Tests for Price Endogeneity in Differentiated Product Models

Kyoo il Kim; Amil Petrin

doi:10.1515/jem-2012-0002

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Tests for Price Endogeneity in Differentiated Product Models

Kyoo il Kim und Amil Petrin

Veröffentlicht/Copyright: 12. März 2014

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Aus der Zeitschrift Journal of Econometric Methods Band 4 Heft 1

Abstract

We develop simple tests for endogenous prices arising from omitted demand factors in discrete choice models. Our approach only requires one to locate testing proxies that have some correlation with the omitted factors when prices are endogenous. We use the difference between prices and their predicted values given observed demand and supply factors. If prices are exogenous, these proxies should not explain demand given prices and other explanatory variables. We reject exogeneity if these proxies enter significantly in utility as additional explanatory variables. The tests are easy to implement as we show with several Monte Carlos and discuss for three recent demand applications.

Keywords: control function; discrete choice demand; endogenous price; LM test; testing; unobserved demand factor; Wald test

JEL Classification: C3; L0

Corresponding author: Kyoo il Kim, Department of Economics, University of Minnesota, 4-101 Hanson Hall, 1925 4th Street South, Minneapolis, MN 55455, USA, Tel.: +612-625-6793, Fax: +612-624-0209, E-mail: kyookim@umn.edu; and School of Economics, Sungkyunkwan University, Seoul, Republic of Korea

Appendix A Asymptotic Distributions of Test Statistics

A.1 Proof of Lemma 5.1

Proof. The unconstrained ML estimator solves the first order condition ∂logLN(ξ^m, θ˜^U)∂θ˜=0. Define Γ^(θ˜, π)=−∂logLN(ξ˜m(π), θ˜)∂θ˜ ∂θ˜′,Γ0(θ˜, π)=limM→∞E[−∂logLN(ξ˜m(π), θ˜)∂θ˜ ∂θ˜′],Γ^(θ˜)=Γ^(θ˜, π^), and Γ0=Γ0(θ˜0, π0). The asymptotic distribution of M(θ˜^U−θ˜0) is obtained from the asymptotic expansion of

(13)

M(θ˜^U−θ˜0)=Γ^−1(θ˜∗){M∂logLN(ξ˜m, θ˜0)∂θ˜+M×(∂logLN(ξ^m, θ˜0)∂θ˜−∂logLN(ξ˜m, θ˜0)∂θ˜)}+op(1), (13)

which is obtained from the element-by-element mean value expansions of the first order condition ∂logLN(ξ^m, θ˜^U)∂θ˜=0 around θ˜0 where θ˜∗ lies between θ˜^U and θ˜0. Therefore the asymptotic variance of M(θ˜^U−θ˜0) contains two variance terms that are from (i) the ML estimation and (ii) the estimation of the controls.

Note that under Assumptions 5.1, 5.2 (i)(a), (v), and (vi), by the uniform law of large numbers and the continuity, we obtain

(14)

Γ^(θ˜∗)→pΓ0 (14)

because

(15)

‖Γ^(θ˜∗)−Γ0‖≤‖Γ^(θ˜∗)−Γ0(θ˜∗, π^)‖+‖Γ0(θ˜∗, π^)−Γ0‖ ≤supθ˜∈Θ˜0, π∈Π0‖Γ^(θ˜, π)−Γ0(θ˜, π)‖+‖Γ0(θ˜∗, π^)−Γ0‖=op(1) (15)

where the first term in (15) converges to zero by the uniform LLN under Assumption 5.2 (vi) and the second term in (15) converges to zero because of the continuity of Γ₀ [which is implied by Assumption 5.2 (v) and (vi) due to the dominated convergence theorem] and because (θ˜∗, π^)→p(θ˜0, π0).

To derive the variance term due to the second term inside {‧} bracket in (13), we approximate the second term in (13) using a first order mean value expansion,

(16)

M(∂logLN(ξ^m, θ˜0)∂θ˜−∂logLN(ξ˜m, θ˜0)∂θ˜)=1N∑m=1M∑i=1Nm∑j″=0JYimj″∑j′=1J∂2logPimj″(v(ξ˜m(π^*)), θ˜0)∂θ˜ ∂vj′(ξ˜m)∑j=1J∂vj′(ξ˜m(π^*))∂ξ˜mj∂ξ˜mj(π^*)∂πjM(π^j−πj0)+op(1) (16)

where π^* lies between π^ and π₀.

