By Dennis Fox
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26) where ηˆi = (ˆ ηi1 , . . , ηˆiT ) . Given an estimate of σI2 , this relationship can be used to obtain 2 . estimates of σ12 , . . 27) ˜ When taking the square where one should include the appropriate ML estimator for Σ. roots of the diagonal elements of this matrix, one obtains the appropriate standard errors. 2 Modeling market shares Log-centering approach The above estimation routine is based on the reduced-form model, which is obtained from reducing the system of equations using the base-brand approach.
C + = ... 38) . 1 − −1 . . 1 − 1 I − I1 I − I1 ... − I1 I Note that the matrix C + satisﬁes H (lc) = C + H (bb) . The above shows that the transformations yield equivalent parameters. For example, assume that the log-centered form of the model is estimated, giving estimates of H (lc) µ, H (lc) Bk and H (lc) ΣH (lc) . By multiplying the estimated system of equations by C we get CH (lc) µ, CH (lc) Bk and CH (lc) ΣH (lc) C as model coeﬃcients. Using the invariance principle of maximum likelihood and the relation CH (lc) = H (bb) , these coeﬃcients are the maximum likelihood estimates of H (bb) µ, H (bb) Bk and H (bb) ΣH (bb) .
The expected market shares with and without a promotion, respectively. To compare the additional proﬁts under the two forecasting methods, we will consider the percentage diﬀerence of the additional proﬁts, that is, (∆Πn − ∆Πs )/∆Πs , with n denoting naive forecasts and s denoting simulated forecasts. Note that this measure does not depend on the price, unit cost or the market size. 8. This model includes as explanatory variables prices, display, feature, these variables one period lagged, and lagged market shares.
Advanced econometric marketing models by Dennis Fox