BLS WORKING PAPERS

U.S. DEPARTMENT OF LABOR

Bureau of Labor Statistics

 

OFFICE OF EMPLOYMENT AND UNEMPLOYMENT

STATISTICS

Estimating Linear Regressions with Mismeasured, Possibly Endogenous, Binary

Explanatory Variables

Harley Frazis, U.S. Bureau of Labor Statistics

Mark A. Loewenstein, U.S. Bureau of Labor Statistics

Working Paper 355

March 2002, Revised September 2002

 

The views expressed are those of the authors and do not necessarily reflect the policies of the U.S. Bureau of Labor

Statistics or the views of other staff members.
Estimating Linear Regressions with Mismeasured, Possibly

Endogenous, Binary Explanatory Variables

Harley Frazis Mark A. Loewenstein

Bureau of Labor Statistics Bureau of Labor Statistics

2 Massachusetts Ave. NE 2 Massachusetts Ave. NE

Suite 4945 Suite 4130

Washington D.C. 20212 Washington D.C. 20212

Phone: 202-691-7394 Phone: 202-691-7385

Frazis_H@bls.gov Loewenstein_M@bls.gov

February 2002, revised September 2002

We thank Dan Black and Tim Erickson for helpful comments, and Chris Foote for use of

his job reallocation data. Anonymous referees’ comments led to a substantial redirection

and improvement of this paper. Opinions expressed are those of the authors and do not

reflect the views of the Bureau of Labor Statistics or those of its staff.
Abstract

This paper is concerned with mismeasured binary explanatory variables in a linear

regression. Modification of a technique in Hausman et al. (1998) allows simple

computation of bounds under relatively weak assumptions. When one has instruments,

we show how to obtain consistent parameter estimates using GMM. We show how to

incorporate the estimated measurement error bounds into the GMM estimates, and we

develop a specification test based on the compatibility of the GMM estimates with the

measurement error bounds. When the mismeasured variable is endogenous, the IV

estimate and the measurement error bounds can be used to bound the mismeasured

variable’s coefficient.
I. Introduction

This paper is concerned with mismeasured binary explanatory variables in a linear

regression. We obtain results that allow us to improve on existing estimators under

different assumptions about the extent of prior information. We examine three different

cases: 1) the mismeasured variable is assumed exogenous, and no instruments are

available; 2) the mismeasured variable is assumed exogenous, and one or more

instruments are available; and 3) the mismeasured variable is not assumed exogenous,

and instruments are available. In the first two cases, we derive bounds or point estimates

under weaker assumptions on prior information than in the previous literature, and in the

third case—which has not to our knowledge been analyzed—we also derive bounds.

In case 1, the traditional approach to the measurement error problem is to use

auxiliary information on the measurement error process. More generally, one may not

have good point estimates of the measurement error parameters, but may nevertheless be

able to bound them—using a validation study, for example. Bollinger (1996) shows how

these bounds can in turn be used to bound the regression coefficient.

Recently Hausman, Abrevaya, and Scott-Morton (1998) (hereafter HAS) have

developed a technique that allows the analyst to bound the measurement error process by

estimating the parameters from the data, without information from validation studies. We

modify their technique to allow simple computation of bounds without functional form

assumptions. Combining the estimated measurement error bounds with the OLS

coefficient yields bounds on the true effect of the mismeasured explanatory variable.

When instruments are available, as in case 2, instrumental variable (IV)

estimation is another common method of dealing with measurement error. However,
2

recent research has shown that IV estimation is upwardly biased when the mismeasured

variable is binary (Loewenstein and Spletzer 1997; Berger, Black and Scott 2000; Kane,

Rouse and Staiger 1999) because measurement error in this case must be negatively

correlated with the true value. Berger, Black, and Scott (2000) (BBS hereafter) and

Kane, Rouse, and Staiger (1999) (KRS hereafter) show that when one has two erroneous

measures, one can obtain a consistent estimate using a generalized method of moments

(GMM) technique.

Two distinct measures of the same variable are not commonly available. We thus

extend the analysis in BBS and KRS to the case where the second measure is replaced by

one or more instruments. We provide a closed-form solution for the GMM parameter

estimates. We also show how to incorporate the estimated measurement error bounds

into the GMM estimates, and we develop a specification test of the measurement error

model based on the compatibility of the GMM estimates with the measurement error

bounds.

Lastly, we show that the GMM technique is not easily extended to the case where

the mismeasured variable is endogenous. However, the IV estimate and estimates of the

measurement error bounds can be used to bound the effect of the mismeasured variable,

analogous to the OLS case without endogeneity.

The paper is organized as follows. Section II outlines the model, summarizes the

HAS technique, and introduces our extension of HAS. Section III then turns to the case

where one or more instruments exist for the mismeasured variable. Section IV considers

the case where there are available instruments, but the mismeasured binary variable is

endogenous. Section V presents an empirical example looking at the returns to on-the-
3

job training and section VI concludes.

II. Bounding the Effect of a Mismeasured Binary Explanatory Variable When No

 Instrument is Available

Our model is:

(1) i i i i Y = c + X γ + βT * +e

for observation i in a random sample of n observations, where Yi, Ti* and ei are scalars, c

is a constant, and Xi is a 1 x k row vector. We assume without loss of generality that β >

0. Dropping the subscript for convenience, the variable T* is a binary variable that takes

on the values of zero and one; T* and the elements of X are not linearly dependent. The

error term e is mean zero and uncorrelated with T* and X. The variable T* is measured

with error. Instead of T*, we observe the binary variable T = T* + U, where U denotes

measurement error (which can take on the values of 1, -1, or 0).

Define the measurement error probabilities Pr( 1| * 0) α 0 ≡ T = T = = Pr(U = 1| T* =

0) and Pr( 0 | * 1) α1 ≡ T = T = = Pr(U = -1| T* = 1). These probabilities are assumed to be

independent of X and e. We should note that this may be a strong assumption in many

applications (see Black, Sanders, and Taylor 2000); in a regression of earnings on

education and (binary) training status, for example, more educated respondents may

better understand the survey questions on training. Modifying this assumption would

require modification of the standard results on the effect of measurement error on

regression coefficients (Aigner 1973, BBS, KRS).

Let p ≡ Pr(T =1) and p* ≡ Pr(T* =1). It is straightforward to show that

(2) cov(T,T*) = p*(1-p*)(1-α0-α1).

Following Bollinger (1996) and others, we impose the restriction that cov(T,T*) > 0 (if

this is not the case, then measurement error is so severe that (1-T) is a better measure of
4

T* than is T), which implies α0 + α1 < 1.

Note that

(3) ) ( | )

( , | ) (1 ˆ lim

Var T X

Cov T U X

p β ols = β − .

Since Cov(T,U|X) > 0, p β ols < β ˆ lim . After some algebra, one can show that

 (4) ( , , , ) ˆ β lim β χ R α 0 α1 p p = ols ,

where 

( )(1 ) (1 )

(1 )(1 )(1 ) ( , , , )

0 1

0 1

0 1

p p Rp p

p p R

p R − α − α − − −

− − − α − α

χ α α ≡ and where

(1 )

( , ) ( ) ( , )’ 1

p p

Cov X T Var X Cov X T R − ≡

−

 is the theoretical R-squared from a regression of T on

X. It is straightforward to show that χ is an increasing function of both α0 and α1. Thus,

if we have available upper bounds max α 0 and max α1 on the measurement error parameters α0

and α1, we can bound β:

 (5) ( , , , ) ˆ lim ˆ lim max

1

max p β ols < β < p β ols χ α 0 α p R .

A Percentile Method for Bounding Measurement Error

	


 0 1. The independence of X and the

measurement error process implies that

(6) Pr(T=1|X) = (1 – α1) Pr(T*=1|X) + α0 (1 – Pr(T*=1|X))

 = α0 + (1 – α0 – α1) Pr(T*=1|X).

If one knows the functional form of Pr(T* = 1|X), one can estimate α0 and α1 from (6) by

maximum likelihood, as in Hausman, Abrevaya, and Scott-Morton (1998) (HAS
5

hereafter). But as we now show, simple n -convergent bounds for α0 and α1 are

available without knowledge of the true functional form.1

Equation (6) has the important implication that 0 1 α ≤ Pr(T =1| X ) ≤1−α for all X.

Thus, for any subset S of the range of X, an estimate of Pr(T=1|X ∈ S) can be used to

bound α0 and α1. Imagine a procedure where one takes a subset S comprising q percent

of the sample in order to use the sample average T over S as a bound. If one has prior

knowledge of the ranking of sample observations according to Pr(T=1|X), one would

obviously obtain the tightest possible bound for α0 (α1) by estimating E(T | X ∈S) over

that subset of sample observations having the lowest (highest) expected value of T - that

is, the set S should consist of observations with percentile rank of Pr(T=1|X) less than q

(greater than 1-q). Without prior knowledge of the ranking of observations by

Pr(T=1|X), one must estimate the ranking from the sample.

If the functional form chosen to estimate the ranking of Pr(T=1|X) is incorrect,

the estimated bounds will not be tight for a given q because some observations will be

misclassified in the limit as having percentile rank above q when their true rank is below

q, and vice versa (and correspondingly for rank 1-q). Since 0 1 1 α ≤ Pr(T =1| X ) ≤ − α

for all X, an incorrect functional form will affect only the tightness of the bounds, not

their validity. In most cases, the percentile ranks of Pr(T=1|X) are unlikely to be

drastically affected by functional form.

 

1

 HAS present another procedure for estimating misclassification probabilities (and

Pr(T*=1|X)) that does not rely on prior knowledge of the functional form of Pr(T*=1|X),

but this procedure is computationally quite complex and converges at a rate slower than

n .
6

Because we require asymptotic results where the functional form of Pr(T=1|X)

may be misspecified, it is convenient to work with quasi-maximum-likelihood (QML)

estimation (see White 1982) in estimating the ranking of Pr(T=1|X). Specifically,

consider estimation of the model E(T|X) = G(X ; δ), where G is a cdf but not necessarily

the true one and δ is a vector of parameters. Let δ

ˆ denote the quasi-maximumlikelihood estimator: ≡ argmax ( ) ≡ ˆδ L δ

∑ ∑ = =

− δ + δ : 0 : 1argmax(1/ )( ln(1 ( ; )) ln ( ; ))

i Ti i

i

i T

i n G x G x , so that the predicted value of T is

simply )ˆ ( ; ˆT = G X δ . We assume sufficient regularity such that the quasi-maximum

likelihood estimator δ

ˆ exists and converges to a limit δ* and such that *) ˆ n(δ − δ

converges to a normal distribution.2

We now introduce some necessary notation. Let δ J denote the cumulative

distribution function of G(X; δ ) and let * κ q ≡ 1

*

−

δ J (q) denote the q-quantile for δ * J .

