Rotated Nelson-Siegel model
Nyholm (2018) introduced the Rotated Dynamic Nelson-Siegel (RDNS) model with the intention of enhancing the interpretability of yield factors through a more conventional framework. The primary emphasis of this model centers on the short rate component.
Since the factor loading matrix is the key distinguishing difference between DNS and RDNS models, I opt for the period-by-period OLS estimation of the Rotated Nelson-Siegel model for simplicity, rather than the Rotated DNS model.
The well-known Nelson-Siegel (NS) model has the following form (B is the NS factor loading matrix).
\[\begin{align} y(\tau) &= \beta_1 + \beta_2 \left( \frac{1-e^{- \tau \lambda }}{\tau \lambda }\right) + \beta_3 \left(\frac{1-e^{- \tau \lambda }}{\tau \lambda }-e^{- \tau \lambda }\right) \\ &= B \times \beta \end{align}\]
In order to rotate the NS factors from [Level, -Slope, Curvature] into [Short Rate, Slope, Curvature], the NS model is augmented with a rotation matrix, denoted by A, which is chosen in such a way that the desired factor-interpretation emerges, and such that \(I=A^{-1}A\), where I is the identity matrix.
Absolutely, keep in mind that the slope factor is defined as the difference between the long-term and short-term components: slope = long-term - short-term.
\[ \begin{align} \begin{bmatrix} \text{ShortRate} \\ \text{Slope} \\ \text{Curvature} \\ \end{bmatrix} = \begin{bmatrix} 1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} \text{Level} \\ -\text{NS Slope} \\ \text{Curvature} \\ \end{bmatrix} \end{align} \]
In sum, the rotation matrix \(A\) turns the NS yield factors into RNS yield factors as follows.
\[\begin{align} G = B \times A^{-1} \end{align}\]
Nyholm (2018) rotates the factor structure to enable direct parametrization of the short rate process.
\[\begin{align} y(\tau) &= \gamma_1\left(1+ \frac{1-e^{- \tau \lambda }}{\tau \lambda }\right) + \gamma_2 \left( \frac{-1+e^{- \tau \lambda }}{\tau \lambda }\right) \\ &+ \gamma_3 \left(\frac{1-e^{- \tau \lambda }}{\tau \lambda }-e^{- \tau \lambda }\right) \\ &= G \times \gamma \\ &= B \times A^{-1} \times \gamma \\ &= B \times \beta \\ &= \beta_1 + \beta_2 \left( \frac{1-e^{- \tau \lambda }}{\tau \lambda }\right) + \beta_3 \left(\frac{1-e^{- \tau \lambda }}{\tau \lambda }-e^{- \tau \lambda }\right) \end{align}\]
Hence, the RDNS model shares identical statistical properties with the DNS model, rendering it observationally equivalent to the DNS model.
NS and Rotated NS model
The following R code estimates the NS and Rotated NS models by using the period-by-period OLS estimation.
#========================================================# # Quantitative Financial Econometrics & Derivatives # ML/DL using R, Python, Tensorflow by Sang-Heon Lee # # https://shleeai.blogspot.com #--------------------------------------------------------# # Nyholm (2018) Rotated Nelson-Siegel Model #========================================================# graphics.off(); rm(list = ls()) setwd('D:/SHLEE/blog/R/Rotated_NS') # data is located in the following address # 'https://github.com/Financial-Times/yield-curve-sonification/blob/7d131970377380f118fb9a0e1626f5ed1ecd35ca/yield-curve-monthly-data.csv' yield <- as.matrix(read.csv('yield-curve-monthly-data.csv')[,-1]) mat <- c(1, 3, 6, 12, 24, 36, 60, 84, 120, 240, 360) vcol <- c("darkgray", "blue", "hotpink") #----------------------------------------------------------- # NS and Rotated NS models #----------------------------------------------------------- # Factor loading matrix NS_factor_loading <-function(la, m) { B <- cbind( rep(1,length(m)), (1-exp(-la*m))/(la*m), (1-exp(-la*m))/(la*m)-exp(-la*m)) return(B) } RNS_factor_loading <-function(la, m) { # short rate, conventional slope, curvature A1 <- c(1, 1, 0) A2 <- c(0,-1, 0) A3 <- c(0, 0, 1) A <- matrix(cbind(A1, A2, A3), 3, 3, byrow = TRUE) G <- NS_factor_loading(la, m)%*%solve(A) return(G) } # fitted yield curve fit_NS <- function(beta, la, m) { return(NS_factor_loading(la, m)%*%beta) } fit_RNS <- function(gamma, la, m) { return(RNS_factor_loading(la, m)%*%gamma) } # RMSE rmse_NS <- function(beta, la, y, m) { return(sqrt(mean((y - fit_NS(beta, la, m))^2))) } rmse_RNS <- function(gamma, la, y, m) { return(sqrt(mean((y - fit_RNS(gamma, la, m))^2))) } # Cross-sectional OLS est_NS_ols <- function(la, y, m) { B <- NS_factor_loading(la, m) beta <- solve(t(B)%*%B)%*%t(B)%*%y; rmse <- rmse_NS(beta, la, y, m) return(list(beta=beta, la=la, rmse=rmse)) } est_RNS_ols <- function(la, y, m) { G <- RNS_factor_loading(la, m) gamma <- solve(t(G)%*%G)%*%t(G)%*%y; rmse <- rmse_RNS(gamma, la, y, m) return(list(gamma=gamma, la=la, rmse=rmse)) } # period-by-period OLS Estimation est_NS_ols_ts <- function(la, y_all, m) { nobs <- nrow(y_all); nmat <- length(m) rmse <- rep(NA,nobs) beta <- matrix(NA, nobs, 3) for(t in 1:nobs) { lt <- est_NS_ols(la,y_all[t,],m) beta[t,] <- lt$beta; rmse[t] <- lt$rmse } return(list(beta=beta, la=la, rmse=rmse)) } est_RNS_ols_ts <- function(la, y_all, m) { nobs <- nrow(y_all); nmat <- length(m) rmse <- rep(NA,nobs) gamma <- matrix(NA, nobs, 3) for(t in 1:nobs) { lt <- est_RNS_ols(la,y_all[t,],m) gamma[t,] <- lt$gamma; rmse[t] <- lt$rmse } return(list(gamma=gamma, la=la, rmse=rmse)) } # Estimation out_NS <- est_NS_ols_ts (0.0609, yield, mat) out_RNS <- est_RNS_ols_ts(0.0609, yield, mat) # plot x11(width = 16/2, height = 9/2.3) matplot(out_NS$beta, type="l", lty=1, lwd=4, col = vcol, main = paste0("NS model : Avg. RMSE = ", round(mean(out_NS$rmse),4))) legend("bottomright", pch = 16, col = vcol, cex = 0.9, bty = "n", legend=c("Level","-Slope","Curvature")) x11(width = 16/2, height = 9/2.3) matplot(out_RNS$gamma, type="l", lty=1, lwd=4, col = vcol, main = paste0("Rotated NS model : Avg. RMSE = ", round(mean(out_RNS$rmse),4))) legend("bottomright", pch = 16, col = vcol, cex = 0.9, bty = "n", legend=c("Short Rate","Slope","Curvature")) | cs |
The following figure displays estimated 3 factors from the NS model.
The presented figure illustrates the estimated three factors derived from the Rotated NS model. Notably, the initial two factors correspond to the short rate and the conventional slope factor, thus enhancing the interpretability of yield factor fluctuations. Indeed, the fitting results, measured by the average root mean square error (RMSE), are identical across both models, given their observational equivalence.
Rotated NS model with 3-month curvature correction
Nyholm (2018) points out that it may not be desirable to define the short rate process at the absolute minimum of all possible maturity values. In fact, it may be preferable to work with a short rate defined at around 3 months maturity. To achieve this, the rotation matrix, denoted as A, is adjusted through a 3-month curvature correction. For further elaboration, please refer to Nyholm (2018).
#----------------------------------------------------------- # Modified Rotated NS model #---------------------------------------------------------- RNSm_factor_loading<-function(la, m, tau) { # short rate, conventional slope, curvature # with modification in Nyholm (2018) A1 <- c(1, (1-exp(-la*tau))/(la*tau), (1-exp(-la*tau))/(la*tau)-exp(-la*tau)) A2 <- c(1, 0, 0) - A1 A3 <- 1 - A1 A <- matrix(cbind(A1, A2, A3), 3, 3, byrow = TRUE) G <- NS_factor_loading(la, m)%*%solve(A) return(G) } fit_RNSm <- function(gamma, la, m, tau) { return(RNSm_factor_loading(la, m, tau)%*%gamma) } rmse_RNSm <- function(gamma, la, y, m, tau) { return(sqrt(mean((y - fit_RNSm(gamma, la, m, tau))^2))) } est_RNSm_ols <- function(la, y, m, tau) { G <- RNSm_factor_loading(la, m, tau) gamma <- solve(t(G)%*%G)%*%t(G)%*%y; rmse <- rmse_RNSm(gamma, la, y, m, tau) return(list(gamma=gamma, la=la, rmse=rmse)) } est_RNSm_ols_ts <- function(la, y_all, m, tau) { nobs <- nrow(y_all); nmat <- length(m) rmse <- rep(NA,nobs) gamma <- matrix(NA, nobs, 3) # period-by-period OLS estimation for(t in 1:nobs) { lt <- est_RNSm_ols(la,y_all[t,],m, tau) gamma[t,] <- lt$gamma; rmse[t] <- lt$rmse } return(list(gamma=gamma, la=la, rmse=rmse)) } # Estimation out_RNSm <- est_RNSm_ols_ts(0.0609, yield, mat, 3) # plot x11(width = 16/2, height = 9/2.3) matplot(out_RNSm$gamma, type="l", lty=1, lwd=4, col = vcol, main = paste0("Modified Rotated NS model : Avg. RMSE = ", round(mean(out_RNSm$rmse),4))) legend("bottomright", pch = 16, col = vcol, cex = 0.9, bty = "n", legend=c("Short Rate","Slope","Curvature")) | cs |
It is not much different from the case without curvature correction, and there is only a slight change. Naturally, the average RMSE remains unaltered.
Concluding Remarks
This post implements the Rotated Nelson-Siegel model of Nyholm (2018), incorporating the application of a rotation matrix. The objective is to enhance the interpretability of yield factors using a more conventional approach.
Reference
Nyholm, K. (2018). A Rotated Dynamic Nelson-Siegel Model, Economic Notes 47-1, 113-124. \(\blacksquare\)
No comments:
Post a Comment