Dynamic Nelson-Siegel Model

#termstructure #economics #finance #SSM #DNS

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.

Main References

Kim, Seung Hyun. (2024). "Asset Pricing and Term Structure Models". WORK IN PROGRESS.

Nelson-Siegel Model

Level, Slope, and Curvature

Suppose the data on the yield curve is given as a panel, where we have yields with maturities from time to . Then, the data is collected in the matrix as where Then, by Litterman and Scheinkman (1991), the first three principle components from the PCA (principal components analysis) of the yield curve can be understood as follows:

Pasted image 20240710223421.png

level factor: represents the average of yields for all maturities, an intercept for the yield curve, or long end of the yield curve.
slope factor: represents yield spread, a difference between long and short term yields. or, the short end of the yield curve.
curvature factor: determines how curved the yield curve is, or the middle end of the yield curve.

Model

In the Nelson-Siegel model, we assume that the yields are determined by three latent factors and the loadings of these factors are assumed so that the three factors always represent the level, slope, and curvature factor.

Proposition (Nelson-Siegel restriction).

In the Nelson-Siegel model, a yield that has periods left to maturity at time is determined as a linear combination of the level factor , the slope factor , and the curvature factor by following manner: where is called as a decay parameter, and represents the factor loadings.

Proof.Since the level factor is already a constant, it is left to show that the second and third factors represent the slope and curvature factor, respectively.

From the first derivative is where the last inequality holds since . Thus is decreasing in for any fixed .

Next, from the first derivative is with respect to .
Now, let be a solution for the equation then since for any , and for any , is maximized when This shows that the decay parameter determines the middle end of the yield curve. □

Note that, as the middle end of the yield curve is usually between 24 month and 36 month, a standard value for the decay parameter presented in Diebold and Li (2006) is 0.0609, which corresponds to a peak of around 30 months.

Estimation of NS

See Factor Models Factor Models for the detailed explanation for the static factor model.

Let be a data matrix of yield curve. Then the yields of time can be written as where contains the measurement error,

Since we only have to estimate a single parameter for the period , it is convenient to first concentrate out the factors, and then obtain an estimator of .

The mean squared error is given as where the last equality holds since and are both scalar.

Now for any , let the minimizer of with respect to be denoted as . Then, the F.O.C. is which are collected for as Then the concentrated objected function is Now it is equivalent to we find our estimator of as the solution for the following problem where is a small positive value that ensures the above problem has a solution . Since below indicates that yields of maturity 10 years or longer be the middle end of the yield curve, in practice, we let .

Dynamic Nelson-Siegel Model

Model

The Dynamic Nelson-Siegel (DNS) model was developed by Diebold and Li (2006), as a special version of the NS model where the dynamics of the factors is specified.

Similar to the previous, setting we have

Proposition 1 (Nelson-Siegel restriction).

and for the we now employ a state-space form as
where

is vector collecting the sample yields at time .
is matrix of factor loadings
is a vector of mean and variance of measurement error terms.
is a vector of mean and variance , where the dimension of is allowed to be smaller than the number of factors .

Estimation of DNS

We use State-Space Model State-Space Model to estimate DNS model, applying the two-step method in Doz, Giannone and Reichlin (2011).

Estimate measurement equation via PC:
We first estimate the decay parameter , factors , and idiosyncratic variance via least squares as follows
Estimate transition equation:
Now using the estimated factors , we estimate the parameters of the transition equation , , and as follows
Smoothed estimates of the factors:
Given the estimated parameters and using State-Space Model > Kalman Filter State-Space Model > Kalman Filter and State-Space Model > Kalman Smoother State-Space Model > Kalman Smoother, we can obtain the smoothed version of the factors and their variance denoted as

Consistency

Remark that and

The detailed proof will be given here here.