Curvature-Calibrated Quasi-Bayesian Updating for Moment-Restricted Models

Moment restrictions provide a flexible basis for quasi-Bayesian inference when a full likelihood is unavailable, but the weighting matrix in a quadratic moment criterion determines both the relative importance of the moments and the information scale of posterior updating. We propose curvature-calibrated quasi-Bayesian updating, which uses the inverse of the covariance (or long-run covariance) of the moment conditions evaluated at a self-consistent quasi-posterior center. The resulting fixed-point procedure alternates between covariance estimation and simulation from a fixed-weight quasi-posterior, thereby avoiding parameter-dependent weighting during each simulation run. Under a Bernstein-von Mises condition for the fixed-weight quasi-posterior at the efficient population weight, we show that the calibration map is locally contractive, that its fixed point is consistent at the standard parametric rate, and that the Gaussian approximation continues to hold under the calibrated data-dependent weight, with covariance given by the inverse Godambe information matrix. Under a uniform fourth-moment condition, the scaled quasi-posterior covariance converges to the same matrix, so quasi-posterior and repeated-sampling uncertainty agree to first order. Simulations show improved covariance calibration and interval coverage after a few updates. An application to longitudinal binary-response data illustrates the method with within-subject dependence and overidentified residual moments.