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Constructing Instruments for Regressions with Measurement Error When No Additional

Variables Are Available: Comment

Timothy Erickson, U.S. Bureau of Labor Statistics

Working Paper 321

June 1999

 

The author thanks Steve Holden, Rob McClelland, Keith Waehrer, and Toni Whited for many valuable comments. The

views expressed are those of the author and do not reflect the policies of the U.S. Bureau of Labor Statistics or the views of

other staff members.
Abstract

Lewbel (1997) has ingeniously shown that linear instrumental variables estimators for

the errors-in-variables model can be constructed using functions of the dependent variable,

proxy, and perfectly measured regressors as instruments. He proves consistency for the

estimator and then asserts that \standard limiting distribution theory for TSLS can now

be applied." In this note I assume that \standard theory" is given by White (1982), the

source of the standard errors used by Professor Lewbel in his empirical application. I show

that when White's formulas are applied to Lewbel's instruments, they give an ine±cient

estimator, an incorrect asymptotic covariance matrix, and an inconsistent covariance matrix

estimator. These results stem from a subtle violation of the familiar instrumental variable

orthogonality condition. SpeciØcally, only one of Lewbel's instruments can be measured from

an arbitrary origin and satisfy the orthogonality condition; the remaining instruments satisfy

orthogonality only if measured as deviations from their population means. The substitution

of sample means therefore generates a nonstandard asymptotic covariance matrix of the type

described by Newey and McFadden (1994) in their discussion of \plug-in" estimators. I apply

the theory for such estimators to Lewbel's instruments to obtain an e±cient estimator, the

correct asymptotic covariance matrix, and a consistent covariance matrix estimator.
I. Introduction

Lewbel (1997) has ingeniously shown that linear instrumental variables estimators for the

errors-in-variables model can be constructed using functions of the dependent variable, proxy,

and perfectly measured regressors as instruments. He proves consistency for the estimator

and then asserts that \standard limiting distribution theory for TSLS can now be applied."

In this note I assume that \standard theory" is given by White (1982), the source of the

standard errors used by Lewbel in his empirical application. I show that when White's

formulas are applied to Lewbel's instruments, they give an ine±cient estimator, an incorrect

asymptotic covariance matrix, and an inconsistent covariance matrix estimator. These results

stem from a subtle violation of the familiar instrumental variable orthogonality condition.

SpeciØcally, under i.i.d. sampling only one of Lewbel's instruments can be measured from an

arbitrary origin and satisfy the orthogonality condition; the remaining instruments satisfy

orthogonality only if measured as deviations from their population means. The substitution

of sample means therefore generates a nonstandard asymptotic covariance matrix of the

type described by Newey and McFadden (1994), who show that the covariance matrix of

a feasible estimator diÆers from that of its infeasible counterpart whenever the consistency

of the feasible estimator requires consistency of the nuisance parameter estimates. I apply

the theory of these authors to Lewbel's instruments to obtain an an e±cient estimator, the

correct asymptotic covariance matrix, and a consistent covariance matrix estimator.

II. The Lewbel estimator and its asymptotic distribution

Lewbel considers the model

Yi = a + b

0

Wi + cXi + ei (1)

Zi = d + Xi + vi; (2)

where only (Wi; Yi; Zi) are observable, c 6= 0, and all variables are scalar except for the J £1

vectors Wi and b. Equations (1)-(2) imply

Yi = Æ + b

0

Wi + cZi + "i; (3)

1
where Æ = a ° cd and

"i = ei ° cvi; (4)

the last equality combining with (2) to generate the correlated error and regressor problem

associated with classical measurement error. Lewbel shows that it is possible to use nonlinear

transformations of the observed variables as instruments. SpeciØcally, if Gi = G(Wi) is

a nonlinear function of Wi, and G; 1 Z1, and Y1 denote sample means, then under certain

conditions

q1i = Gi ° G1

q2i = (Gi ° G1)(Zi ° Z1)

q3i = (Gi ° G1)(Yi ° Y1 )

are instrumental variables that yield consistent Two Stage Least Squares (TSLS) estimates

of Æ, b, and c. Lewbel notes that these new instruments can be combined with the longrecognized instruments

q4i = (Yi ° Y1 )(Zi ° Z1)

q5i = (Zi ° Z1)

2

q6i = (Yi ° Y1 )

2

;

the last two being applicable if E (e3

i) = E (v3i) = 0.

