Asymptotic Efficiency of Joint Estimator Relative to Two-Stage Estimator Under Misspecified Likelihoods

Doosoo Kim

doi:10.1515/snde-2023-0009

Article

Asymptotic Efficiency of Joint Estimator Relative to Two-Stage Estimator Under Misspecified Likelihoods

Doosoo Kim

Published/Copyright: May 27, 2024

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 29 Issue 3

Abstract

The two-stage estimator is often more tractable when there are nuisance parameters that can be separately estimated and plugged into an objective function. The joint estimator tends to bear the higher computational cost since it estimates all parameters in one stage by optimizing the sum of objective functions used in the two stages. It is well-known that the joint estimator is asymptotically more efficient than the two-stage estimator if the objective function is the true log-likelihood. When the objective function is not the true log-likelihood, I show that the relative asymptotic efficiency of the joint estimator still holds under a finite number of testable moment conditions. The implications of the main result on models based on quasi-limited information likelihoods are discussed.

Keywords: quasi-likelihood; control function; information matrix

PACS: C13; C21; C26; C31

Corresponding author: Doosoo Kim, Department of Economics, 7984 Toronto Metropolitan University , Toronto, Canada, E-mail: doosoo@torontomu.ca, www.doosoo.kim

Acknowledgment

I sincerely thank Jeffrey M. Wooldridge for his valuable guidance and support. I appreciate the helpful comments from Cathy Ning, Leo Michelis, Kyoo il Kim, and Peter Schmidt. I also thank an anonymous referee for constructive comments.

Appendix A: Assumptions

1. w _i is i.i.d. 2. Θ ⊂ c p t R p . 3. q 1 : Θ × W → R and q 2 : Θ 2 × W → R where w _i ∈ W. 4. θ o ∈ i n t Θ and let N be a neighborhood of θ _o. 5. With probability one, q 1 w i , θ 1 , θ 2 and q 2 w i , θ 2 are continuously differentiable at each θ ∈ Θ and twice continuosly differentiable in N . 6. E sup θ ∈ Θ ∂ q 1 + q 2 / ∂ θ ∂ q 2 / ∂ θ 2 < ∞ . 7. Each element of ∂ q 1 w i , θ o 1 , θ o 2 / ∂ θ and ∂ q 2 w i , θ o 2 / ∂ θ 2 has finite second moment. 8. E sup θ ∈ N ∂ q 1 + q 2 / ∂ θ ∂ θ ′ ∂ q 2 / ∂ θ 2 ∂ θ ′ < ∞ . 9. θ o = θ ∈ Θ : E ∂ q 1 + q 2 / ∂ θ = 0 . 10. θ o = θ ∈ Θ : E ∂ q 1 θ / ∂ θ 1 ∂ q 2 θ 2 / ∂ θ 2 = 0 . 11. E ∂ 2 q 1 + q 2 θ o / ∂ θ ∂ θ ′ and V ∂ 2 q 1 + q 2 θ o / ∂ θ are invertible. 12. E ∂ 2 q 1 θ o / ∂ θ 1 ∂ θ ′ ∂ 2 q 2 θ o 2 / ∂ θ 2 ∂ θ ′ and V ∂ q 1 θ o / ∂ θ 1 ∂ q 2 θ o 2 / ∂ θ 2 are invertible.

Appendix B: Proofs

B.1 An Example: A Maximal Linearly Independent Moment Function as an Efficiency Benchmark

The asymptotic variances of efficient GMM estimator based on (5) and the efficient GMM estimator in Proposition 1 of Penaranda and Sentana (2012; PS) based on (6) are the same.

