766 Microeconometrics: Methods and Developments
Microeconometrics applications rarely use multiple imputation methods, in part
due to concern that missingness may be for nonignorable reasons, such as endoge-
nous stratification discussed in section 14.7.1. Wooldridge (2007) considers use of
inverse-probability weighting estimators when data are missing and provides a link
with the framework of Rubin (1976).
14.7.3 Measurement error
Standard results for measurement error consider the linear regression model with
classical measurement error in regressors. OLS coefficients are then inconsistent
and understate the magnitude of the true coefficient. More recent work has con-
sidered nonlinear regression models and, in some cases, nonclassical measurement
error.
Supposey = βx∗+u, with erroruuncorrelated withx∗, but we observex
rather thanx∗and regressyonx. Then, from Angrist and Krueger (1999), the
OLS estimator̂β=[
∑
ix2
i]− 1 ∑
ixiyi=[∑
ix2
i]− 1 ∑
ixi(βx∗
i+ui)is in general
inconsistent as:
plim̂β=[V[x]]−^1 Cov[x,x∗]β=λβ, (14.52)
whereλ=Cov[x,x∗]/V[x]is the reliability ratio ofxas a measure ofx∗, and we have
assumed that plimN−^1
∑
ixiui=0. This assumption thatxis uncorrelated withu
requires the additional assumption thatuis uncorrelated with the measurement
errorv=x−x∗, in addition to the usual assumption that the model erroruis
uncorrelated withx∗.
The size of the inconsistency depends on the size of the reliability ratio, which
has been measured in various survey validation studies. Angrist and Kruger (1999,
p. 1346) present a summary table with reliability ratios for log annual earnings,
annual hours and years of schooling ranging from 0.71 to 0.94. Bound, Brown and
Mathiowetz (2001, pp. 3749–830) summarize many validation studies for labor-
related data that also indicate that measurement error is large enough to lead to
appreciable bias in OLS coefficients.
Result (14.52) makes few assumptions beyond independence of measurement
error and model error. Textbook treatments of measurement error emphasize the
classical measurement error model, a more restrictive model that assumes:
y=βx∗+u,u∼iid[
0,σu^2]x=x∗+v,v∼iid[
0,σv^2]
and x∗∼iid[
0,σx^2 ∗]. (14.53)
Then plim̂β=λβ, whereλ=σx^2 ∗/(σx^2 ∗+σv^2 )= 1 /( 1 +s)and wheres=σv^2 /σx^2 ∗
is the noise-to-signal ratio. Sinces≥0,̂βis downward biased asymptotically
towards zero, a bias called attenuation bias. The attenuation bias is reduced
if additional (correctly measured) regressors are included, and is increased if
panel data are used with estimation by differencing methods such as the within-
estimator.