Imbens, Lecture Notes 5, ARE213 Spring ’06
1
ARE213
Econometrics
Spring 2006 UC Berkeley Department of Agricultural and Resource Economics
Ordinary Least Squares V:
Measurement Error (W 4.4)
Here we look at the eﬀect of measurement error on least squares estimates. In addition
to classical measurement error we look at measurement error that is uncorrelated with the
reported value. In addition we look at measurement error in the regressors as well as in the
outcome variable. In all cases the key equation is the omitted variable bias formula.
Let
X
*
denote the true value of a variable of interest, and
X
the recorded value. The
measurement error is the diﬀerence between the recorded and true value:
ε
≡
X

X
*
.
(1)
The standard Classical Measurement Error (CME) model, assumes that the measurement
error is independent of the true value (
ε
⊥
X
*
). Assuming that the measurement error
has mean zero, this implies
E
[
ε

X
*
] = 0. This model is typically defended by reference
to physical measurement models where often passive recording of measurements based on
imprecise measuring instruments takes place.
The alternative model, which we refer to as the Optimal Prediction Error (OPE) model,
is based on the assumption that the measurement error is independent of the reported value
(
ε
⊥
X
). This implies
E
[
ε

X
] = 0. An argument in support of this model views the
agent reporting the data as fully aware of the lack of precision of the measuring instrument.
Suppose the agent is asked to provide the value of some variable. The agent has no way of
ascertaining the true value
X
*
of this variable, but has available a Fawed or noisy measure,
˜
X
=
X
*
+
η
X
, with the measurement error
η
X
independent of the true value of the variable,
exactly as in the CME model. However, suppose that the agent is aware of the lack of
precision of the measurement, and corrects for this by reporting the best estimate of the
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2
underlying true value
X
*
based on this measurement
˜
X
. To operationalize this we interpret
“best” in terms of a quadratic loss function, which implies the agent would report the
expected value of the true value given the agent’s information set. An alternative would be
to assume absolute value loss, in which case the agent would report the median of
X
*
given
the information set. For most of the illustrative calculations below the mean and median
will give the same answers because we assume normality. Under this interpretation the error
ε
=
X

X
*
should have mean zero given the information set of the agent. Since the reported
value is clearly in the information set, this implies that the error has mean zero given the
reported value.
A crucial ingredient in the OPE model is the information set. It may be that the respon
dent only has a single unbiased measurement of the underlying true variable. Alternatively,
other variables, which themselves may enter the econometric model of interest, may be used
to produce this estimate. (For example, Ashenfelter and Krueger (1994) survey twins and
ask each sibling to report both their own education and their sibling’s education. To the ex
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 Spring '06
 IMBENS
 Regression Analysis, Ope, measurement error, regressor

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