Example: Linear IV

As a first example, suppose that $D_{i}=\left (Y_{i},X_{i},Z_{i}\right )$ for $Y_{i}\in \mathbb {R}$ a scalar outcome, $X_{i}\in \{0,1\}$ a binary endogenous treatment, and $Z_{i}\in \mathbb {R}^{k}$ a vector of k mean-zero exogenous variables, $E[Z_i]=0$ . We assume that these data are generated from a potential outcomes model, where the potential outcomes $Y_i(x,z)$ may in general depend on both $X_i$ and $Z_i,$ and the potential treatments $X_i(z)$ may depend on $Z_i.$ The parameter of interest $\theta \in \mathbb {R}$ is the average treatment effect (ATE):

$$\begin{align*}\theta=E[Y_i(1,Z_i)-Y_i(0,Z_i)], \end{align*}$$

which captures the average effect on $Y_i$ from changing $X_i$ from zero to one.

If the researcher wants to estimate a constant-effect linear instrumental variables model with excluded instrument $Z_i$ , this can be justified by assuming that $Z_i$ is excluded from $Y_i, Y_i(x,z)=Y_i(x,z')$ for all $(x,z,z')$ (so one may denote the potential outcome as $Y_i(x)$ ), that the instrument is randomly assigned $Z_i\perp \!\!\!\!\perp Y_i(\cdot ),X_i(\cdot ),$ and that treatment effects are constant, $Y_i(1)-Y_i(0)=\theta $ for all i. Under these assumptions, $Y_i$ follows the linear model $ Y_{i}=X_{i}\theta +\varepsilon _{i},$ where $\varepsilon _i=Y_i(0)$ and $E\left [Z_{i}\varepsilon _{i}\right ]=0$ . Consequently, $\theta $ solves the moment condition (1) for

$$\begin{align*}Y\left(P\right)=E_{P}\left[Z_{i}Y_{i}\right],\,\,X\left(P\right)=E_{P}\left[Z_{i}X_{i}\right], \end{align*}$$

which are (up to pre-multiplication by $E_P[Z_iZ_i']^{-1}$ ) equal to the reduced-form and first-stage coefficient vectors in the linear IV model, respectively. $\triangle $

Example: Logit Model

As a second example, suppose that $D_i = (Y_i, X_i)$ for $Y_i \in \{0, 1\}$ a binary outcome and $X_i = (1, \tilde {X}_{i})\in \mathbb {R}^2$ an exogenous variable, where $\tilde {X}_{i} \in \{x_1, \ldots , x_J\}$ . If the researcher assumes a logistic regression (i.e., logit) model for $Y_i,$

$$\begin{align*}Y_i = 1\{X_i' \psi> \varepsilon_i\}, \end{align*}$$

where $\varepsilon _i \sim \textrm {Logistic}(0,1)$ is independent of $X_i$ and $\psi $ is a conformable vector of coefficients, then, under this model,

$$\begin{align*}E_P[Y_i|X_i=x] = \Psi(x'\psi) \end{align*}$$

for $\Psi (u)=\frac {e^u}{1+e^u}$ the logistic function or, equivalently,

$$\begin{align*}\Psi^{-1}(E_P[Y_i|X_i=x]) = x'\psi \end{align*}$$

for $\Psi ^{-1}(u)=\log \left (\frac {u}{1-u}\right )$ the logit function.

We suppose that the object of interest $\theta \in \mathbb {R}^2$ parameterizes the conditional mean of Y given two distinct as-yet-unobserved values of ${X}_{i}:$

$$\begin{align*}\theta=(\theta_1,\theta_2)'=(\Psi^{-1}(E[Y_i|X_i=(1,x_1^*)]),\Psi^{-1}(E[Y_i|X_i=(1,x_2^*)]))', \end{align*}$$

where $x_1^*,x_2^*\not \in \{x_1, \ldots , x_J\}.$ The model implies that $\theta $ solves (1) for

$$\begin{align*}Y\left(P\right)&= \begin{pmatrix} \Psi^{-1}(E_{P}\left[Y_i|X_i=(1,x_1)\right]) \\ \vdots \\ \Psi^{-1}(E_{P}\left[Y_i|X_i=(1,x_J)\right]) \end{pmatrix}, \\X\left(P\right)& = \begin{pmatrix} 1 & x_1 \\ \vdots & \vdots \\ 1 & x_J \end{pmatrix} \begin{pmatrix} \frac{x_2^*}{x_2^* - x_1^*} & -\frac{x_1^*}{x_2^* - x_1^*} \\ -\frac{1}{x_2^* - x_1^*} & \frac{1}{x_2^* - x_1^*} \end{pmatrix}. ~~~ \triangle \end{align*}$$

