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edited Feb 5, 2019 at 5:29

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.

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^*$$\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.

The purpose of the model misspecification framework is not to ensure yourmodel model is validvalidity but rather to ensure reliability. That is, ensure that the sampling error associated with your your parameter estimates, confidence intervals, hypothesis tests, and so on are 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.

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".

The purpose of the model misspecification framework is not to ensure your model is valid but rather to 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.

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".

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.

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created Feb 5, 2019 at 5:24

RMG

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".

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.