5
I hope I understood you correctly and that the following thoughts help you a bit.
Reference point: Univariate curve fitting using splines
With a univariate function $f(x)$ you can perform 1D spline interpolation and require for each (inner) $x_i$-node that:
$$
\begin{align}
\left.f_{i-1}(x)\right|_{x=x_i}&=\left.f_i(x)\right|_{x=x_i} \quad \mathrm{...
2
Is it correct to use the YTM?
Maybe. These days, for accounting reasons, a lot of bonds out there are callable, and the call is in the money, and the YTW is very different from the YTM. You should check whether your bonds are callable before assuming that you can use YTM.
Is it correct to graph it vs. the maturity in months
You introduce unnecessary noise ...
2
I think different researchers might have different thresholds for what they perceive to be "stable." FWIW, the picture below provides our beta estimates going back to 1992 for the US Treasury market:
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