Statistical Inference Under Misspecification

Question

The classical treatment of statistical inference relies on the assumption that that a correctly specified statistical is used exists. That is, the distribution $\mathbb{P}^*(Y)$ that generated the observed data $y$ is part of the statistical model $\mathcal{M}$: $$\mathbb{P}^*(Y) \in \mathcal{M}=\{\mathbb{P}_\theta(Y) :\theta \in \Theta\}$$ However, in most situations we cannot assume that this is really true. I wonder what happens with statistical inference procedures if we drop the correctly specified assumption.

I have found some work by White 1982 on ML-estimates under misspecification. In it is argued that the maximum likelihood estimator is a consistent estimator for the distribution $$\mathbb{P}_{\theta_1}=\arg \min_{\mathbb{P}_\theta \in \mathcal{M}} KL(\mathbb{P}^*,\mathbb{P}_\theta)$$ that minimizes the KL-divergence out of all distributions within the statistical model and the true distribution $\mathbb{P}^*$.

What happens to confidence set estimators? Lets recapitulate confidence set estimators. Let $\delta:\Omega_Y \rightarrow 2^\Theta$ be a set estimator, where $\Omega_Y$ is the sample space and $2^\Theta$ the power set over the parameter space $\Theta$. What we would like to know is the probability of the event that the sets produced by $\delta$ include the true distribution $\mathbb{P}^*$, that is $$\mathbb{P}^*(\mathbb{P}^* \in \{P_\theta : \theta \in \delta(Y)\}):=A.$$

However, we of course don't know the true distribution $\mathbb{P}^*$. The correctly specified assumption tells us that $\mathbb{P}^* \in \mathcal{M}$. However, we still don't know which distribution of the model it is. But, $$\inf_{\theta \in \Theta} \mathbb{P}_\theta(\theta \in \delta(Y)):=B$$ is a lower bound for the probability $A$. Equation $B$ is the classical defintion of the confidence level for a confidence set estimator.

If we drop the correctly specified assumption, $B$ is not necessarily a lower bound for $A$, the term that we are actually interested in, anymore. Indeed, if we assume that the model is misspecied, which is arguably the case for most realistic situations, $A$ is 0, because the true distribution $P^*$ is not contained within the statistical model $\mathcal{M}$.

From another perspective one could think about what $B$ relates to when the model is misspecified. This a more specific question. Does $B$ still have a meaning, if the model is misspecified. If not, why are we even bothering with parametric statistics?

I guess White 1982 contains some results on these issues. Unluckily, my lack of mathematical background hinders me from understanding much that is written there.

Answer 1

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.

Answer 2

Firstly, let me say that this is a really fascinating question; kudos to Julian for posting it. As I see it, the fundamental problem you face in this kind of analysis is that any inference of any subset of $\Theta$ is an inference over the restricted class of probability measures in the model $\mathcal{M}$, so when you start asking about probabilities of inferring the true model, under the model, this degenerates down to a trivial question of whether or not there is misspecification to begin with. White gets around this by looking at how close the model gets to the true probability measure, using an appropriate distance metric. This leads him to the probability measure $\mathbb{P}_{\theta_1}$, which is the closest proxy for $\mathbb{P}^*$ in $\mathcal{M}$. This method of looking at $\mathbb{P}_{\theta_1}$ can be extended to give interesting quantities relating to your question about the confidence sets.

Before getting to this, it is worth pointing out that the values $A$ and $B$ are mathematically well-defined in your analysis (i.e., they exist), and they still have a meaning; it is just not necessarily a very useful meaning. The value $A$ in your analysis is well-defined; it is the true probability that the inferred set of probability measures includes the true probability measure. You are correct that $\mathbb{P}^* \notin \mathcal{M}$ implies $A = 0$, which means that this quantity is trivial in the case of misspecification. Following White's lead, it is perhaps more interesting to look at the quantity:

$$A^* \equiv A^*(Y) \equiv \mathbb{P}^* (\mathbb{P}_{\theta_1} \in \{P_\theta | \theta \in \delta(Y) \} ).$$

Here we have replaced the inner occurrence of $\mathbb{P}^*$ with its closest proxy in the model $\mathcal{M}$, so that the quantity is no longer rendered trivial when $\mathbb{P}^* \notin \mathcal{M}$. Now we are asking for the true probability that the inferred set of probability measures includes the closest proxy for the true probability measure in the model. Misspecification of the model no longer trivialises this quantity, since we have $\mathbb{P}_{\theta_1} \in \mathcal{M}$ by construction.

White analyses misspecification by showing that the MLE is a consistent estimator of $\mathbb{P}_{\theta_1}$. This is valuable because it tells you that even if there is misspecification, you still correctly estimate the closest proxy to the true probability measure in the model. A natural follow-up question concerning confidence sets is whether or not a particular inference method $\delta$ imposes any lower bound on the quantity $A^*$ or any convergence result in the limit as $n \rightarrow \infty$. If you can establish a (positive) lower bound or a (positive) convergence result, this gives you some value in guaranteeing that even if there is misspecification, you still correctly estimate the closest proxy with some probability level. I would recommend that you explore those issues, following the kind of analysis done by White.