A draft of a paper by Beenstock and Reingewertz has surfaced in the blogosphere, but there seems to be confusion about what unit roots and cointegration are, and I can’t find anywhere on the web that explains them simply for the average Joe. Given one can’t understand the paper without a good grasp of these concepts; I am going to do a few posts in an attempt to make their argument clearer.

The ‘differential diagnosis’ of Beenstock that GHG’s are not giving plant Earth a fever, consists of advanced analysis of time series, showing surface temperature and greenhouse gasses have different integration orders I(n). Specifically, they claim temperature is I(1) and irradiance due to GHGs (rfGHGs) are I(2) and so ‘asymptotically independent’.

It follows from ‘asymptotic independence’ that a doubling of rfGHG cannot cause a permanent increase in temperature. All this sounds complicated but it is really quite simple. They essentially say the two have fundamentally different units, and so even though they may correlate over a limited period, they ultimately do not have an equilibrium relationship.

As an example, consider random buckets of water F, pouring into (and out of) a tank Y. The level in the tank at time t is Yt, and could be calibrated in units of gallons. The buckets pouring water into the tank could be in units of gallons per hour. The flow into the tank is called integration order I(0), while the level in the tank is integration order one, or I(1). The tank integrates the flow of gallons per hour into gallons.

In the short run, one would expect some correlation over time between F and Y. For example, a run of buckets poured in would result in a higher level. However, this run of inflows could occur when the water level is high or low. They are ‘asymptotically independent’.

In essence, Beenstock claims that comparing temperature and rfGHG is a case of inappropriate mixture of units, i.e. apples and oranges. One of the main requirements of a physical model is that the units are consistent. If not, it cannot be a physically valid model, and is essentially a ‘heuristic’ or empirical relationship. So you can see that it is ill-informed call these models ‘unphysical’. In fact, the orders integration I(0) and I(1) have a very obvious physical interpretation in this example: as a rate and a quantity.

They also find that solar radiation is I(1) and so can be related to surface temperature. Even though both rfGHG and Solar irradiance are expressed in the same units (Watts/meter^3) there must be some difference between them producing the apparently different behaviours.

What that difference is, is not clear. It could be that while changes in solar radiation change surface temperature to maintain overall radiative balance, but forcing by rfGHG in the atmosphere are compensated within the atmosphere in some way. In fact, they go on to claim that changes in GHG’s are ‘felt’ by the surface temperature, but the effect is temporary, lasting less than a year.

This is a simplified explanation. Some issues arise from limited data, so that its hard to be sure of the exact order I(n). There could also be a ‘drift’, for example due to evaporation from the tank. Seasonality, autocorrelation and structural breaks can all affect the tests, leading to difficulty in applying them correctly. This problem is well known by Beenstock who tries to eliminate them. However, this could also be why there are differences of opinion in the literature.