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I − 1. One hypothesis corresponds to the parameter restriction µ ˜i = µ can also focus on the effect of a single specific marketing instrument, like the relative (1) (2) price. In that case, the test concerns β˜kji = β˜kji , i = 2, . . , I − 1 and j = 2, . . , I for a specific marketing instrument k. Finally, we can test for constant variance of the unexplained attractions of the incumbent brands. As stated before, the covariances can be interpreted as measuring the similarity of brands. Testing for the constancy of the covariance structure can therefore be interpreted as testing for constancy of the overall brand positioning.

I 1 ln MIt 1 1 1 j=1 ln Mjt I −I −I . . 1 − I = H (lc) ln Mt , where H (lc) , with rank I − 1, denotes the transformation matrix corresponding to the log-centering approach. 34) k=1 as H (lc) iI = 0I×I . Due to the reduced rank of H (lc) , the equations, but it only has I − 1 independent equations. 1 ables in the reduced-form model      1 0 ... 0 ln MIt ln M1t 0 1 . . 0   ..    ..  −  .  =  ..  .. . . ln MI−1,t ln MIt 0 0 ... 34) contains I gives as the dependent vari −1 −1  ..

One can now simply use simulated values for lagged market shares, thereby automatically taking into account the uncertainty in these lagged variables. Note that we assume that the marketing efforts of all market players are known. It is possible to also model these efforts and use the estimated model to obtain market share forecasts that also account for that uncertainty. The models describing the marketing efforts can be used to simulate future values of the levels of the marketing instruments. 44).

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