Example: Linear IV (Continued)

Suppose we maintain the exclusion and independence assumptions for the instruments $Z_i,$ but allow treatment effects to be heterogeneous across units, $\text {Var}(Y_i(1)-Y_i(0))>0.$ If this treatment effect heterogeneity is correlated with heterogeneity in the first-stage effect $X_i(z)-X_i(z'),$ the results of Imbens and Angrist (Reference Imbens and Angrist1994) imply that the linear IV moments are not in general zero when evaluated at the ATE $\theta .$ Instead, for $\beta $ the vector of one-instrument-at-a-time IV estimands (i.e., the IV coefficient using the first instrument by itself, the second by itself, and so on), $\iota \in \mathbb {R}^k$ the vector of ones, and $\circ $ the elementwise product, the implied value of $\eta $ is

$$\begin{align*}\eta = E_{P}\left[Z_{i}Y_{i}\right] - E_{P}\left[Z_{i}X_{i}\right] \theta = (\beta - \theta \cdot \iota) \circ E_P[Z_iX_i] \neq 0. \end{align*}$$

Hence, the model is misspecified in the sense we consider whenever the one-at-a-time IV estimands differ from the ATE. Note that the IV model can thus be misspecified even when we have only a single instrument, $k=1$ : if, in this case, we further impose the Imbens and Angrist (Reference Imbens and Angrist1994) monotonicity assumption, the IV model will be misspecified if and only if the local average treatment effect (LATE) differs from the ATE. This is an instance of what Andrews, Chen, and Tecchio (Reference Andrews, Chen and Tecchio2025) term econometric, rather than statistical, misspecification.

In this example, we focus on misspecification arising from treatment effect heterogeneity, but our framework is sufficiently general to accommodate many other ways in which the researcher’s assumptions could fail. For instance, the exclusion restriction may fail, with $Y_i(x,z)\neq Y_i(x,z')$ for some $(x,z,z'),$ or independence may fail with $Z_i \ \ /\kern-11pt \perp \!\!\!\!\perp (Y_i(\cdot ),X_i(\cdot )).$ Each of these failures will imply a particular form for $\eta .\ \triangle $

Example: Logit Model (Continued)

The logit model may be misspecified for a variety of reasons, for instance, because the linear threshold model is incorrect and $X_i$ in fact enters nonlinearly, $Y_i=1\{h(X_i)>\varepsilon _i\},$ or because the linear threshold model is correct but the residual $\varepsilon _i$ does not follow a logistic distribution. Whatever the reason for misspecification, we will have

$$\begin{align*}\eta = \begin{pmatrix} \Psi^{-1}\left(E_{P}\left[Y_i|X_i=(1,x_1)\right]\right) \\ \vdots \\ \Psi^{-1}\left(E_{P}\left[Y_i|X_i=(1,x_J)\right]\right) \end{pmatrix} - \begin{pmatrix} 1 & x_1 \\ \vdots & \vdots \\ 1 & x_J \end{pmatrix} \begin{pmatrix} \frac{x_2^*}{x_2^* - x_1^*} & -\frac{x_1^*}{x_2^* - x_1^*} \\ -\frac{1}{x_2^* - x_1^*} & \frac{1}{x_2^* - x_1^*} \end{pmatrix} \theta\neq 0, \end{align*}$$

where the conditional expectations $E_P[Y_i|X_i=(1,x_j)]$ will depend on the precise form of misspecification. $\triangle $

As these examples highlight, we consider settings where the true parameter $\theta $ is well-defined (e.g., based on some counterfactual or causal effect) even when the researcher’s model is wrong in the sense that the true $\theta ,P$ pair is incompatible with the model. Of course, well-defined does not mean estimable, and if we allow $\eta \neq 0$ and impose no other restrictions, identification of $\theta $ is hopeless since any value of $\theta $ is compatible with any distribution P. To avoid such a pessimistic conclusion, one route pursued in the literature, including in Conley, Hansen, and Rossi (Reference Conley, Hansen and Rossi2012), Manski and Pepper (Reference Manski and Pepper2018), Masten and Poirier (Reference Masten and Poirier2020), Armstrong and Kolesár (Reference Armstrong and Kolesár2021), and Rambachan and Roth (Reference Rambachan and Roth2023), is to consider bounded relaxations of the model.

