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Ln MI−1,t ln MIt 0 0 ... 34) contains I gives as the dependent vari −1 −1  ..  ln Mt . 35) 24 Modeling market shares with H (bb) as the relevant transformation matrix. 34). This system contains only I − 1 equations. The 1-to-1 relation between the parameters in the two approaches follows from the fact that the equation CH (lc) = H (bb) yields a unique solution C, given by   1 0 . . 0 −1 0 1 . . 0 −1    ..  . C =  ... 37)  .   0 0 . . 1 −1 1 1 ... 1 1 Hence, the matrix C relates the “log-centered” parameters to the “base-brand” parameters.

C. Nielsen. The data span 124 weeks from July 1986 to December 1988. We have information on 14 data sets containing market shares, prices, and two promotional 0/1 dummy variables (display and feature). 4. For each of these markets we consider our in-sample model selection strategy. Model selection and estimation is based on the first 111 observations, so as to have 13 observations (a quarter of a year) for out-of-sample forecasting. 3: Forecast rank based on the Root Mean Squared Prediction Error for each brand (first 3 sets of columns) and on the determinant of the covariance matrix of the forecast errors (last 3 columns) Brand 1 NR RC RE Data Generating Process Brand 2 Brand 3 NR RC RE NR RC RE 2 3 1 3 2 1 Tuna market 1 3 1 2 2 2 1 1 3 3 3 2 1 1 2 3 2 1 3 3 2 1 1 2 3 Tuna market 2 2 3 1 2 1 2 2 1 3 1 3 3 3 2 1 1 2 3 2 1 3 3 2 1 Model NR RC RE 2 3 1 3 2 1 3 2 1 Model NR RC RE 1 2 3 2 1 3 3 2 1 All brands NR RC RE RC=restricted competition, RE=restricted elasticities, NR=no restrictions.

K=1 ln AIt µI βk1I the natural logarithm of attraction becomes     βk21 . . βkI1  ln zk1t ε1t βk22 . . βkI2    ..   ..  +     .  ... ... 30) βk2I . . βkII K Bk ln zkt + εt . 29) implies that ln Mit = ln Ait − ln vector notation this gives   ln M1t I   ln Mt :=  ...  = ln At − iI ln Ajt , I j=1 Ajt . 31) j=1 ln MIt where iI denotes a (I × 1) unity vector. 31) cannot be estimated directly due to the nonlinear dependence of ln( Ij=1 Ajt ) on the model parameters, a reduced-form model should be considered.

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