Up until a few days ago I was thinking that the following two forms of the Fisher Information are "always" equivalent: $$(1) \quad \mathcal{I(\theta)}= E_\theta [\frac{\partial \log \ell(y;\theta)}{\partial \theta} \frac{\partial \log \ell(y; \theta)}{\partial \theta'}],$$ $$(2) \quad \mathcal{I(\theta)}= E_\theta [-\frac{\partial^2 \log \ell(y;\theta)}{\partial \theta\partial \theta'} ],$$ where $\theta$ is the parameter vector, $y$ is the vector of observations, and $\ell$ is the likelihood function.
I heard that (1) is the original definition of the Fisher Information, however (2) only holds when the model is not "misspecified".
I consulted some sources, among them Statistics and Econometrics Models (Gourieroux and Monfort 1995, Vol 1, p83 property 3.8), but there is no mention of any regularity conditions needed to hold for the second definition of $\mathcal{I}(\theta)$ to be equivalent to the original definition (1).
How does misspecification of a model cause this equivalency to break?