Letting [ ]’ ’ κ

q

θ ≡ δ , one can define the function, ( ) ( | ( ; ) ) 0 A θ q ≡ E T G X δ ≤ κ q . In

addition, let ( ) ) ˆ ˆ min( | c J ˆ c q q ≡ ≥ δ κ denote the sample q-quantile for Tˆ , where δ

ˆ ˆ

J is the

empirical cdf of Tˆ . Let ( ) 0

q I θ denote the set of all sample observations such that

G X ≤ κq ( ;δ) and similarly, define ( ) 1

1

q I θ − as the set of all sample observations such

that G X ≥ 1−q ( ;δ ) κ .

 

2

 See White (1982) for details. The results in the text and the appendix go through, with

appropriate substitutions, for any estimator δ

ˆ such that *) ˆ n(δ −δ has the same limiting

distribution as ) ( , ; ) ( n

u T X K n ∑ δ , where u is a mean zero function of the data

fulfilling the conditions for the central limit theorem and K is a constant vector. Least
7

We now present our bounds estimates. Let 

)) ˆ #( (

) ˆ ( ˆ 0

) ˆ ( 0 0

q

i I

i

q

I

T

A q

θ

θ θ

∑∈

≡ be average

observed T in ) ˆ ( 0

q I θ , where [ ]’

' ˆ ˆ ˆ q θ ≡ δ κ and let 

)) ˆ #( (

(1 )

) ˆ ( ˆ

1

1

) ˆ (

1

1 1 1

q

i I

i

q

I

T

A q

−

∈

−

∑−

−

≡ θ

θ θ

. The

statistics ) ˆ ( ˆ 0 A θ q and ) ˆ ( ˆ 1

1 A θ −q are obvious upper bound estimates for α0 and α1 and can

be used to bound β . The following bounding results are straightforward:

Proposition 1:

(a) α0 ≤ plim ) ( ) ˆ ( ˆ 0 0 * A θ q = A θ q

(b) α1 ≤ plim ) ( ) ˆ ( ˆ *

1

1

1

1 A θ −q = A θ −q

(c) ( ( ), ( ), , ) ˆ lim ˆ lim ˆ ( ˆ ( ˆ ), ˆ ( ˆ ), , ) lim *

1

0 * 1

1

0 1

p β ols < β < p β olsχ A θ q A θ −q p R = p β olsχ A θ q A θ −q p R ,

where [ ]’ * * *’

q q θ ≡ δ κ .

We now discuss the asymptotic distributions of ) ˆ ( ˆ 0 A θ q and ) ˆ ( ˆ 1

1 A θ −q and how to

use the estimates in constructing asymptotic confidence intervals for α0, α1, and β. We

discuss ) ˆ ( ˆ 0 A θ q ; the properties of ) ˆ ( ˆ 1

1 A θ −q are symmetrical.

Decompose ) ( )) ˆ ( ˆ ( 0 0 * A θ q − A θ q into − )) + ˆ ) ( ˆ ( ˆ ( 0 0 A θ q A θ q ) ( )) ˆ ( ( 0 0 * A θ q − A θ q . The

asymptotic variance of the second term depends on the gradient vector of A0

 at * θ q , and is

not straightforward to estimate. However, note that ) ˆ ( 0 A θ q (the population mean T for

the set defined by T κ q ˆ ˆ ≤ ) is a valid bound for α0. We can therefore treat ) ˆ ( 0 A θ q as an

 

squares regression is one example of such an estimator.
8

object of estimation in its own right, and deal only with the first term in the

decomposition. We show in the appendix, under weak additional regularity conditions,

that the fact that θ q

ˆ is estimated has no effect on the limiting distribution of

)) ˆ ) ( ˆ ( ˆ ( 0 0 A q A q n θ − θ , which takes the simple form ))/ ) ˆ )(1 ( ˆ (0, ( 0 0 A A q θ q − θ q . In effect,

we can treat θ q

ˆ as fixed.

Constructing a confidence interval for ) ˆ ( 0 A θ q is immediate. It is also

straightforward to construct a conservative confidence interval for the parameter of

interest, α0. We summarize our results in the following proposition.

Proposition 2:

a A s n z A A s n z r q r q q r n − − ≤ ≤ + − = − →∞ ) ) 1 ˆ ) ( , ˆ ( ˆ ) ˆ ) ( ˆ ) ( , ˆ ( ˆ (7 ) lim Pr( 1 / 2

0 0

1 / 2

0 θ θ θ θ θ

b A s n z r q r n ≤ ≤ + − ≥ − →∞ ) ) 1 ˆ ) ( , ˆ ( ˆ (7 ) lim Pr(0 1

0 α 0 θ θ .

where zr denotes the rth percentile of the standard normal cdf and

nq

A A

s n q q )) ˆ ( ˆ )(1 ˆ ( ˆ

)ˆ ( ,

0 0 θ θ

θ − = .

Let ( , , , ) ˆ lim max

1

max β Ubound ≡ p β ols χ α 0 α p R denote the theoretical upward bound on

β, as determined by the measurement error bounds max α 0 and max α1 . Using the delta

method, one can derive a confidence interval for β Ubound similar to (7a) from the joint

distribution of β ols

ˆ , p, R, ) ˆ ( ˆ ), and ˆ ( ˆ 1

0 1 A θ q A θ −q . The joint distribution can be estimated

using a stacked regression. For this purpose, it is convenient to let 2 σ T|X denote the

average variance of T conditional on X and to replace R in (4) with 2 σ T|X using the
9

relation 

(1 )

1

2

p p

R T

− = − σ . Regressing the squared residual, 2

Te , from a regression of T on X

on a constant yields the coefficient 2 σ T|X . In addition, let I0 and I1 be indicators of

membership in the sets ) ˆ ( 0

q I θ and ) ˆ ( 1

1

q I θ − , respectively, and let (1 ) (1 2 ) 0 1 V ≡ − I − I − q .

It can be verified that regressing T on V, I0 – qV, and I1 – qV yields the coefficients p,

( ) 0 A θ q , and ( ) 1

1 A θ −q . This gives us the stacked regression:





















+













































−





















= − −





















2

2

|

1

0

0 1

2

1

0 0 0 0 0 0 1

0 0 0 0

1 0 0 0 0

Te

T

T X

T

e

A

A

p

c

V I qV I qV

X T

e

T

Y

ε

ε

σ

β

γ

 .

To construct a w-percent confidence interval for β analogous to (7b), note that

( β ols β Ubound

ˆ , ˆ ) are distributed bivariate normal. Let Φ(c, d,r) denote the probability that

X1 ≤ c and X2 ≤ d, where X1 and X2 are standard normal random variables with

correlation r. Let cq and dq satisfy Φ(cw , d w ,−ρ) = w , where ρ is the correlation between

β Ubound β ols

ˆ and ˆ . We then have

 (8) ) ˆ ˆ lim Pr( ˆ

ols w ols lim ˆols , lim Ubound Ubound w Ubound n w = β − c σ ≤ p β p β ≤ β + d σ →∞

 ) ˆ ˆ lim Pr( ols w ols Ubound w Ubound n ≤ β − c σ ≤ β ≤ β + d σ →∞ ,

where σ denotes the standard error of the subscripted estimator. A minimum length wpercent confidence interval can be constructed by choosing cw and dw to solve the

following minimization problem:
10

min ( )subject to ( , , ) . , d w Ubound cw ols cw d w w cw dwσ + σ Φ −ρ =

The preceding analysis treats q as fixed. )) ˆ ( ˆ ( 0 Var A θ q decreases with q since the

subsample used to compute ) ˆ ( ˆ 0 A θ q increases with q. However, Pr(T*=1), and

consequently, ) ˆ ( ˆ 0 A θ q will in general be increasing in q. The choice of q thus involves a

trade-off between the tightness and the variance of the bounds. We leave the optimal

choice of q as a topic for research.

Our upper bound on β does not take into account the constraint that the variance

of e is bounded below by zero. Whether this constraint binds can be checked by

examining the “reverse regression” coefficient generated by regressing T on Y and X. Let

B denote the coefficient on Y and β

ˆ

rev ≡ 1/ B the slope implied by the reverse regression.

Bollinger (1996) shows that the constraint binds only if β rev A θ q A θ q β Ubound

ˆ )) ˆ ) ( ˆ (1 ( ˆ 0 1 < − − .

We refer the reader to the bounds in Bollinger (1996) for this case (which is unlikely to

occur for low and moderate values of R2 in the main regression).

Stronger prior information will of course yield tighter bounds. We noted above

that knowledge of the functional form of Pr(T* = 1|X) can be used to estimate α0 and α1

from (6) by maximum likelihood, as in HAS. 3 We compare the percentile method with

HAS at several points below. Without knowledge of Pr(T* = 1|X), but with knowledge

 3

 More precisely, one can estimate upper bounds for α0 and α1. As HAS

themselves note, their model is indistinguishable from one where T* = 1 for a fraction of

the population α0, and T* = 0 for a fraction α1, independent of X. More generally, one

can envision a mixture of these two extreme cases, where the proportions α0 and α1 are

composed partly of those whose response is independent of X and partly of those who

misreport, so true measurement error is bounded below by zero and above by the

estimated values of α0 and α1.
11

of a functional form h(X; δ) such that Pr(T=1|X) = G(h(X; δ)) for an unknown cdf G--

that is, a single-index condition--one can consistently estimate δ up to a multiplicative

constant as in Ichimura (1986) and Han (1987), and thus estimate bounds that are tight

for a given q. Finally, if there is a set S such that it is known a priori that Pr(T*=1| X ∈

S) = 0 (for example, knowledge that the training program whose impact is of interest is

not offered at a given location), then Pr(T=1| X ∈ S) = α0, and correspondingly for α1.

III. A GMM Estimator for Estimating the Effect of a Mismeasured Binary

Explanatory Variable When Instruments are Available

Where instruments are available, instrumental variable estimation is a commonly

prescribed fix to measurement error in a regressor. We begin this section by showing that

IV estimation by itself is inconsistent for the coefficient of the mismeasured variable but

consistent for other variables (assumed to be correctly measured). We next derive a

consistent closed-form GMM estimator and then show how to incorporate the estimated

measurement error bounds developed in the previous section into the GMM estimates.

We also develop a specification test for the measurement error model.

IV Estimation

When cov(T*,U) = 0, any variable Z which is correlated with T* and independent

of e and the measurement error process can be used as an instrument. However, if T* is

binary, T* and U will be negatively correlated for any (non-degenerate) distribution of T*

and U. Because the classical assumption that T* and U are uncorrelated cannot hold

when T* is binary, Z will not be a valid instrument. To be more precise, let Z be a vector

of variables such that Cov(Z,T*) ≠ 0, Cov(Z, e)=0, and Cov(Z,U|T*)=0. The last equality

 
12

captures the idea that Z is independent of the measurement error process conditional on

T*, but since Cov(T*,U) ≠ 0, this does not imply that the unconditional covariance

Cov(Z,U)=0.