To see this result hold in an example, consider probit model in Example 1 and 2. Given the simplified quasi-likelihoods

q 1 θ 1 , θ 2 = 1 − y 1 log 1 − Φ w θ + y 1 ⁡ log Φ w θ q i 2 θ 2 = − k 2 ln ⁡ 2 ⁡ π − 1 2 ln Σ 22 − 1 2 v i 2 δ 2 Σ 22 − 1 v i 2 δ 2 ′

where θ 1 = α ′ , δ 1 ′ , η ′ ′ , θ 2 = vec δ 2 ′ , vech Σ 22 ′ ′ , x = y 2 z 1 v 2 and w θ = x θ 1 , taking derivatives, quasi-scores can be expressed as

∂ q 1 ∂ θ 1 = y 1 − Φ w θ 1 − Φ w θ Φ w θ ϕ w θ y 2 ′ z 1 ′ v 2 ′ ∂ q 1 ∂ θ 2 = − y 1 − Φ w θ 1 − Φ w θ Φ w θ ϕ w θ − η ⊗ z ′ 0 r r + 1 2 × 1 ∂ q 2 ∂ θ 2 = I r ⊗ z ′ Σ 22 − 1 v 2 δ 2 ′ 1 2 L r vec Σ 22 − 1 v i 2 δ 2 ′ v i 2 δ 2 Σ 22 − 1 − Σ 22 − 1

There are two cases that require separate consideration. If η = 0, then ∂ q 1 ∂ θ 2 is a zero vector. If η ≠ 0, then only the part of ∂ q 1 ∂ θ 2 with k 2 − r excluded instruments can be linearly independent. For simplicity, put r = 1. The first case with η = 0 is trivial. In the case with η ≠ 0, we can take Π θ t = 0 1 η I k 1 − δ 1 ′ 0 − 1 I k 1 0 k 1 × k 2 0 k 2 − 1 × k 1 δ 2 ′ η I k 2 × k 2 0 0 0 k 1 + k 2 × p 2 0 0 1 × k 1 0 0 1 × k 1 + k 2 1 0 1 × p 2 , which is continuously differentiable in the subset of the parameter space considered i.e. η ≠ 0 . Then, in both cases, by the argument right below < 10 > of Penaranda and Sentana (2012), it is implied that the asymptotic variance of the efficient GMM estimator in Proposition 1 Penaranda and Sentana (2012) is equal to the usual GMM formulae with the inverse replaced by a genearlized inverse, and that the optimal GMM effectively use the maximal linearly independent moment conditions in each case, which coincides with the moment conditions in (5).

Moreover, Lemma C.1. of Penaranda and Sentana (2012) implies that the result holds in general. To show redundancy of ∂ q 1 ∂ θ 21 , take the moment functions in (6) as h ₁ and ∂ q 1 ∂ θ 21 as h ₂. Then, g ₂ will be a zero vector a.s. due to the linear dependence. Hence the result trivially holds.

B.2 Proof of Proposition 1

Under regularity conditions, Theorem 3.4 of Newey and McFadden (1994) implies the result.

B.3 Proof of Theorem 1

Denote V est r S ≡ Avar N θ ̂ S , est r − θ o S for partition θ ≡ θ S , θ − S . Note GIMEs imply the condition in (c) of lemma below.

Lemma 1.

Assume that Assumptions 1–12 hold and that θ ₂₂ is nonempty. Then, we have (a) V _QGMM ⪯ V _J and V _QGMM ⪯ V _TS, (b) V _QGMM = V _TS if and only if

E o ∂ 2 q i 1 o ∂ θ 22 ∂ θ ′ = cov o ∂ q i 1 o ∂ θ 22 , ∂ q i 1 o ∂ θ 1 ′ , ∂ q i 2 o ∂ θ 2 ′ ′ V o ∂ q i 1 o ∂ θ 1 ′ , ∂ q i 2 o ∂ θ 2 ′ ′ − 1 E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ ′ ∂ 2 q i 2 o ∂ θ 2 ∂ θ ′

, (c) V _QGMM = V _J if and only if

E o ∂ 2 q i 1 o ∂ θ 2 * ∂ θ ′ = cov o ∂ q i 1 o ∂ θ 2 * , ∂ q i o ∂ θ V o ∂ q i o ∂ θ − 1 E o ∂ q i o ∂ θ ∂ θ ′

where θ 2 * is a subvector of θ ₂ such that ∂ q 1 o ∂ θ 2 * , ∂ q i 1 o ∂ θ 1 and ∂ q i o ∂ θ 2 are maximal linearly independent. (d) V QGMM S = V J S iff

