A Stochastic Response Model With Application to Brand Choice
A Stochastic Response Model With Application to Brand Choice
David Bruce Montgomery
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Recall that by our axioms the observed responses will be cross- sectionally and temporally independent under the model hypothesis. Under this hypothesis tables for and t (t = 1, ... , T) will be independent and by the additivity property of independent chi squares we may achieve a sufficient number of degrees of freedom to test the model by summing the chi squares and degrees of freedom for several such tables. If we have a sequence of T + 1 Page 30. responses from M independent respondents, we... may form the statistic T (6-10) x^ = 2 X^CO. T), t=l where x^(0>t) is given by (6-9). When we estimate the parameters P(0, 0), a, and k by minimizing the statistic (6-10), x^ will be approximately chi square with 2T - 3 degrees of freedom. As noted above the estimates of P(0, 0), a, and k will have desirable statistical properties and we may test the "goodness of fit" of the CEM to the data by comparing (6-10) to a chi square statistic having 2T - 3 degrees of freedom. It remains to justify our use of Q(0) as an estimate of the parameter 34 P(0).
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