To outline one approach along these lines, note that, for a P-dependent positive-definite weighting matrix $W\left (P\right ),$ the model is correctly specified if and only if the W-weighted norm of the misspecification parameter $\eta $ is equal to zero

$$\begin{align*}\left\Vert \eta\right\Vert{}_{W}=\sqrt{\eta'W\left(P\right)\eta} = 0. \end{align*}$$

To allow the possibility of misspecification, the researcher could instead assume only that $\left \Vert \eta \right \Vert {}_{W}$ is bounded above by some known constant d. The identified set for $\theta $ is then the set of values $\theta $ such that the minimum distance objective function

$$\begin{align*}Q_W(\theta;P)=\Vert g(\theta;P)\Vert_W^2 \end{align*}$$

takes a value no larger than $d^2$

(3) $$ \begin{align} \Theta_{I}\left(P,d\right)=\left\{ \theta:Q_W(\theta;P)\le d^2\right\}. \end{align} $$

While this approach requires the researcher to specify the norm bound $d,$ the data do contain some information about this quantity. Specifically, when $d^2<J_W(P)$ for $ J_W(P) = \min _{\theta }Q_W(\theta ;P), $ the population analog of the J-statistic (Hansen, Reference Hansen1982), $\Theta _{I}\left (P,d\right )$ is empty, so the data reject the assumption that $\|\eta \|_W<d.$ Thus, as discussed in, for example, Armstrong and Kolesár (Reference Armstrong and Kolesár2021), the data imply lower, but not upper, bounds on the degree of misspecification.

Another common practice when we are concerned with model misspecification is to focus on pseudo-true parameter values (cf. White, Reference White1982, Schennach, Reference Schennach2007, Müller, Reference Müller2013, Hansen and Lee, Reference Hansen and Lee2021, and Andrews and Kwon, Reference Andrews and Kwon2024), which are defined as the minimizers of a population objective function. In our setting, the pseudo-true parameter corresponds to the value of $\theta $ at which $J_W(P)$ is attained and is equal to the coefficient from a generalized least squares regression of $Y(P)$ on $X(P)$ weighting by $W(P):$

$$\begin{align*}\theta_{W}\left(P\right)= {\mathrm{arg\,min}}_{\theta}Q_W(\theta;P)=(X(P)'W(P)X(P))^{-1}X(P)'W(P)Y(P). \end{align*}$$

Example: Linear IV (Continued)

In the linear IV model with treatment effect heterogeneity, it is common to focus attention on the two-stage least squares (TSLS) estimand, which corresponds to the pseudo-true value using the TSLS weighting matrix $W(P) = E_P[Z_i Z_i']^{-1},$ and can be interpreted as a LATE under appropriate assumptions (Angrist and Imbens, Reference Angrist and Imbens1995). $\triangle $

Example: Logit Model (Continued)

In the logit model with misspecification, we cannot choose $\theta $ to match the full set of observed conditional means $E_P[Y_i| X_i=x].$ The weighting matrix governs how we prioritize matching different elements of this vector, and one option is to take $W(P)$ to be the diagonal matrix with the jth diagonal element equal to the probability that $\tilde {X}_{i}=x_j$ , $E_P[1\{\tilde {X}_{i}=x_j\}]$ , which prioritizes matching behavior at $X_i$ values which are more common in the population. $\triangle $

The pseudo-true parameter corresponds to the identified set with $d^2=J_W(P):$

$$\begin{align*}\Theta_I(P,J_W(P)^{1/2})=\{\theta_{W}\left(P\right)\}. \end{align*}$$

Hence, if a researcher assumes that the true parameter value is equal to the pseudo-true parameter, this is the same as assuming that the degree of misspecification, measured in the norm $\|\cdot \|_W$ , is as small as it can possibly be given the observed distribution P.Footnote ³ If they instead allow the possibility that $\theta $ and $\theta _W(P)$ are different, then as discussed in the introduction, it is not obvious how, if at all, the pseudo-true value $\theta _W(P)$ relates to the economic questions that motivate the analysis in the first place. Consequently, it is unclear when we would want to estimate pseudo-true values. The following two sections address this question by providing conditions under which optimal decisions depend on the data through (i) the population minimum distance objective $Q_W(\theta ;P)$ and (ii) the pseudo-true value $\theta _W(P)$ in particular.

