Identification of ATSMs

#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.

Here, we address the problem of identification of the factors defined in the Affine Term Structure Models Affine Term Structure Models.

Identification Problem

Invariant Affine Transformation

Consider a Affine Term Structure Models > ^24c2e6 Affine Term Structure Models > Definition 5 (Gaussian ATSM) given as and note that by Affine Term Structure Models > ^beb7fd Affine Term Structure Models > Proposition 6 (Ricatti equation for ATSM), the first two equations are sufficient to derive the bond prices and yields, while the last equation implies that the factor follows a VAR(1) process.

Remark that the time yield of a period bond is given as where and have been driven by Affine Term Structure Models > ^72d83f Affine Term Structure Models > Corollary 7 (Ricatti equations for GASTM).

Definition (invariant affine transformation).

An invariant affine transformation of the factors is defined as for some and a non-singular matrix .
Note that under , the short rate, risk-neutral, and physical dynamics are given as and the yields are by the bond pricing formula.

Without any restrictions, we are not able to figure out whether the yields were generated from the original GASTM model, or from the transformed model. In such situation, we call that the model is underidentified.

Note that under ^c86440 Definition 1 (invariant affine transformation), the new parameters can be driven as follows:

from short rate dynamics: thus
from risk-neutral dynamics: and thus we have
similar to physical dynamics, we have

Identification Steps

Our goal is to identify the model against invariant affine transformations, by imposing restrictions on the model parameters so that there are only one set of factors that generate the yields and simultaneously satisfy such restrictions.

This means, for then we have and .

In general, the affine term structure models are identified in two steps:

Impose identification restrictions: in either the two of the short rate, risk-neutral, or physical dynamics, impose restrictions.
Equivalence of true model to canonical form: suppose the model is given as then, there exists an invariant affine transformation invariant affine transformation of such that can formulate the dynamics given identification restrictions. This form of the model is called canonical form.
Uniqueness of canonical form: let be factors under identification restrictions. Then if is an affine transformations of that also satisfies the identification restrictions, then .

Dai-Singleton Canonical Model

This model is first introduced in Dai and Singleton (2000), which imposes restrictions on the physical factor dynamics. Instead, we look into Singleton (2006), which is an equivalent version of the model with the restrictions on the risk-neutral dynamics, which restrictions are consistent with the JSZ Model JSZ Model and AFNS model AFNS model.

Definition (Dai-Singleton model).

Consider a Gaussian ATSM Gaussian ATSM stated as The Dai-Singleton model imposes the identification restrictions of

all elements of are non-negative
is lower triangular with no eigenvalues equal to , i.e. no unit roots. and the diagonal entries of are also distinct and ordered in decreasing order

The proof of the model is given here here.

Note that we only state the short rate and risk-neutral dynamics, since the physical factor dynamics are left unrestricted. Also, remark that Hamilton and Wu (2012) pointed out that without the third restrictions, the model is unidentified.

If the risk-neutral factor dynamics include a unit root, then we may impose restrictions on the physical factor dynamics instead, i.e. and be lower triangular with no unit roots and decreasing diagonal entries.
However, if both risk-neutral and physical dynamics contain unit roots, then we must impose additional zero restrictions on or in order to identify the model. This is what we opt for in the FS-ZLB model.

Another problem of ^026429 Definition 2 (Dai-Singleton model) is that it does not encompass ATSM where or have complex eigenvalues, since there may not exists a decomposition of with real lower triangular and real orthogonal . The JSZ Model JSZ Model, which provides an alternative means of identifying Gaussian ATSMs that is robust under both the presence of a (single) unit root and complex eigenvalues in the mean reversion parameters.

JSZ Model

Here, we look JSZ model introduced in Joslin, Singleton, and Zhu (2011) that offers a seminal identification scheme for gaussian ATSMs.

Definition (JSZ model).

Consider a Gaussian ATSM Gaussian ATSM stated as where is the (positive definite and nonrandom) conditional variance.
The JSZ model imposes the identification restrictions of

is lower triangular, and usually it is taken to be the Cholesky factor of the conditional variance matrix.
and .
is in ordered Jordan form, that is and for each where is real and . The blocks are ordered so that . Also, we allow for at most one single unit root, or an eigenvalue equal to .
the last entries of are , i.e.

Proof of Canonical JSZ model Proof of Canonical JSZ model

Yields from Canonical JSZ model

From Proof of Canonical JSZ model Proof of Canonical JSZ model, the JSZ model JSZ model is observationally equivalent to an ATSM of where is lower triangular matrix with positive diagonal entries, and thus the model is identified against invariant affine transformations. In other words, the risk-neutral dynamics can be summarized in terms of the parameters of Solving for bond prices under canonical JSZ form, by Affine Term Structure Models > ^72d83f Affine Term Structure Models > Corollary 7 (Ricatti equations for GASTM) we yields are formulated as Firstly, since then thus we have Next, since we have

Below, we continue to work using this simpler model with the canonical factors .