As others have demonstrated, the fact that Cov(Z,U)≠0 means that the use of Z as

instruments will result in an inconsistent estimate of β.

4

 Interestingly, as we now show,

the IV estimate of γ is consistent. In anticipation of the discussion to follow, it is

convenient to frame the analysis in terms of GMM estimation.

As is well known, the IV estimator 















iv

iv

γ

β

ˆ

ˆ is equivalent to the GMM estimator





 





γ

β

ˆ

ˆ

1 whose probability limit is given by

(9) [ ][ ]) ~

, ~ ) ( ~, ~ ) ’ ( ~, ~ ) ( ~, ~ argmin ( lim ˆ

ˆ lim

~ 1 1 2 1 1 1 2 1 , ~

1

1

 = β γ β γ β γ β γ



 





γ

β

β γ

m m A m m

p

p ,

where

 (10a) ) ~ ~ ) ( , ~, ~

(1 1 1 m β γ = Cov Z Y − β T − Xγ

(10b) ) ~ ~ ) ( , ~, ~( 2 1 1 m β γ = Cov X Y − β T − Xγ ,

 and A is a weighting matrix (in the case of TSLS, 1 ( )

− A =Var Z X ).5

 (The subscript on

the estimate for β will be convenient in the discussion below where we obtain other

estimates using additional moment conditions.) We now prove

 

4

 We follow previous papers (Loewenstein and Spletzer 1997, Berger, Black and Scott

1997, Kane, Rouse and Staiger 1999) in referring to estimators using the elements of Z as

instruments as IV estimators, in spite of the fact that technically the Z variables are not

instruments.

5

 Naturally, in carrying out the GMM estimation, the population moments are replaced by

the sample moments.
13

Proposition 3: β1 1β ˆ p lim = k and γ = γ iv p lim ˆ , where (1 )

1

0 1

1 −α −α

k = .

Proof:

It follows immediately from equation (1) that

(11a) Cov(Z,Y) = βCov(Z,T*) + Cov(Z, X )γ

(11b) Cov(X,Y) = βCov(X,T*) +Var(X )γ .

Letting rAW denote the coefficient of correlation between two random variables A

and W, and rAW⋅V the partial correlation between A and W conditional on V, the coefficient

of correlation between X and U conditional on T* can be expressed as (Gujarati 1978, p.

112)

 (1 )(1 ) 2

*

2

*

* *

*

XT UT

XU XT UT

XU T

r r

r r r

r

− −

− ⋅ =

.

Using the fact that rXU⋅T* = 0, one finds that Var(T*)Cov(X,U) =

Cov(T*,U)Cov(X,T*). This result together with (2) and the fact that Cov(T*,U) =

− p*(1- p*)(α0+α1) gives us

(12) ( , ) ( ) ( , ). Cov X U = − α 0 +α1 k1Cov X T

From (12) and its analogue for Z, we have Cov(Z,T*)/Cov(Z,T) = Cov(X,T*)/Cov(X,T) =

k1. Substituting into (11) yields

(13a) ( , ) β ( , ) ( , )γ Cov Z Y = k1 Cov Z T + Cov Z X

(13b) ( , ) β ( , ) ( )γ Cov X Y = k1 Cov X T +Var X .

 Substituting (13) into (10), we see that

) ~ ) ( , ) ( )( ~ ) ( ~, ~ (12 ’) (

) ~ ) ( , ) ( , )( ~ ) ( ~, ~ (12 ’) (

2 1 1 1

1 1 1 1

β γ β β γ γ

β γ β β γ γ

= − − −

= − + −

b m k Cov X T Var X

a m k Cov Z T Cov Z X ,
14

which implies that β1 1β ˆ p lim = k and γ = γ 2 p lim ˆ . Q.E.D.

Although IV yields an inconsistent estimate of β, the proof of proposition 3

suggests a way that one might be able to obtain a consistent estimate. The GMM

estimator using the moment conditions (11) yields an estimate of 1

ˆ

β that is a simple

function of β and the measurement error parameters α0 and α1. If one can use additional

moments that allow determination of α0 and α1, then one can solve for β.

GMM Estimation

Consider for the moment the case without covariates. KRS and BBS analyze a

model with two mismeasured indicators T1 and T2 of T*. They note that seven moments

are observable: ( | , ) 1 2 E Y T = i T = j , Pr( , ) 1 2 T = i T = j , i,j={0,1} (one of the cell

probabilities is redundant). This allows the identification of the seven parameters c, β,

p*, α0k , α1k , k={1,2}. Note that knowledge of ( | , ) 1 2 E Y T = i T = j and

Pr( , ) 1 2 T = i T = j is equivalent to knowledge of the following sets of moments: E(Y),

Cov(Tk,Y), Cov(T1T2,Y), E(Tk), and Cov(T1,T2). In our analysis, an instrument takes the

place of one of the alternate measures, so that the moments are E(Y ), Cov(T,Y ),

Cov(Z,Y), Cov(ZT,Y ), E(T), E(Z ), and Cov(Z,T ). Thus, our model is identified and can

be estimated using GMM.

Turning our attention back to the full model (1), note that we have already used

the covariances between Z and Y in the moment conditions (13a) and the covariances

between X and Y in the moment conditions (13b). To use Cov(T,Y ) and Cov(ZT,Y ), we

note from (1) that
15

β γ

β γ

(13 ) ( , ) ( , ) ( , )

(13 ) ( , ) ( ) ( , )

3

2

d Cov W Y k Cov W T Cov W X

c Cov T Y k Var T Cov T X

= +

= +

where W ≡ (Z − Z ) ⋅T , ( )

( , *)

2 Var T

Cov T T k ≡ , and ( , )( , *)

3 Cov W T

Cov W T k ≡ .

6

 Algebra

establishes that:

(14) (1 )(1 )

( )(1 )

0 1

0 1

2

α α

α α

− − −

− − − = p p

p p k ;

 (1 )(1 )

(1 )

0 1

1 0

3

α α

α α

− − −

− − + = p

p k .

To estimate β 2 ≡ k2β and β 3 ≡ k3β , we expand the IV moment conditions (10) to

include:

 (10c) ) ~ ~ ) ( , ~, ~( 3 2 2 m β γ = Cov T Y − β T − Xγ

 (10d) ) ~ ~ ) ( , ~, ~( 4 3 3 m β γ = Cov W Y − β T − Xγ .

Since the factors k2 and k3 are both functions of p as well as α0 and α1, we need an

estimate of p to close the model, so we add a final moment condition:

(10e) m E T p

~ ( ) 5 = − .

 Substituting (13c) and (13d) into (10) yields

. ~ ) ( ) ~ (12 ’) (

) ~ ) ( , ) ( , )( ~ ) ( ~, ~ (12 ’) (

) ~ ) ( ) ( , )( ~ ) ( ~, ~ (12 ’) (

) ~ ) ( , ) ( )( ~ ) ( ~, ~ (12 ’) (

) ~ ) ( , ) ( , )( ~ ) ( ~, ~ (12 ’) (

5

4 3 3 3

3 2 2 2

2 1 1 1

1 1 1 1

e m p E T p

d m k Cov W T Cov W X

c m k Var T Cov T X

b m k Cov X T Var X

a m k Cov Z T Cov Z X

= −

= − + −

= − + −

= − − −

= − + −

β γ β β γ γ

β γ β β γ γ

β γ β β γ γ

β γ β β γ γ

 This leads to

 

6

 Note the definition of W. The formula for k3 below is correct only if Z is normalized to

zero before multiplying by T.
16

Proposition 4: Let 1

ˆ

β , 2

ˆ

β , 3

ˆ

β , and pˆ be GMM estimates using moments (12).

Then

(15a) 2

1 2 3 1 ) ˆ ˆ ˆ ((1 ˆ) ˆ ˆ 4 ˆ(1 ˆ) ˆ

β = p − p β β + − p β − pβ

(15b) 

1

1 3

0 ˆ 2

ˆ ˆ (1 ˆ) ˆ ˆ ˆ β

β β β α

+ − − = p p

and

(15c) 

1

1 3

1 ˆ 2

ˆ ˆ (1 ˆ) ˆ (2 ˆ) ˆ β

β β β α − − − − = p p

are consistent estimators for β, α0 and α1.

7

Proof:

From (12a’) - (12e’), it is clear that any GMM estimator using moment conditions

(10a) – (10e) will have:

(16 ) lim ˆ .

ˆ (16 ) lim

ˆ (16 ) lim

(16 ) lim ˆ lim ˆ

ˆ (16 ) lim ˆ lim

3 3

2 2

1 1

e p p p

d p k

c p k

b p p

a p p k

iv

iv

=

β = β

β = β

γ = γ = γ

β = β = β

Substituting (14) and Proposition 3 into (16) and solving for β, α0 and α1 yields the

consistency result. Q.E.D.

An optimal GMM estimator can be derived as follows. Let

 

7

 Note that )) sgn( ) ˆ )) sgn( lim( ˆ )) sgn( lim( ˆ sgn( lim( p β1 = p β 2 = p β 3 = β since k1, k2, and

k3		

		 

2

1 1 2 3 1 ) ˆ ˆ ˆ ((1 ˆ) ˆ ˆ ) 4 ˆ(1 ˆ) ˆ sgn( β

ˆ = β p − p β β + − p β − pβ .
17

























≡

0 0 0 0 1

0 0 0 0

0 0 0 0

0 0 0

w

t

z x

H , 

























Ω ≡

y

y

y

y

, 

























Γ ≡

0 0 0 0 1

0 0 0

0 0 0

0 0 0

t x

t x

t x

, 





































Π ≡

0

0

0

3

0

2

0

1

0

p

γ

β

β

β

, and ˆ ’[ ] 0 ≡ H Ω − ΓΠ ,

where 0 Π is some initial consistent estimate of Π and where z, x, y, and t devote the

deviations of Z, X, Y, and T from their sample means. Write the sample moments (10) in

stacked form:

(17) m ≡ (1/ n)H’[Ω − ΓΠ].

The optimal GMM estimator minimizes mV m−1 , where V is the asymptotic

variance matrix of m (Hansen 1982). Letting [ ]’ [ ]’ 0 1 2 3 4 Ω − ΓΠ ≡ ε ε ε ε , note that

i

 is not generally homoscedastic. For example,

, ~ ~ ~ ~ 1

1

*

1

1

i i i i i i i i ε = y − β t − x γ = βt − β t − x γ + e so that ( | ) ( *| ) 1 2 Var ε t = β Var T T = t + T

+Var(e) will not in general be constant across t. The optimal GMM estimator given the

moment conditions (10) and taking into account heteroscedasticity is

= Γ Γ Γ Ω − − − ˆ ( ’ ’ ) ’ ’ 1 1 1 π HS H HS H , where = ∑( ' ˆ ˆ ’ ) Hi i i Hi S ε ε ; see Wooldridge (1996).

