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Forecasting Yield Curves using Dynamic Nelson-Siegel model with R code

sang-heon lee
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< !--shlee --> This post explains how to forecast yield curves using Dynamic Nelson-Siegel model given information of estimated parameters.
< !--
–>

Forecasting of Dynamic Nelson-Siegel model



We estimated parameters of dynamic Nelson-Seigel model in the previous post below.


Using estimated parameters in the previous post, let’s forecast yield curves.



Forecast


Forecasting equations of DNS model (\(h=1,…,H\)) consist of the state and measurement equations as follows.

\[\begin{align} X_{t+h} &= \psi_0 + \psi_1 X_{t+h-1} \\ y_{t+h} (\boldsymbol{\tau}) &= B(\boldsymbol{\tau}) X_{t+h} \end{align}\]
Here, \(X_t (h=1)\) is the filtered estimate of a state vector at time \(T\) (end of sample).



Estimated Parameters


From the previous post, we can get the following the filtered state estimate and estimated parameters.

\[\begin{align} \lambda &= 0.4336145 \\ A &= \begin{bmatrix} 0.9909972 & 0 & 0 \\ 0 & 0.9614324 & 0 \\ 0 & 0 & 0.9027326 \end{bmatrix} \\ \mu_X &= \begin{bmatrix} 0.0334947 \\ -0.0114954 \\ -0.0066945 \end{bmatrix} \\ X &= \begin{bmatrix} 0.01743225 \\ -0.002418071 \\ -0.0040720452 \end{bmatrix} \end{align}\]

R code


The R code below implements the above forecasting procedure using estimated parameters of DNS model. Forecasts are performed during the 120-month horizon.

< !--shlee -->
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#========================================================#
# Quantitative ALM, Financial Econometrics & Derivatives 
# ML/DL using R, Python, Tensorflow by Sang-Heon Lee 
#
# https://kiandlee.blogspot.com
#——————————————————–#
# Forecasting in Dynamic Nelson-Siegel model
#========================================================#
 
graphics.off()  # clear all graphs
rm(list = ls()) # remove all files from your workspace
 
# DNS factor loading matrix
NS.B <– function(lambda, tau)
{
    col1 <– rep.int(1,length(tau))
    col2 <– (1–exp(–lambda*tau))/(lambda*tau)
    col3 <– col2–exp(–lambda*tau) 
    return(cbind(col1,col2,col3))
}
 
#—————————    
# estimated parameters
#—————————    
    
    # AR matrix
    A  <– diag(c(0.9909972, 0.9614324, 0.9027326))   
    
    # mean vector
    MU <– c(0.0334947, –0.0114954, –0.0066945) 
    
    lambda <– 0.4336145
 
    Phi0 <– (diag(3)–A)%*%MU 
    Phi1 <– A
    
    # factor loading matrix
    v.mat <– c(0.25, 0.5, 0.75, 1, 1.5, 2,  
               2.5, 3, 4, 5, 7, 10, 20)
    nmat  <– length(v.mat)
    B     <– NS.B(lambda,v.mat)
 
#—————————————–    
# filtered state estimate at time T
#—————————————– 
    
    # final state 
    prevX <– c(0.01743225, –0.002418071, –0.0040720452)
    
#—————————————–    
# forecasting H-horison
#—————————————– 
    # forecasting horizon
    nhor <– 120 # 5-year
    
    # placeholder for predicted states 
    # and yield curve forecasts
    Xf <– matrix(NA, nhor, 3)
    yf <– matrix(NA, nhor, nmat)
    
    # dynamic forecasting
    for(h in 1:nhor) {
        
        # prediction
        Xhat <– Phi0+Phi1%*%prevX 
        
        # forecast yield curve
        y_fit  <– B%*%Xhat
        
        # use forecasted state at h 
        # as previous state at h+1
        prevX <– Xhat
        
        # save
        Xf[h,] <– Xhat 
        yf[h,] <– y_fit
    }
    
    yf
    Xf
 
cs


< !--shlee --> Running this R code produces the following output table of factors and yields forecast.


Factor forecasts are done recursively according to its vector autoregressive process. The following figure draws a conbined graph for the historical factor estimates and their forecasts.


Yields forecasts are calculated based on factor forecasts using the measurement equation. The following figure draws a conbined graph for the historical yield curves and their forecasts.



Concluding Remarks


This post shows how to forecast yield curves using DNS model based on its estimated parameters. For those who are not familiar with this procedure, I hope this will help. \(\blacksquare\)


< !--shlee -->
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