Model with Observable Factors

One of the contribution of the JSZ model JSZ model is that it allows us to derive an observationally equivalent model with the observable factors.

Suppose the followings:

yields of maturities, collected in dimensional random vector of theoretical yields
latent factors
measurement errors cause the observed yields .

Thus we have which allows us to write the model as a state-space model.

For simplicity, suppose that portfolios of yields are observed without error, meaning that there exist of an random matrix and an dimensional random vector such that or, Below, we introduce the two most popular choices for this affine transformation. First, using principle components to derive and , and the second is to assume that some yields are perfectly priced.

1) Using Principle Components

Here, we choose and that make through the principle components of the demeaned yields. Formally, we have where is the sample mean of the yields, and the rows of are the orthonormal eigenvectors corresponding to the largest eigenvalues of the sample covariance matrix such that Note that to apply the principle components method, the sample covariance matrix should be consistent with the true covariance matrix, or the weak stationarity and variance ergodicity of the yield process . Furthermore, the parameters and do not depend on the model parameters in the both cases.

2) Using Perfectly Priced Yields

An alternative approach is to assume that some yields are perfectly priced, while the others are observed with error. Denote the yields under the first case as , and the later case as , which means By letting and be collected in the first rows of and , we have so that This means that are observed without error, so that , and and are functions of the parameters .

While this approach is more restrictive than the 1) Using Principle Components 1) Using Principle Components, it allows the case of non-stationary yields, adding more generality compared to the previous approach.

Solutions with Observable Factors

Whether 1) Using Principle Components 1) Using Principle Components or 2) Using Perfectly Priced Yields 2) Using Perfectly Priced Yields, we obtain where is the JSZ canononical factors and is an invariant affine transformation of provided that is nonsingular.

Then, an observantly equivalent Gaussian ASTM with factors has the dynamics of where Thus the yields can be represented in terms of as so in JSZ, we can formulate GASTM with factors that depends on as few parameters as possible.

Arbitrage-free Nelson-Siegel Model

The arbitrage-free Nelson-Siegel (AFNS) model, presented by Christensen, Diebold, and Rudebusch (2011) and adapted for a discrete time by Niu and Zeng (2012), is a special case of the JSZ Model JSZ Model.

While Dynamic Nelson-Siegel Model Dynamic Nelson-Siegel Model fits the yield curve very well, its main shortcoming is that it is an empirical model that does not impose the No-Arbitrage Condition No-Arbitrage Condition. Thus Nelson-Siegel model allows for arbitrage opportunities to arise, which brings up the need for an theoretically consistent model.

Definition (AFNS model).

Consider a Gaussian ATSM Gaussian ATSM stated as The arbitrage-free Nelson-Siegel (AFNS) model imposes the identification restrictions of

and
in JSZ model.
is in ordered Jordan form where the diagonal term is restricted as i.e.

Note that we do not impose any restrictions on the physical dynamics.
Thus, the short-rate and risk-neutral factor dynamics of AFNS model are given as

Proposition (prices and yields in AFNS Model).

Under the restriction given in ^6231fe Definition 4 (AFNS model), we have the bond yields satisfying Dynamic Nelson-Siegel Model > ^0a7421 Dynamic Nelson-Siegel Model > Proposition 1 (Nelson-Siegel restriction), i.e. where

Proof.Note that by Affine Term Structure Models > ^beb7fd Affine Term Structure Models > Proposition 6 (Ricatti equation for ATSM), the prices and the yields of the bonds in Gaussian ASTM model are given as where the loadings are given from Affine Term Structure Models > ^72d83f Affine Term Structure Models > Corollary 7 (Ricatti equations for GASTM), From the restriction and we have which satisfies the N-S restriction. Note that the third row is given by For the intercept term, as we have and , then is not necessarily zero. □

Here, unlike the original Dynamic Nelson-Siegel Model Dynamic Nelson-Siegel Model, ^6231fe Definition 4 (AFNS model) contains the intercept term . This suggest that the baseline N-S model does not satisfy the no-arbitrage condition. Heuristically, the intercept term presented in AFNS model can be interpreted as a correction for yields of various maturities to satisfy the no-arbitrage conditions.

Compared to the previous models, the AFNS model assume from the outset and is identified with consistent estimates of the three factors (level, slope and curvature) against invariant affine transformations.