To state the asymptotic distribution for the TSLS estimators, let

μ =

0

@

Æ

b

c

1

A Ri =

0

@

1

Wi

Zi

1

A Qi(m) =

0

@

1

Wi

qi(m)

1

A ; (5)

where qi(m) is a column vector containing one or more of the following functions of m =

(mG; mZ; mY )0

:

Gi ° mG (6)

(Gi ° mG)(Zi ° mZ) (7)

(Gi ° mG)(Yi ° mY ) (8)

2
(Yi ° mY )(Zi ° mZ) (9)

(Zi ° mZ)

2 (10)

(Yi ° mY )

2

: (11)

In this notation (3) is

Yi = R0

iμ + "i; (12)

and TSLS estimators can be written as

^μ =

3

M1 0AnM1

¥°1

M1 0An

Ã

1

n

Xn

i=1

Q^iYi

!

; (13)

where An is any appropriately-dimensioned positive deØnite matrix, M1 ¥ (1=n)

Pn

i=1 Q^iR0

i,

Q^i ¥ Qi(^1), and ^1 ¥ °G; 1 Z; 1 Y1 ¢0.

Assumption 1: (i) (Wi; Xi; ei; vi) is i:i:d: for i = 1;:::;n, the errors ei and vi are independent of each other and of (Wi; Xi) and satisfy E (ei) = E (vi) = E (e3

i) = E (v3i)=0,

and (Gi; Wi; Xi; ei; vi) has Ønite moments of every order; (ii) An converges in probability to

a positive deØnite matrix A.

Note that part (i) satisØes the weaker assumptions given by Lewbel. Let Ãi(m) ¥

((Gi ° mG);(Zi ° mZ);(Yi ° mY ))0

, 1 ¥ (E(Gi); E(Zi); E(Yi))0, Ãi ¥ Ãi(1), Qi ¥ Qi(1),

M ¥ E £QiR0

i

§

, and

D = E

Ã

@Qi(m)"i

@m

Ø

Ø

Ø

Ø

m=1

!

: (14)

Proposition 1 If Assumption 1 holds, then pn

3

^μ ° μ

¥ d °!

N °0;(M0AM)°1M0A­AM(M0AM)°1

¢

, where ­ = var [Qi"i + DÃi].

1

The proof is in the appendix, where one can see that ­ is the covariance matrix of the

limiting distribution for (1=

pn)

Pn

i=1 Q^i"i. The above result is therefore \nonstandard" in

1The object Ãi is called the in°uence function for 1^, because it gives the in°uence of

the i-th observation on 1^ up to an op(1) remainder term. Newey and McFadden (1994)

discuss in°uence functions and their use in deriving the asymptotic distributions of \plugin" estimators.

3
the sense that constructing the instruments as deviations from ^1 rather than 1 has generated

the term DÃi appearing in ­. Note that D is generally nonzero if any of (7) through (11)

are included in Qi(m). For example, if (7) is included, then (14) includes the element

E

"Ã @(Gi ° mG)(Zi ° mZ)

@mG

Ø

Ø

Ø

Ø

m=1

!

"i

#

= E [° (Zi ° E(Zi)) "i]

= E [° (Xi ° E(Xi) + vi) (ei ° cvi)]

= cE °v2

i

¢

;

where the second equality is obtained by substituting from (2) and (4), and the last equality

follows from the assumed independence of ei, vi, and (Wi; Xi). The reason D is nonzero is

that the instrumental variable orthogonality conditions associated with (7) through (11) are

satisØed if and only if m = 1. To see this for the above example, note that

E [(Gi ° mG)(Zi ° E(Zi))"i] can be rewritten as °cE(Gi °mG)E(v2

i), which vanishes if and

only if mG = E(Gi). In contrast, (6) does not generate nonzero elements in D because the

orthogonality condition E [(Gi ° mG)"i] = 0 holds for all mG. This situation exempliØes

Newey and McFadden's comment that the covariance matrix of a feasible estimator requires

adjustment whenever consistency of the feasible estimator depends on consistency of the

\plugged-in" estimate.

Users of standard theory will estimate ­ with White's formula ­^ STD = (1=n)

Pn

i=1 "^2

iQ^iQ^0

i,

where ^"i ¥ Yi ° R0

i

^μ. The following result shows that ­^ STD is not consistent for ­:

Proposition 2 If Assumption 1 holds, then ­^ STD

p °! var [Qi"i] :

Using ­^STD to estimate ­ therefore produces asymptotically invalid inferences. Also, since

an e±cient estimator of μ is obtained by choosing An to be a consistent estimator of ­°1

(see Newey (1994), p. 1368), it follows that An = ­^ °1

STD gives an ine±cient estimator of

μ. Valid inferences and e±cient estimation are made possible by the following result, where

Ã^i ¥ Ãi(^1) and

D^ = 1

n

Xn

i=1 Ã

@Qi(m)^"i

@m

Ø

Ø

Ø

Ø

m=^1

!

