We derive asymptotic properties of an estimator for supply and demand models extended with additional moments relating consumer demographics to the characteristics of purchased products. We clarify the structure of a practical sampling scheme in which the extended estimator is consistent, asymptotically normal, and more efficient than the original estimator. We provide conditions guaranteeing the asymptotic theorems hold for the random coefficient logit model of demand with oligopolistic suppliers. Extensive simulation studies demonstrate significant benefits of the additional moments in estimating the random coefficient logit model. * Manuscript Furthermore, substituting the ω(θ, s n , P R ) = (ω 1 (θ, s n , P R ), . . . , ω J (θ, s n , P R )) obtained from evaluating (7) at ξ = ξ(θ d , s n , P R ) and P = P R for ω j (θ, s 0 , P 0 ) in (11) givesNotice that the first two moments G d J and G c J in (22) are sample moments averaged over products j = 1, . . . , J , although the third moment G a J,T is averaged over consumers t = 1, . . . , T . Also, note that in the expression G J,T (θ, s n , P R , η N ), there exist five distinct randomnesses: one from the draws of the product characteristics (x 1j , ξ j , w 1j , ω j ), two from the sampling processes of consumers for s n and η N (not controlled by the econometrician), and two from the empirical distributions P R and P T (employed by the econometrician). The impact of these randomnesses on the estimate of θ are decided by the relative sizes of the samples-J , n, N, R, and T . Now, we are going to operationalize the sampling and the simulation errors. † 1 , . . . , ξ † J ) is a set of intermediate vectors between ξ(θ d , s n , P R ) and ξ(θ d , s 0 , P R ), and so are ξ ‡where J,T has full column rank.
ASSUMPTION B3.For all sequences of positive numbers δ J,T such that δ J,T → 0, (a) supThis matrix is of full-column rank if the components ∂G d J (θ d , s 0 , P 0 )/∂θ d and ∂G c J (θ, s 0 , P 0 )/∂θ c are, respectively, of full-column rank, regardless of the value of ∂G c J (θ, s 0 , P 0 )/∂θ d . Moreover, we know that ∂G c J (θ, s 0 , P 0 )/∂θ c = −J −1 Z c W by the definition of the cost side moment in Section 2.3 and the assumed linear dependence of ω on W in (7). By properly choosing cost side instruments Z c and cost shifters W, we can construct ∂G c J (θ, s 0 , P 0 )/∂θ c to be of full-column rank for all J . Therefore, we only need to check ∂G d J (θ d , s 0 , P 0 )/∂θ d :we have E [G J,T (θ, s 0 , P 0 , η 0 )] = E [G J,T (θ)]. Thus, condition (ii) follows from Assumption B2. The condition (iii)(a) can be shown as follows: sup ||θ−θ 0 ||≤δ J,T ||G J (θ) − E [G J (θ)] − G J (θ 0 )|| J − 1 2 + ||G J (θ)|| + || E [G J (θ)]|| ≤ sup ||θ−θ 0 ||≤δ J,T J 1 2 ||G J (θ) − E [G J (θ)] − G J (θ 0 )|| ≤ sup ||θ d −θ 0 d ||≤δ J,T J 1 2