Note that the asymptotic distribution for 















−

−

γ γ

β β

ˆ

ˆ n is

N(0, ) 0 1

0 ’( ’ ’ ) 0 10 1 1





 



  Γ Γ



 



 − − d

HS H

d , where 



 





∂

∂

∂

∂

∂

∂

∂

∂ ≡ p

d ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ ' 1 2 3

β

β

β

β

β

β

β can be

obtained from (15a). If desired, cˆ may be calculated as − βˆ * − γˆ Y ˆ p X .

Finally, we might point out that there is one substantive difference between the

situation in which there is more than one imperfect measure of a binary explanatory

variable and that where one or more instruments take the place of one of the alternative
18

measures. In the former case, Cov(X,T1)Cov(T2,T*) = Cov(X,T2)Cov(T1,T*) since

Cov(X,Ti) = Cov(X,T*)Cov(Ti,T*) for i = 1,2. KRS use this restriction to obtain

additional identifying information. Since Cov(X,Z) is unrestricted, there is no analogous

restriction when instruments take the place of one of the measures.8

Incorporating Restrictions on the Measurement Error Parameters Into the GMM

Estimation

Estimates of α0 and α1 based on (15b-c) are not guaranteed to be between 0 and

1. Additionally, they are not guaranteed to be less than bounds derived from the HAS or

percentile methods in Section II. Bounds on the measurement error parameters can be

incorporated into the GMM estimation procedure.9

 We first discuss imposing bounds

using the percentile method, than briefly discuss using the HAS bounds.

Let ) ˆ ( ˆ lim q

i Ai ≡ p A θ , Di ≡ Ai −α i , and q ii Di A θ ) αˆ ˆ ( ˆ ˆ ≡ − The restriction that the

measurement error parameters be between zero and the bounds Ai can be expressed as L ≥

0, where [ ] L ≡ D0 D1 α 0 α1 . Note that these restrictions imply that that ≤1 α i .

Let Π denote the vector of parameters [π A0 A1]’. One can directly incorporate the

percentile bounds into the estimation procedure by extending the moment conditions (10)

to include a regression of T on a constant over the subset of the data with Xδ κ q ˆ ˆ ≤ to

estimate A0, and a regression of 1 − T on a constant over the subset of the data with

 

8

 If one makes the stronger assumption that e is not just uncorrelated with, but is

independent of X and T*, additional moment conditions analogous to (13d) can be

imposed using Cov(xT ,Y). More broadly, if any interactions between elements j of X and

T* can be excluded from (1), Cov(xjT, Y) can be used as in (13d).

9

 BBS do not take into account the bounds on α0 and α1 in their GMM procedure. KRS

account for it by parameterizing α0 and α1 to be between zero and one, resulting in a nonlinear GMM procedure.
19

X ≥ 1−q ˆ δ

ˆ κ to estimate A1, similar to the stacked regressions above. Denote this extended

moment set ma.

The classical approach to point estimation is to estimate Π by minimizing the

weighted sum of squares ma’Va

-1

ma subject to [ ˆ ˆ ] 0 ˆ ˆ ˆ L ≡ D0 D1 α 0 α1 ≥ . Note that the

constraints are non-linear since the α i ˆ estimates are non-linear functions of the

regression parameters. Incorporating inequality constraints into classical inference

presents challenges. The asymptotic distribution depends on whether the true parameter

is on the boundary of the feasible set. If the true parameter is in the interior of the

feasible set, then the asymptotic distribution is equal to that of the equivalent

unconstrained estimator, but this may be a poor guide to finite-sample behavior. The

bootstrap, a common method of improving finite-sample performance of variance

estimates, is inconsistent in inequality-constrained problems (Andrews 2000).

We believe that Geweke’s (1986) Bayesian method presents the simplest

satisfactory approach to this problem. Geweke shows that if the prior for the parameter

vector is diffuse over the feasible set, the posterior distribution is the portion of the

estimated sampling distribution of the unrestricted parameter estimate in the feasible

region. The posterior mean can be evaluated by taking random draws from the sampling

distribution and averaging over draws in the feasible region. Applied to our problem, one

first uses GMM to obtain an unconstrained estimate Πˆ . One then takes draws from the

distribution )) ˆ , ( ˆ N(Π V Π and averages over those draws where Lˆ ≥ 0.

A Specification Test for the Measurement Error Model.

The foregoing assumes that the data were generated by (1) (i.e., that e is

uncorrelated with T* and X), that the measurement error is uncorrelated with X and Z
20

conditional on T*, and that the instruments Z are valid. If these assumptions are violated,

it is not necessarily the case that 0 ˆ p lim L ≥ . Thus we can use a test of the null hypothesis

that L ≥ 0 as a specification test of the measurement error model.

Let δ denote the 4 x 1 vector )’ ˆ ˆ (L L R − , where R Lˆ denotes the inequality-restricted

estimator of L. Letting δA denote the subvector of δ corresponding to all the binding

constraints and letting )ˆ VA (L denote the corresponding elements of the variance matrix

of Lˆ , construct the test statistic

(18) δ A VA L δ A τ 1 )ˆ ' ( − = .

We now show that asymptotically the 1−q percentile of the chi-square distribution with 2

degrees of freedom is a valid critical value for a test with significance level q.

Proposition 5:

lim Pr( | 0) ( ) 2

2 c L c n τ ≤ ≥ ≥ χ →∞ and lim Pr( | 0) ( ) 22 c L c n τ ≤ = = χ →∞

Proof:

The ) ˆ ( ˆ q

i A θ parameters must be non-negative since they are sample means of nonnegative variables. Thus, no more than two out of the four elements of Lˆ can violate the

inequality constraints and δA will never have more than two elements.

All four constraints are satisfied with equality when α0 = A0 = α1 = A1 = 0.

Asymptotically, )) ˆ ( ˆ ( q

i Var A θ = Ai (1 − Ai) = 0 at this point, so that the distribution of Di

ˆ

converges to the distribution of α i − ˆ and the restrictions collapse to

[ ] ˆ ˆ ˆ ˆ 0 −α 0 −α1 α 0 α1 ≥ , or equivalently, ˆ ˆ 0 α 0 =α1 = . Whatever combinations of
21

constraints are violated in the sample, the statistic τ thus equals

[] [] ˆ ˆ ( ˆ) ˆ ˆ ’ 0 1

1 α 0 α1 α α α − Var , which is asymptotically distributed ) (2 2 χ .

10

Now consider the case where two constraints are satisfied with equality, one

corresponding to α0 and one to α1. If αi = 0, let λi =αi and μi =Di. Conversely, if Di = 0,

let λi =Di and μi =αi. As the sample becomes large, the probability that the inequality

constraints involving λi are violated approaches zero. We therefore need only consider

cases where the constraints involving αi are violated in the sample. If no constraints are

violated in the sample, τ = 0. If one constraint i is violated, ) ˆ / ( ˆ 2 = λi Var λi τ , which is

distributed χ2

(1). If both constraints are violated, [ ] R R

0 0 1 1 ˆ ˆ ˆ ˆ τ = λ − λ λ − λ

[ ]′ − − − R R Var 0 0 1 1

1 ˆ ˆ ˆ ˆ )ˆ (λ λ λ λ λ , which is precisely Wolak’s (1987) test statistic W. Wolak

(1987) shows that W is equal to the distance between the inequality-restricted and

unrestricted estimates, evaluated in the norm of the covariance matrix of the unrestricted

estimates. This distance must be less than or equal to the distance ∆ between the

unrestricted estimates and the estimates obtained imposing the equality restrictions

0. λ0 = λ1 = ∆ is distributed ). (2 2 χ Thus τ is a mixture of random variables all of

which are less than or equal to ). (2 2 χ

Applying a similar argument for cases with other combinations of constraints

holding with equality, we can conclude that the point where all four constraints are

 

10 Note that the more traditional chi-square statistic δ δ1 )ˆ ' ( − Var L , as in Wolak (1991),

does not exist at this point, as )ˆ Var(L is singular.
22

satisfied with equality is the least favorable point for the purpose of computing a

distribution of the test statistic under the null hypothesis.11, 12 Q.E.D.

Applying the HAS bounds

The application of HAS bounds to the Geweke technique is immediate. However,

the distribution of the test statistic τ under the null is not easily derived under the HAS

method. Negative estimates of A0 and A1 are possible, so all four constraints may be

violated at the same time. As explained in Wolak (1991), the distribution of τ at the point

where all constraints bind is a weighted mixture of chi-squares from zero to four degrees

of freedom, with the weights depending on )ˆ Var(λ . Since )ˆ Var(λ will vary across

points in the null due to the non-linearity of the constraints, one can no longer show that

this point is the least favorable point in the null.

IV. Bounding the Effect of a Mismeasured Endogenous Binary Explanatory

Variable

Now suppose the residual e in (1) is uncorrelated with X and Z, but correlated

with T*. Projecting T on T* and Z, we have: , (1 ) * T = α0 + − α0 − α1 T +η where η is

orthogonal to T* and Z. Thus, ( , ) (1 ) ( *, ) 0 1 Cov T e = −α −α Cov T e and Cov(W , e) =

 

11 Our constraints involve 0 αˆ and 1 αˆ , which are nonlinear functions of , ˆ , ˆ , ˆ

β1 β 2 β 3 and pˆ .

While it is always true for linear constraints that the least favorable point in the null

occurs where all constraints are satisfied with equality, we needed to demonstrate this

here because it is not always true when the constraints are non-linear; see Wolak (1991).

12 


 	
	

 R Lˆ subject to non-linear constraints, is in

many cases easy to calculate. Cases with zero or one constraint binding were dealt with

in the text. If two constraints are violated, and if each constraint is still violated after

imposing the other constraint as an equality and calculating the constrained optimum,

then R Lˆ = 0. One can also show that the constraint that 0 α0 ≥ is equivalent to the linear

constraint β2 ≤ β3.
23

(1 ) ( *, ) 0 1 − α − α Cov zT e , which means that the population moment conditions (13c) and

(13d) become:

(13 ’) ( , ) ( , ) ( , ) (1 ) ( *, ).

(13 ’) ( , ) ( ) ( , ) (1 ) ( *, )

3 0 1

2 0 1

d Cov W Y k Cov W T Cov W X Cov zT e

c Cov T Y k Var T Cov T X Cov T e

= β + γ + − α − α

= β + γ + − α − α

By assumption, Cov(T*,e) is not equal to zero. What about Cov(zT*,e)?

