# 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.