Example: Linear IV (Continued)

Suppose the decision-maker needs to set a tax or subsidy $a\in \mathbb {R}$ for the treatment, where $a>0$ denotes a subsidy, while $a<0$ denotes a tax, and that the loss is $L(a,\theta )=(a-\theta )^2$ . That is, the optimal subsidy level is equal to the ATE, while the loss increases quadratically as the subsidy departs from the ATE. This loss arises naturally when demand for treatment varies linearly with the subsidy amount. $\triangle $

Example: Linear IV (Continued)

Focusing on the case where $W(P)$ is the TSLS weighting matrix $W(P)=E_P[Z_iZ_i']^{-1},$ Lemma 1 implies that the decision-maker’s posterior for the ATE depends on the data only through the reduced-form and first-stage regression coefficients, together with the covariance matrix of the instruments. This rules out, for instance, situations where the decision-maker’s beliefs about the ATE are informed by higher moments of P. $\triangle $

Assumption 1 further implies that the posterior density $\pi (\theta | P)$ can be expressed in terms of the minimum distance moments (1). In particular, the first part of Assumption 1 implies that the conditional priors for $\theta $ given $(W(P),X(P))$ and $\eta $ given $( W(P), X(P),\theta )$ are continuous. For brevity, let $\pi _\theta (\theta )=\pi (\theta |W(P),X(P))$ and $\pi _\eta (\eta | \theta )=\pi (\eta |W(P),X(P),\theta )$ denote their densities with respect to Lebesgue measure. The posterior density of $\theta $ given P is then

(4) $$ \begin{align} \pi(\theta | P) = \frac{\pi_\theta(\theta)\pi_\eta(Y(P)-X(P)\theta | \theta)}{\int \pi_\theta(\theta)\pi_\eta(Y(P)-X(P)\theta | \theta)d\theta}=\frac{\pi_\theta(\theta)\pi_\eta(g(\theta;P)| \theta)}{\int \pi_\theta(\theta)\pi_\eta(g(\theta;P) | \theta)d\theta}. \end{align} $$

This resembles the posterior in a finite-sample problem with parameter $\theta ,$ prior $\pi _\theta ,$ and likelihood $\pi _\eta .$ Consistent with this resemblance, the posterior $\pi (\theta |P)$ will be non-degenerate with non-trivial uncertainty about the true value of $\theta $ even though the data distribution P in our problem is perfectly known. This reflects the fact that $\theta $ is not point-identified once $\eta $ is unknown, so when the conditional prior $\pi _\eta $ is non-dogmatic the decision-maker remains uncertain about $\theta $ even after observing $P.$

Under restrictions on the misspecification prior $\pi _\eta (\eta |\theta )$ , Assumption 1 further implies that the prior density at $\eta $ conditional on $(\theta ,W(P))$ depends only on $\eta 'W(P)\eta $ , $\pi _\eta (\eta |\theta )\propto f(\eta 'W(P)\eta |\theta , W(P))$ .

Assumption 2(i) requires that the conditional prior on misspecification be sufficiently smooth. Assumption 2(ii) further requires that it not depend on the Jacobian matrix $X(P)$ , and thus that $X(P)$ and $\eta $ be conditionally independent given $(\theta ,W(P))$ under the prior. In the context of linear IV, for instance, this requires that the researcher not update about the violation of the model’s overidentifying restrictions based on the observed first stage. Assumption 2(iii) further requires that $\eta $ have conditional mean zero under the prior given $W(P),$ so the researcher thinks that the model is “right on average” even after observing the weighting matrix. To simplify the proof, we further assume that we can freely vary $(Y(P),X(P)).$

