A General Correction for Measurement Error
Zijing Hu
November 13, 2022
Hu (2008) shows that one can recover the true value of the interested variable using three measure-
ments (one of them needs to be binary) under certain assumptions.
Suppose that X, Y , and Z are observed variables (with measurement errors) that are dependent on
X
⋆
. X and Z need to be discretized if they are continuous measurements. Y ∈ {0, 1} is binary.
Assumption 1: X ⊥ Y ⊥ Z | X
⋆
or X ⊥ Y ⊥ Z | X
⋆
, W , where W are covariates, making the
assumption less restrictive. With this assumption, we have
P r(X, Y, Z) =
X
X
⋆
P r(X, Y, Z, X
⋆
)
=
X
X
⋆
P r(X | Y, Z, X
⋆
)P r(Y | Z, X
⋆
)P r(Z | X
⋆
)P r(X
⋆
)
=
X
X
⋆
P r(X | X
⋆
)P r(Y | X
⋆
)P r(Z | X
⋆
)P r(X
⋆
) given the assumption
Thus, we can rewrite above formula in a matrix form
M
XY Z
= M
X|X
⋆
D
Y |X
⋆
M
′
ZX
⋆
,
where
M
XY Z
=
P r(Y, X = x
1
, Z = z
1
) · · · P r(Y, X = x
1
, Z = z
k
)
.
.
.
.
.
.
P r(Y, X = x
k
, Z = z
1
) · · · P r(Y, X = x
k
, Z = z
k
)
,
M
X|X
⋆
=
P r(X = x
1
| X
⋆
= x
⋆
1
) · · · P r(X = x
1
| X
⋆
= x
⋆
k
)
.
.
.
.
.
.
P r(X = x
k
| X
⋆
= x
⋆
1
) · · · P r(X = x
k
| X
⋆
= x
⋆
k
)
,
D
Y |X
⋆
=
P r(Y | X
⋆
= x
⋆
1
)
.
.
.
P r(Y | X
⋆
= x
⋆
k
)
,
M
ZX
⋆
=
P r(Z = z
1
, X
⋆
= x
⋆
1
) · · · P r(Z = z
1
, X
⋆
= x
⋆
k
)
.
.
.
.
.
.
P r(Z = z
k
, X
⋆
= x
⋆
1
) · · · P r(Z = z
k
, X
⋆
= x
⋆
k
)
We also have
1
P r(X, Z) =
X
X
⋆
P r(X, Z, X
⋆
)
=
X
X
⋆
P r(X | Z, X
⋆
)P r(Z | X
⋆
)P r(X
⋆
)
=
X
X
⋆
P r(X | X
⋆
)P r(Z | X
⋆
)P r(X
⋆
) given the assumption
Therefore, the matrix form of the above formula is
M
XZ
= M
X|X
⋆
M
′
ZX
⋆
.
Assumption 2: M
ZX
⋆
is a full rank matrix (invertible), which can be tested. Then we have
M
−1
XZ
= M
′
−1
ZX
⋆
M
−1
X|X
⋆
M
XY Z
M
−1
XZ
= M
X|X
⋆
D
Y |X
⋆
M
′
ZX
⋆
M
′
−1
ZX
⋆
M
−1
X|X
⋆
= M
X|X
⋆
D
Y |X
⋆
M
−1
X|X
⋆
= BΛB
−1
,
where Λ and B can be obtained using eigendecomposition.
Computational tricks
• In practice, Y need to be fixed, i.e., Y = y.
• We use control variable W to ensure that Assumption 1 holds.
• One need make reasonable assumptions, e.g., P r(y | X
⋆
= 1, w) > P r(y | X
⋆
= 0, w), to re-rank
diagonal elements in Λ and match them with true values.
• Elements in the same column in B need to be normalized to sum 1 as the columns of matrix B
are probabilistic distributions. An alternative way to constrain parameters is to use extremum
estimator:
ˆ
θ = arg min
θ
c
M
XyZ|w
c
M
−1
XZ|w
− M
X|X
⋆
,w
D
y |X
⋆
,w
M
−1
X|X
⋆
,w
s.t. (i) θ
i
∈ [0, 1] ∀i
(ii) P r(y | X
⋆
, w) is a monotonic function of X
⋆
(iii) P r(X = x | X
⋆
= i, w) = P r(X = x | X
⋆
= j, w) ∀ i = j
where
c
M
XyZ|w
j,k
=
n
X
i=1
1
(X
i
= x
j
, Y
i
= y, Z
i
= z
k
)
1
(W
i
= w) /
n
X
i=1
1
(W
i
= w)
c
M
XZ|w
j,k
=
n
X
i=1
1
(X
i
= x
j
, Z
i
= z
k
)
1
(W
i
= w) /
n
X
i=1
1
(W
i
= w)
References
Hu, Yingyao (2008). “Identification and estimation of nonlinear models with misclassification error
using instrumental variables: A general solution”. In: Journal of Econometrics 144(1), pp. 27–61.
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