Statistical Methods in Customer Relationship Management

C.2 Cointegration Tests: Does a Long-Run Equilibrium Exist between Evolving Series?

Evolving variables are said to be cointegrated when a linear combination exists between them that results in stable residuals. Consider, without loss of generality, a three-variable example where a brand's sales (S), marketing support (M), and its competitors' marketing support (CM) are all evolving (i.e., they all have a unit root). The existence of a perfect equilibrium relationship between these three variables would imply (see Powers et al. [4] for a more in-depth discussion)

(C.6)

In practice, however, we are unlikely to observe a perfect equilibrium in every single period. A more realistic requirement is that its deviations are mean reverting (stable) around zero, that is, in Equation C.7 below should no longer be evolving, even though each of the other variables in the equation is:

C.7

For a detailed review of the test procedure for cointegration, readers may refer to Dekimpe and Hanssens [1].