Conditional Gauss-Hermite Filtering With Application to Volatility Estimation

The conditional Gauss–Hermite filter (CGHF) utilizes a decomposition of the filter density p(y1, y2) into the product of the conditional density p(y1|y2) with p(y2) where the state vector y is partitioned into (y1, y2). In contrast to the usual Gauss–Hermite filter (GHF) it is assumed that the product terms can be approximated by Gaussians. Due to the nonlinear dependence of φ(y1|y2) from y2, quite complicated densities can be modeled, but the advantages of the normal distribution are preserved. For example, in stochastic volatility models, the joint density p(y, σ) strongly deviates from a bivariate Gaussian, whereas p(y|σ)p(σ) can be well approximated by φ(y|σ)φ(σ). As in the GHF, integrals in the updates can be computed by Gauss–Hermite quadrature. We obtain recursive update fomulas for the conditional moments E(y1|y2), Var(y1|y2) and E(y2), Var(y2).