Microinformation, nonlinear filtering, and granularity

The recursive prediction and filtering formulas of the Kalman filter are difficult to implement in nonlinear state space models since they require the updating of a function. The aim of this paper is to consider the situation of a large number n of individual measurements, called microinformation, and to take advantage of the large cross-sectional size to get closed-form prediction and filtering formulas at order 1/n. The state variables have a macrofactor interpretation. The results are applied to maximum likelihood estimation of a macroparameter and to computation of a granularity adjusted Value-at-Risk (VaR) for large portfolios. The granularity adjustment for VaR is illustrated by an application of the value of the firm model Merton, 1974, Journal of Finance 29, 449-470) taking into account both default and loss given default.