Proof of Dai-Singleton Model

#termstructure #economics #finance #atsm

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.

Main References

Kim, Seung Hyun. (2024). "Asset Pricing and Term Structure Models". WORK IN PROGRESS.

Proof.Consider a Gaussian ATSM specified as where it is assumed that has real and distinct eigenvalues within the unit circle. Below we show that this ATSM is observationally equivalent to GASTM satisfying restrictions on Identification of ATSMs > ^026429 Identification of ATSMs > Definition 2 (Dai-Singleton model), and that this canonical model is identified against affine transformations.

Equivalence of true model to canonical form

First, define and under this rotation, the dynamics are given as where Since and are similar, they share the same eigenvalues. And since we are assuming that has the real eigenvalues within the unit circle, also does.

Now consider the Schur decomposition (see Schur's lemma Schur's lemma) of where is a complex Hermitian matrix and is an upper triangular matrix with diagonals equal to the eigenvalues of . As the eigenvalues of has real-valued eigenvalues, and are both real-valued (implying that ).

Also, by implication, is an orthogonal matrix (i.e. ), and we have the equality of where is a lower triangular matrix.

Define the rotation and similarly, the dynamics become where since due to the orthogonality of . This means that the variance remains unchanged under compared to , implying that Now finally, consider the translation where is well-defined since none of the eigenvalues of are equal to . Under , the dynamics become where Thus we have where we have , , and is real-valued lower triangular matrix. Lastly to make every elements of be non-negative, we can rotate the factors one last time so that if is negative, then is multiplied by . Therefore the resulting model satisfies the restrictions of Identification of ATSMs > ^026429 Identification of ATSMs > Definition 2 (Dai-Singleton model).

Uniqueness of canonical form

Suppose we have Dai-Singleton canonical form under as where is a real-valued lower triangular matrix.
Now let be an invariant affine transformation of that also satisfy the restrictions of Identification of ATSMs > ^026429 Identification of ATSMs > Definition 2 (Dai-Singleton model), given as and suppose we have where the parameters are given as We show the uniqueness of the canonical form by showing that and , sequentially.

() Note that from and the assumption of , we have where the first equality holds as also satisfies the Dai-Singleton restrictions Dai-Singleton restrictions. Then, is orthogonal matrix, and becomes the schur decomposition of . Then, as the diagonal entries of are ordered in decreasing order, the orthogonal matrix be the identity matrix.

() Note that from where by the Dai-Singleton restrictions Dai-Singleton restrictions. Then we have and if is non-singular, then . □

Note that we cannot exclude the case when is singular, which means is not always guaranteed. This is called as the local identification issue pointed out in Hamilton and Wu (2012), and the reason we need an additional assumption of the eigenvalues within the unit circle.