Define ΛjN,M(θ˜, ξ˜m)=1N∑m=1M∑i=1Nm∑j″=0JYimj″∑j′=1J∂2logPimj″(v(ξ˜m), θ˜)∂θ˜∂vj′(ξ˜m)∂vj′(ξ˜m)∂ξ˜mj∂ξ˜mj∂πj. Then we can rewrite (16) as

M(∂logLN(ξ^m, θ˜0)∂θ˜−∂logLN(ξ˜m, θ˜0)∂θ˜)=∑j=1JΛjN,M(θ˜0, ξ˜m(π^*))ϖj+op(1)→dN(0, ∑j, kΛ0jCov(ϖj, ϖk)Λ′0k)=N(0, V1)

where Λ0j=limM→∞E[ΛjN,M(θ˜0, ξ˜m)] and the asymptotic normality result holds by the continuous mapping theorem and by the Lindeberg-Feller CLT under Assumption 5.1. In the above we can show ΛjN,M(θ˜0, ξ˜m(π^*))→pΛ0j by following similar steps to (14) under Assumptions 5.1 and 5.2.

Then by M∂logLN(ξ˜m, θ˜0)∂θ˜→dN(0, Γ0⋅limM→∞MN) [due to the Lindeberg-Feller CLT under Assumption 5.2 (vii)], (14), the Slutsky theorem, and the continuous mapping theorem, from (13) we obtain

(17)

M(θ˜^U−θ˜0)→dN(0, Γ0−1(Γ0⋅limM→∞MN+V1)Γ0−1). (17)

■

A.2 Asymptotic Distribution of the LM Test

Proof. We show that the feasible LM test statistic has the same asymptotic distribution with the corresponding Wald test statistic.

Element-by-element mean value expansions of ∂logLN(ξ^m, θ˜^R)∂θ˜ around θ˜0 yield

(18)

M∂logLN(ξ^m, θ˜^R)∂θ˜=M∂logLN(ξ^m, θ˜0)∂θ˜ +∂logLN(ξ^m, θ˜∗)∂θ˜ ∂θ˜′M(θ˜^R−θ˜0)+op(1) (18)

where θ˜∗ lies between θ˜^R and θ˜0. Note that H(θ˜^R−θ˜0)=0 by construction of H. Write Γ^(θ˜∗)=−∂logLN(ξ^m, θ˜∗)∂θ˜ ∂θ˜′. Then by multiplying HΓ^(θ˜∗)−1 to both sides of (18), it follows that

MHΓ^(θ˜∗)−1∂logLN(ξ^m, θ˜^R)∂θ˜=MHΓ^(θ˜∗)−1∂logLN(ξ^m, θ˜0)∂θ˜+op(1)=MHΓ0−1∂logLN(ξ^m, θ˜0)∂θ˜+op(1)=MHΓ0−1(∂logLN(ξ˜m, θ˜0)∂θ˜+(∂logLN(ξ^m, θ˜0)∂θ˜−∂logLN(ξ˜m, θ˜0)∂θ˜))+op(1)

by the similar argument with (14) and the Slutsky theorem. Therefore, under the null hypotheses (7), MHΓ^(θ˜∗)−1∂logLN(ξ^m, θ˜^R)∂θ˜ follows the same asymptotic distribution with MH(θ˜^U−θ˜0) and we obtain from (17)

(19)

MHΓ̂(θ˜̂R)−1∂logLN(ξ̂m,θ˜̂R)∂θ˜→dℕ0,HΓ0−1Γ0limM→∞MN+V1Γ0−1H′ (19)

by the Lindeberg-Feller CLT and the continuous mapping theorem and because Γ^(θ˜^R)→pΓ0, Γ^(θ˜∗)→pΓ0. Then by (19), Γ^(θ˜^R)→pΓ0,V^1(θ˜^R)→pV1, and the continuous mapping theorem it follows that

LM˜M,N=N∂logLN(ξ^m, θ˜^R)∂θ˜'Γ^(θ˜^R)−1H′V˜M,N−1(θ˜^R)HΓ^(θ˜^R)−1∂logLN(ξ^m, θ˜^R)∂θ˜={MHΓ^(θ˜^R)−1∂logLN(ξ^m, θ˜^R)∂θ˜}′×{HΓ^(θ˜^R)−1(Γ^(θ˜^R)MN+V^1(θ˜^R))Γ^(θ˜^R)−1H'}−1×{MHΓ^(θ˜^R)−1∂logLN(ξ^m, θ˜^R)∂θ˜}→dχ2(dim((λ′, γ˜′, γ˜′p)′)).

Therefore, LM˜M,N follows the same asymptotic distribution as that of the Wald test statistic T˜M,N under the null hypothesis of the price exogeneity.■

A.3 Consistency of the LM Test

Define h¯M,N(v(ξ˜m), θ˜)=1N∑m=1M∑i=1Nmhim(v(ξ˜m), θ˜) and E¯[him(v(ξ˜m), θ˜)]=1N∑m=1M∑i=1NmE[him(v(ξ˜m), θ˜)]. Assume the following to show the consistency of the LM test.

Assumption A.1(i)θ˜^R→pθ˜RAunder the alternative hypothesis against (7); (ii)–(viii) each corresponding Assumption of Assumption 5.2 (ii) to (viii) holds replacingθ˜0withθ˜RAandΘ˜0withΘ˜AwhereΘ˜Adenotes a neighborhood ofθ˜RA.