Lemma 2. Under Assumptions 1 and 2, the conditional prior density $\pi _\eta (\eta | \theta )$ is rotationally invariant with respect to $W(P)$ , taking the form

$$\begin{align*}\pi_\eta(\eta | \theta) = f(\eta'W(P)\eta| \theta, W(P)).\end{align*}$$

Lemma 2 implies that the prior density $\pi _\eta (\eta | \theta )$ is invariant to rotation of $W(P)^{\frac {1}{2}}\eta ,$ in the sense that for any $\eta , \tilde \eta $ such that $W(P)^{\frac {1}{2}}\eta =O W(P)^{\frac {1}{2}}\tilde \eta $ for an orthonormal matrix $O,$ the prior density is the same at $\eta $ and $\tilde \eta $ , $\pi _\eta (\eta | \theta )=\pi _\eta (\tilde \eta | \theta )$ . The density $\pi _\eta (\eta | \theta )$ is thus constant on the ellipsoids $\{\eta :\eta 'W(P)\eta =C\}$ for all C, from which it follows that $\pi _\eta (\eta | \theta )$ is an elliptically contoured distribution (see Muirhead, Reference Muirhead1982, Chap. 1.5).

Example: Linear IV (Continued)

Recall that $\eta =(\beta - \theta \cdot \iota ) \circ E_P[Z_iX_i]$ measures the difference between the ATE and the vector of one-instrument-at-a-time IV estimands. One example of an elliptically contoured distribution in this setting takes $\eta |\theta ,W(P)\sim N(0,W(P)^{-1})$ , which corresponds to $f(u|\theta , W(P))\propto \exp (-\frac {1}{2}u)$ . Under this prior, each single-instrument estimand is equal to the ATE plus a mean-zero noise term, where the covariance matrix of the noise is determined by $W(P)$ and the first stage $E_P[Z_iX_i].$ There are many other rotation-invariant priors, however, including a multivariate t prior with $\nu $ degrees of freedom, where $ f(u|\theta , W(P))\propto \left (1+\frac {1}{\nu }u\right )^{-\frac {\nu +k}{2}}.\ \triangle $

Lemma 2 implies that, under the conditions of Assumption 2, the posterior density $\pi (\theta |P)$ is proportional to a function of $(Q_W(\theta ;P),W(P),\theta ),$ as required by Assumption 1, only if the (conditional) prior on $\eta $ is rotation-invariant. In fact, rotation-invariance is both necessary and sufficient for Assumption 1 in settings where Assumption 2 holds.

Proposition 1. Under Assumption 2, Assumption 1 holds if and only if for all $P\in \mathcal {P},$ the conditional prior $\pi (Y(P),\theta |X(P),W(P))$ is absolutely continuous and

$$\begin{align*}\pi(\theta | P) = \frac{\pi_\theta(\theta)f(Q_W(\theta;P)|\theta, W(P))}{\int \pi_\theta(\theta)f(Q_W(\theta;P)|\theta, W(P))d\theta}, \end{align*}$$

for a nonnegative function $f(u|\theta , W(P))$ with $\int f(\eta '\eta |\theta , W(P))d\eta < \infty .$

We have thus shown that, under the conditions on $\eta |\theta ,W(P),X(P)$ imposed by Assumption 2, the minimum distance objective is sufficient in the sense of Assumption 1 if and only if our misspecification priors $\pi _\eta $ are rotation-invariant. Hence, for such priors, it is natural to focus on the minimum distance objective and, conversely, a focus on the minimum distance objective is most reasonable under such priors. Justifying a focus on the pseudo-true value, that is, on the argmin of the minimum distance objective, requires further restrictions, which we turn to next.

Example: Linear IV (Continued)

If $\pi _{\eta }(\eta |\theta )$ is an $N(0,W(P)^{-1})$ density, then since $\exp \left (-\frac {a}{2}u\right )/\exp \left (-\frac {1}{2}u\right )=\exp \left (\frac {1-a}{2}u\right )$ and $a>1,$ Assumption 3 holds. $\triangle $

Under sequences of priors that (i) satisfy Assumptions 1–3 and (ii) take the degree of misspecification to be small a priori, the posterior concentrates on the pseudo-true value. Formally, we consider a scale family of priors on the misspecification parameter $\eta , \pi _{\eta ,c}(\eta |\theta )\propto f(\frac {1}{c}\eta 'W(P)\eta ),$ where the scale parameter c controls the magnitude of misspecification and the prior variance of $\eta $ scales with c. Our main result in this section considers posterior behavior as the scale parameter becomes small, $c\to 0$ , corresponding to priors that assume a vanishing degree of misspecification.

