An Optimal Orthogonal Variance Decomposition

22 Pages Posted: 16 Apr 2008 Last revised: 10 Nov 2008

See all articles by Woon K. Wong

Woon K. Wong

IMRU, Cardiff Business School

Date Written: November 8, 2008


Let e and Sigma be respectively the vector of shocks and its variance covariance matrix in a linear system of equations in reduced form. This article shows that a unique orthogonal variance decomposition can be obtained if we impose a restriction that maximizes the trace of A, a positive definite matrix such that Az=e where z is vector of uncorrelated shocks with unit variance. Such a restriction is meaningful in that it associates the largest possible weight for each element in e with its corresponding element in z. It turns out that A=Sigma^1/2, the square root of Sigma.

Keywords: Variance decomposition, Cholesky decomposition, square root matrix, stock returns, bond returns

JEL Classification: C10, G10, M40

Suggested Citation

Wong, Woon K., An Optimal Orthogonal Variance Decomposition (November 8, 2008). Available at SSRN: or

Woon K. Wong (Contact Author)

IMRU, Cardiff Business School ( email )

Cardiff CF10 3EU
United Kingdom

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