Projecting T* onto z, X, and e, we have: T* = δ 0 + δ 1X + δ 2 z + δ 3e +ω , where ω is

orthogonal to X, Z, and e. So

 (19) ( *, ) ( , *) ( ) ( ) ( ) ( , ) 2

3

2 Cov zT e = Cov ze T = δ 1E zXe + δ 2E z e + δ E ze + Cov ze ω

Independence of X and Z with e, not just orthogonality, is required to guarantee that the

first three terms on the right hand side of (19) are zero. Cov(ze, ) will not in general be

zero except under the strong assumption that E(T*| X, Z, e) is linear in X, Z, and e--that is,

that the linear probability model applies to T*.

We can conclude that endogeneity of T* adds two more sets of unknown

parameters to the moment conditions (13) -- Cov(T*,e) and Cov(zT*,e). Consequently,

the GMM method described above is now underidentified. However, note that

Proposition 3 still holds, so that p βiv

ˆ lim still equals (1 ) β −α 0 −α1 (and limγˆ = γ ) iv p .

Under the maintained assumption that α0 +α1 < 1, β iv

ˆ is asymptotically still an upper

bound, and zero is a lower bound. This lower bound can be tightened by employing the

HAS or the percentile method to obtain upper bounds max α0 and max α 1 for the

measurement error parameters α0 and α1.

13 Specifically, one has:

p β iv 1 α 0 α1 β p β iv

ˆ lim ˆ ( ) lim max max − − < < .

 

13 Naturally, the instruments Z can be used in estimating Pr(T=1|X, Z) (for the percentile
24

 Confidence intervals for the lower bound can be generated as in Section II.

V. Empirical Example

We now illustrate the use of our measurement error techniques with an analysis of

the effect of training incidence on wage growth. Employee training is a particularly

interesting application of these techniques because there is evidence that it is measured

with a great deal of error. Using a survey of matched employer-employee responses to

the same training questions, Barron, Berger, and Black (1997) find that the correlation

between worker reported training and employer reported training is only .32 for the

incidence of on-site formal training, and .38 for off-site formal training.

We use data from the National Longitudinal Survey of Youth 1979 Cohort

(NLSY79). NLSY79 is a dataset of 12,686 individuals who were aged 14 to 21 in 1979.

These youth have been interviewed annually since 1979, and the response rate has been

90 percent or greater in each year. We use data from the 1987 through 1994 surveys.

Our dependent variable is the change in real log wages between interviews. We exclude

job changers, so all wage growth is within-job. We also exclude the military subsample,

observations with real wages below $1 or above $100 in 1982-84 dollars, and

observations where the respondent is an active member of the armed forces, selfemployed, in a farm occupation, or enrolled in school. Finally, we exclude observations

where variables have missing values (except for the cases noted below where we use

missing indicators).

 

method) or Pr(T*=1|X, Z) (for HAS) to help tighten the measurement error bounds.
25

Our training measure equals one if the respondent reports completing a training

program on the current job since the last interview (the training may have started before

the last interview) and zero otherwise. Our control variables are age, tenure, experience,

the Armed Forces Qualifying Test measure of cognitive skills (AFQT),14 and dummies

for female, black, Hispanic, ever married, one-digit occupational categories, collective

bargaining, part-time status, and calendar year. In addition, there are dummies for

missing AFQT, collective bargaining status, and part-time status, with the variables set

equal to zero if their missing indicators equal one. Our final sample has 20,300

observations from 8,031 jobs and 6,345 individuals. The observed incidence of training

is 12.9 percent.

We use two instruments. The first is a measure of gross job destruction and

creation by 2 digit industry created from Michigan unemployment insurance data.15 The

second is years of completed schooling. Our first instrument can be justified by the fact

that the employer’s return to a given training-induced increase in worker productivity is

higher for a longer-lived job match (see Royalty 1996). Job creation and destruction

rates are plausibly related to the magnitude of exogenous demand shocks and hence

exogenously shift turnover rates. The second instrument is motivated by the consideration

that years of school is an indicator of trainability. In this context, note that the inclusion

of AFQT and one-digit occupation dummies in the wage growth equation arguably

controls for the direct effect of schooling on productivity growth. We also include runs

where the reallocation variable serves as the sole instrument (rows denoted “1

instrument” in the table).

 

14 Specifically, the residual from a regression of AFQT on dummies for year of birth.
26

To illustrate the effects of our measurement error bounds on estimates of γ , we

also show OLS bounds and IV estimates for the coefficient on AFQT. Our uncorrected

OLS results indicate a statistically significant link between wage growth and AFQT.

Given that we find (as is common in this literature) a strong effect of AFQT on the

probability of receiving training, it is of interest to what extent the observed AFQT effect

on wage growth may be due to measurement error in training.

Means and standard deviations for the variables of interest are shown in table 1.

Point estimates and bounds for the wage return to training, β, for the measurement

parameters, α0, α1, and for γafqt are shown in table 2. All standard errors are from a panel

version of the White heteroscedasticity-consistent estimator (see Froot 1989 and Rogers

1993). Table 3 shows minimum length confidence intervals as in (8).

The OLS results indicate that training during a period raises next period wages by

1.9 percent. IV estimation raises the training coefficient dramatically, to between 13 and

14 percent. This increase is consistent with the hypothesis of substantial measurement

error.

To generate percentile bounds, we first estimate a probit of training incidence on

the independent variables and instruments and then observe the incidence of training

below the 5th and above the 95th percentiles of the distribution of predicted training. The

bound for α0 is relatively tight at 2.4 percent. The bound for α1, 70 percent, is much

higher, indicating the potential for a great deal of measurement error. The relative

magnitudes of these bounds are intuitively plausible, as it seems more likely that

respondents would forget or neglect to report training spells rather than to report training

 15 We thank Chris Foote for supplying these data. See Foote (1998) for more details.
27

that did not occur. We generate HAS bounds by estimating (6) taking F to be the normal

cdf. The HAS bounds are slightly tighter than the percentile bounds.

Applying the bounds on α0 and α1 to the OLS results using (5) yields upper

bounds on β of about 4 percent for both the percentile and HAS methods, roughly double

the OLS estimate. 16 The confidence intervals from (8) are similar to, but slightly to the

left of what one would get by adding 1.96 σUbound to the upper bound estimate and

subtracting 1.96 σols from the OLS estimate. Because σUbound > σols, taking a smaller

multiple of σUbound and a larger multiple of σols allows us to shorten the confidence

interval slightly.

GMM results based on (17) using both instruments give a point estimate of 4

percent, similar to the bounds.17 However, the GMM estimates of α0 and α1 are

infeasible, with α0 negative and α1 above the percentile (and hence HAS) bound. We

apply the Geweke technique, generating 10,000 draws in the feasible region, to produce

feasible estimates of α0 and α1. Relative to GMM, the Geweke method does not affect

the point estimate of β very much, but reduces the standard error from .008 to .007.18

The results are similar when only the reallocation variable is used as an instrument.

However, the improvement in precision from using the Geweke technique is more

 

16 The reverse-regression upper bound on β from Bollinger (1996), referred to in Section

II, is 22.2, corresponding to values of (α0, α1) of (0.123, 0). The much tighter bound

produced by our method stems from the fact that, as explained in Bollinger (1996, p.

396), information reducing misclassification from the larger to the smaller group – in this

case, α0 - is particularly powerful in reducing the upper bound.

17 Adding an overidentifying instrument generates two overidentifying moments (15a)

and (15d). The test statistic for the conventional GMM overidentification test (Greene

2000, p. 482) is 1.64, far from the χ2(2) critical values.

18 Of course, the GMM estimator and the Geweke estimator are not strictly comparable

unless one considers GMM as a Bayesian estimator combining the data and a diffuse
28

dramatic, as the GMM standard error for β with only one instrument is .012 while the

Geweke standard error is still .007 to three decimal places.

To test whether the measurement error model underlying the GMM and Geweke

estimates is compatible with the data, we calculate the test statistic τ in (18) (using the

percentile bounds). With two instruments, τ is 2.68, which is not significant at

conventional levels using a χ2(2) distribution (the 5% critical value is 5.99). Using one

instrument, τ is zero, as the GMM values are feasible.

The results for γafqt show a substantial effect of measurement error. The OLS

coefficient is statistically significant at the 1 percent level whether highest grade

completed is included in the regression or not. However, IV estimation reduces the

coefficient by at least two-thirds; the estimated coefficient is less than its standard error.

Recall that γ is consistently estimated by IV. The OLS bounds using the estimated α

parameters are consistent with the low IV estimates, with the percentile bounds slightly

lower than the IV estimates and the HAS bounds quite close.

Finally, if we allow for endogeneity, the lower bounds for β from IV estimation

range from .036 to .041 across the different specifications, though with relatively large

standard errors. These lower bounds are similar to the upper bounds from OLS and to the

point estimates from the GMM and Geweke methods. The length of the confidence

intervals reflects the large standard errors, σLbound, on the lower bounds estimates; as

above, they are to the left of intervals generated by subtracting 1.96 σLbound to the bound

and adding 1.96 σiv to the IV estimate.

 

prior.
29

Our prior expectation was that training incidence may be positively associated

with unobservable determinants of wage growth. However, the bounds estimates indicate

that true training is either exogenous or negatively correlated with the wage growth

residual, as the lower bound estimates allowing for endogeneity coincide with the point

estimates when one takes training to be exogenous. We conclude that our evidence is

consistent with measurement error cutting the OLS estimate of the return to training in

half, but we cannot rule out an additional downward bias to OLS due to endogeneity.

Much of the apparent effect of AFQT on wage growth appears to be due to measurement

error in training.

VI. Conclusion

This paper has explored techniques for dealing with a mismeasured binary

explanatory variable in a linear regression. If the binary variable is measured with error,

is uncorrelated with the error term in the regression, and there is no instrument available,

then its true coefficient, β, can be bounded by combining the least squares coefficient

with the HAS or percentile method for bounding measurement error presented in Section

II. If an instrument is available, IV is inconsistent, but β can be consistently estimated by

the GMM estimator in Section III. The estimated measurement bounds can be

incorporated into the GMM estimates, and the specification can be tested by comparing

the GMM estimates with the measurement error bounds. Finally, if the mismeasured

binary explanatory variable is correlated with the error term in the regression, the GMM

estimator is inconsistent, but β can be bounded by applying the HAS or percentile

measurement error bounds to the IV estimate.
30

References

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the Boundary of the Parameter Space.” Econometrica 68:2, pp. 399-405.

Barron, John M., Mark C. Berger, and Dan A. Black. 1997. "How Well do We Measure

 Training?" Journal of Labor Economics 15:3 (part 1), pp. 507-528.

