By Dennis Fox
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The optimal value of P turns out to be 1 for all data sets, except for Sugar market 2, where P appears to be two. The ﬁnal six rows show the restrictions (if any) on the dynamic structure, the covariance matrix, and the exogenous variables resulting from the model selection strategy. We notice that in most cases (11 out of 14), models with common dynamics [CD] are preferred. Also, the restricted covariance matrix restriction is preferred in 10 out of 14 cases. This is quite interesting as this restriction is rarely (if ever) imposed in practice!
In a Bayesian setting one could even derive posterior probabilities for the proposed models. One may select the model with the highest posterior probability or one can combine several models. For example, to construct forecasts, one can use the posterior probabilities to weight forecasts generated by the diﬀerent models. Another strategy is to start with a general model and try to simplify it using statistical 26 Modeling market shares test. In this chapter we opt for this general-to-simple model selection strategy, following Hendry (1995).
The average of the forecasts over all generated parameter vectors constitutes unbiased forecasts of the market shares under uncertain parameters. That is, we calculate E[Mit ] as L1 Ll=1 E[Mit | θˆ(l) ]. It is not necessary to use many simulation rounds conditional on the parameters. Theoretically it suﬃces to use one round for every θˆ(l) . In a classical setting we have to rely on bootstrapping techniques to account for parameter uncertainty. A Bayesian analysis of market share models would have the advantage that it provides a more natural approach to account for parameter uncertainty.
Advanced econometric marketing models by Dennis Fox