsUpWithThat.com site went so far as to say:

Where the hell do these guys get off using “nonstationary time series” and “methodology of polynomial cointegration”? Looking through their paper, they say things like “The method of cointegration is designed to test hypotheses with time series data that are non-stationary to the same order, and to avoid the pitfall of spurious regression.” So what. How was it designed? Under what circumstances will this design succeed in modeling reality and how might it fail? And this is supposed to be a Nature paper of general interest? How are they establishing causality where others have failed? There is no discussion nor any proof. They simply assert that they are correct on the strength of tests that they don’t explain.

Ignorance on the behalf of the commenter neither invalidates the methodology nor the results of the study. Of course the main assertion by the authors in the article abstract was bound to touch a nerve: “We show that although greenhouse gas forcings share a common stochastic trend, this trend is empirically independent of the stochastic trend in temperature and solar irradiance. Therefore, greenhouse gas forcings, global temperature and solar irradiance are not polynomially cointegrated, and AGW is refuted.” This may have seemed a reasonable statement to a couple of economists but them's fightin' words to a climate scientist.

It is no wonder that the econometrics camp and the climate change camp are talking past each other. This is unfortunate, because what the cointegration analysis has shown is that the relationship between CO₂ and temperature is not a straightforward causal one. There is still room for CO₂ to be involved in climate regulation but a claim that CO₂ levels directly control temperature has been shown to be statistically invalid. A conscientious scientist in any field would feel obliged to investigate a claim of refutation closely. But sadly, scientists are human and it is human nature to dismiss such charges out of hand, while throwing in a bit of name calling for good measure.

Climate science has reached a crisis, its old methods and habits have proven unequal to the task required of them. The 20^th century philosopher of science Thomas Kuhn recognized that old, outmoded majority points of view are not easily changed. Chaos and disorder within a field are necessary for a major change to take place. Kuhn would have identified the current chaos in climate science as signaling an immanent paradigm shift—a radical restructuring of accepted climate science dogma. In essence, a climate science revolution.

When asked how long a period of cooling global temperatures would be necessary for him to reconsider the AGW paradigm, NASA's Gavin Schmidt replied: “Long term trends from the forcing are expected to be around 0.2 - 0.3 deg/decade. Therefore you need to be able to get uncertainties down to well below those values in order to find a clear discrepancy. Judging from the last thirty years, that period is around a decade.” So a decade's reversal of the previously observed warming trend should give global warming supporters pause.

Dr. Phil Jones, former head of CRU, has publicly admitted that in the last 15 years there has been no “statistically significant” warming. Even when judged by its ardent supporters' own standards, anthropogenic global warming is failing. The only thing that can save climate science from itself is a paradigm shift—a revolutionary change in the basic assumptions within its ruling theory. The theory of AGW will not go without a fight, but go it will—Viva La Revolución!

Be safe, enjoy the interglacial and stay skeptical.