Proposition 2. Suppose Assumptions 1–3 hold. For any continuous $\pi _\theta (\theta )$ with $\pi _\theta (\theta _W(P))>0,$ the posterior

$$\begin{align*}\pi_c(\theta | P) = \frac{\pi_\theta(\theta) f(\frac{1}{c}Q_W(\theta;P))}{\int \pi_\theta(\theta) f(\frac{1}{c}Q_W(\theta;P)) d\theta} \end{align*}$$

concentrates on $\theta _W(P)$ as $c\to 0$ , in that for $B_\varepsilon (\theta _W(P))=\left \{\theta :\Vert \theta _W(P)-\theta \Vert <\varepsilon \right \}$ ,

$$\begin{align*}\lim_{c\to 0}\int 1\{\theta \notin B_\varepsilon(\theta_W(P))\}d\pi_c(\theta | P) = 0 \text{ for all }\varepsilon>0. \end{align*}$$

Proposition 2 shows that for priors satisfying Assumptions 1–3 where the degree of misspecification is small, the posterior distribution concentrates on $\theta _W(P).$ This is entirely expected when $J_W(P)=0$ , since, in this case, the data provide no evidence of misspecification and priors with $c\to 0$ assume the model is nearly correct. When $J_W(P)>0,$ by contrast, the data imply non-trivial misspecification but Proposition 2 shows that the posterior continues to concentrate.

Concentration of $\pi _c(\theta |P)$ implies convergence of the Bayes decision rule

$$\begin{align*}\delta_{\pi_c}(P)\in {\mathrm{arg\,min}}_{a\in \mathcal{A}}\int L(a, \theta)d\pi_c(\theta | P), \end{align*}$$

under conditions on the decision problem.

Proposition 3 shows that for bounded loss functions that are Lipschitz in a and continuous in $\theta ,$ Bayes decision rules corresponding to the priors we study converge to plug-in decision rules based on the pseudo-true value. This result is useful for a number of reasons. First, it shows that plug-in decision rules using the pseudo-true parameter value correspond to the limit of a sequence of Bayes decision rules for a large class of loss functions, providing one justification for such rules. Second, it shows that the pseudo-true parameter value $\theta _W(P)$ is a sufficient statistic for communication with an audience whose priors take the limiting form we consider: a researcher looking to summarize the data for such an audience is justified in reporting only the pseudo-true parameter value, since it allows audience members to compute the optimal decision for whatever loss function they have, provided that loss satisfies the conditions of Proposition 3.

The conditions in Proposition 3 are somewhat restrictive and rule out squared error loss on an unbounded domain. These conditions are sufficient but not necessary, however, and we can obtain convergence of decision rules in many other settings by using additional structure for the loss function and prior.

Example: Linear IV (Continued)

Proposition 3 does not apply in this example, because the loss $L(a,\theta )=(a-\theta )^2$ is neither bounded nor Lipschitz. Nonetheless, if $\pi _\eta (\eta |\theta )$ corresponds to an $N\left (0,\left (\frac {1}{c}W(P)\right )^{-1}\right )$ distribution while the prior on $\theta $ is flat, the posterior density is

$$ \begin{align*} \pi_c(\theta | P) &= N\left(\theta_W(P), c\cdot(E_P[Z_iX_i]'W(P)E_P[Z_iX_i])^{-1}\right) \\ &= N\left(\theta_W(P), c\cdot(X(P)'W(P) X(P))^{-1}\right). \end{align*} $$

Hence, for all $c,$ the posterior distribution is, normal centered at $\theta _W(P)$ with variance proportional to c. Consistent with Proposition 2, this posterior converges weakly to a point mass at $\theta _W(P)$ as $c\to 0$ . Moreover, despite the conditions of Proposition 3 not holding in this example, the Bayes decision rule $\delta _{\pi _c}(P)$ is equal to $\theta _W(P)$ for all $c.\ \triangle $

