I(0), I(1) or I(2)? What does it all mean? Below is a visual presentation of CO2 concentration from Mauna Loa and global temperature from GISS, demonstrating the difference in their order of integration.

The blue series is the increasing level of CO2 in annual steps. Differencing means to successively subtract the previous value at each step, giving the change at each step (the delta or dCO2 shown in magenta). Differencing again gives ddCO2 shown in yellow. After these two differences the values are clearly oriented around the zero line (or stationary).

The red series is the global temperature in annual steps. The purple series is the first difference. The first difference is clearly oriented on the zero line.

Series that are oriented on zero are called I(0) — no differencing is required for them to be stationary. Series like temperature that require one differencing to become stationary are called I(1). Series like CO2 that require two differences to become stationary are called I(2).

One of the central ideas around cointegration is that series that require differencing to become stationary are very prone to ‘spurious regression’. That is they look like they are correlated because they have the same trend, but will tend to wander further away from each other over time. Special methods have been developed to distinguish when series are really related, when simple correlation will be fooled.

Another idea is that series with different orders of integration can always wander arbitrarily far away from each other. In order for two series to be related, they must be of the same order. This is the basis for the claim that the first difference of CO2, or change in CO2, is related to temperature and not absolute level of CO2.