For the setting of multiple regression with measurement error in a single regressor, we present some very simple formulas to assess the result that one may expect when correcting for measurement error. It is shown where the corrected estimated regression coefficients and the error variance may lie, and how the t-value behaves.

Meijer, E.

Wansbeek, T.

English

Research Papers in Economics

Measurement error in a single regressor
Erik Meijer and Tom Wansbeek
June 22, 2000
SOM-theme F Interactions between consumers and ﬁrms
Abstract
For the setting of multiple regression with measurement error in a single regressor, we present
some very simple formulas to assess the result that one may expect when correcting for
measurement error. It is shown where the corrected estimated regression coefﬁcients and the
error variance may lie, and how the t-value behaves.
Keywords: attenuation, t-statistic, CALS
JEL classiﬁcation: C21; C52
We thank Paul Bekker, Theo Dijkstra, and Ton Steerneman for their valuable comments.
 Department of Econometrics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The
Netherlands Tel.: +31 50 3633793; fax: +31 50 3633720; e-mail: e.meijer@eco.rug.nl.
11 Introduction
In the applied econometrics literature of the cross-sectional type, one often meets the
case of a regression equation being estimated where the regressor set includes a number
of control variables to account for population heterogeneity in addition to one regressor
focal to the topic under study. This variable is often theory-based and is not directly
observable, for conceptual or practical reasons. Examples are, among many others,
consumption models relating consumption to permanent income, labor supply models
relating hours to wage per hour, models assessing the policy impact of central bank
independence, and investment models relating investment to Tobin’s q. The case of
multiple regression with measurement error in a single but interesting regressor may
hence be considered almost generic.
It is generally appreciated that in such cases OLS gives an inconsistent result. Most
textbooks discuss this and come up with solutions. For example, when the measurement
error variance is known, the OLS results can be adapted to construct a consistent
estimator. However, the approach most frequently mentioned is the use of instrumental
variables. Instruments can sometimes be found by further scrutinizing the available data
set but they can also be constructed from the data already employed for estimating the
model. For example, when the variables are non-normal, the model y D x C u, with
x subject to measurement error, can be consistently estimated with x  y as instrument,
as was shown by Lewbel (1997), who elaborated on this simple idea to suggest a wide
set of IV’s.
Be it as it is, before entering on such a course it may be worthwhile to know
what to expect when applying methods that correct for measurement error in a single
regressor in a multiple-regression setting; see e.g. Wansbeek and Meijer (in press)
for an overview. To that end we group, in this note, a number of partly new results
that are very simple to apply and that are intended to provide guidance to the applied
researcher. These results build on the output of OLS estimation of the model and show
how the results change when different values of the measurement error variance ()a r e
considered.
Throughout, we will assume that the measurement error is stochastically
independent from the value of the underlying explanatory variable, from the other
explanatory variables, and from the equation error. This is known as the standard
errors-in-variables situation. Krasker and Pratt (1986, 1987) showed that if this is not
the case, then even in the limit we can frequently not be sure of the signs of regression
coefﬁcients.
2The set-up is as follows. After summarizing, in section 2, the basic results for
regression with measurement error in multiple regression, we narrow the results down
for the case of measurement error in a single variable in section 3. It is shown where
the estimate of the regression coefﬁcients and of the residual error variance may lie
depending on . The situation is depicted graphically in section 4. A consistent
estimator of the asymptotic variance is given in section 5.
As stated above, the mismeasured variable is often the core variable in the research,
and that makes an investigation of the correction procedure for its coefﬁcient estimate
especially interesting. As section 6 shows, increasing values of  correspond with
increasing values of the estimate of its asymptotic variance. This increase outpaces
the increase in coefﬁcient estimate and the t-value is monotonously decreasing.
The relationship between the measurement error variance and the t-value is given
explicitly below. This relationship can be of direct use in applied work, where t-values
are assigned a dominant role, since the impact of  on the t-value can be assessed
directly. In particular, it can be seen at what level of noise in a variable it ‘disappears
into insigniﬁcance’.
2 Properties of the measurement error model
The standard linear multiple regression model can be written as
y D 4 C "; (1)
where y is an observable N-vector and " an unobservable N-vector of random variables,
assumed i.i.d. with zero expectation and variance  2
" .T h e g -vector  is ﬁxed but
unknown. The N  g-matrix 4 contains the regressors, assumed independent of ".
If there are errors of measurement in the explanatory variable, 4 is not observable.
Instead, we observe the matrix X D 4CV,w h e r eV( Ng) is a matrix of measurement
errors. Its rows are assumed to be i.i.d. with zero expectation and covariance matrix 
(g  g) and uncorrelated with 4 and ".
Deﬁne the sample second-moment matrices of 4 and X:
KN 
1
N
4
04I AN 
1
N
X
0X:
Note that AN is observable but KN is not.
We can interpret (1) in two ways. It is either a functional or a structural model.
Under the former interpretation, we do not make explicit assumptions regarding the
distribution of 4, but consider its elements as unknown ﬁxed parameters. Under the
3latter interpretation, the elements of 4 are supposed to be random variables. The
assumption plimKN D K, with K a positive deﬁnite g  g-matrix, is meant to cover
both cases. As a consequence, A  plimAN D K C .
Let b  .X0X/−1X0y and s2
"  1
Ny0.IN − X.X0X/−1X0/y be the usual estimators
of  and  2
" , neglecting measurement error. As is well-known, these estimators are
inconsistent:
plimb D A
−1.A − /
plims2
" D  2
" C 0.plimb/:
If  is known, Slutsky’s theorem implies that
O   .AN − /
−1ANb (2)
O  2
"  s2
" − O 0b (3)
are consistent. The asymptotic distribution of these consistent estimators for both the
functional and the structural model is given by
p
N

