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 canâ€™t find anywhere on the web that explains them simply for the average Joe. Given one canâ€™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 â€˜differential diagnosisâ€™ of Beenstock that GHGâ€™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 â€˜asymptotically independentâ€™.

It follows from â€˜asymptotic independenceâ€™ 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 â€˜asymptotically independentâ€™.

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 â€˜heuristicâ€™ or empirical relationship. So you can see that it is ill-informed call these models â€˜unphysicalâ€™. 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 â€˜feltâ€™ 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 â€˜driftâ€™, 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.

“Some issues arise from limited data, so that its hard to be sure of the exact order I(n).”Well, yes. But they sound very sure. And give no test statistics to say how certain you can be that temp is I(1) and not I(2), and CO2 conversely.But how can you ever deduce “asymptotic independence” from a finite set of data? All you have is the time period you see.“Some issues arise from limited data, so that its hard to be sure of the exact order I(n).”Well, yes. But they sound very sure. And give no test statistics to say how certain you can be that temp is I(1) and not I(2), and CO2 conversely.

But how can you ever deduce “asymptotic independence” from a finite set of data? All you have is the time period you see.

Please check:”The flow into the tank is called integration order I(0), while the level in the tank is integration order one, or I(0).”Do you mean “The flow into the tank is called integration order zero, or I(0), while the level in the tank is integration order one, or I(1).”

Please check:

“The flow into the tank is called integration order I(0), while the level in the tank is integration order one, or I(0).”

Do you mean

“The flow into the tank is called integration order zero, or I(0), while the level in the tank is integration order one, or I(1).”

I know this series has moved on but like a number of layman I'm still trying to get my head around the significance between integration 1(1) and 1(2), being the difference between solar and CO2. It occurs to me that the difference is reflected in the distinction between effective temperature, 255K, and greenhouse temperature, 33K. Variations in solar determine effective temperature; variations in CO2, partially, affect greenhouse temperature; partially because water is the biggest determinator of greenhouse temperature and CO2 is governed by the log decline; given the quantitative proportions solar effects dwarf CO2 effects and produce the cointegration differences.

Perhaps its like this. Temperature vaies with solar as they are the same order. But GHGs accumulate with increases in temperature. However, the changes in CO2 also feedback to affect temperature, driving it a bit higher when temperature goes up. But ultimately the absolute level of CO2 does not affect temperature. For CO2 to keep temperature elevated at current levels, it must keep increasing at a constant rate.

I know this series has moved on but like a number of layman I’m still trying to get my head around the significance between integration 1(1) and 1(2), being the difference between solar and CO2. It occurs to me that the difference is reflected in the distinction between effective temperature, 255K, and greenhouse temperature, 33K. Variations in solar determine effective temperature; variations in CO2, partially, affect greenhouse temperature; partially because water is the biggest determinator of greenhouse temperature and CO2 is governed by the log decline; given the quantitative proportions solar effects dwarf CO2 effects and produce the cointegration differences.

Perhaps its like this. Temperature vaies with solar as they are the same order. But GHGs accumulate with increases in temperature. However, the changes in CO2 also feedback to affect temperature, driving it a bit higher when temperature goes up. But ultimately the absolute level of CO2 does not affect temperature. For CO2 to keep temperature elevated at current levels, it must keep increasing at a constant rate.

That being the case the notion of Equilibrium and Transient sensitivity and CO2 residency time are irrelevant.

That being the case the notion of Equilibrium and Transient sensitivity and CO2 residency time are irrelevant.

Yes, exactly.

Yes, exactly.

There is a good popular account of co-integration here. http://www-stat.wharton.upenn.edu/~steele/Cours…

There is a good popular account of co-integration here. http://www-stat.wharton.upenn.edu/~steele/Courses/434/434Context/Co-integration/Murray93DrunkAndDog.pdf

This is a rather bad generalization. Its not that you cant make a correlation between an I(0) and an I(1) it that you have to use polynomial cointegration. When that is done the CO2 to temperature correlation falls flat.

This is a rather bad generalization. Its not that you cant make a correlation between an I(0) and an I(1) it that you have to use polynomial cointegration. When that is done the CO2 to temperature correlation falls flat.

This is a rather bad generalization. Its not that you cant make a correlation between an I(0) and an I(1) it that you have to use polynomial cointegration. When that is done the CO2 to temperature correlation falls flat.

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