Bias of the Quasi Score Estimator of a Measurement Error Model Under Misspecification of the Regressor Distribution

In a structural error model the structural quasi score (SQS)
estimator is based on the distribution of the latent regressor
variable. If this distribution is misspecified the SQS estimator is
(asymptotically) biased. Two types of misspecification are
considered. Both assume that the statistician erroneously adopts a
normal distribution as his model for the regressor distribution. In
the first type of misspecification the true model consists of a
mixture of normal distributions which cluster round a single normal
distribution, in the second type the true distribution is a normal
distribution admixed with a second normal distribution of low
weight. In both cases of misspecification the bias, of course,
tends to zero when the size of misspecification tends to zero.
However, in the first case the bias goes to zero very fast so that
small deviations from the true model lead only to a negligible
bias, whereas in the second case the bias is noticeable even for
small deviations from the true model.