4.2.1 Fragility to Prior Contamination

If we mix the concentration-inducing priors studied in the previous section with any fixed, full-support prior for $\eta |\theta $ , posterior concentration fails when $J_W(P)>0.$

Proposition 4. Consider conditional priors of the form

$$\begin{align*}\pi_{\eta, c}^\phi(\eta | \theta) = (1-\phi)\pi_{\eta, c}(\eta | \theta) + \phi\pi^*_\eta(\eta | \theta) \end{align*}$$

for any conditional prior $\pi ^*_\eta (\eta |\theta )$ with strictly positive density and $\phi \in (0,1)$ . Suppose Assumption 3 holds. If $J_W(P)>0$ , then, for any $\pi _\theta (\theta ),$ the resulting posterior satisfies

$$\begin{align*}\lim_{c\to 0}\pi_c^\phi (\theta | P) = \frac{\pi_\theta(\theta)\pi_\eta^*(Y(P)-X(P)\theta| \theta)}{\int \pi_\theta(\theta)\pi_\eta^*(Y(P)-X(P)\theta| \theta)d\theta}. \end{align*}$$

If instead $J_W(P)=0$ , $\pi _\theta (\theta )$ is continuous, and $\pi _\theta (\theta _W(P))>0,$ then

$$\begin{align*}\lim_{c\to 0}\int 1\{\theta \notin B_\varepsilon(\theta_W(P))\}d\pi_c^{\phi}(\theta | P) = 0 \text{ for all }\varepsilon>0. \end{align*}$$

Proposition 4 shows that if $J_W(P)>0$ and we contaminate the concentration-inducing prior $\pi _{\eta ,c}(\eta |\theta ),$ to an arbitrarily small extent, with any full-support prior $\pi _\eta ^*$ for $\eta |\theta ,$ then the posterior converges to the same limit as if we had set $\pi _\eta (\eta |\theta )=\pi ^*_\eta (\eta |\theta )$ . By contrast, when $J_W(P)=0,$ the contaminated posterior continues to concentrate on $\theta _W(P)$ as $c\to 0.$ In this case, the concentration-inducing component $\pi _{\eta ,c}(\eta |\theta )$ receives asymptotically all of the posterior weight, despite the contamination by $\pi _\eta ^*(\eta |\theta )$ .

Proposition 4 can be interpreted in terms of pre-testing for model specification: in the case where $J_W(P)>0,$ the data imply that the model is non-trivially misspecified. While the priors $\pi _{\eta ,c}(\eta |\theta )$ imply that the model is misspecified with probability one, as $c\to 0,$ they imply that the degree of misspecification is arbitrarily small. When we allow the possibility that $\eta $ is instead drawn from a fixed full-support distribution $\pi ^*_\eta ,$ for c sufficiently small, the data provide arbitrarily strong support for $\pi ^*_\eta (\eta |\theta )$ over $\pi _{\eta ,c}(\eta |\theta )$ . Loosely speaking, the concentration-inducing prior $\pi _{\eta ,c}(\eta |\theta )$ is rejected in favor of the full-support prior $\pi ^*_\eta (\eta |\theta )$ .Footnote ⁴

4.2.2 Fragility to Heavy Tails

Even under prior sequences such that the degree of misspecification goes to zero, the concentration obtained in Proposition 2 relies on thin tails for $\pi _\eta (\eta |\theta )$ . To illustrate this point, we show that posterior concentration around the pseudo-true value fails in two examples with heavy-tailed priors.

Example: Posterior Non-Concentration with t Prior

Suppose that our prior on $\eta $ corresponds to a multivariate t distribution centered at zero with scale matrix $W(P)^{-1}$ and $\tilde \nu $ degrees of freedom, $f(u) \propto \left (1 + \frac {1}{\tilde \nu }u\right )^{-\frac {\tilde \nu + k}{2}}$ . Provided $J_W(P)>0,$ if we define $\nu =\tilde \nu + k - p$ and

$$\begin{align*}\Sigma(P) = J_W(P) \left({\nu} X(P)'W(P)X(P)\right)^{-1}, \end{align*}$$

one can show that

$$ \begin{align*} \lim_{c\to0} \pi_c(\theta | P) &\propto \pi_\theta(\theta)\left(1 + {\nu}^{-1}(\theta - \theta_W(P))'\Sigma(P)^{-1}(\theta-\theta_W(P))\right)^{-({\nu}+p)/2}, \end{align*} $$

where the second term is the density for a multivariate t distribution centered at $\theta _W(P)$ , with scale matrix $\Sigma (P)$ and $\nu $ degrees of freedom. Consequently, the $c\to 0$ limiting posterior corresponds to updating the prior $\pi _\theta $ based on observing $\theta _W(P)\sim t_{{\nu }}(\theta ,\Sigma (P)).$