E o ∂ 2 q i 2 o ∂ θ 22 ∂ θ S ′ − cov o ∂ q i 2 o ∂ θ 22 , ∂ q i 1 o ∂ θ 1 ∂ q i 1 o ∂ θ 2 + ∂ q i 2 o ∂ θ 2 W o * E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ S ′ ∂ 2 q i 1 o ∂ θ 2 ∂ θ S ′ + ∂ 2 q i 2 o ∂ θ 2 ∂ θ S ′ = E o ∂ 2 q i 2 o ∂ θ 22 ∂ θ − S ′ − cov o ∂ q i 2 o ∂ θ 22 , ∂ q i 1 o ∂ θ 1 ∂ q i 1 o ∂ θ 2 + ∂ q i 2 ∂ θ 2 W o * E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ − S ′ ∂ 2 q i 1 o ∂ θ 2 ∂ θ − S ′ + ∂ 2 q i 2 o ∂ θ 2 ∂ θ − S ′ × E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ − S ′ ∂ 2 q i 1 o ∂ θ 2 ∂ θ − S ′ + ∂ 2 q i 2 o ∂ θ 2 ∂ θ − S ′ ′ W o * E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ − S ′ ∂ 2 q i 1 o ∂ θ 2 ∂ θ − S ′ + ∂ 2 q i 2 o ∂ θ 2 ∂ θ − S ′ − 1 × E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ − S ′ ∂ 2 q i 1 o ∂ θ 2 ∂ θ − S ′ + ∂ 2 q i 2 o ∂ θ 2 ∂ θ − S ′ ′ W o * E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ S ′ ∂ 2 q i 1 o ∂ θ 2 ∂ θ S ′ + ∂ 2 q i 2 o ∂ θ 2 ∂ θ S ′

where W o * = V o ∂ q i 1 o ∂ θ 1 ∂ q i 1 o ∂ θ 2 + ∂ q i 2 o ∂ θ 2 − 1 , and (e) V QGMM S = V T S S iff

E o ∂ 2 q i 1 o ∂ θ 22 ∂ θ S ′ − cov o ∂ q i 1 o ∂ θ 22 , ∂ q i 1 o ∂ θ 1 ∂ q i 2 o ∂ θ 2 V o ∂ q i 1 o ∂ θ 1 ∂ q i 2 o ∂ θ 2 − 1 E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ S ′ ∂ 2 q i 2 o ∂ θ 2 ∂ θ S ′ = E o ∂ 2 q i 1 o ∂ θ 22 ∂ θ − S ′ − cov o ∂ q i 1 o ∂ θ 22 , ∂ q i 1 o ∂ θ 1 ∂ q i 2 o ∂ θ 2 V o ∂ q i 1 o ∂ θ 1 ∂ q i 2 o ∂ θ 2 − 1 E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ − S ′ ∂ 2 q i 2 o ∂ θ 2 ∂ θ − S ′ × E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ − S ′ ∂ 2 q i 2 o ∂ θ 2 ∂ θ − S ′ ′ V o ∂ q i 1 o ∂ θ 1 ∂ q i 2 o ∂ θ 2 − 1 E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ − S ′ ∂ 2 q i 2 o ∂ θ 2 ∂ θ − S ′ − 1 E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ − S ′ ∂ 2 q i 2 o ∂ θ 2 ∂ θ − S ′ ′ V o ∂ q i 1 o ∂ θ 1 ∂ q i 2 o ∂ θ 2 − 1 E o ∂ 2 q i 1 o ∂ θ 1 ∂ θ S ′ ∂ 2 q i 2 o ∂ θ 2 ∂ θ S ′

Proof.

(a): V _QGMM ⪯ V _TS is trivial. To see V _QGMM ⪯ V _J, note first that, at true parameter value,

∂ ∂ θ 1 q 1 θ 1 , θ 2 ∂ ∂ θ 2 q 1 θ 1 , θ 2 + ∂ ∂ θ 2 q 2 θ 2

is linearly independent by Assumptions 7 and 11. Thus, there exists an extension to a basis

(22) ∂ ∂ θ 1 q 1 θ 1 , θ 2 ∂ ∂ θ 2 q 1 θ 1 , θ 2 + ∂ ∂ θ 2 q 2 θ 2 ∂ ∂ θ 2 * q 1 θ

which is an invertible linear transformation of (6). Hence, the result follows. (b): Apply BQSW redundancy condition to (6). (c): Apply BQSW redundancy condition to (22). (d) and (e): Apply BQSW partial redundancy results to (6) and (22), respectively. □

