On the Robustness of Coefficient Estimates to the Inclusion of Proxy Variables

Appendix: Proofs

Let

V(Z1iZ2i)=[V1CC′V2].

The matrix V₁ is the k×k variance matrix for Z_1i, C is the k×1 covariance, and V₂ is the scalar variance of Z_2i. Let δ be an arbitrary l×1 vector such that δ′ρ=γ>0 for some given value γ. Let θ=β/γ.

The next three Lemmas establish key results for Proposition 3.

Lemma 1Expressions for (α–a) and (θ–t).

Then

[at]=[V1γCγC′γ2V2+(δ′Σδ)]−1[V1α+θγCγC′α+θγ2V2].

Rewriting yields

[V1γCγC′γ2V2+(δ′Σδ)][at]=[V1α+θγCγC′α+θγ2V2],

which is equivalent to

[V1γCγC′γ2V2]−1[V1γCγC′γ2V2+(δ′Σδ)][at] =[V1γCγC′γ2V2]−1[V1α+θγCγC′α+θγ2V2].

This yields

[Iγ(V1−CV2−1C)−1C(1−(γ2V2)−1(γ2V2+(δ′Σδ)))0(γ2(V2−C′V1−1C))−1(γ2V2+(δ′Σδ)−γ2C′V1−1C)][at]=[αθ].

Noting that V₂, γ, and (δ′Σδ) are all scalars, this can be written as

[I−γ(V1−CV2−1C)−1C((δ′Σδ)γ2V2)0′(1+(δ′Σδ)(γ2(V2−C′V1−1C)))][at]=[αθ].

Rearranging gives

[a−γ(V1−CV2−1C)−1C((δ′Σδ)γ2V2)t(1+(δ′Σδ)(γ2(V2−C′V1−1C)))t]=[αθ].

Thus

(a−a)=γ(V1−CV2−1C)−1CV2−C′V1−1CV2(d′Σd)(γ2(V2−C′V1−1C))+(d′Σd)θ=(V1−CV2−1C)−1CV2−C′V1−1CV2(d′Σd)(γ2(V2−C′V1−1C))+(d′Σd)β,

and

(t−θ)=((γ2(V2−C′V1−1C))(γ2(V2−C′V1−1C))+(δ′Σδ))θ−θ=−((δ′Σδ)(γ2(V2−C′V1−1C))+(δ′Σδ))θ.

QED.

Lemma 2The term ((δ′Σδ)(γ2(V2−C′V1−1C))+(δ′Σδ)) is positive and increasing in (δ′Σδ).

The term (γ2(V2−C′V1−1C))+(δ′Σδ) is positive provided that γ≠0 and Σ is positive semi-definite. The term V2−C′V1−1C is the determinant of the V(Z_1i, Z_2i), and so is, by necessary assumption, positive. The term (δ′Σδ) will be non-negative provided Σ is positive semi-definite. The derivative with respect to the term (δ′Σδ) is

(γ2(V2−C′V1−1C))+(δ′Σδ)−(δ′Σδ)((γ2(V2−C′V1−1C))+(δ′Σδ))2   =γ2(V2−C′V1−1C)((γ2(V2−C′V1−1C))+(δ′Σδ))2>0.

Hence the inconsistency in both are increasing in (δ′Σδ). QED.

Lemma 3The solution to min_δ(δ′Σδ) s.t.δρ=γ isδ=γΣ^–1ρ(ρ′Σ^–1ρ)^–1.

The Lagrangian is

(δ′Σδ)−λ(δ′ρ−γ).

FOC are

2Σδ−λρ=0δ′ρ−γ=0.

Solving:

δ=12λΣ−1ρ12ρ′Σ−1ρλ=γ.

Substitution yields

δ=γΣ−1ρ(ρ′Σ−1ρ)−1λ=2γ(ρ′Σ−1ρ)−1.

QED.

Proof. The proof of proposition 1 follows from the details in the text combined with the above lemmas.   ■

Proof. Proof of Corollary 1. Substitution of the results from proposition 1 into the expressions in Lemma 1 yields

(δ′Σδ)=γ2ρ′Σ−1ΣΣ−1ρ(ρ′Σ−1ρ)2=γ2(ρ′Σ−1ρ).

From Lemma 1 we have that

(t−θ)=−θ((δ′Σδ)(γ2(V2−C′V1−1C))+(δ′Σδ))

Alternatively,

t=θ(1−((δ′Σδ)(γ2(V2−C′V1−1C))+(δ′Σδ)))=βγ(γ2(V2−C′V1−1C)(γ2(V2−C′V1−1C))+(δ′Σδ)).

Substitute the optimal choice of δ from proposition 1 which yields

t=βγ(γ2(V2−C′V1−1C)γ2(V2−C′V1−1C)+γ2(ρ′Σ−1ρ)−1)=βγ((V2−C′V1−1C)(V2−C′V1−1C)+(ρ′Σ−1ρ)−1).

