parametric assumption required to obtain a logit form conditional probability, it requires that εijt
be uncorrelated with price and thus that no unobserved heterogeneity of products be correlated
with consumer tastes and with price. As some product characteristics might be omitted in the
specification of utility (18), like for instance, product advertising, and be correlated with the prices
of products, Petrin and Train (2010) propose a control function approach to solve this endogeneity
problem of prices. This method consists in estimating a first stage regression of prices on observed
cost shifters as follows :
Pjt = λb(j') + λr(j) + yWjt + ηjt
where λb(j) and λr(j) are brand specific and retailer specific effects and Wjt represents a vector of
cost shifters like input prices and ηj∙t is a random shock defined as the residual of the orthogonal
projection of pj-t on ʌb(j), ʌr(j), Wjt. Then, introducing the estimated term τjj∙t in the specification
of the consumer utility Uijt makes the assumption of orthogonality of the residual consumer utility
deviations (denoted Uijt) with price more plausible. This method amounts to assume that the
consumer utility can be written as follows : Uijt = βb(j) + βr(j) + δtXj — atpjt + ττjjt + utjt where
by definition uijt = εijt — τηjjt with the maintained assumption that uijt is orthogonal to pjt. With
this random utility, we assume that consumer i chooses alternative j{i,t) if Uij(ijt)t > Uijt for all
j = 1,.., J and Uij(ijt)t > Uijt for some j.
This method allows to estimate consistently the demand price elasticities even if time varying
unobserved characteristics (correlated with rjjt) affect consumer tastes and are correlated with
price (like advertising), provided that the residual or the projection of these unobservables on
rjjt be uncorrelated with the price Pjt. Remark that such specification also implies that policy
simulations have to be taken cautiously. Actually, the endogenous determination of this unobserved
heterogeneity is not modelled and is thus unknown under possible counterfactual situations to be
simulated (for example like a merger), unless we maintain that this unobserved heterogeneity does
not change in the counterfactual situation.
Then, instead of making a parametric assumption on ε⅛t, we assume that the idiosyncratic
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