## SFSbootband (R 2.13.1)

SFSbootband compares the asymptotic and bootstrapped confidence band of the conditional quantile curve for Bank of America and Citigroup weekly returns

Tue, July 31 2012 by Dedy Dwi Prastyo

`-`

`alpha`- (1-confidence level)
`n`- number of observations
`q`- quantile between 0 and 1

- The quantile curve, asymptotic and bootstrapped confidence band

Description: The example is produced for the values q=0.05, alpha=0.95

```rm(list=ls(all=TRUE))
library(quantreg)
library(KernSmooth)
library(foreign)
N=546 ### 260 weekly values = 5 years (52 weeks per year)
seuil=0 ### value from which we start
citi_trunc=numeric(0)
boa_trunc=numeric(0)
for(i in 1:N){boa_trunc[i] <- boa[(length(boa[,1])-N-seuil+i),1]}
for(i in 1:N){citi_trunc[i] <- citi[length(citi[,1])-N-seuil+i,1]}
q1<-numeric(0)
q1band<-numeric(0)
q2<-numeric(0)
q2band<-numeric(0)
VAR1_trunc=numeric(0)
VAR1_trunc <- c(1:N)/N
VAR2_trunc=numeric(0)
VAR2_trunc <- boa_trunc[order(citi_trunc)]
f<-approxfun(density(citi_trunc)\$x,density(citi_trunc)\$y, method="linear")
n <- 546 # number of observations
gridn <- n
q <- 0.05
alpha <- 0.05; # (1-alpha)*100% significance level for CI
bound <- c(min(VAR2_trunc),max(VAR2_trunc))
yuv <- sort(VAR1_trunc) # just sort them for later use # regressor
yur <- VAR2_trunc[order(VAR1_trunc)] # regressand
h2 <- dpill(yuv, yur, gridsize = gridn)
qrh2 <- 2*h2*((q*(1-q)/(dnorm(qnorm(p=q))^2))^(1/5) )
fit2bis <- lprq(VAR1_trunc, VAR2_trunc, h=qrh2, m=n, tau=q)
cc = 1/4 # this is for normal kernel, if quartic kernel, value is  3/2
lambda= 1/2/sqrt(pi) # this is for normal kernel, if quartic kernel, value is    5/7
delta=-log(qrh2)/log(n)
dd = sqrt(2*delta*log(n))+(2*delta*log(n))^(-1/2)*log(cc/2/pi)
h12=(0.5*(1-0.5)/dnorm(qnorm(0.5))^2)^0.2 *2.42*sd(yuv)*n^(-0.2)
if (h12<1) {b2=10*max(h12^5/((q*(1-q)/dnorm(qnorm(q))^2)^0.2 *2.42*sd(yuv)*n^(-0.2))^3, (q*(1-q)/dnorm(qnorm(q))^2)^0.2 *2.42*sd(yuv)*n^(-0.2)/10)} else
b2=10*h12^4/((q*(1-q)/dnorm(qnorm(q))^2)^0.2 *2.42*sd(yuv)*n^(-0.2))^3
fxd<-bkde(yuv, gridsize = gridn, range.x = c(fit2bis\$xx[1], fit2bis\$xx[gridn]),truncate=TRUE)
fl2bis <- vector(length= gridn, mode="numeric")
kernelq <- function(u){dnorm(u,mean=0,sd=1)}    # or, quadratic kernel:    ){(15/16)*((1-u^2)^2)*(-1 <= u)*(u <= 1)}
for (k in 1: gridn){
fl2bis[k]= sum ((kernelq((yuv - fit2bis\$xx[k])/qrh2)*kernelq((yur - fit2bis\$fv[k])/b2)/qrh2/b2))/sum(kernelq((yuv - fit2bis\$xx[k])/qrh2)/qrh2)
}
bandt2bis=(fxd\$y)^(1/2)*fl2bis
cn=log(2)-log(abs(log(1-alpha)))
band2bis=(n*qrh2)^(-1/2)*sqrt(lambda*q*(1-q))*bandt2bis^(-1)*(dd+cn*(2*delta*log(n))^(-1/2))
qrh2
q1<-fit2bis\$fv
q1band<-band2bis
B<-500
"lprq2" <- function(x, y, h, tau, x0) # modified from lprq, s.t. we can specify where to estimate quantiles
{       xx <- x0
fv <- xx
dv <- xx
for(i in 1:length(xx)) {
z <- x - xx[i]
wx <- dnorm(z/h)
r <- rq(y~z, weights=wx, tau=tau, ci=FALSE)
fv[i] <- r\$coef[1.]
dv[i] <- r\$coef[2.]
}
list(xx = xx, fv = fv, dv = dv)}
"lprq3" <- function(x, y, h, x0) # modified from lprq, s.t. we can specify where to estimate quantiles, but "random quantiles"
{       xx <- x0
fv <- xx
dv <- xx
for(i in 1:length(xx)) {
z <- x - xx[i]
wx <- dnorm(z/h)
r <- rq(y~z, weights=wx, tau=runif(1), ci=FALSE)  # "random quantiles" is seen at tau=runif(1)
fv[i] <- r\$coef[1.]
dv[i] <- r\$coef[2.]
}
list(xx = xx, fv = fv, dv = dv)}
fitover<-lprq2(yuv,yur,h=qrh2*n^(4/45),tau=q,x0=yuv)
d <- vector(length= B, mode="numeric") # initilize the bootstrap maximum #e <- vector(length= B, mode="numeric") # initilize the bootstrap maximum
for(jj in 1: B){
fiterror <- lprq3(yuv,(yur-fit2bis\$fv),h=qrh2,x0=yuv)
ystar <- fitover\$fv + fiterror\$fv
fitstar <- lprq(yuv, ystar,h= qrh2,tau=q, m=gridn) #    e[jj] <- max(abs(fitstar\$fv - fitover2\$fv))
d[jj] <- max(abs(bandt2bis*(fitstar\$fv - fitover\$fv)))
print(jj)
}
dstar <- quantile(d, probs = 1-alpha)
dstar <- dstar*bandt2bis^(-1)
plot(citi_trunc, boa_trunc,pch=20,cex=1.2, xlab="C Returns",ylab="BOA Returns",cex.axis=1.2, cex.lab=1.3, lab = c(3,3,0), ylim = c(min(VAR2_trunc)-0.2,max(VAR2_trunc)+0.10),main="") #Returns scatter plot and Quantile Function
lines(sort(citi_trunc), q1/sqrt(f(sort(citi_trunc))), col = "blue2", lwd=3)
lines(sort(citi_trunc), (q1-q1band)/sqrt(f(sort(citi_trunc))),col = "magenta", lty = 2, lwd=4)
lines(sort(citi_trunc), (q1+q1band)/sqrt(f(sort(citi_trunc))),col = "magenta", lty = 2, lwd=4)
lines(sort(citi_trunc), (q1+dstar)/sqrt(f(sort(citi_trunc))),col = "red", lty = 4, lwd=4)
lines(sort(citi_trunc), (q1-dstar)/sqrt(f(sort(citi_trunc))),col = "red", lty = 4, lwd=4)
```