Hence, by choosing

γ=(V2−C′V1−1C)+(ρ′Σ−1ρ)−1(V2−C′V1−1C)=1+1(ρ′Σ−1ρ)(V2−C′V1−1C),

we have t=β: no inconsistency in the coefficient on X^δ. QED   ■

Lemma 4(Sherwin-Morrison_Woodbury Matrix Inversion Lemma): If A and B are non-singular matrices, and X is conformable, then (A+XBX′)^–1=A^–1–A^–1X(B^–1+X′A^–1X)^–1X′A^–1.

Proof. Proof of Proposition 2:

The linear regression of y_i on Z_1i and X_i yields slope coefficients consistent for

(ab)=[V1Cρ′ρC′(ρρ′V2+Σ)]−1[V1α+CβρC′α+ρV2β].

Rewriting yields

[V1Cρ′ρC′(ρρ′V2+Σ)][ab]=[V1α+CβρC′α+ρV2β],

which is equivalent to

[V1Cρ′ρC′Iρ′ρV2]−1[V1Cρ′ρC′(ρρ′V2+Σ)][ab]=[V1Cρ′ρC′Iρ′ρV2]−1[V1α+CβρC′α+ρV2β],

where I is the identity matrix of appropriate dimensions. The inverse of the leading matrix (a partitioned matrix) can be written as

[(V1−Cρ′(Iρ′ρV2)−1ρC′)−1−(V1−Cρ′(Iρ′ρV2)−1ρC′)−1Cρ′(Iρ′ρV2)−1−(Iρ′ρV2−ρC′V1−1Cρ′)−1ρC′V1−1(Iρ′ρV2−ρC′V1−1Cρ′)−1].

Since ρ′ρV₂ is a scalar, this reduces to

[(V1−CV2−1C′)−1−(Iρ′ρV2−ρC′V1−1Cρ′)−1ρC′V1−1 −(V1−CV2−1C′)−1Cρ′(ρ′ρV2)−1(Iρ′ρV2−ρC′V1−1Cρ′)−1].

Substitution and simplification yields

[I−(V1−CV2−1C′)−1(Cρ′(ρ′ρV2)−1Σ)0(Iρ′ρV2−ρC′V1−1Cρ′)−1(ρ(V2−C′V1−1C)ρ′+Σ)][ab] =[α(Iρ′ρV2−ρC′V1−1Cρ′)−1ρ(V2−C′V1−1C)β],

or

[a−(V1−CV2−1C′)−1(Cρ′(ρ′ρV2)−1Σ)b(Iρ′ρV2−ρC′V1−1Cρ′)−1(ρ(V2−C′V1−1C)ρ′+Σ)b] =[α(Iρ′ρV2−ρC′V1−1Cρ′)−1ρ(V2−C′V1−1C)β].

We can write

b=(ρ(V2−C′V1−1C)ρ′+Σ)−1ρ(V2−C′V1−1C)β,

and

a=a+(V1−CV2−1C′)−1(Cρ′(ρ′ρV2)−1Σ)×(ρ(V2−C′V1−1C)ρ′+Σ)−1ρ(V2−C′V1−1C)β.

Turning first to the term a and applying the Sherwin-Morrison_Woodbury Matrix Inversion Lemma:

a=a+(V1−CV2−1C′)−1(Cρ′(ρ′ρV2)−1Σ)×(Σ−1−Σ−1ρ((V2−C′V1−1C)−1+ρ′Σ−1ρ)−1ρ′Σ−1)ρ(V2−C′V1−1C)β.

Simplification yields

a=a+(V1−CV2−1C′)−1C(V2−C′V1−1C)V2×1−ρ′Σ−1ρ(V2−C′V1−1C)−1+ρ′Σ−1ρβ

or

=a+(V1−CV2−1C′)−1C(V2−C′V1−1C)V2×(ρ′Σ−1ρ)−1(V2−C′V1−1C)+(ρ′Σ−1ρ)−1β,

which is the expression for a when the error-variance-minimizing choice of δ is used to construct X^δ (See Corollary 2).

Turning now to b, consider

ρ′b=ρ′(ρ(V2−C′V1−1C)ρ′+Σ)−1ρ(V2−C′V1−1C)β.

Again using the Sherwin-Morrison_Woodbury Matrix Inversion Lemma,

ρ′b=ρ′(Σ−1−Σ−1ρ((V2−C′V1−1C)−1+ρ′Σ−1ρ)−1ρ′Σ−1)ρ(V2−C′V1−1C)β=(ρ′Σ−1ρ−ρ′Σ−1ρ((V2−C′V1−1C)−1+ρ′Σ−1ρ)ρ′Σ−1ρ)(V2−C′V1−1C)β=(V2−C′V1−1C)−1(ρ′Σ−1ρ)+(ρ′Σ−1ρ)2−(ρ′Σ−1ρ)2(V2−C′V1−1C)−1+ρ′Σ−1ρ(V2−C′V1−1C)β=(V2−C′V1−1C)(V2−C′V1−1C)+(ρ′Σ−1ρ)−1β.

This is equal to the expression for a when the error variance minimizing choice of δ is used to construct X^δ in Corollary 1 if γ=1.QED   ■