Bayesian Statistics from Methods to Models and Applications by Sylvia Frühwirth-Schnatter Angela Bitto Gregor Kastner & Alexandra Posekany

Author:Sylvia Frühwirth-Schnatter, Angela Bitto, Gregor Kastner & Alexandra Posekany
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(7.4)

From here we can derive the smoothing density, or conditional posterior density of θ 0: T . We use the method of [18], based on [21], for drawing from this density, called the mixed Cholesky factor algorithm (MCFA) by [23]. The following derivation closely follows Appendix C of [23]. The full conditional density of θ 0: T can be written as

where

Then g has the form where K is some constant with respect to θ 0: T , Ω is a square, symmetric matrix of dimension and ω is a column vector of dimension . This gives . Further, Ω is block tridiagonal since there are no cross product terms involving θ t and θ t+k where | k |  > 1. Because of this, the Cholesky factor and thus inverse of Ω can be efficiently computed leading to the Cholesky factor algorithm (CFA) [21]. Instead of computing the Cholesky factor of Ω all at once before drawing θ 0: T as in the CFA, the same technology can be used to draw θ T , then θ t  | θ (t+1): T recursively in a backward sampling structure, resulting in the MCFA. In simulations, the MCFA has been found to be significantly cheaper than Kalman filter based methods and often cheaper than the CFA [18].

In order to implement the algorithm, we need to first characterize the diagonal and off diagonal blocks of Ω and the blocks of ω:

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