This article explains how to estimate parameters of the dynamic Nelson-Siegel (DNS) model (Diebold and Li;2006, Diebold, Rudebusch, and Aruoba;2006) using Kalman filter. We estimate not only parameters but also filtered latent factor estimates such as level, slope, and curvature using R code.


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Dynamic Nelson-Siegel model

1. DNS model

The dynamic Nelson Siegel model can be expressed as the state state representation which consists of both measurement and state equation as follows.
\begin{align} y_t (\boldsymbol{\tau}) &= B(\boldsymbol{\tau}) X_t + \epsilon_t (\boldsymbol{\tau}) \\ X_t – \mu_X &= \phi_X (X_{t-1}-\mu_X) + \eta_t \end{align}
\[ B(\boldsymbol{\tau}) = \begin{bmatrix} 1 & \displaystyle \frac{1-e^{- \tau_1 \lambda }}{\tau_1 \lambda } & \displaystyle \frac{1-e^{- \tau_1 \lambda }}{\tau_1 \lambda }-e^{- \tau_1 \lambda } \\ 1 & \displaystyle \frac{1-e^{- \tau_2 \lambda }}{\tau_2 \lambda } & \displaystyle \frac{1-e^{- \tau_2 \lambda }}{\tau_2 \lambda }-e^{- \tau_2 \lambda } \\ ⋮&⋮&⋮\\ 1 & \displaystyle \frac{1-e^{- \tau_N \lambda }}{\tau_N \lambda } & \displaystyle \frac{1-e^{- \tau_N \lambda }}{\tau_N \lambda }-e^{- \tau_N \lambda } \end{bmatrix}, \] \[ y_t (\boldsymbol{\tau}) = \begin{bmatrix} y _{t} ( \tau _{1} )\\y _{t} ( \tau _{2} )\\ ⋮ \\y _{t} ( \tau _{N} )\end{bmatrix}, \epsilon_t (\boldsymbol{\tau}) = \begin{bmatrix} \epsilon_{t} ( \tau _{1} )\\\epsilon_{t} ( \tau _{2} )\\ ⋮ \\\epsilon_{t} ( \tau _{N} )\end{bmatrix}, \] \[ \phi_X = \begin{bmatrix} \phi_L & 0 & 0 \\ 0 & \phi_S & 0 \\ 0 & 0 & \phi_C \end{bmatrix}, \mu_X = \begin{bmatrix} \mu_L \\ \mu_S \\ \mu_C \end{bmatrix} \]
where \(y_{t}(\tau)\) is continuously compounded spot rates of maturity \(\tau\) at time \(t\). \( L_t, S_t, C_t \) are level, slope, curvature factors respectively and its unconditional means and autoregressive coefficients are denoted as \( \mu_L, \mu_S, \mu_C \) and \( \phi_L, \phi_S, \phi_C \) sequentially. \( \lambda \) is an exponential decay parameter. \( \epsilon_{t} \sim N( 0_{N \times 1}, [\sigma_{\tau_1}^2,\sigma_{\tau_2}^2,…,\sigma_{\tau_N}^2] · I_{N \times N}) \) and \( \eta_{t} \sim N( 0_{3 \times 1}, [\sigma_L^2,\sigma_S^2,\sigma_C^2]·I_{3 \times 3}) \) are the multivariate normal distributed random disturbances.

DNS model as a state space model is linear with respect to factors, we can use Kalman filter to estimate parameters with numerical optimization. If we let \( \psi_0 = (I – \phi_X) \mu_X \) , \( \psi_1 = \phi_X\) for initialization, Kalman filtering is represented as the following recursive calculations. \begin{align} X_{t|t-1} &= \psi_0 + \psi_1 X_{t-1|t-1} \\ V_{t|t-1} &= \psi_1 V_{t-1|t-1} \psi_1^{‘} + \Sigma_{\eta} \\ e_{t|t-1} &= y_t – B(\boldsymbol{\tau}) X_t \\ ev_{t|t-1} &= B(\boldsymbol{\tau}) V_{t|t-1} B(\boldsymbol{\tau})^{‘} + \Sigma_{\epsilon}\\ X_{t|t} &= X_{t|t-1} + K_t e_{t|t-1} \\ V_{t|t} &= V_{t|t-1} – K_t B(\boldsymbol{\tau}) V_{t|t-1} \end{align} where \(K_t = V_{t|t-1} B(\boldsymbol{\tau})^{‘} ev_{t|t-1}^{-1}\) is the Kalman gain which reflects the uncertainty of prediction and is used as a weight for updating the time \(t-1\) prediction after observing time \(t\) data.

Since Kalman fitler is an iterative procedure, we use \(X_{1|0} = \mu_X \) and \(V_{1|0} = (I-\phi_X \phi_X^{‘})^{-1} \Sigma_{\eta} \) for initial guess. We use numerical optimization algorithm to search parameters for maximizing the log likelihood function that is constructed from conditional prediction errors (\(e_{t|t-1}\)) and its uncertainties (\(ev_{t|t-1}\)). \begin{align} ln L_t (\boldsymbol{\theta}) &= -\frac{NT}{2} ln(2\pi) – \frac{1}{2} \sum_{t=1}^{T} ln |ev_{t|t-1}| \\ &\quad – \frac{1}{2} \sum_{t=1}^{T} e_{t|t-1}^{‘}(ev_{t|t-1})^{-1}e_{t|t-1} \end{align}

2. R code for DNS model

Now we can implement the DNS model using R code as follows. Our estimation use monthly KTB (Korean Treasury Bond) term structure of zero coupon bonds (spot rates) from January 2011 to December 2019. The data are end-of-month, zero-coupon yields at thirteen maturities: 0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 4, 5, 7, 10, and 20 years.
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3. Estimated DNS model

Running the above R code for the DNS model, we can get the estimated parameters and the latent factor estimates(\(L, S, C\)). The following figure prints out the convergence of the log-likelihood function and estimated parameters with standard errors, t-statistics, and p-values. We use the delta method for statistical inference.


We can also plot the time-varying filtered estimates of latent factors (level, slope, curvature factors).


Reference

Diebold, F., Li, C., 2006. Forecasting the term structure of government bond yields. J. Econ. 130 (2), 337–364.

Diebold, F.X., Rudebusch, G.D., Aruoba, S.B., 2006. The macroeconomy and the yield curve: a dynamic latent factor approach. J. Econ. 131 (1–2), 309–338
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