O  − 
O  2
" −  2
"

d
−! N g C1

0;

γK−1AK−1 C !!0 −2γ!
−2γ! 0 2γ2

; (4)
where γ  2
" C 0 and !  K−1, cf. Kapteyn and Wansbeek (1984).
3 Measurement error in a single regressor
We now consider the case where there is measurement error in a single regressor only.
Without loss of generality we take this to be the ﬁrst regressor. Then
 D e1e
0
1; (5)
with e1 the ﬁrst unit vector, and  the variance of the measurement error in the ﬁrst
regressor. To elaborate this case, the following notation is used:
a  A
−1
N e1
  e0
1A
−1
N e1
 
1
1 − 
:
4Hence ANa D e1 and
 D 1 C  (6a)

2 D 1 C . C1/ (6b)
Ig C ae
0
1 D.AN − e1e
0
1/
−1AN (6c)
a D.AN − e1e0
1/−1e1I (6d)
(6c) and (6d) follow by multiplying both sides by AN −e1e0
1. We assume that the data
relate to the values of  deemed relevant such that >0; this holds anyhow in the limit.
Substitution of (5) in (2) using (6c) gives
O  D .AN − e1e
0
1/
−1ANb D .Ig C ae
0
1/b
D b C .b1/a D b C a; (7)
with b1 the ﬁrst element of b and
  b1: (8)
In particular, the ﬁrst element O 1 of O  can be written as
O 1 D .1 C /b1 Db1: (9)
We assume b1 > 0 without loss of generality, hence O 1  0. So (9) gives the correction
for the downward bias when estimating 1 by OLS if there is measurement error, and
(7) shows, for all elements of  jointly, that this correction is along a line in -space,
from b in the direction given by the ﬁrst column of the inverse of X0X.
We now consider the estimation of  2
" . Combining (3), (5), (8), and (9) gives
O  2
" D s2
" −  O 0e1e0
1b D s2
" −  O 1b1
D s
2
" − b
2
1 D s
2
" −b1 (10)
as a consistent estimator given a value for .
Restricting this estimator to nonnegative values imposes an upper limit on the values
of  that one may wish to consider. Putting (10) to zero and solving gives
max D
1
 C b2
1=s2
"
(11)
since then
max D
1
1 − max
D 1 C

b2
1=s2
"
;
5hence
max D maxmaxb1 D
s2
"
b1
; (12)
which solves (10). This gives
O 1maxD maxb1 D b1 C

b1
s2
"
as the upper bound on the estimator of 1.
4 A graphical illustration
Adapting from Bekker, Kapteyn, and Wansbeek (1984) we can illustrate the above
graphically. Let us ﬁrst consider the case of general   0, and assume that the values
considered for  satisfy <A N. (Again, this holds anyhow in the limit.) Hence
.AN − /
−1 − A
−1
N  0; (13)
with equality holding only if  D 0. An implication of (2) is
O 
0ANb D b
0AN.AN − /
−1ANb:
Then, on using (13),
. O  − b/
0ANb D b
0AN
 
.AN − /
−1 − A
−1
N

ANb  0;
with equality holding only if b D 0. So, whatever  may be, the corrected estimators
lie in -space beyond the plane . O  − b/0ANb D 0, as seen from the origin. This is the
plane through b perpendicular to ANb.
We now consider the implication of O  2
"  0o r
O  0b  s2
"; (14)
cf. (3). Rewriting (2) as O 0 D . O  − b/0AN and substituting this in (14) gives
. O  − b/0ANb  s2
": (15)
So the measurement-error corrected estimator cannot lie beyond the plane . O  −
b/0ANb D s2
". This plane is parallel to the plane . O  − b/0ANb D 0, further away
from the origin. The situation is illustrated in ﬁgure 1. The set of O ’s compatible with
 D e1e0
1 is given by the line segment between the two planes, ranging from b to
b C maxa.
6. O  − b/0ANb D 0
. O  − b/0ANb D s2
"

