Nelson-Siegel model and rjags R package
We installed and used rjags R package and performed a Bayesian estimation of multiple linear regression as an example in the previous post
In an interesting and practical example, let's run a Bayesian estimation of the Nelson-Siegel yield curve model.
Nelson-Siegel model
The Nelson-Siegel model is widely used in practice for fitting the cross-sectional term structure of interest rates in a parsimonious way. It is so famous and lots of extended models have been emerged based on the this model. It assumes that an instantaneous forward rate is of the following form.
\[\begin{align} f(\tau) = \beta_0 + \beta_1e^{- \tau \lambda } + \beta_2 \tau \lambda e^{- \tau \lambda } \end{align}\]
Averaging through an integration of \(f(\tau)\) results in a continuously compounded spot rate.
\[\begin{align} y(\tau) &= \frac{1}{\tau} \int_{0}^{\tau} f(u) du \\ &= \beta_0 + \beta_1 \left( \frac{1-e^{- \tau \lambda }}{\tau \lambda }\right) + \beta_2 \left(\frac{1-e^{- \tau \lambda }}{\tau \lambda }-e^{- \tau \lambda }\right) \end{align}\]
Here, \(\tau\) is a maturity and \(y(\tau)\) is a continuously compounded spot rate. \(\beta_0, \beta_1, \beta_2\) are coefficients and \(\lambda\) is the shape or time decay parameter.
Likelihood
The likelihood of the Nelson-Siegel model is constructed from the following model specification.
\[\begin{align} Y_t &\sim \text{Normal}(\mu, \psi) \\ \mu &= \beta_0 + \beta_1 \left( \frac{1-e^{- \tau \lambda }}{\tau \lambda }\right) + \beta_2 \left(\frac{1-e^{- \tau \lambda }}{\tau \lambda }-e^{- \tau \lambda }\right) \\ \sigma &= 1/\sqrt{\psi} \end{align}\]
Here, \(\psi\) in \(\text{Normal}(\mu, \psi)\) is not a variance but a precision which is the reciprocal of a variance (\(\psi = \frac{1}{\sigma^2}\)).
Prior
\(\beta_0, \beta_1, \beta_2\) are assumed to have normal priors but \(\psi\) is non-negative and is assumed to have a gamma prior. \(\lambda\) is also non-negative and is assumed to have a uniform prior.
\[\begin{align} \beta_0 &\sim \text{Normal}(y(\tau_{max}), 0.01) \\ \beta_1 &\sim \text{Normal}(y(\tau_{min})-y(\tau_{max}), 0.01) \\ \beta_2 &\sim \text{Normal}(2y(\tau_{mid})-y(\tau_{min})-y(\tau_{max}), 0.01) \\ \lambda &\sim \text{Unit}(0.02,0.15) \\ \psi &\sim \text{Gamma}(0.1,0.001) \end{align}\]
R code
The R code below performs Bayesian estimation of Nelson-Siegel model using a cross-section of yields with the help of rjags R package.
#========================================================# # Quantitative ALM, Financial Econometrics & Derivatives # ML/DL using R, Python, Tensorflow by Sang-Heon Lee # # https://shleeai.blogspot.com #--------------------------------------------------------# # Nelson-Siegel fitted yield curve using rjags #========================================================# graphics.off() # clear all graphs rm(list = ls()) # remove all files from your workspace library(rjags) library(coda) # Prepare data mat <- c(3, 6, 9, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 144, 180, 240) yield <- c(5.638781, 5.857775, 6.062339, 6.232739, 6.533613, 6.586592, 6.604274, 6.672092, 6.643833, 6.636791, 6.621937, 6.608467, 6.608286, 6.631803, 6.640675, 6.569184) vnm <- c("beta0","beta1","beta2","la","sigma") Data <- list(x = mat, y = yield) # for use inside model Data$y_short <- yield[1] Data$y_long <- yield[16] Data$y_mid <- yield[6] # Model string jags.script <- " model{ # priors beta0 ~ dnorm(y_long,0.01) beta1 ~ dnorm(y_short-y_long,0.01) beta2 ~ dnorm(2*y_mid-y_short-y_long,0.01) la ~ dunif(0.02,0.15) psi ~ dgamma(0.1,0.001) # transform sigma <- 1 / sqrt(psi) # likelihood for( i in 1:length(x[])) { y[i] ~ dnorm(mu[i], psi) mu[i] <- beta0 + beta1*((1-exp(-la*x[i]))/(la*x[i]))+ beta2*((1-exp(-la*x[i]))/(la*x[i])-exp(-la*x[i])) } } " mod1 <- jags.model(textConnection(jags.script), data = Data, n.chains = 4, n.adapt = 1000) update(mod1, 4000) mod1.samples <- coda.samples(model = mod1, variable.names = vnm, n.iter = 2000) # plot trace and posterior density for each parameter x11(); plot(mod1.samples[][,c(1,2,3)]) x11(); plot(mod1.samples[][,c(4,5)]) # descriptive statistics sm <- summary(mod1.samples) # correlation cor(mod1.samples[[1]]) # means of posteriors of parameters b0 <- sm$statistics[vnm[1], "Mean"] b1 <- sm$statistics[vnm[2], "Mean"] b2 <- sm$statistics[vnm[3], "Mean"] la <- sm$statistics[vnm[4], "Mean"] c(b0, b1, b2, la) # in-sample fit ypred <- b0 + b1*((1-exp(-la*mat))/(la*mat)) + b2*((1-exp(-la*mat))/(la*mat) - exp(-la*mat)) # graph x11(width = 16/2, height = 9/2); plot(mat, yield, col = "red", lwd=3, pch=16, ylab = "Yield(%)",xlab = "Maturity(month)") lines(mat, ypred, col = "blue", lwd = 3) | cs |
Running this R code produces the following estimation results.
Estimation results also contain of the following posterior distributions for each parameters.
We can generate the fitted yield curve with average coefficients.
Concluding Remarks
This post shows how to use rjags R package to perform Bayesian MCMC estimation of Nelson-Siegel model. Of course, as my setting for priors is not an answer but an alternative proposal, I think more elaborations on setting priors are needed. \(\blacksquare\)
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