This article explains how to estimate parameters of the Arbitrage-Free dynamic Nelson-Siegel (AFNS) model (Christensen, Diebold, and Rudebusch; 2009, Christensen, Diebold, and Rudebusch; 2011) using Kalman filter. We estimate not only parameters but also filtered latent factor estimates such as level, slope, and curvature using R code.

1. AFNS model

AFNS model can be expressed as a state state model which consists of measurement equation and state equation as follows.
\begin{align} y_t (\boldsymbol{\tau}) &= B(\boldsymbol{\tau}) X_t + \frac{A(\boldsymbol{\tau})}{\boldsymbol{\tau}} + \epsilon_t (\boldsymbol{\tau}) \\ dX_t &= K^P (\theta^P – X_t) dt + \Sigma dW_t^P \end{align}
\[ B(\boldsymbol{\tau}) = \begin{bmatrix} 1 & \displaystyle \frac{1-e^{- \tau_1 \lambda }}{\tau_1 \lambda } & \displaystyle \frac{1-e^{- \tau_1 \lambda }}{\tau_1 \lambda }-e^{- \tau_1 \lambda } \\ 1 & \displaystyle \frac{1-e^{- \tau_2 \lambda }}{\tau_2 \lambda } & \displaystyle \frac{1-e^{- \tau_2 \lambda }}{\tau_2 \lambda }-e^{- \tau_2 \lambda } \\ ⋮&⋮&⋮\\ 1 & \displaystyle \frac{1-e^{- \tau_N \lambda }}{\tau_N \lambda } & \displaystyle \frac{1-e^{- \tau_N \lambda }}{\tau_N \lambda }-e^{- \tau_N \lambda } \end{bmatrix}, \] \[ y_t (\boldsymbol{\tau}) = \begin{bmatrix} y _{t} ( \tau _{1} )\\y _{t} ( \tau _{2} )\\ ⋮ \\y _{t} ( \tau _{N} )\end{bmatrix}, \epsilon_t (\boldsymbol{\tau}) = \begin{bmatrix} \epsilon_{t} ( \tau _{1} )\\\epsilon_{t} ( \tau _{2} )\\ ⋮ \\\epsilon_{t} ( \tau _{N} )\end{bmatrix}, \] \begin{align} K^P &= \begin{bmatrix} \kappa_{11}^P & 0& 0 \\ 0& \kappa_{22}^P &0 \\ 0& 0& \kappa_{33}^P \end{bmatrix}, \theta^P = \begin{bmatrix} \theta_{1}^P \\ \theta_{2}^P \\ \theta_{3}^P \end{bmatrix}, \\ \Sigma &= \begin{bmatrix} \sigma_{11} & 0 & 0 \\ \sigma_{21} & \sigma_{22} & 0 \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix}, dW_t^P = \begin{bmatrix} dW_{1t}^P \\ dW_{2t}^P \\ dW_{3t}^P \end{bmatrix} \end{align} Christensen et al. (2009, 2011) introduces the yield adjustment term to ensure the no-arbitrage condition for bond price. This yield adjustment term looks very complicated but can be calculated using estimated parameters such as \(\tau\), \(\lambda\), \(\Sigma\) (see Christensen et al. (2009, 2011) for the detailed derivation).
\begin{align} &\scriptstyle\frac{A( τ )} {τ } =A \frac{τ ^{2}} {6} \\ &\scriptstyle+B \left[ \frac{1}{2 λ ^{2}} – \frac{1}{λ ^{3}} \frac{1-e ^{- λ τ }}{τ } + \frac{1}{4 λ ^{3}} \frac{1-e ^{-2 λ τ }}{τ } \right]\\ &\scriptstyle+C \left [ \frac{1}{2 λ ^{2}} + \frac{1}{λ ^{2}} e ^{- λ τ } – \frac{1}{4 λ } τ e ^{-2 λ τ } – \frac{3}{4 λ ^{2}} e ^{-2 λ τ } – \frac{2}{λ ^{3}} \frac{1-e ^{- λ τ }}{τ } + \frac{5}{8 λ ^{3}} \frac{1-e ^{-2 λ τ }}{τ } \right ]\\ &\scriptstyle+D \left [ \frac{1}{2 λ } τ + \frac{1}{λ ^{2}} e ^{- λ τ } – \frac{1}{λ ^{3}} \frac{1-e ^{- λ τ }}{τ } \right ]\\ &\scriptstyle+E \left [ \frac{3}{λ ^{2}} e ^{- λ τ } + \frac{1}{2 λ } τ + \frac{1}{λ } τ e ^{- λ τ } – \frac{3}{λ ^{3}} \frac{1-e ^{- λ τ }}{τ } \right ]\\ &\scriptstyle+F \left [ \frac{1}{λ^2} + \frac{1}{λ ^{2}} e ^{- λ τ } – \frac{1}{2λ ^{2}} {e ^{- 2λ τ }} – \frac{3}{λ ^{3}} \frac{1-e ^{- λ τ }}{τ } + \frac{3}{4λ ^{3}} \frac{1-e ^{-2 λ τ }}{τ } \right ] \end{align} where \(A=coef(1,1)\),\(B=coef(2,2)\),\(C=coef(3,3)\),\(D=coef(1,2)\),\(E=coef(1,3)\),\(F=coef(2,3)\) if we denote \(coef(i,j) = σ_{i1}σ_{j1}+σ_{i2}σ_{j2}+σ_{i3}σ_{j3}\).

