A Note On The Dai-Singleton Canonical Representation Of Affine Term Structure Models
Dai and Singleton (2000) study a class of term structure models for interest rates that specify the short rate as an affine combination of the components of an N-dimensional affine diffusion process. Observable quantities in such models are invariant under regular affine transformations of the underlying diffusion process. In their canonical form, the models in Dai and Singleton (2000) are based on diffusion processes with diagonal diffusion matrices. This motivates the following question: Can the diffusion matrix of an affine diffusion process always be diagonalized by means of a regular affine transformation? We show that if the state space of the diffusion is of the form D = R-+(m) x RN-m for integers 0 <= m <= N satisfying m <= 1 or m >= N - 1, there exists a regular affine transformation of D onto itself that diagonalizes the diffusion matrix. So in this case, the Dai-Singleton canonical representation is exhaustive. On the other hand, we provide examples of affine diffusion processes with state space R-+(2) x R-2 whose diffusion matrices cannot be diagonalized through regular affine transformation. This shows that for 2 <= m <= (N - 2), the assumption of diagonal diffusion matrices may impose unnecessary restrictions and result in an avoidable loss of generality.