Yield-Only Identification in Maximum-Likelihood Affine Term Structure Models

Summary

The document outlines an affine term structure model in which log bond prices are linear functions of latent factors. With as many observed yields as factors, the factor vector can be recovered from yields by inverting the model’s loading matrix, assuming that inversion is valid. The paper being discussed adds further yields, measured with error, because the initial set alone is insufficient to identify all parameters under both the physical and risk-neutral measures.

The central issue is how many additional yields are needed: the question asks whether one extra maturity suffices, whether the number must match the factor count, or whether more can be included. No answer or estimation procedure is provided, so the document does not resolve the minimum or maximum number of extra observations. Its useful contribution is to frame the identification problem and distinguish factor recovery from identifying the full model parameters; conclusions require the specific paper’s assumptions and rank conditions.

Key ideas

• An affine term structure model expresses log bond prices as functions of latent factors and maturity-specific loadings.
• A set of yields equal in number to the factors can recover the factors when the loading matrix is invertible.
• Recovering latent factors does not by itself identify all parameters under the physical and risk-neutral measures.
• Additional yields are treated as noisy observations to support parameter identification.
• The document poses but does not answer how many extra yields are required or allowed.

Tags

# Estimation of Affine Term Structure Model


# Estimation of Affine Term Structure Model












In this paper the estimation of Affine Term Structure models via ML is discussed. In the Affine $N$-factors model the price of the bond is

$$ P(X_t,t,T;\theta) = \exp(-\gamma_0(T-t;\theta)-\gamma(T-t;\theta)^{\prime}X_t)\Rightarrow\\ g(X_t,t,T;\theta)\equiv-\ln(P(X_t,t,T;\theta)) = \gamma_0(T-t;\theta)+\gamma(T-t;\theta)^{\prime}X_t,\quad X_t\in\mathbb{R}^N $$

The paper assume to have $N$ observed yields with maturity $\tau_1,...,\tau_N$ and hence

$$ \left[\begin{array}{c} g(X_t,t,t+\tau_1;\theta)\\ \vdots\\ g(X_t,t,t+\tau_N;\theta) \end{array} \right] = \left[\begin{array}{c} \gamma_0(\tau_1;\theta)\\ \vdots\\ \gamma_0(\tau_N;\theta) \end{array} \right]+\left[\begin{array}{c} \gamma(\tau_1;\theta)^{\prime}\\ \vdots\\ \gamma(\tau_N;\theta)^{\prime} \end{array} \right]\,\left[\begin{array}{c} X_{1,t}\\ \vdots\\ X_{N,t} \end{array} \right] $$

so that $X_t$ can be derived as a function of the vector of yields $(g(X_t,t,t+\tau_1;\theta),..,,g(X_t,t,t+\tau_N;\theta))^{\prime}$ by a proper matrix inversion. The authors says that, in order to identify all the parameters of the model under both the $P$ and $Q$ measures, only $N$ yields are not enough, so one needs $H$ furthers yields.

My problem. I do not clearly understand which is the minimum value for $H$ ($H=1$? $H=N$?) and if $H$ can be arbitrarily large. It is said that the additional $H$ yields must be assumed to be observed with errors. I am lost in all these assumptions since I do not find a rigorous treatment in the literature.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.