Let $y_1, \ldots, y_n$ be the observed data which is presumed to be a realization of a sequence of i.i.d. random variables $Y_1, \ldots, Y_n$ with common probability density function $p_e$ defined with respect to a sigma-finite measure $\nu$. The density $p_e$ is called Data Generating Process (DGP)
density.

In the researcher's probability model
${\cal M} \equiv \{ p(y ; \theta) : \theta \in \Theta \}$ is a collection
of probability density functions which are indexed by a parameter vector
$\theta$. Assume each density in ${\cal M}$ is a defined with respect to
a common sigma-finite measure $\nu$ (e.g., each density could be a probability
mass function with the same sample space $S$).

It is important to keep the density $p_e$ which actually generated the
data conceptually distinct from the probability model of the data. In 
classic statistical treatments a careful separaration of these concepts
is either ignored, not made, or it is assumed right from the beginning
that the probability model is correctly specified.

A correctly specified model ${\cal M}$ with respect to $p_e$ is defined
as a model where $p_e \in {\cal M}$ $\nu$-almost everywhere. When 
${\cal M}$ is misspecified with respect to $p_e$ this corresponds
to the case where the probability model is not correctly specified.

If the probability model is correctly specified, then there exists
a $\theta^*$ in the parameter space $\Theta$ such that
$p_e(y) = p(y ; \theta^*)$ $\nu$-almost everywhere. Such a parameter
vector is called the "true parameter vector". If the probability model
is misspecified, then the true parameter vector does not exist.

Within White's model misspecification framework the goal is to find the parameter estimate $\hat{\theta}_n$ that minimizes 
$\hat{\ell}_n({\theta}) \equiv (1/n) \sum_{i=1}^n \log p(y_i ; { \theta})$ over some compact parameter space $\Theta$. It is assumed that
a unique strict global minimizer, $\theta^*$, of the
expected value of $\hat{\ell}_n$ on $\Theta$ is located in the interior of $\Theta$. In the lucky case where the probability model is correctly specified, $\theta^*$ may be interpreted as the "true parameter value".

In the special case where the probability model is correctly
specified, then $\hat{\theta}_n$ is the familiar maximum likelihood estimate.
If we don't know have absolute knowledge that the probability model
is correctly specified, then $\hat{\theta}_n$ is called a quasi-maximum
likelihood estimate and the goal is to estimate $\theta^*$. 
If we get lucky and the probability model is 
correctly specified, then the quasi-maximum likelihood estimate reduces as
a special case to the familiar maximum likelihood estimate and 
$\theta^*$ becomes the true parameter value.

Consistency within White's (1982) framework corresponds to convergence
to $\theta^*$ without requiring that $\theta^*$ is necessarily the true
parameter vector. Within White's framework, we would never estimate
the probability of the event that the sets produced by δ include the TRUE distribution P*. Instead, we would always estimate the probability distribution P** which is the probability of the event that the sets
produced by δ include the distribution specified by the density
$p(y ; \theta^*)$.

Finally, a few comments about model misspecification. It is easy to find
examples where a misspecified model is extremely useful and very predictive.
For example, consider a nonlinear (or even a linear) regression model
with a Gaussian residual error term whose variance is extremely small
yet the actual residual error in the environment is not Gaussian.

It is also easy to find examples where a correctly specified model
is not useful and not predictive. For example, consider a random walk
model for predicting stock prices which predicts tomorrow's closing
price is a weighted sum of today's closing priced and some Gaussian
noise with an extremely large variance.

The purpose of the model misspecification framework is not to ensure model
validity but rather to ensure reliability. That is, ensure that the sampling error associated with your parameter estimates, confidence intervals, hypothesis tests, and so on are correctly estimated despite the presence of either a small or large amount of model misspecification. The quasi-maximum likelihood
estimates are asymptotically normal centered at $\theta^*$ with a covariance matrix estimator which depends upon both the first and second derivatives of the negative log-likelihood function. In the special case where you get lucky and the model is correct then all of the formulas reduce to the familiar classical statistical framework where the goal is to estimate the "true" parameter values.