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Such an approach could turn out to be useful in a category in which expansion effects play an important role. However, in a sales model, one needs to model seasonal effects and the category expansion or category contraction explicitly. In order to analyze changes in actual behavior of a brand manager, that is, changes in the use of instruments like price and display, one needs methods to test for structural changes in time series variables. Upon doing so, we build on the findings in Srinivasan et al.

Note that the reduced-form model in this case still contains I equations. To demonstrate the equivalence of parameters obtained through the log-centering technique of Cooper and Nakanishi (1988) and those using the base-brand approach, we show that there exists an exact relationship between these sets of parameters. The parameters for the base-brand specification can uniquely be determined from the parameters for the log-centering specification and vice versa. Given the 1-to-1 relationship the likelihoods are the same, that is, the discussed feasible GLS estimator yields the same maximum value of the likelihood as we can use the invariance principle of maximum likelihood, see for example Greene (1993, page 115).

5 evaluates the out-of-sample performance of the six considered models. The appropriate evaluation criterion is again the log of the determinant of the covariance matrix of the forecast errors. The models selected by our strategy (model VI) deliver the best forecasts in 7 out of 14 cases for both horizons, where for each forecast 25,000 replications are used. Furthermore, the average rank across models is lowest. Additionally, for each model we calculate the ratio of the forecast error and the forecast error of the best performing model.

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