:

4
Proposition 3 If Assumption 1 holds, then ­^ ¥ n°1 Pn

i=1 3

Q^i"^i + D^Ã^i

¥ 3Q^i"^i + D^Ã^i¥0 p °!

­.

III. A bibliographic note

Professor Lewbel also shows that high-order moments can be used to estimate regressions

that are quadratic in a mismeasured regressor. It should be noted that this is a special case

of the analysis of polynomial regressions given by Geary (1942, pp. 73-75) in his seminal

paper on estimation using high-order cumulants.

Appendix: Proofs

Lemma 4.3 of Newey and McFadden (1994), henceforth Lemma 4.3NM, will be used repeatedly in proving Propositions 1, 2, and 3. Lemma 1 below veriØes a condition assumed by

Lemma 4.3NM. For notation, let kAk¥kvec (A)k, where A is a matrix and k¢k is the Euclidean norm. Two easily veriØed facts used below are kAbk·kAk¢kbk and kab0

k = kak¢kbk,

where a and b are column vectors and k¢k can denote norms for diÆerently dimensioned

vectors.

Lemma 1 There exist neighborhoods N1 of 1 and N of (10

; μ0) such that

(i) E

h

supm2N1 kQi(m)R0

ik2

i

< 1; (ii) E

h

supm2N1

°

°

°

@Qi(m)"i

@m0

°

°

°

i

< 1;

(iii) E £sup(m;t)2N kQi(m) (Yi ° R0

it)k2§

< 1; (iv) E £sup(m;t)2N kÃi(m)Qi(m)0 (Yi ° R0

it)k

§

<

1.

Proof: To prove (i), Ørst note that supm2N1 kQi(m)R0

ik2

= supm2N1 kQi(m)k2¢kRik2. In view

of (5), it therefore su±ces to show supm2N1 kqi(m)k2 is integrable. Since kqi(m)k2 is a sum

of terms of the form °(Gi ° mG)°(Zi ° mZ)±(Yi ° mY )ø

¢2

, where °; ±; and ø are nonnegative

integers summing to 1 or 2, it su±ces to show

sup

m2N1

(Gi ° mG)

2°(Zi ° mZ)2±

(Yi ° mY )

2ø (15)

is integrable. Let N1 = BG £ BZ £ BY , where Bk is an open interval of length 2 ̧k centered

5
at E(ki), k = G; Z; Y . Then (15) equals

sup

mG2BG

(Gi ° mG)

2° sup

mZ 2BZ

(Zi ° mZ)

2± sup

mY 2BY

(Yi ° mY )

2ø , (16)

which is less than or equal to

(jGi ° E(Gi)j +  ̧G)

2°(jZi ° E(Zi)j +  ̧Z)2±

(jYi ° E(Yi)j +  ̧Y )

2ø : (17)

Assumption 1 and equations (1)-(2) imply that (17), and therefore (16) is integrable, establishing (i). The proof of (ii) follows by a similar argument, after noting that

supm2N1

°

°

°

@Qi(m)"i

@m0

°

°

° = supm2N1

°

°

°

@Qi(m)

@m0

°

°

°¢j"ij and that

°

°

°

@Qi(m)

@m0

°

°

°

2

is a sum of terms of the form

°

£(Gi ° mG)°(Zi ° mZ)±(Yi ° mY )ø

¢2

, where (£; °; ±; ø ) are nonnegative integers. To prove

(iii), let N = N1 £ Bμ, where Bμ is an open ball of radius  ̧μ centered at μ, and note that

sup(m;t)2N kqi(m) (Yi ° R0

it)k2

= supm2N1 kqi(m)k2supt2Bμ jYi ° R0

itj

2

. The right hand side

of the last equality is less than or equal to the product of (17) and (jYi ° R0

iμj +  ̧μ kRik)

