Estimation and Forecast of Dynamic Nelson-Siegel model
I found the following useful Python library for estimating and forecasting the DNS or Svensson (DNSS) model.
In the Jupyter notebook, this library can be installed by running the following command.
1 | !pip install Dynamic-Nelson-Siegel-Svensson-Kalman-Filter | cs |
Correlated-factor DNS model
This Python DNS library use the specification of correlated-factor DNS model instead of the independent-factor model more generally. The correlated-factor DNS model can be expressed as the state space representation which consists of both measurement and state equation as follows.
\begin{align} y_t &= B(\boldsymbol{\tau}) X_t + \epsilon_t \\ X_t - \mu &= A (X_{t-1}-\mu) + \eta_t \end{align}
\[ B(\boldsymbol{\tau}) = \begin{bmatrix} 1 & \displaystyle \frac{1-e^{- \tau_1 \lambda }}{\tau_1 \lambda } & \displaystyle \frac{1-e^{- \tau_1 \lambda }}{\tau_1 \lambda }-e^{- \tau_1 \lambda } \\ 1 & \displaystyle \frac{1-e^{- \tau_2 \lambda }}{\tau_2 \lambda } & \displaystyle \frac{1-e^{- \tau_2 \lambda }}{\tau_2 \lambda }-e^{- \tau_2 \lambda } \\ ⋮&⋮&⋮\\ 1 & \displaystyle \frac{1-e^{- \tau_N \lambda }}{\tau_N \lambda } & \displaystyle \frac{1-e^{- \tau_N \lambda }}{\tau_N \lambda }-e^{- \tau_N \lambda } \end{bmatrix}, \] \[ y_t = \begin{bmatrix} y _{t} ( \tau _{1} )\\y _{t} ( \tau _{2} )\\ ⋮ \\y _{t} ( \tau _{N} )\end{bmatrix}, \epsilon_t = \begin{bmatrix} \epsilon_{t} ( \tau _{1} )\\\epsilon_{t} ( \tau _{2} )\\ ⋮ \\\epsilon_{t} ( \tau _{N} )\end{bmatrix}, \eta_{t} = \begin{bmatrix} \eta_{t}(L) \\ \eta_{t}(S)\\ \eta_{t}(C) \end{bmatrix}, \] \[ X_t = \begin{bmatrix} L_t \\ S_t \\ C_t \end{bmatrix}, \mu = \begin{bmatrix} \mu_L \\ \mu_S \\ \mu_C \end{bmatrix}, A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \]
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 time decay parameter. \( \epsilon_{t} \) follows a multivariate normal distribution with only diagonal variances, in other words, a diagonal covariance matrix.
Taking correlated-factors into account, \( \eta_{t} \) has a non-diagonal variance-covariance matrix \(Q=q \times q^{'}\). \[ q = \begin{bmatrix} q_{11} & q_{12} & q_{13} \\ 0 & q_{22} & q_{23} \\ 0 & 0 & q_{33} \end{bmatrix} \]
Python Jupyter Notebook Code
From the web page above, sample data can be obtained, which is the Diebold paper's monthly U.S yield curve dataset.
Initial guesses for parameters are prepared and this is a rounded version of initial guesses which are provided by the above web page. I redefine an objective function (objf) based on kalman() function in this library to reduce the number of arguments to be passed. I also define a callback function (callbackF) which prints function evaluations periodically to check the progress of this optimization.
The dynamic Nelson-Siegel-Svensson (DNSS) model can also be estimated by setting model = 'S'.
Optimization is done by using optimize.minimize() function of scipy library with 'L-BFGS-B' algorithm. Of course, there are many alternative optimization algorithm such as 'Nelder-Mead', 'BFGS', and 'CG' to name a few. The callback function prints the number of iteration and the function evaluations.
After many iterations, we can get the following results of estimation.
Parameter estimates are stored at x of the resulted object (dns_opt.x).
Concluding Remarks
This post gives a guided-tour of how to use a Python library for estimation of dynamic Nelson-Siegel model. This also can be useful when the dynamic Svensson model needs to be estimated. \(\blacksquare\)
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