Since AFNS model have a continuous-time representation, we should express AFNS model as the discretized form for estimating parameters using discrete time series data as follows. \begin{align} y_t (\boldsymbol{\tau}) &= B(\boldsymbol{\tau}) X_t + \frac{A(\boldsymbol{\tau})}{\boldsymbol{\tau}} + \epsilon_t (\boldsymbol{\tau}) \\ X_t &= (I – e^{-K^P \Delta t}) \theta_P + e^{-K^P \Delta t} X_{t-1} + \eta_{t} \end{align} where \(\Delta t\) is 1/12 for monthly data and 1/52 for weekly data.

AFNS model as a state space model is linear with respect to factors, we can use Kalman filter to estimate parameters. If we use \( \psi_0 = (I – e^{-K^P \Delta t}) \theta_P \) , \( \psi_1 = e^{-K^P \Delta t} \) 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 – A(\boldsymbol{\tau})/\boldsymbol{\tau} \\ 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.

The conditional covariance (\(\Sigma_{\eta}\)) and unconditional covariance (\(\Sigma_{\eta,0}\)) are defined as \begin{align} \Sigma_{\eta} &= \int_{0}^{Δt} exp⁡(-K^p s)ΣΣ^{‘} exp⁡(-K^p s) ds \\ \Sigma_{\eta,0} &= \int_{0}^{\infty} exp⁡(-K^p s)ΣΣ^{‘} exp⁡(-K^p s) ds \end{align} Christensen and Rudebusch (2015) explain the way to calculate the analytical covariance as follows. \begin{align} \Sigma_{\eta} = V \left( \frac{\omega_{ij}}{\lambda_i + \lambda_j}(1-e^{-(\lambda_i + \lambda_j)\Delta t}) \right)_{n \times n} V^{‘} \end{align} where \(V\) and \(\Lambda\) are the eigenvector matrix and the diagonal matrix with eigenvalues respectively from the diagonalization of \(K^P = V\Lambda V^{-1}\). \(\lambda_i\) is the (\(i,i\))-th element of \(\Lambda\) and \(\omega_{ij}\) is the (\(i,j\))-th element of \(\Omega = V^{-1}ΣΣ^{‘}(V^{-1})^{‘}\).

From the equation for the conditional covariance matrix, we can eaily derive the unconditional covariance matrix. \begin{align} \Sigma_{\eta,0} = V \left( \frac{\omega_{ij}}{\lambda_i + \lambda_j} \right)_{n \times n} V^{‘} \end{align} Since Kalman fitler is an iterative procedure, we use \(X_{1|0} = \theta_P \) and \(V_{1|0} = \Sigma_{\eta,0} \) 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 AFNS model

Now we can implement the AFNS model using R code as follows. Our estimation is based on monthly KTB (Korean Treasury Bond) 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.

3. Estimated AFNS model

Running the above R code for the AFNS 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

Christensen, J. H. E., F. X. Diebold, and G. D. Rudebusch, 2009, An Arbitrage-free Generalized Nelson-Siegel Term Structure Model, Econometrics Journal 12, 33~64.

Christensen, J. H. E., F. X. Diebold, and G. D. Rudebusch, 2011, The Arbitrage-free Class of Nelson-Siegel Term Structure Models. Journal of Econometrics 164, 4-20.

Christensen, J. H. E. and G. D. Rudebusch (2015), Analytical Formulas for the Second Moment in Affine Models with Stochastic Volatility, working paper
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