B.4 Proof of Theorem 2

To show (i), define Schur complements as A / A 22 + C 22 ≡ A 11 − A 12 A 22 + C 22 − 1 A 21 and A / A 11 ≡ A 22 + C 22 − A 21 A 11 − 1 A 12 where A = A 11 A 12 A 21 A 22 + C 22 , A 11 = E − ∂ 2 q i 1 o ∂ θ 1 ∂ θ 1 ′ , A 12 = A 21 t = E − ∂ 2 q i 1 o ∂ θ 1 ∂ θ 2 ′ , A 22 = E − ∂ 2 q i 1 o ∂ θ 2 ∂ θ 2 ′ and C 22 = E − ∂ 2 q i 2 o ∂ θ 2 ∂ θ 2 ′ . Assume GIMEs i.e. V ∂ q i 1 o ∂ θ 1 = τ 1 A 11 , c o v ∂ q i 1 o ∂ θ 1 , ∂ q i 1 o ∂ θ 2 = τ 1 A 12 , V ∂ q i 1 o ∂ θ 2 = τ 1 A 22 , V ∂ q i 2 o ∂ θ 2 = τ 2 C 22 and cov ∂ q i 1 o ∂ θ , ∂ q i 2 o ∂ θ 2 = 0 . Then, the claim is V T S θ 1 − V J θ 1 = A 11 − 1 A 12 τ 2 W 1 + τ 1 − τ 2 W 2 A 21 A 11 − 1 where W 1 = C 22 − 1 − [ A / A 11 ] − 1 and W 2 = − [ A / A 11 ] − 1 A 22 − A 21 A 11 − 1 A 12 [ A / A 11 ] − 1 .

First, the variance difference V T S θ 1 − V J θ 1 is equal to

A 11 − 1 B 2 A 11 − 1 − A 11 − A 12 A 22 + C 22 − 1 A 21 − 1 B 1 A 11 − A 12 A 22 + C 22 − 1 A 21 − 1

where

B 1 = V ∂ q i 1 o ∂ θ 1 − A 12 A 22 + C 22 − 1 ∂ q i 1 o ∂ θ 2 + ∂ q i 2 o ∂ θ 2 = τ 1 A 11 + A 12 A 22 + C 22 − 1 τ 1 A 22 + τ 1 C 22 − τ 1 C 22 + τ 2 C 22 A 22 + C 22 − 1 A 21 − 2 τ 1 A 12 A 22 + C 22 − 1 A 21 = τ 1 A 11 − A 12 A 22 + C 22 − 1 A 21 + τ 2 − τ 1 A 12 A 22 + C 22 − 1 C 22 A 22 + C 22 − 1 A 21

and B 2 = V ∂ q i 1 o ∂ θ 1 − A 12 C 22 − 1 ∂ q i 2 o ∂ θ 2 = τ 1 A 11 + τ 2 A 12 C 22 − 1 A 21 .

Since A 11 − A 12 A 22 + C 22 − 1 A 21 − 1 = A 11 − 1 + A 11 − 1 A 12 [ A 22 + C 22 ] − A 21 A 11 − 1 A 12 − 1 A 21 A 11 − 1 , the difference V T S θ 1 − V J θ 1 is equal to

τ 2 A 11 − 1 A 12 C 22 − 1 − A 22 + C 22 − A 21 A 11 − 1 A 12 − 1 A 21 A 11 − 1 + τ 1 − τ 2 A 11 − A 12 A 22 + C 22 − 1 A 21 − 1 A 12 D A 21 A 11 − A 12 A 22 + C 22 − 1 A 21 − 1 − τ 1 − τ 2 A 11 − 1 A 12 A 22 + C 22 − A 21 A 11 − 1 A 12 − 1 A 21 A 11 − 1

where D = A 22 + C 22 − 1 C 22 A 22 + C 22 − 1 . Now, it suffices to show

A 11 − A 12 A 22 + C 22 − 1 A 21 − 1 A 12 D A 21 A 11 − A 12 A 22 + C 22 − 1 A 21 − 1 − A 11 − 1 A 12 A 22 + C 22 − A 21 A 11 − 1 A 12 − 1 A 21 A 11 − 1 = − A 11 − 1 A 12 A / A 11 − 1 A 22 − A 21 A 11 − 1 A 12 A / A 11 − 1 A 21 A 11 − 1

