diagonalization used in linear factor analyses, we do not work with random vectors. Instead,
we work with their densities. This approach offers the advantage that the problem remains
linear even when the random vectors are related nonlinearly.
The conditional independence requirement of Assumption 2 is weaker than the full in-
dependence assumption traditionally made in standard linear factor models as it allows for
heteroscedasticity. Assumption 3 requires θ, Z1 , Z2 to be vectors of the same dimensions,
while Assumption 4 can be satisfied even if Z3 is a scalar. The minimum number of mea-
surements needed for identification is therefore 2L + 1, which is exactly the same number of
measurements as in the linear, classical measurement error case.
Versions of Assumption 3 appear in the nonparametric instrumental variable literature
(e.g., Newey and Powell, 2003; Darolles et al., 2002). Intuitively, the requirement that
pZ1 |Z2 (Z1 |Z2 ) forms a bounded complete family requires that the density of Z1 vary suffi-
ciently as Z2 varies (and similarly for pθ∣Z1 (θ∣Z1)).19
Assumption 4 is automatically satisfied, for instance, if θ is univariate and a3 (θ, ε3) is
strictly increasing in θ. However, it holds much more generally. Since a3 (θ, ε3) is nonsepa-
rable, the distribution of Z3 conditional on θ can change with θ, thus making it possible for
Assumption 4 to be satisfied even if a3 (θ, ε3) is not strictly increasing in θ.
Assumption 5 specifies how the observed Z1 is used to determine the scale of the un-
observed θ. The most common choices of the functional Ψ would be the mean, the mode,
the median, or any other well-defined measure of location. This specification allows for non-
classical measurement error. One way to satisfy this assumption is to normalize a1 (θ, ε1) to
be equal to θ + ε1 , where ε1 has zero mean, median or mode. The zero mode assumption
is particularly plausible for surveys where respondents face many possible wrong answers
but only one correct answer. Moving the mode of the answers away from zero would there-
fore require a majority of respondents to misreport in exactly the same way— an unlikely
scenario. Many other nonseparable functions can also satisfy this assumption. With the
distribution of pθ (θ) in hand, we can identify the technology using the analysis presented
below in Section 3.4.
Note that Theorem 2 does not claim that the distributions of the errors εj or that the
functions aj∙ (∙, ∙) are identified. In fact, it is always possible to alter the distribution of εj∙ and
the dependence of the function αj∙ (∙, ∙) on its second argument in ways that cancel each other
out, as noted in the literature on nonseparable models.20 However, lack of identifiability of
19In the case of classical measurement error, bounded completeness assumptions can be phrased in terms of
primitive conditions requiring nonvanishing characteristic functions of the distributions of the measurement
errors as in Mattner (1993). However, apart from this special case, very little is known about primitive
conditions for bounded completeness, and research is still ongoing on this topic. See d’Haultfoeuille (2006).
20See Matzkin (2003, 2007).
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