Theorem A.1Suppose Assumptions 5.1 and A.1 hold. Then the LM testLM˜M,Nis consistent underlimM→∞||E¯[him(v(ξ˜m), θ˜RA)]||≠0.

Proof. Under the alternative hypothesis against (7), using a mean value expansion, we obtain

(20)

Mh¯M,N(v(ξ^m), θ˜^R)= 1N∑m=1M∑i=1Nm(him(v(ξ˜m), θ˜RA)−E[him(v(ξ˜m), θ˜RA)])×MN (20)

(21)

+M(h¯M,N(v(ξ^m), θ˜RA)−h¯M,N(v(ξ˜m), θ˜RA)) (21)

(22)

+∂h¯M,N(v(ξ^m), θ˜A∗)∂θ'M(θ˜^R−θ˜RA)+ME¯[him(v(ξ˜m), θ˜RA)], (22)

where θ˜A∗ lies between θ˜^R and θ˜RA. Now we analyze each term one by one below.

For (21) applying the mean value expansion around π₀, we obtain

(23)

M(h¯M,N(v(ξ^m), θ˜RA)−h¯M,N(v(ξ˜m), θ˜RA)) =∂h¯M,N(v(ξ˜m(π∗)), θ˜RA)∂π′M(π^−π0) (23)

where π^* lies between π^ and π₀. Let ΓπA=limM→∞E[∂h¯M,N(v(ξ˜m(π∗)), θ˜RA)∂π]. We then have

M(h¯M,N(v(ξ^m), θ˜RA)−h¯M,N(v(ξ˜m), θ˜RA))=Γ′πAϖ+op(1)

by Assumption 5.1 and because under E[supπ∈Π0‖∂him(v(ξ˜m(π)), θ˜RA)∂π‖2]<∞ for all m [which holds under Assumption A.1 (viii)], we have ∂h¯M,N(v(ξ˜m(π∗)), θ˜RA)∂π→pΓπA by the uniform Law of Large numbers and because E[∂h¯M,N(v(ξ˜m(π)), θ˜RA)∂π] is continuous at π=π₀ [which is implied by Assumption A.1 (viii) due to the dominated convergence theorem].

Next to analyze the first term in (22) let the inverse of the asymptotic variance matrix of the unconstrained estimator θ˜^U be B=Γ0(Γ0⋅limM→∞MN+V1)−1Γ0 and define a matrix ℳ=I−B−1/2H′(HB−1H′)−1HB−1/2. Then the asymptotic distribution of the constrained estimator θ˜^R is given by M(θ˜^R−θ˜RA)→dN(0, B−1/2ℳB−1/2)≡Z2A (see Newey and McFadden 1994, 2217–2220). Then we obtain

∂h¯M,N(v(ξ^m), θ˜A∗)∂θ˜′M(θ˜^R−θ˜RA)=Γ′θAZ2A+op(1)

where M(θ˜^R−θ˜RA)→dZ2A and ΓθA=limM→∞E[∂h¯M,N(v(ξ^m), θ˜A∗)∂θ˜] because under Assumption A.1 (vi)–(v), we have ∂h¯M,N(v(ξ^m), θ˜A∗)∂θ˜→pΓθA by the uniform Law of Large numbers, because E[∂h¯M,N(v(ξ˜m(π)), θ˜R)∂θ˜] is continuous at θ˜R=θ˜RA and π=π₀ [which is implied by Assumption A.1 (vi)–(v) due to the dominated convergence theorem], and because θ˜A∗→pθ˜RA and π^→pπ0.

Next we consider the term in (20). Note that under E[∥him(v(ξ˜m), θ˜RA)∥4]<∞ for all m [which holds under Assumption A.1 (vii)], by the Lindeberg-Feller CLT, we have

∑m=1M∑i=1Nm(him(v(ξ˜m), θ˜RA)−E[him(v(ξ˜m), θ˜RA)])/N→dZ1A ≡N(0, VhA)

where VhA=limM→∞1N∑m=1M∑i=1NmE[{him(v(ξ˜m), θ˜RA)−E[him(v(ξ˜m), θ˜RA)]}⋅{him(v(ξ˜m), θ˜RA)−E[him(v(ξ˜m), θ˜RA)]}′]. Combining these results, we obtain

Mh¯M,N(v(ξ^m), θ˜^R)=Z1AMN+Γ′πAϖ+Γ′θAZ2A +ME¯[him(v(ξ˜m), θ˜RA)]+op(1)=Op(1)+ME¯[him(v(ξ˜m), θ˜RA)].

Therefore, we obtain ||Mh¯M, N(v(ξ^m), θ˜^R)||→∞ if limM→∞||E¯[him(v(ξ˜m), θ˜RA)]||≠0 under the alternative against (7). This implies LM˜M, N→∞ under the alternative if limM→∞||E¯[him(v(ξ˜m), θ˜RA)]||≠0. Therefore the LM test for price endogeneity is consistent.■

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control function; discrete choice demand; endogenous price; LM test; testing; unobserved demand factor; Wald test