The first term on the LHS:

A 11 − A 12 A 22 + C 22 − 1 A 21 − 1 A 12 D A 21 A 11 − A 12 A 22 + C 22 − 1 A 21 − 1 = A 11 − 1 A 11 + A 12 A / A 11 − 1 A 21 A 11 − 1 A 12 D A 21 A 11 − 1 A 11 + A 12 A / A 11 − 1 A 21 A 11 − 1 = A 11 − 1 A 12 A / A 11 − 1 A / A 11 + A 21 A 11 − 1 A 12 D A / A 11 + A 21 A 11 − 1 A 12 A / A 11 − 1 A 21 A 11 − 1 = A 11 − 1 A 12 A / A 11 − 1 A 22 + C 22 D A 22 + C 22 A / A 11 − 1 A 21 A 11 − 1 = A 11 − 1 A 12 A / A 11 − 1 C 22 A / A 11 − 1 A 21 A 11 − 1

The second term on the LHS:

A 11 − 1 A 12 A 22 + C 22 − A 21 A 11 − 1 A 12 − 1 A 21 A 11 − 1 = A 11 − 1 A 12 A / A 11 − 1 A / A 11 A / A 11 − 1 A 21 A 11 − 1

Hence the result.

To see positive semi-definiteness of W ₁, note that following statements are equivalent when A ₁₁ ≻ 0: (1) C 22 − 1 − A / A 11 − 1 ⪰ 0 , (2) A / A 11 − C 22 = A 22 − A 21 A 11 − 1 A 12 ⪰ 0 (3) A 11 A 12 A 21 A 22 ⪰ 0 where the equivalence between (2) and (3) is from Schur complement condition for positive semi-definiteness. Since A 11 = 1 τ 1 V o ∂ q i 1 o ∂ θ 1 ≻ 0 by linear independence of elements in ∂ q i 1 ∂ θ 1 at θ _o, and A 11 A 12 A 21 A 22 = 1 τ 1 V o ∂ q i 1 o ∂ θ ⪰ 0 , the statement is shown. Negative semi-definiteness of W ₂ is also proved similarly.

To show (ii), let V E s t r θ 2 τ 1 , τ 2 denote the asymptotic variance of an estimator for θ ₂ where τ 1 , τ 2 are the scaling factors in GIMEs (12) and (13), respectively. The first term τ ₂ W ₁ is equal to the asymptotic variance difference of joint and two-stage estimators for θ ₂ under GIMEs with a common scaling factor τ ₂ i.e. τ 2 W 1 = V T S θ 2 τ 2 , τ 2 − V J θ 2 τ 2 , τ 2 . Since the second term W ₂ is the negative of the asymptotic variance of joint estimator for θ ₂ ignoring ∂ q 2 ∂ θ 2 in the outer product, the whole middle part of (15) is actually equal to V T S θ 2 τ 2 , τ 2 − V J θ 2 τ 1 , τ 2 . Since V T S θ 2 τ 2 , τ 2 = V T S θ 2 τ 1 , τ 2 , we have result. To formally write this, we can show

V J θ 2 = A / A 11 − 1 A 21 A o − 1 V 11 1 A o − 1 A 12 − V 21 1 A o − 1 A 12 − A 21 A o − 1 V 12 1 + V 22 1 A / A 11 − 1 V T S θ 2 = C 22 − 1 V 22 2 C 22 − 1

where V 11 1 = V o ∂ q i 1 o ∂ θ 1 , V 12 1 = V 21 1 t = c o v o ∂ q i 1 o ∂ θ 1 , ∂ q i 1 o ∂ θ 2 , V 22 1 = V o ∂ q i o ∂ θ 2 and V 22 2 = V o ∂ q i 2 o ∂ θ 2 . Then, the variance difference is