ANb
b C maxa
b
Figure 1: Admissible values of O .
5 Estimating the asymptotic variance
For  as in (5) with given , we can derive the asymptotic variance of the estimators
(9) and (10) by elaborating (4) for this special case. More interestingly, we elaborate
consistent estimation of this variance for O  based on the consistent estimators for  and
 2
" . We consider the various terms in (4) in turn.
First, using (6a) and (8), a consistent estimator of γ D  2
" C 0 D  2
" C 2
1 is
given by
O  2
" C  O 2
1 D s2
" − b1 C 2b2
1 D s2
" −b1 C b1
Ds
2
" C. − 1/b1 D s
2
" C b1 Ds
2
" C
2:
Next, using (6d) and (9), ! D K−1 D .A−e1e0
1/−1e1e0
1 is consistently estimated
by
.AN −e1e0
1/−1e1e0
1 O  D  O 1a D2b1a D a;
and, using (6a)—(6c), K−1AK−1 D .A − e1e0
1/A.A − e1e0
1/is estimated by
.AN − e1e
0
1/
−1AN.AN − e1e
0
1/
−1 D .Ig C ae
0
1/A
−1
N .Ig C ae
0
1/
DA
−1
N C. C1/aa
0:
7Putting these results together gives for O :
d .asy:var: O / D .s2
" C 2/.A
−1
N C . C1/aa0/ C 22aa0; (16)
which of course reduces to s2
"A
−1
N when there is no measurement error.
6T h e t -value for the ﬁrst regression coefﬁcient
For the estimator of the coefﬁcient of the ﬁrst regressor, taking the upper-left element in
(16) gives
d .asy:var: O 1/ D .s2
" C 2/. C . C1/2/ C 222
D 
2.s
2
" C2
2/:
The t-statistic if there is no measurement error is
t0 D
b1
p
N
p
s2
"
;
so
s
2
" D
b2
1N
t2
0
: (17)
The t-statistic corresponding with a measurement error of size  is, using (9) and (17),
given by
t D
O 1
p
N
q
d .asy:var: O 1/
D
b1
p
N

p

p
s2
" C22
D
t0 q
1C 2
Nh2

; (18)
with h  t0=b1 D . − 1/t0. After some straightforward calculations there appears
to hold
@t
@
D−
2
N

 
t
3
2 <0:
8Sot has derivative 0 in  D 0a n d− 
p
N=2i nD1 =, and decreases monotonically.
Figure 2 illustrates the behavior of t ,f o rD1a n dND100, for various values of t0.
Substitution of (17) in (11) gives
max D
1

 
1 C t2
0=N
;
so the relevant parts of the curves in Figure 2 end to the left of  D 1= D 1, and the
minimal t-values are above 0.
0 1

0
1
2
3
4
5
t
Figure 2: t as a function of  and t0.
Sometimes it is of interest to see which level of  corresponds with a given t-value.
Let this value be t. This corresponds with  D  satisfying
t D
t0 q
1 C 2
Nh2

; (19)
cf. (18). Let
w 
(
N
2
 
t0
t
2
− 1
!)1=2
;
9then (19) implies h D w or
 D
1

w
w C t0
:
In particular, one may be interested to see, by putting t at the conventional level
of 2, what level of noise in a variable makes its estimated coefﬁcient disappear into
insigniﬁcance.
References
Bekker, P. A., Kapteyn, A., and Wansbeek, T. J. (1984). Measurement error and
endogeneity in regression: bounds for ML and 2SLS estimates. In T. K. Dijkstra
(Ed.), Misspeciﬁcation analysis (pp. 85–103). Berlin: Springer.
Kapteyn, A., and Wansbeek, T. J. (1984). Errors in variables: Consistent Adjusted
Least Squares (CALS) estimation. Communications in Statistics — Theory and
Methods, 13, 1811–1837.
Krasker, W. S., and Pratt, J. W. (1986). Bounding the effects of proxy variables on
regression coefﬁcients. Econometrica, 54, 641–655.
Krasker, W. S., and Pratt, J. W. (1987). Bounding the effects of proxy variables on
instrumental-variables coefﬁcients. Journal of Econometrics, 35, 233–252.
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.
Wansbeek, T. J., and Meijer, E. (in press). Measurement error and latent variables. In
B. H. Baltagi (Ed.), Companion in theoretical econometrics. Oxford, UK: Basil
Blackwell.
10

Bounding the effects of proxy variables on instrumental-variables coefﬁcients.

Bounding the effects of proxy variables on regression coefﬁcients.

Constructing instruments for regressions with measurement error when no additional data are available, with an application to patents and

Errors in variables: Consistent Adjusted Least Squares (CALS) estimation.

Measurement error and endogeneity in regression: bounds for ML and 2SLS estimates. In

Measurement error in a single regressor

Measurement error in a single regressor

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