Berger, Mark, Dan Black, and Frank Scott. 2000. "Bounding Parameter Estimates with

Non-Classical Measurement Error.” Journal of the American Statistical Association

95:451, pp. 739-48.

Black, Dan, Seth Sanders, and Lowell Taylor. 2000. “Measurement of Higher Education

in the Census and CPS”. Paper presented at Southern Economic Association

Meetings, Arlington VA.

Bollinger, Christopher R. 1996. “Bounding Mean Regressions When a Binary Regressor

is Mismeasured.” Journal Of Econometrics 73:2, pp. 387-399.

Foote, Christopher. 1998. "Trend Employment Growth and the Bunching of Job Creation

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Geweke, John. 1986. “Exact Inference in the Inequality Constrained Normal Linear

Regression Model.” Journal of Applied Econometrics 1:2, pp. 127-141.

Greene, William H. 2000. Econometric Analysis, 4th edition. Upper Saddle River, New

Jersey: Prentice Hall.

Gujarati, Damodar. 1978. Basic Econometrics. New York: McGraw-Hill.

Han, A. K. 1987. "Non-parametric analysis of a generalized regression model." Journal

of Econometrics 35, pp. 303-16.

Hansen, L. 1982. “Large Sample Properties of Generalized Method of Moments

Estimators.” Econometrica 50, pp. 1029-54.
31

Hausman, J. A., Abrevaya, Jason, and F. M. Scott-Morton. 1998. "Misclassification of

 the Dependent Variable in a Discrete-Response Setting." Journal Of Econometrics 87,

 pp. 239-69.

Ichimura, H. 1993. "Semiparametric Least Squares (SLS) and Weighted SLS Estimation

of Single Index Models." Journal of Econometrics 58, pp. 71-120.

Kane, Thomas J., Cecilia Elena Rouse and Douglas Staiger. 1999. “Estimating Returns

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Koenker, Roger, and Gilbert Bassett Jr. 1982. “Robust Tests for Heteroscedasticity

Based on Regression Quantiles.” Econometrica 50:1, pp. 43-61.

Loewenstein, Mark A. and James. R. Spletzer. 1997. “Belated Training: The

 Relationship Between Training, Tenure, and Wages.” Unpublished paper, Bureau of

 Labor Statistics.

Rogers, W. H. 1993. “Regression Standard Errors in Clustered Samples.” Stata

Technical Bulletin 13, pp. 19-23.

Royalty, Anne Beeson. 1996. “The Effects of Job Turnover on the Training of Men and

Women.” Industrial and Labor Relations Review 49:3, pp. 506-21.

White, Halbert. 1982. “Maximum Likelihood Estimation of Misspecified Models.”

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Wolak, Frank A. 1987. “An Exact Test for Multiple Inequality and Equality Constraints

in the Linear Regression Model.” Journal of the American Statistical Association

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Wolak, Frank A. 1991. “The Local Nature of Hypothesis Tests Involving Inequality

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Wooldridge, Jeffrey M. 1996. “Estimating Systems of Equations with Different

Instruments for Different Equations.” Journal of Econometrics 74, pp. 387-40
32

Table 1

Descriptive Statistics, Selected Variables

Variable Mean Std.

Deviation

Min. Max.

Change in Ln Wage 0.025 0.225 -1.492 1.487

Training Incidence 0.129 0.335 0 1

AFQT 0.7 20.3 -65.5 45.9
33

Table 2

Estimates and Bounds for Selected Coefficients and Measurement Error Parameters

Parameter β α0 α1 γafqt (x 10-4)

Specification

Point Estimates

OLS, 2 instruments .019

(.005)

3.24

(0.81)

OLS, 1 instrument .019

(.005)

2.37

(0.87)

IV, 2 instruments .140

(.045)

0.72

(1.22)

IV, 1 instrument .133

(.064)

0.79

(1.27)

GMM(2) .040

(.008)

-.047

(.046)

.757

(.029)

Geweke(2) percentile1 .038

(.007)

.008

(.006)

.634

(.082)

Geweke(2) HAS .038

(.007)

.006

(.004)

.636

(.072)

GMM(1) .039

(.012)

.020

(.053)

.688

(.091)

Geweke(1) percentile1 .037

(.007)

.014

(.008)

.599

(.124)

Geweke(1) HAS .036

(.006)

.009

(.006)

.587

(.124)

Upper Bounds2

Percentile1 .024

(.005)

.702

(.017)

HAS .015

(.003)

.695

(.008)

OLS percentile bound1 (2) .043

(.011)

0.39

(1.67)

OLS percentile bound1 (1) .043

(.011)

0.44

(1.36)

OLS HAS bound (2) .038

(.009)

0.88

(1.10)

OLS HAS bound (1) .038

(.010)

0.79

(1.16)

Lower Bounds

Endogenous T*, IV(2)

percentile bound1

.038

(.012)

Endogenous T*, IV(2) HAS

bound

.041

(.013)

Endogenous T*, IV(1)

percentile bound1

.036

(.017)

Endogenous T*, IV(1) HAS

bound

.039

(.018)

1

 Standard error conditional on θ

ˆ (see Section II).

2

 Lower bound for γafqt corresponding to upper bound for β.
34

Table 3

Minimum Length 95% Confidence Intervals for Training Coefficient, Selected

Specifications

Specification Interval

OLS (2), Percentile Bounds [.009, .063]

OLS (1), Percentile Bounds [.008, .063]

Endogenous T*, IV (2),

Percentile Bounds

[.014, .219]

Endogenous T*, IV (1),

Percentile Bounds

[.002, .244]

OLS (2), HAS Bounds [.009, .055]

OLS (1), HAS Bounds [.008, .056]

Endogenous T*, IV (2),

 HAS Bounds

[.009, .217]

Endogenous T*, IV (1),

HAS Bounds

[-.004, .240]
35

Appendix

In the text, we showed that ) ( ) ˆ ( ˆ 0 0 *

q

p A θ q → A θ . In this appendix, we show that

convergence occurs at rate nq . As the analysis below makes clear, the argument is

complicated by a small sample effect arising from the fact that θ q

ˆ depends on the

realized values of Ti over the entire sample. After demonstrating that

) ( )) ˆ ( ˆ ( 0 0 * A q A q nq θ − θ converges to a normal distribution, we go on to establish the useful

result that )) ˆ )(1 ( ˆ )) (0, ( ˆ ) ( ˆ ( ˆ ( 0 0 0 0

q q

d

nq A θ q − A θ q → N A θ − A θ .

In the following, let h(X) denote the density of X and let pT(X) denote the

conditional probability Pr(T=1|X). Letting ) J (c δ ≡ Pr(G(X; δ)≤ c) denote the cumulative

distribution function of G(X;δ), we assume that for all δ in some neighborhood of δ*,

(i) ) J (c δ is continuous and differentiable in δ and (ii) ( ) ( ) ’ J c j c δ ≡ δ is continuous and

positive everywhere that 0 < Jδ(c) < 1. Let κ (δ ) q ≡ −1

δ J (q) denote the q-quantile for

J (c) δ and let ( ) )), ˆ ˆ ( ) min(c | J c q κ q δ ≡ δ ≥ where δ J

ˆ is the empirical cdf for G(X; δ) for a

given δ.

As a preliminary, we establish the asymptotic normality of κ q ˆ and thus θ

ˆ :

Lemma 1: ( ˆ *) (0, ) n κ κ N Vκˆ

d

q − q → , where Vκˆ =V1 +V2 +V3 ,

δ

κ δ

δ δ

δ

δ

δ

δ

δ

δ δ

δ

δ

κ δ

∂

∂

∂ ∂

∂

∂

∂

∂

∂

∂ ∂

∂

∂

∂ = − − ( *) ) ) ’

( *) ) ( ( *) ( *) ) ( ’( *) ’( ( ( *) 1

’ 2

1

2

1

q L q E L L E L V E , 2

*

2 ( *)

(1 )

q j

q q V

δ κ

− = ,
36

and Cov V3 = 2 ( ) ( *)

( * *) ) , ’ ( ( *)

*

’

1

2 ’

q

q q

j

L L X

E

κ

ψ δ κ

δ δ δ δ

κ δ

δ

−

∂

∂

∂ ∂

∂

∂

∂ − , and where L is the loglikelihood and 







− − ≤

> = (1 ) 00

( )

q if Y

q if Y

ψ Y .

) (0, ) ˆ ( ˆ , ˆ ˆ

, ˆ ˆ ˆ *















 − →

δ κ κ

δ δ κ θ θ C V

V C

n N d

q q , where

1

’ 2

1

2

ˆ ) ’

( *) ) ( ( *) ( *) ) ( ’( *) ( − −

∂ ∂

∂

∂

∂

∂

∂

∂ ∂

∂ = δ δδ

δ

δ

δ

δ

δ δ

δ

δ

L E L L E L V E and

) ( *)

( * *) ), ( *) ) ( ’

( *) ( ( ( *)

*

’

1

2

ˆ

'

, ˆ ˆ

q

q q

j

L L X

C V Cov E

κ

ψ δ κ

δ

δ

δ δ

δ

δ

κ δ

δ

δ κ δ

−

∂

∂

∂ ∂

∂

+

∂

∂ = − .

Proof: We can decompose ( ˆ *) q q n κ −κ as:

 (A1) ( ˆ *) q q n κ −κ = ) ˆ ( *)) ( ˆ ( *) *). ˆ ( ˆ ( n κ q δ −κ q δ + n κ q δ −κ q

Koenker and Bassett (1982) show that under our continuity assumptions,

 (A2) (1) ( ( ; *) ( *))

( ( *))

1 ( ˆ ( *) ( *))

*

p

q

q

q q o

n

G X

n

j

n + − − = ∑ψ δ κ δ

κ δ

κ δ κ δ

δ

.

To determine the distribution of the first term of (A1), we now show that

) ˆ ( *)) ˆ n(κˆ q (δ −κ q δ has the same limiting distribution as ) ( *)) ˆ n(κ q (δ −κ q δ . From

(A2) and its analogue for δ

ˆ , we have:

(1). ( ( ; *) ( *))

( ( *))

1

)) ˆ ) ( ˆ ( ( ;

)) ˆ ( (

1 )) ( ˆ ( *) ( *)) ˆ ) ( ˆ ( 3) ( ˆ (

*

ˆ

p

q

q

q

q

q q q q

o

n

G X

n

j

n

G X

n

j

A n n

+ − −

− − − − =

∑

∑

ψ δ κ δ

κ δ

ψ δ κ δ

κ δ

κ δ κ δ κ δ κ δ

δ

δ
37

Since )) ( ( *)) ˆ ( ( ˆ κ δ δ * κ δ δ q

p

q j → j , it follows from(A3) that

)) ( ˆ ( *) ( *)) ˆ ) ( ˆ n(κˆ q (δ − κ q δ − n κ q δ −κ q δ has the same limiting distribution as

) )) ( ( ; *) *) ˆ ) ( ˆ ( ( ;( ( *)

1

* n

G X

n

G X

n

j

q q

q

∑ ∑ − − ψ δ − κ δ ψ δ κ

δ κ .