V T S θ 2 − V J θ 2 = τ 2 C 22 − 1 − τ 2 A / A 11 − 1 A 22 + C 22 − A 21 A o − 1 A 12 A / A 11 − 1 + τ 2 − τ 1 A / A 11 − 1 A 22 − A 21 A o − 1 A 12 A / A 11 − 1 = τ 2 W 1 + τ 1 − τ 2 W 2

B.5 Proof of Theorem 4

Under Condition A, we can directly show V J ζ 1 = V J ζ 2 = A 1 − 1 B 1 A 1 − 1 where A 1 = E ∂ m i 1 ζ o 1 , ζ o 2 ∂ ζ 1 ′ and B 1 = V m i 1 ζ o 1 , ζ o 2 . The case with condition B is implied by Lemma 1.

B.6 Proof of Corollary 1

Consider linear model in Example 1. For notational convenience, define the following v i 2 δ 2 ≡ y i 2 − z i δ 2 , e i θ ≡ y i 1 − y i 2 α − z i 1 δ 1 − v i 2 δ 2 Σ 22 − 1 Σ 21 , σ 11 | 2 θ ≡ Σ 11 − Σ 12 Σ 22 − 1 Σ 21 , and h i θ ≡ e i θ 2 − σ 11 | 2 θ where parameters are defined comformably. From the quasi-log-likelihoods

q i 1 θ 1 , θ 2 = − 1 2 ln ⁡ 2 ⁡ π − 1 2 ln σ 11 | 2 θ − 1 2 e i θ 2 σ 11 | 2 θ − 1 q i 2 θ 2 = − k 2 ln ⁡ 2 ⁡ π − 1 2 ln Σ 22 − 1 2 v i 2 δ 2 Σ 22 − 1 v i 2 δ 2 ′ ,

quasi-scores can be derived as follows

∂ q 1 ∂ θ 2 = − σ 11 | 2 θ − 1 e i θ Σ 22 − 1 Σ 21 ⊗ z ′ L r Σ 22 − 1 ⊗ Σ 22 − 1 D * θ

∂ q 2 ∂ θ 2 = I r ⊗ z ′ Σ 22 − 1 v 2 δ 2 ′ 1 2 L r vec Σ 22 − 1 v i 2 δ 2 ′ v i 2 δ 2 Σ 22 − 1 − Σ 22 − 1

where D * θ = [ Σ 21 ⊗ Σ 21 ] 1 2 σ 11 | 2 ( θ ) − 2 h i ( θ ) − Σ 21 ⊗ v i 2 δ 2 ′ σ 11 | 2 ( θ ) − 1 e i ( θ ) , θ 1 = α ′ , δ 1 ′ , Σ 21 ′ , Σ 11 ′ , θ 2 = vec δ 2 ′ , vech Σ 22 ′ ′ , L r is a r r + 1 2 × r 2 elimination matrix (defined in Section 5.7.3, Turkington 2013).

First, consider following reparameterizations with η ≡ Σ 22 − 1 Σ 21 and σ 11 | 2 ≡ Σ 11 − Σ 12 Σ 22 − 1 Σ 21 where θ 1 = α ′ , δ 1 ′ , η ′ , σ 11 | 2 ′ , θ 2 = vec δ 2 ′ , vec Σ 22 ′ . Now, the quasi-scores of q _i1 are modified to ∂ q i 1 ∂ θ 1 = σ 11 | 2 − 1 e i θ x i ′ 1 2 σ 11 | 2 − 2 h i θ and ∂ q i 1 ∂ θ 2 = − σ 11 | 2 − 1 e i θ η ⊗ z i ′ 0 r r + 1 2 × 1 where x i = [ y i 2 z i 1 v i 2 δ 2 ] and e i θ , h i θ is defined correspondingly. If η = 0, the result is trivial. Suppose that there exists at least one nonzero element of η and θ ₂₂ is nonempty. By substitution of y _i2 and an invertible linear transformation, moment functions for LIML and CF can be transformed into following expressions

