parameter vector, rather than for every point of the parameter space. Note that (local) underidentification is not related to the sample at hand; it is a property of the population model.

While local underidentification causes a rank-deficient Jacobian, the converse is *not* true [@shapiro_investigation_1983]: when the Jacobian is rank deficient, it is not necessarily the case that, for all samples, $\boldsymbol{\sigma}(\boldsymbol{\theta}) = \boldsymbol{\sigma}(\boldsymbol{\theta}^\prime)$ (the model is underidentified). Generally, this can occur because the reverse implication holds only when the loss function is twice differentiable and the Jacobian has constant rank in a neighborhood around the parameter vector. When there are single parameter points at which the rank becomes deficient, this last assumption is violated. A well known example is the two-factor model with two indicators for each factor, which has a rank deficiency when the correlation among factors is zero. Since the probability of finding such a rank-deficient point *exactly* equals zero, the model is said in the literature to be locally identified “almost everywhere”, i.e. everywhere except in points with probability measure zero [@shapiro_identifiability_1985].

To illustrate intuitively how it is possible to have a rank-deficient information matrix (second derivative) at a singular point but no identification problem, consider Figure 