Let c

q q s I ) (( )) ˆ ) ( ˆ ( 0 *1 δ ≡ θ ∩ θ , ) ( *) ˆ ) ( ˆ ( 02 δ θ q θ c

q s ≡ ∩ I , )) ˆ ) Pr( ( ˆ ( p1 δ ≡ X ∈ s1 δ , and

)) ˆ ) Pr( ( ˆ ( p2 δ ≡ X ∈ s2 δ . Note that

,

)ˆ (

)) ˆ )) #( ( ˆ #( ( ) )) ( ( ; *) *) ˆ ) ( ˆ ( ( ;

( 2 1

n

b

n

n

s s

n

G X

n

G X

n

i

i

q q

∑

∑ ∑

=

− = − − −

δ

ψ δ κ δ ψ δ κ δ δ

where 















− ∈

∈

=

 0 otherwise.

), ˆ 1 if (

), ˆ 1 if (

)ˆ ( 2

1

δ

δ

δ X s

X s

b i

i

i

By the central limit theorem, ( ))

)ˆ (

( i

i

i

E b

n

b

n −

∑ δ

 converges in distribution to

(0, ( )) i N Var b . Note that )ˆ ) ( ˆ ( ) ( E bi = p1 δ − p2 δ and

)) . ˆ ) ( ˆ ) ( ( ˆ ) ( ˆ ( ) ( 2 Var bi = p1 δ + p2 δ − p1 δ − p2 δ

Since ) 0 ˆ (1 →p p δ , ) 0 ˆ ( 2 →p p δ , we can conclude that 0

)ˆ (

→

∑

i p

i

n

b

n

δ

. It

follows that )) ( ˆ ( *) ( *)) 0 ˆ ) ( ˆ ( ˆ ( − − − →p

q q q q n κ δ κ δ n κ δ κ δ , which implies that

) ˆ ( *)) ˆ n(κˆ q (δ −κ q δ has the same limiting distribution as ) ( *)) ˆ n(κ q (δ −κ q δ .
38

A standard delta-method argument establishes that ) *) ˆ ( ( q q n κ δ − κ has the same

limiting distribution as *) ˆ ( ( *) δ δ

δ

κ δ − ∂∂

n q , which in turn, as shown by White (1982),

has the same limiting distribution as ) '

( *) ) ) ( '( *) ( ( ( *) 1

2

n

L

n

L E q

∑ ∂

∂

∂ ∂

∂ − ∂

∂ −

δ

δ

δ δ

δ

δ

κ δ

, where L

is the log-likelihood. A multivariate central limit theorem applies to





 





−

−

ˆ ( *) *

) ˆ ( *) ˆ ˆ (

q q

q q n

κ δ κ

κ δ κ δ , implying that the sum ( ˆ *) q q n κ −κ =

) ˆ ( *)) ( ˆ ( *) *) ˆ ( ˆ ( q q q q n κ δ − κ δ + n κ δ −κ is asymptotically normal with the specified

variance. 

Similarly, *) ˆ n(θ −θ is asymptotically normal, with variance 



 



 =

δ κ κ

δ δ κ

θ ˆ , ˆ ˆ

, ˆ ˆ ˆ

ˆ C V

V C

V .

Q.E.D.

Lemma 2 below is key to the argument demonstrating the convergence of

) ( )) ˆ ( ˆ ( 0 0 * A q A q nq θ − θ :

Lemma 2: Let 

#( ( )) ( ) ˆ 0 *

( ) 0 * 0 *

q

i I

i

q I

T

A q

θ

θ θ

∑∈

≡ . Then ( )) (0, ’ ) ˆ ) ˆ ( ˆ ( ˆ

0 0 * − → ∆ ∆

q

n A A N V d

q q θ θ θ ,

where A θ q θ q ∆ ≡ ∂ ( ) / ∂ 0 * denotes the gradient vector of 0 A at * θ q .

Proof: Let c

q q q S I ) I ( ) ˆ ) ( ˆ ( 0 0 *1 θ ≡ θ ∩ θ , cq q q S I I ) ˆ ) ( ) ( ˆ ( 0 * 02 θ ≡ θ ∩ θ ,

)) ˆ ) Pr( ( ˆ ( 1 q 1 q P θ ≡ X ∈S θ , and )) ˆ ) Pr( ( ˆ ( 2 q 2 q P θ = X ∈S θ . One can write

 (A4) )

)) #( ( )) ˆ #( (

( )) ( ˆ ) ˆ ( ˆ ( 0 *

( )

0

) ˆ ( 0 0 * 0 0 *

q

i I

i

q

i I

i

q q I

T

I

T

n A A n q q

θ θ

θ θ θ θ

∑ ∑ ∈ ∈

− = −
39

 )

)) ˆ #( (

)) ˆ #( ( )) #( (

#( ( )) #( ( ))

( 0

0 * 0

0 *

) ˆ (

0 *

) ( ) ˆ ( 0 0 * 0

q

q q

q

i I

i

q

i I

i

i I

i

I

I I

I

T

I

T T

n q q q

θ

θ θ

θ θ

θ θ θ − +

−

=

∑ ∑ ∑ ∈ ∈ ∈

 = )

)) ˆ #( (

)) ˆ )) #( ( ˆ # ( (

#( ( )) # ( ( ))

( 0

2 1

0 *

) ˆ (

0 *

0

1 2

q

q q

q

i I

i

q

i S

i

i S

i

I

S S

I

T

I

T T

n q q

θ

θ θ

θ θ

θ − +

∑ − ∑ ∑ ∈ ∈ ∈

 )

) ˆ (

)) ˆ #( (

) ˆ (

(

#( ( )) 0

) ˆ (

0 *

0

n

B

I

T

n

T B

I

n

n i

i q

q

i I

i

i

i i q

q

q

∑ ∈

∑ ∑ = −

θ

θ

θ

θ

θ

where















− ∈

∈

=

otherwise

if X S

if X S

B i q

i q

i q

0

) ˆ 1 (

) ˆ 1 (

) ˆ ( 2

1

θ

θ

θ .

In determining the asymptotic distribution of

)

) ˆ (

)) ˆ # ( (

) ˆ (

(

#( ( )) 0

) ˆ (

0 *

0

n

B

I

T

n

T B

I

n

n i

i q

q

i I

i

i

i i q

q

q

∑ ∈

∑ ∑ −

θ

θ

θ

θ

θ

, one must take into account the small

sample effect arising from the fact that θ q

ˆ depends on the realized values of Ti over the

entire sample. Thus, let )) ˆ ) Pr( 1| ( ˆ ( T1 q 1 q p θ ≡ T = X ∈S θ , and let

) ˆ ) ( ) ( ) / ( ˆ ), ˆ ) Pr( 1| ( ˆ ( ~ 1

) ˆ (

1

1

1 q

X S

pT q T X S q q pT X h X dX P

q

θ θ θ θ

θ

∫

∈

≡ = ∈ ≡ denote the probability that T

= 1 when ) ˆ ( 1 q X ∈S θ and θ q

ˆ is taken to be exogenous. For convenience, let

) ˆ ( ~ ) ˆ ) ( ˆ ( 1 q T1 q T1 q a θ ≡ p θ − p θ . Similarly, let )) ˆ ) Pr( 1| ( ˆ ( T2 q 2 q p θ ≡ T = X ∈S θ and write

) ≡ ˆ ( T2 q p θ ) ˆ ) ( ˆ ( ~

T2 q 2 q p θ + a θ ,
40

where ) ˆ ) ( ) ( ) / ( ˆ ), ˆ ) Pr( 1| ( ˆ ( ~ 2

) ˆ (

2

2

2 q

X S

pT q T X S q q pT X h X dX P

q

θ θ θ θ

θ

∫

∈

≡ = ∈ ≡ and ) ˆ ( 2 q a θ is a

small sample effect. Since ) ˆ ( 1 q a θ and ) ˆ ( 2 q a θ approach 0 as θ q

ˆ approaches * θ q and

ˆ *

q

p θ q →θ , ) 0 ˆ ( 1 →p

q a θ and ) 0 ˆ ( 2 →pq a θ .

Note that

))) ˆ ) ( ˆ ) ( ˆ ) ( ˆ ( (

) ˆ (

(

#( ( ))

) ˆ (

# ( ( )) ( 5) 0 * 0 * T1 q 1 q T 2 q 2 q

i

i i q

q

i

i i q

q

p P p P

n

T B

n

I

n

n

T B

I

n A n θ θ θ θ

θ

θ

θ

θ = − −

∑ ∑

 

)). ˆ ) ( ˆ ( ~ ) ˆ ) ( ˆ ( ~(

#( ( ))

)) ˆ ) ( ˆ ) ( ˆ ) ( ˆ ( (

#( ( ))

0 * 1 2 2

1 1 2 2 0 *

T1 q q T q q

q

q q q q

q

n p P p P

I

n

n a P a P

I

n

θ θ θ θ

θ

θ θ θ θ

θ

+ −

+ −

Substituting (A5) into (A4) and adding and subtracting

)) ˆ ) ( ˆ ( (

)) ˆ #( ( )) #( ( 2 1 0

) ˆ (

0 *

0

q q

q

i I

i

q

P P

I

T

I

n

n qθ θ

θ θ

θ

−

∑∈

, one obtains

)))) ˆ ) ( ˆ ) ( ˆ ) ( ˆ ( (

) ˆ (

(

#( ( )) ( )) ˆ ) ˆ ( ˆ ( 6) ( 0 * 1 1 2 2

0 0 *

T q q T q q

i

i i q

q

q q p P p P

n

T B

n

I

n A n A A θ θ θ θ

θ

θ

θ − θ = − −

∑

 

)). ˆ ) ( ˆ ( ~ ) ˆ ) ( ˆ ( ~(

#( ( ))

)) ˆ ) ( ˆ ) ( ˆ ) ( ˆ ( (

#( ( ))

0 * 1 2 2

1 1 2 2 0 *

T1 q q T q q

q

q q q q

q

n p P p P

I

n

n a P a P

I

n

θ θ θ θ

θ

θ θ θ θ

θ

+ −

+ −

 )) ˆ ) ( ˆ ))) ( ( ˆ ) ( ˆ ( (

) ˆ (

(

)) ˆ #( ( )) #( ( 1 2 1 2 0

) ˆ (

0 *

0

q q q q

i

i q

q

i I

i

q

P P P P

n

B

I

T

I

n

n qθ θ θ θ

θ

θ θ

θ

− − − + − ∈

∑ ∑ .