(23) E y i 1 − y i 2 α − z i 1 δ 1 δ 22 ′ z i 2 ′ y i 1 − y i 2 α − z i 1 δ 1 z i 1 ′ y i 1 − y i 2 α − z i 1 δ 1 − v i 2 η v i 2 ′ y i 1 − y i 2 α − z i 1 δ 1 − v i 2 η 2 − σ 11 | 2 vec z i 1 ′ y i 2 − z i δ 2 Σ 22 − 1 vec z i 2 ′ y i 2 − z i δ 2 Σ 22 − 1 − vec z i 2 ′ σ 11 | 2 − 1 y i 1 − y i 2 α − z i 1 δ 1 − v i 2 η η ′ L r vec Σ 22 − 1 v i 2 δ 2 ′ v i 2 δ 2 Σ 22 − 1 − Σ 22 − 1 = 0

and

(24) E y i 1 − y i 2 α − z i 1 δ 1 δ 22 ′ z i 2 ′ y i 1 − y i 2 α − z i 1 δ 1 z i 1 ′ y i 1 − y i 2 α − z i 1 δ 1 − v i 2 η v i 2 ′ y i 1 − y i 2 α − z i 1 δ 1 − v i 2 η 2 − σ 11 | 2 vec z i ′ y i 2 − z i δ 2 Σ 22 − 1 L r vec Σ 22 − 1 v i 2 δ 2 ′ v i 2 δ 2 Σ 22 − 1 − Σ 22 − 1 = 0

repectively. Then the result follows by Theorem 4.

B.7 Proof of Corollary 2

We state the following lemma without providing its proof.

Lemma 2.

Assume that Assumptions 1–12 hold. Suppose there exists l 1 + l 2 nuisance parameters λ = λ 1 , λ 2 such that E o ∂ 2 q i 1 θ o , λ o / ∂ θ ∂ λ ′ = 0 p × l 1 + l 2 and E o ∂ 2 q i 2 θ o 2 , λ o 2 / ∂ θ 2 ∂ λ 2 ′ = 0 p 2 × l 2 . Then, V J θ and V T S θ are not affected by treating λ as known and redefining q ̃ i 1 θ = q i 1 θ , λ o and q ̃ i 2 θ 2 = q i 2 θ 2 , λ o 2 . Moreover, if QGMM moment function (6) contains exactly l 1 + l 2 scores regarding λ, then V QGMM θ is also not affected by the redefinition. □

Note that E ∂ 2 q i 1 o / ∂ θ 1 ∂ vech Σ 22 ′ = 0 and E ∂ 2 q i 2 o / ∂ vec δ 2 ∂ vech Σ 22 ′ = 0 . Thus we can treat Σ₂₂ as a known value by Lemma 2. Then, redefined q ̃ i 2 is also a member of linear exponential family (Gouriéroux, Monfort and Trognon 1984). Now, it suffices to show GLM variance assumptions imply GIMEs with corresponding scaling factor in linear exponential family. Let m i θ ≡ G y i 2 , z i 1 , v i 2 , θ 1 . Based on general form of score and Hessian (Wooldridge 2010), we have

E o ∂ q 1 o ∂ θ ∂ q 1 o ∂ θ ′ = E o E o y i 1 − m i θ o 2 y i 2 , z i V q y i 1 y i 2 , z i 2 ∂ m i θ o ∂ θ ∂ m i θ o ∂ θ ′ = τ 1 E o − ∂ 2 q i 1 o ∂ θ ∂ θ ′

and

V o ∂ q ̃ 2 o ∂ vec δ 2 = E o I r ⊗ z i ′ Σ o 22 − 1 v i 2 ′ v i 2 Σ o 22 − 1 I r ⊗ z i = τ 2 E o − ∂ 2 q ̃ 2 o ∂ vec δ 2 ∂ vec δ 2 ′

Orthogonality of scores holds under correct specification of conditional means since

E o ∂ q 1 o ∂ θ ∂ q 2 o ∂ θ 2 ′ = E o E o y i 1 − m i θ o y i 2 , z i V q y i 1 y i 2 , z i ∂ m i θ o ∂ θ ∂ q i 2 o ∂ θ 2 ′ = 0

Then, by Theorem 2, QLIML is efficient relative to CF for θ ₁.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2023-0009).

Received: 2023-02-03

Accepted: 2024-05-03

Published Online: 2024-05-27

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https://doi.org/10.1515/snde-2023-0009

Keywords for this article

quasi-likelihood; control function; information matrix