**Freakonometrics - Tag - R-english**, and kindly contributed to R-bloggers)

In a recent post (here, by @teramonagi), Teramonagi mentioned the use of PCA to model yield curve, i.e. to obtain the three factor, “parallel *shift*“, “*twist*” and “*butterfly*“. As in Nelson & Siegel, if *m* is maturity, is the yield of the curve at maturity *m, *assume that

where β_{0}, β_{1}, β_{2} and τ, are parameters to be fitted

- β
_{0}is interpreted as the long run levels of interest rates (the loading is 1, it is a constant that does not decay) - β
_{1}is the short-term component (it starts at 1, and decays monotonically and quickly to 0);

- β
_{2}is the medium-term component (it starts at 0, increases, then decays to zero); - τ is the decay factor: small values

produce slow decay and can better fit the curve at long maturities,

while large values produce fast decay and can better fit the curve at

short maturities; τ also governs where β_{2}achieves its maximum.

(see e.g. here). Those factors can be obtained using PCA,

term.structure = read.csv("C:\tmp\FRB_H15.csv",

stringsAsFactors=FALSE)

term.structure = tail(term.structure,1000)

term.structure = term.structure[,-1]

label.term = c("1M","3M","6M","1Y","2Y","3Y","5Y"

,"7Y","10Y","20Y","30Y")

colnames(term.structure) = label.term

term.structure = subset(term.structure,

term.structure$'1M' != "ND")

term.structure = apply(term.structure,2,as.numeric)

term.structure.diff = diff(term.structure)

term.structure.princomp = princomp(term.structure.diff)

factor.loadings = term.structure.princomp$loadings[,1:3]

legend.loadings = c("First principal component",

"Second principal component","Third principal component")

par(xaxt="n")

matplot(factor.loadings,type="l",

lwd=3,lty=1,xlab = "Term", ylab = "Factor loadings")

legend(4,max(factor.loadings),

legend=legend.loadings,col=1:3,lty=1,lwd=3)

par(xaxt="s")

axis(1,1:length(label.term),label.term)

> summary(term.structure.princomp)

Importance of components:

Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8

Standard deviation 0.2028719 0.1381839 0.06938957 0.05234510 0.03430404 0.022611518 0.016081738 0.013068448

Proportion of Variance 0.5862010 0.2719681 0.06857903 0.03902608 0.01676075 0.007282195 0.003683570 0.002432489

Cumulative Proportion 0.5862010 0.8581690 0.92674803 0.96577411 0.98253486 0.989817052 0.993500621 0.995933111

using Teramonagi’s

R code, When

generating stress scenarios, the idea is to generate independently

those factors (or components) and then to aggregate them (using the expression given above). With Principal Component Analysis, PCA, we get *orthogonal* components, while with Independent Component Analysis, ICA, we get *independent* components. And independence and null correlation can be rather different. We recently discussed that idea in a paper with Christophe Villa (available soon here).

Consider the following sample

ns=10000

X=runif(ns)

Y=runif(ns)

I=(Y<.25)*(Y<3*X)*(Y>X/3) +

(Y>.75)*(Y<X/3+3/4-1/12)*(Y>3*X-2)+

(Y>.25)*(Y<.75)*(Y<3*X)*(Y>3*X-2)

FACT1=X[I==1]

FACT2=Y[I==1]

DATA=data.frame(FACT1,FACT2)

PCA<-princomp(DATA)

op <- par(mfrow = c(1, 2))

plot(FACT1[1:2000],FACT2[1:2000],main="Principal component analysis",col="black",cex=.2,xlab="",ylab="",xaxt="n",yaxt="n")

arrows(.5,.5,.8,.8,type=2,col="red",lwd=2)

arrows(.5,.5,.2,.8,type=2,col="red",lwd=2)

plot(PCA$scores,cex=.2,main="Principal component analysis",

xaxt="n",yaxt="n")

The PCA obtain the following projections on the two components (drawn in red, below)

> X=PCA$scores[,1];

> Y=PCA$scores[,2];

> n=length(FACT1)

> x=X[sample(1:n,size=n,replace=TRUE)]

> y=Y[sample(1:n,size=n,replace=TRUE)]

> PCA$loadings

Loadings:

Comp.1 Comp.2

FACT1 -0.707 0.707

FACT2 -0.707 -0.707

Comp.1 Comp.2

SS loadings 1.0 1.0

Proportion Var 0.5 0.5

Cumulative Var 0.5 1.0

> F1=0.707*x-.707*y

> F2=0.707*x+.707*y

Hence, with PCA, we have two components, orthogonal, with a triangular distribution, so if we generate them independently, we obtain

which is quite different, compared with the original sample. On the other hand, with ICA,we obtain factors that are really independent….

library(fastICA)

nt=10000

ICA<-fastICA(DATA,2)

m=ICA$K

x=ICA$S[,1]

y=ICA$S[,2]

plot(ICA$S,cex=.2,main="Independent component analysis",

xlab="",ylab="",xaxt="n",yaxt="n")

see below for the graphs, and here for more details,

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