Note that the degrees of freedom in the “likelihood,” $\nu $ , are equal to the degrees of freedom in the misspecification prior $\pi _\eta $ plus the degree of over-identification, so a higher degree of over-identification leads to thinner tails for the posterior, all else equal. The scale parameter in the “likelihood,” $\Sigma (P) = J_W(P) \left ({\nu } X(P)'W(P)X(P)\right )^{-1}$ , is increasing in the J-statistic so cases where the moment conditions are observed to be more badly violated lead to a more uncertain posterior, all else equal. $\triangle $

The tail thickness of $f(\cdot )$ matters because it determines beliefs about the total level of misspecification conditional on a given value of the J-statistic. To see this, let us again assume that $\eta |\theta \sim \pi _{\eta ,c}(\eta |\theta )$ and consider the conditional distribution of $\|\eta \|_W$ conditional on the norm of $\eta $ exceeding a threshold $\tau $ , $\|\eta \|_W|\left (\tau \leq \|\eta \|_W\right )$ . For thin-tailed priors (i.e., those satisfying Assumption 3), one can show that, for all $a>1$ and all positive constants $\tau>0$ ,

(5) $$ \begin{align} \lim_{c\to 0}Pr_{\pi_{\eta,c}}\left\{\|\eta\|_W\ge a\cdot \tau| \left(\tau\le \|\eta\|_W\right)\right\} = 0. \end{align} $$

Hence, even in the case where the model is known to be misspecified, as $c\to 0,$ thin-tailed priors imply that the degree of additional misspecification, beyond that implied by the lower bound $\tau $ , is negligible. Recall, however, that the J-statistic is itself a lower bound on the total degree of misspecification, $\|\eta \|_W^2\ge J_W(P)$ . Consistent with this, when $c\to 0,$ thin-tailed priors imply that the total degree of misspecification must not be much larger than that suggested by the J-statistic, and thus that $\theta $ must be close to $\theta _W(P),$ the unique parameter value compatible with $\|\eta \|_W^2=J_W(P).$

By contrast, for f corresponding to a multivariate t distribution, we have that

(6) $$ \begin{align} \lim_{c\to 0}Pr_{\pi_{\eta,c}}\left\{\|\eta\|_W\ge a\cdot \tau| \left(\tau \le \|\eta\|_W \right)\right\} = p(a), \end{align} $$

for a fixed, nonzero function $p(\cdot )$ that does not depend on $\tau $ . Consequently, in this case, the researcher’s belief about the total degree of misspecification is non-degenerate and, once c is sufficiently small, scales proportionally with $\tau .$

While t-distributed priors for $\eta |\theta $ correspond to updates via a t likelihood in the $c\to 0$ limit, if we instead consider a multivariate power law prior for $\eta |\theta ,$ then dependence on c vanishes entirely.

Example: Posterior Non-Concentration with Power Law Prior

Suppose $f(x) \propto x^{-\alpha }$ for $\alpha> p/2$ (noting that this is improper). Then for $\nu =2\alpha -p$ ,

$$\begin{align*}\Sigma(P)= J_W(P)\left(\nu X(P)'W(P)X(P)\right)^{-1}, \end{align*}$$

and for all $c,$

$$ \begin{align*} \pi_c(\theta | P) \propto \pi(\theta)\left(1 + \nu^{-1}(\theta-\theta_W(P))'\Sigma(P)^{-1}(\theta - \theta_W(P))\right)^{-(\nu + p)/2}, \end{align*} $$

which corresponds to the posterior distribution from observing $\theta _W(P)\sim t_\nu (\theta ,\Sigma (P))$ . We again see that a higher degree of over-identification leads to thinner tails for the posterior, while a larger J-statistic leads to a more dispersed posterior. $\triangle $