Parameter Constraints for Nelson–Siegel Yield Curve Models
Summary
The document discusses fitting Nelson–Siegel and Nelson–Siegel–Svensson yield curves by minimizing squared differences between observed and fitted yields. It describes the decay parameters as positive, while the beta coefficients are generally unrestricted real values because they govern the curve’s level, slope, and curvature. It notes that a positive long-run level is common and that an additional condition on the short-end combination of coefficients may be used when positive short rates are required.
For the Svensson extension, the account adds another curvature coefficient and a second positive decay parameter. These are presented as practical restrictions rather than universal rules: constraints can depend on the curve and market, and difficult curves may be hard to fit due to local optima and collinearity. The answer recommends considering alternative numerical optimization methods for such cases, but gives no comparative results or detailed implementation. The guidance is therefore a starting point for calibration, not a guarantee of a stable or economically appropriate fit.
Key ideas
- Nelson–Siegel calibration can minimize squared errors between observed and fitted yields.
- The decay parameter is constrained to be positive, while the beta coefficients are generally allowed to take real values.
- A positive long-run level is common, and extra restrictions may be used to enforce positive short rates.
- The Svensson model adds a curvature coefficient and a second positive decay parameter.
- Local optima and collinearity can complicate fitting, so constraints and optimization methods may need to reflect the curve being fitted.
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Full text
# Parameters in Nelson-Siegel model and Nelson-Siegel-Svensson model
# Parameters in Nelson-Siegel model and Nelson-Siegel-Svensson model
I am trying to determine the parameters for the Nelson Siegel and Nelson Siegel Svensson model and try to solve
SE=$\sum_{i=1}^{n_{i}}(y_{t_{i}}-\hat{y}_{t_{i}}(X))^{2}$
where $y_{t_{i}}$ denotes the actual yield for the period $i$ and $\hat{y}_{t_{i}}$ denotes approximated value for period $i$.
How am I supposed to set the parameters restriction of Nelson Siegel model and Nelson Siegel Svensson model? I read several articles, but restrictions were set differently. My yields are in percents.
I know that $\lambda$ shoud be positive, but what about others?
## Answer by Hans-Peter Schrei (score 7, accepted)
https://quant.stackexchange.com/a/75297
The Nelson-Siegel model has four parameters: $\beta_0$, $\beta_1$, $\beta_2$, and $\lambda$. These parameters have the following restrictions:
- $\beta_0$, $\beta_1$ and $\beta_2$ can be any real numbers. They don't have specific restrictions, as they determine the level, slope, and curvature of the term structure. However, $\beta_0$ is usually positive since it represents the long-term yield level. Additionally, $\beta_1 + \beta_2 > 0$ might be required in order to guarantee that short term rates are positive.
- $\lambda$ should be positive, as you mentioned, to ensure the model's stability and smoothness.
The Nelson-Siegel-Svensson model extends the Nelson-Siegel model by introducing two additional parameters ($\beta_3$ and $\lambda_2$) to improve the model's flexibility. The restrictions for this model are:
- $\beta_0$, $\beta_1$, $\beta_2$ and $\beta_3$ can be any real numbers. Similar to the Nelson-Siegel model, $\beta_0$ is usually positive. $\beta_3$ determines the additional curvature of the term structure. As above, $\beta_0$ is usually positive since it represents the long-term yield level. Additionally, $\beta_1 + \beta_2 > 0$ might be required in order to guarantee that short term rates are positive.
- $\lambda_1$ and $\lambda_2$ should both be positive. They ensure the stability and smoothness of the model.
In practice, you can use optimization techniques (e.g., nonlinear least squares) to estimate the model parameters by minimizing the sum of squared errors (SE) between the actual yields and the approximated yields. The restrictions on $\lambda$ (in Nelson-Siegel) and $\lambda_1$ and $\lambda_2$ (in Nelson-Siegel-Svensson) can be enforced during the optimization process. In some cases, you may find it useful to apply additional constraints on the parameters based on the specific characteristics of the yield curve or market.
Update
In the comments, you give an example of a particular term structure with an outlier on the first maturity. If I am not mistaken, your question is how to constrain Nelson-Siegel or Nelson-Siegel-Svensson in such a way that it is possible to calibrate it to problematic curves such as the given one.
However, constraining the model for such a custom situation will likely not succeed due to the difficulty of determining an initial starting point that does not lead to a local optimum and collinearity considerations. Rather the way forward lies in a different methodology.
For further reading on this, I would highly recommend Numerical Methods and Optimization in Finance, Chapter 14, specifically section 14.1.2 (1). An implementation is described in (2).
This implementation gives the following fit with the constraints mentioned above:
(1) Manfred Gilli, Dietmar Maringer, and Enrico Schumann. (2019). Numerical Methods and Optimization in Finance (2nd ed.). Academic Press.
(2) https://cran.r-project.org/web/packages/NMOF/vignettes/DEnss.pdfShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.