Letting ∫ ≤

≡ ≤ ≡

X G X q

q q G X h X dX

δ κ

δ κ

) ˆ ˆ : ( ;

) ˆ ) ( ) ˆ ˆ Pr( ( ; , one can write
41

 (A7) 

q

p X h X dX

q

p X h X dX

A A q X G X q

T

X G X

T

q q

∫ ∫ ≤ ≤

− = − * ) ˆ : ( ; *) ˆ : ( ; 0 0 *

( ) ( )

ˆ

( ) ( )

) ( ) ˆ (

δ κ δ κ

θ θ

 = 

qq

q p X h X dX p X h X dX

q q X S

T

X S

T

ˆ

( ( ) ( ) ( ) ( ) )

) ˆ ) ( ˆ ( 1 2

∫ ∫ ∈ ∈

−

θ θ

 qq

q q p X h X dX

X G X q

T

ˆ

( ˆ) ( ) ( ) * : ( ; )*

∫ ≤

−

+

δ κ

 = 

q

pT q P q pT q P q q q A q

ˆ

) ( ˆ) ( ) ˆ ) ( ˆ ( ~ ) ˆ ) ( ˆ ( ~ 0 *

1 2 2 1 θ θ − θ θ + − θ ,

which implies that

)) ( ) ˆ ) ( ˆ ) ( ) ( ( ˆ ) ˆ( ( ˆ ) ( ˆ ( ~ ) ˆ ) ( ˆ ( ~ ( 8) 0 *

2 1

0 0 *

T1 q 1 q T 2 q 2 q A q A q P q P q A q A p θ P θ − p θ P θ = q θ − θ + θ − θ θ

.

Substituting (A8) into (A6) and rearranging yields:

 (A9) ) ˆ ) ( ))) ( , ˆ ( ˆ( (

# ( ( )) ( )) ˆ ) ˆ ( ˆ ( 0 0 *

0 *

0 0 *

q q q

q

q q n q A A X n

I

n

n A A θ θ θ

θ

θ − θ = − +

where

 (A10) ) ˆ ) ( ˆ ) ( ˆ ) ( ˆ ( (

) ˆ (

(

#( ( )) ) ˆ ( , 0 * T1 q 1 q T 2 q 2 q

i

i i q

q

q p P p P

n

T B

n

I

n X n θ θ θ θ

θ

θ

θ = − −

∑

))

 + )) ˆ ) ( ˆ ) ( ˆ ) ( ˆ ( (

#( ( )) 0 * 1 q 1 q 2 q 2 q

q

n a P a P

I

n θ θ θ θ

θ −

 ))) ˆ ) ( ˆ ( (

) ˆ (

(

)) ˆ # ( ( )) #( ( 1 2 0

) ˆ (

0 *

0

q q

i

i q

q

i I

i

q

P P

n

B

I

T

I

n

n qθ θ

θ

θ θ

θ

− − − ∈

∑ ∑
42

 )). ˆ ) ( ˆ ( ))( (

)) ˆ #( (

(

#( ( )) 1 2

0 *

0

) ˆ (

0 *

0

q q q q

q

i I

i

q

A P P

I

T

I

n

n qθ θ θ

θ θ

θ

− − −

∑∈

 As n becomes large, )ˆ X (n,θ converges in probability to zero. To see this, note

that by the central limit theorem, )) ˆ ) ( ˆ ) ( ˆ ) ( ˆ ( (

) ˆ (

( T1 q 1 q T 2 q 2 q

q

i

i i

p P p P

n

T B

n θ θ θ θ

θ

− −

∑

)

converges in distribution to ))) ˆ ), ( ˆ (0, ( ( N χ P1 θ q P2 θ q , where

)) . ˆ ) ( ˆ ) ( ˆ ) ( ˆ )( 1 ( ˆ )) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ )(1 ( ˆ )) ( ˆ ), ( ˆ ( ( 2

2 1 1 2 2

2

1 q 2 q 1 q T1 q 1 q T 2 q 2 q q T q q T q P q χ P θ P θ = P θ − p θ P θ + p θ P θ + P θ − − p θ P θ + p θ θ

Since ) 0 ˆ ( 1 →p P θ q , ) 0 ˆ ( 2 →p P θ q , and q

n

I q →p

#( ( )) 0 * θ ,

))) 0 ˆ ) ( ˆ ) ( ˆ ) ( ˆ ( ( (

#( ( )) 0 * − 1 1 − 2 2 →

∑

p

T q q T q q

i

i i

q

p P p P

n

T B

n

I

n θ θ θ θ

θ . Similarly, the third term

on the right hand side of (A7), ))) ˆ ) ( ˆ ( (

) ˆ (

(

)) ˆ #( ( )) #( ( 2 1 0

) ˆ (

0 *

0

q q

i

i q

q

i I

i

q

P P

n

B

I

T

I

n

n qθ θ

θ

θ θ

θ

− − ∈

∑ ∑ ,

converges in probability to zero. Also, note that ) ˆ (

#( ( )) 0 * 1 q

q

nP

I

n θ

θ

 and

) ˆ (

#( ( )) 0 * 2 q

q

nP

I

n θ

θ converge asymptotically to stable (normal) distributions. Since

) ˆ ( 1 q a θ and ) ˆ ( 2 q a θ both converge in probability to 0,

)) 0 ˆ ) ( ˆ ) ( ˆ ) ( ˆ ( (

)) ˆ #( ( )) #( ( 1 1 2 2 0

) ˆ (

0 *

0

− →

∑∈ p

q q q q

q

i I

i

q

n a P a P

I

T

I

n q

θ θ θ θ

θ θ

θ

.
43

Finally, consider the fourth term on the right hand side of (A10). Since

( ) ( )

*

2

* P1 θ q = P θ q = 0 and ) ˆ ( *

q q n θ −θ is asymptotically normal,

)) (0,( ( )/ ) ( ( ) / )) ( ( )/ ) ( ( ) / )). ˆ ) ( ˆ ( ( *

2

*

ˆ 1

*

2

*

1 θ 2 θ 1 θ θ θ θ θ θ θ θ θ − → ∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ q q q q

d

n P q P q N P P V P P q

 Since ( )

)) ˆ #( (

0 *

0

) ˆ ( 0

q

p

q

i I

i

A

I

T

q

θ

θ

θ

→

∑∈

 and q

n

I q →p

#( ( )) 0 * θ

, it follows that

))) 0 ˆ ) ( ˆ ( ))( (

)) ˆ # ( (

(

#( ( )) 1 2

0 *

0

) ˆ (

0 *

0

− − →

∑∈ p

q q q q

q

i I

i

q

A P P

I

T

n

I

n q

θ θ θ

θ θ

θ

.

We can conclude that

 (A11) ) ( )) ˆ (( (

# ( ( )) ( ))) ˆ ) ˆ ( ˆ ( 0 0 *

0 *

0 0 *

q q

q

d

q q q n A A

I

n

n A A θ θ

θ

θ − θ → − , which

implies that

( )) (0, ’ ) ˆ ) ˆ ( ˆ ( ˆ

0 0 * − → ∆ ∆

q

n A A N V d

q q θ θ θ Q.E.D.

Note that in order to estimate the variance of ( ) ˆ ) ˆ ( ˆ 0 0 * A θ q − A θ q from the formula in

the lemma, one would need to estimate the derivative vectors ∆, , ( *)

δ

κ δ

∂

∂ q and the

density ( *) * q jδ κ .

Using the above lemma, it is easy to demonstrate that ) ( )) ˆ ( ˆ ( 0 0 * A q A q nq θ − θ

converges to a normal distribution with mean zero.

Theorem 1: ) ( )) ˆ ( ˆ ( 0 0 * A q A q nq θ − θ converges asymptotically to a normal

distribution with zero mean.

Proof:

Decompose ) ( )) ˆ ( ˆ ( 0 0 * A q A q nq θ − θ as follows:
44

 (A12) ( )) ˆ ) ˆ ( ˆ ( ) ( )) ( ˆ ) ( )) ( ˆ ( ˆ ( 0 0 * 0 * 0 * 0 0 *

q q q q A q A q nq A θ − A θ = nq A θ − A θ + nq θ − θ .

Note that ( ) ( )) ˆ ( 0 * 0 * A q A q nq θ − θ ( ( )) 0 * ( ) 0 *

q

i I

i

LD

A

nq

T

nq q

θ θ= −

∑∈

. From the preceding

lemmas, we know that ( )) ˆ ) ˆ ( ˆ ( 0 0 * A q A q nq θ − θ has the same asymptotic distribution as a

linear combination of ) ’ ( n

L

nq ∑∂

∂

δ

 and 

n

G X

nq ∑ψ ( ( ;δ *) − κ q (δ*)) , both of which

are mean zero. By the multivariate central limit theorem, the components of (A12) are

multivariate normal. Thus, the sum ( )) ˆ )ˆ ( ˆ ( ) ( )) ( ˆ ( 0 * 0 * 0 0 *

q q A A q nq A θ − A θ + nq θ − θ is

asymptotically normal with mean zero. Q.E.D.

Corollary 1: )) (0, ( )(1 ( )) ˆ ) ( ˆ ( ˆ ( 0 0 0 * 0 *

q q

d

nq A θ q − A θ q → N A θ − A θ .

Proof:

Instead of (A12), one can decompose )) ˆ ) ( ˆ ( ˆ ( 0 0 A q A q nq θ − θ as

 (A13) ) ( )) ˆ )) ( ( ˆ ) ( ˆ ( ˆ ) ( )) ( ˆ ( ˆ ( 0 0 * 0 0 0 0 *

q q q q A q A q nq A θ − A θ = nq A θ − A θ + nq θ − θ

Substituting (A9) and (A12) into (A13) and rearranging terms yields

 (A14) ) ˆ ( ) ( )) ( , ˆ )) ( ˆ ) ( ˆ ( ˆ ( 0 0 0 * 0 *

q q q q q nq A θ − A θ = nq A θ − A θ + qX n θ

We have already shown than ) 0 ˆ ( , →p X n θ and

( ) ( )) (0, ( )(1 ( ))) ˆ ( 0 * 0 * 0 * 0 *

q q

d

nq A θ q − A θ q → N A θ − A θ . Q.E.D.

Corollary 1 and the fact that ) ( ) ˆ ( 0 0 *

q

p A θ q → A θ immediately give us

Corollary 2: )) ˆ )(1 ( ˆ )) (0, ( ˆ ) ( ˆ ( ˆ ( 0 0 0 0

q q

d

nq A θ q − A θ q → N A θ − A θ .