2

, a

product that is integrable by Assumption 1. Part (iv) is proven similarly, after noting that

kÃi(m)qi(m)0 (Yi ° R0

it)k = kÃi(m)k¢kqi(m)k¢jYi ° R0itj, supm2N1 (kÃi(m)k¢kqi(m)k) ·

supm2N1 kÃi(m)k supm2N1 kqi(m)k, and supm2N1 kÃi(m)k2= supmG2BG (Gi ° mG)2+

supmZ 2BZ (Zi ° mZ)2 + supmY 2BY (Yi ° mY )2. •

Proof of Proposition 1: Expressions (12) and (13) imply

pn

3

^μ ° μ

¥

=

3

M1 0AnM1

¥°1

M1 0An

1

pn

Xn

i=1

Qi(^1)"i: (18)

Applying a mean value theorem gives

1

pn

Xn

i=1

Qi(^1)"i = 1

pn

Xn

i=1

Qi(1)"i +

Ã

1

n

Xn

i=1

@Qi(1§)"i

@m0

! pn(^1 ° 1); (19)

where 1§ is a weighted average of 1 and ^1. (Abusing notation in the usual way, 1§ diÆers for

each individual equation in the vector equation (19).) Lemma 1(ii) and Lemma 4.3NM imply

n°1 Pn

i=1 (@Qi(1§)"i=@m0

) p °! D. The assumption of i.i.d. sampling with Ønite moments

and the fact that ^1 is a vector of sample means implies pn(^1 ° 1) converges in distribution

to a normal distribution. These facts together with Slutsky's theorem imply

1

pn

Xn

i=1

Qi(^1)"i = 1

pn

Xn

i=1

Qi(1)"i + Dpn(^1 ° 1) + op(1); (20)

6
where op(1) denotes a random vector that converges in probability to zero. Substituting the

easily veriØed equation pn(^1 ° 1) = n°1=2 Pn

i=1 Ãi + op(1) into (20) gives

1

pn

Xn

i=1

Qi(^1)"i = 1

pn

Xn

i=1

Qi(1)"i + D 1

pn

Xn

i=1

Ãi + op(1)

= 1

pn

Xn

i=1

(Qi(1)"i + DÃi) + op(1):

The assumption of Ønite moments implies Qi(1)"i + DÃi has a Ønite second moment, so

(1=

pn)

Pn

i=1 Qi(^1)"i

d °! N(0; ­) by the Lindeberg-Levy central limit theorem. Proposition

1 follows from this result together with equation (18), the assumption An

p °! A, and the

fact that Lemma 1(i) and Lemma 4.3NM imply M1 p °! M. •

Proof of Proposition 2: Since ­^ STD = 1

n

Pn

i=1 1⁄2

Qi(^1)(Yi ° R0

i

^μ)

h

Qi(^1)(Yi ° R0

i

^μ)

i0

æ

and °

°Qi(m)(Yi ° R0

it) [Qi(m)(Yi ° R0it)]0

°

° = kQi(m) (Yi ° R0

it)k2

, Lemmas 1(iii) and 4.3NM

imply ­^STD

p °! E ©

Qi(1)(Yi ° R0

iμ) [Qi(1)(Yi ° R0iμ)]0

TM ¥ var [Qi(1)"i]. •

Proof of Proposition 3: Note that ­ ¥ var [Qi"i]+2DE [ÃiQ0

i"i]+D avar(^1)D0 and that

­^ ¥ ­^ STD + 2D^

31

n

Pn

i=1 Ã^iQ^0

i"^i

¥

+ D^

31

n

Pn

i=1 Ã^iÃ^0

i

¥

D^0

. Lemmas 1(iv) and 4.3NM imply

1

n

Pn

i=1 Ã^iQ^0

i"^i

p °! E [ÃiQ0

i"i]. The proposition follows from this result, the limit D^ p °! D

established above, the standard result 1

n

Pn

i=1 Ã^iÃ^0

i

p °! avar(^1), and Proposition 2. •

References

Geary, R. C. (1942): \Inherent Relations Between Random Variables," Proceedings of

the Royal Irish Academy A, 47, 63-76.

Lewbel, A. (1997): \Constructing Instruments for Regressions With Measurement Error

When No Additional Data Are Available, With an Application to Patents and R&D,"

Econometrica, 65, 1201-1213.

Newey, W. (1994): \The Asymptotic Variance of Semiparametric Estimators," Econometrica, 62, 1349-1382.

Newey, W. and D. McFadden (1994): \Large Sample Estimation and Hypothesis

Testing," in Handbook of Econometrics, Vol. IV, ed. by R. Engle and D. McFadden.

Amsterdam: Elsevier, 2111{2245.

White, H. (1982): \Instrumental Variables Regression with Independent Observations,"

Econometrica, 50, 483-499.

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