New Confidence Intervals Stay Reliable Even Under Misspecification

When a model is misspecified, its best-fitting parameter can still point you in the wrong direction. That is the problem with pseudo-true values: they minimize the model’s own objective, but they may not match the quantity you actually care about. This paper studies that gap in a linear population minimum distance problem, where Bayesian decision-makers use priors motivated by the same objective. The authors characterize which sequences of priors make posteriors concentrate on the pseudo-true value, and they find that this convergence is fragile: small changes in the prior can break it. That means pseudo-true values are useful for decision-making only in special cases. The paper also gives a constructive result. It derives simple confidence intervals that guarantee correct average coverage for the true parameter across every prior in the class studied, even when there is no bound on how large the misspecification is. In other words, the paper draws a line between a misleading target and a dependable way to report uncertainty.

Economic models often begin with a tidy lie: linearity where the world bends, identical behavior where people differ, or clean error terms where reality is noisy. When that happens, the estimate that wins inside the model is not always the quantity you care about outside it. Instead, the model tends toward a pseudo-true value, the point that best fits its own rules. This paper asks a hard question about that number. If a Bayesian decision-maker uses a prior that fits the same minimum-distance logic, does the posterior still lock onto the pseudo-true target, and can that target safely guide decisions? The answer is unsettling because it depends on details that look small but matter a lot.

Why the pseudo-true target slips

The main result is a warning, not a rescue. Within a class of priors built to match the minimum-distance problem, the posterior lands on the pseudo-true value only for carefully chosen sequences of priors. Change those priors a little, and the concentration can disappear. That fragility is the point: the pseudo-true value is not a sturdy compass that every reasonable Bayesian procedure will find. It matters only in special cases, because its pull depends on prior details that can shift with a small tweak. The result does not stop there. In the same setting, it also builds simple confidence intervals that keep correct average coverage for the true parameter across every prior in the class, even when the misspecification is not bounded.

How the argument is built

The setting is a linear population minimum-distance problem, which means the model picks the parameter value that makes its population loss as small as possible. That gives a clean way to talk about the pseudo-true value: it is the point the model would settle on if it had to answer using only its own objective. The Bayesian part enters through priors motivated by that same objective. The authors then trace which prior sequences make the posterior pile up around the pseudo-true value, and which ones do not. From that logic, they build intervals for the true parameter that do not depend on a bound for how wrong the model may be.

Why honest intervals beat a false target

This matters because the comfort of a best-fitting estimate can be a trap. If the pseudo-true value only appears under narrow prior conditions, then it is too easy to mistake a model artifact for insight. The confidence intervals do something more modest and more useful: they keep average coverage right for the true parameter, so repeated use gives the right long-run share of successes. That makes them a safer way to report uncertainty in settings where assumptions like linearity, identical agents, or tidy treatment rules are shaky. The payoff is not perfect certainty. It is a way to keep inference honest when the model itself cannot be trusted to name the right destination.

What to trust when the target moves

The practical consequence is simple: you do not have to ask a shaky pseudo-true estimate to act like a final answer. In this linear minimum-distance setting, the paper shows that you can still report intervals that keep average coverage correct for the true parameter, even when misspecification may be arbitrarily large. So the surprise is not that models can be wrong; it is that a dependable uncertainty statement can survive that wrongness while the target itself wobbles. That makes the interval result the safer thing to carry forward, because it tells you what range of values still deserves belief when the model's neat center does not.