Partial yield curve inversion using linearity in some parameters

Yields ${\mathsf y}$ are affine in state variables $\xi$ and constant risk premia $\lambda^1$. (As usual ${\mathsf y}$ represents the yield curve as a vector of yields at selected maturities.) Write again $t\,{\mathsf y}= \mathsf U\,\xi-{\mathsf v}$. Linearity of ${\mathsf y}$ in $\lambda^1$ enters through ${\mathsf v}$. Rewrite ${\mathsf v}$ by splitting it into the parts linear and constant in $\lambda^1$,

$${\mathsf v}= (a+\lambda^1)^\prime\,\mathsf q+\mathsf c\quad .$$ (37)

Then altogether

$$t\,{\mathsf y}= \mathsf U\,\xi-{\lambda^1}^\prime\,\mathsf q- a^\prime\,\mathsf q-\mathsf c\quad ,$$ (38)

and the following linear problem has to be solved:

$$t\,{\mathsf y}= \left(\mathsf U, \mathsf q^\prime\right) ... ...1 \end{array} \right) -a^\prime\,\mathsf q-\mathsf c\quad .$$ (39)

Depending on the number of maturities observed the inversion can either be done exactly or a linear fit can be sought. Some entries of $(\xi, \lambda^1)$ can also be prespecified and be moved into the constant part.