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true_roberto.R
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true_roberto.R
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#setwd("~/Desktop/DTCC_Liquidity")
setwd("C:/Users/rstrepparava/Documents/Projects_2017/DTCC_liquidity_model_development/DTCC_liquidity_results/SARIMAX_Model")
library(forecast)
library(CombMSC)
library(tseries)
library(expm)
library(matrixcalc)
################################################
###### read the data ######
################################################
data = read.csv("cover1.csv",header=TRUE)
data = data[,c(2,3,5,6,7)]
data[,1] = as.Date(data[,1])
total_raw = data[,3] + data[,4] + data[,5]
new = data.frame(data[,1],total_raw)
colnames(new) = c("date","total_raw")
####remove "Sandy"
sandy = which(new[,2]==0)
new = new[-206,]
plot(new,type="l",main="true data",ylab="RLN")
#new = ts(new)
##### center
new[,2] = as.vector(scale(new[,2],center=TRUE,scale=FALSE))/(10^10)
plot(new,type="l",main="true data",ylab="centered RLN")
################################################
###### seperate seasonal component ######
################################################
first = 500
total = nrow(new)-first+1
sigmavec = rep(0,total)
#for (i in 1:length(sigmavec)){
# print(i)
# nstart = first+(i-1)
# newcut = ts(new[1:nstart,2],frequency = 21)
for (i in 1:total){
print(i)
nstart = first+(i-1)
newcut = ts(new[i:nstart,2],frequency = 21)
################################################
###### seperate seasonal component ######
################################################
dnew = stl(newcut, "periodic")
seasonal <- dnew$time.series[,1]
trend <- dnew$time.series[,2]
random <- dnew$time.series[,3]
dseason = newcut-seasonal
#plot(dseason,type="l")
dseason2 = ts(dseason,frequency=1)
#plot(dseason2,type="l")
######################################################
###### represent into AR infinity model ######
######################################################
#armafit <- arma(dseason2, order = c(1, 1))
armafit = arima(dseason2, order=c(1,1,3))
#########Method1: directly use the estimated arma coefficients
arorder = armafit$coef[1]
maorder = armafit$coef[2:4]
#########Method2: use the values from acf and pacf (not correct)
#Switch the phi and the theta in the ARMAtoMA module to get the PIs of the infinite AR process
n = 10
coeff = ARMAtoMA(ar = c(maorder), ma = c(arorder), lag.max=n)
######################################################
###### estimate epsilon t ######
######################################################
x = new[i:nstart,2]
ct = seasonal[length(seasonal)]
pastn = as.matrix(x[(length(seasonal)-n):(length(seasonal)-1)])
et = x[length(x)]-t(as.matrix(coeff))%*%pastn-ct
######################################################
###### construct vector and matrix ######
######################################################
cvec = as.matrix(c(ct,rep(0,(n-1))))
C = rbind(coeff,cbind(diag((n-1)),0))
evec = as.matrix(c(et,rep(0,(n-1))))
######################################################
###### estimation ######
######################################################
#H = -logm(C,method="Higham08")
H = -logm(C,method="Eigen")
mu = solve(diag(n)-C) %*% cvec
StS = C %*% cov(evec %*% t(evec)) %*% t(C)
###vec function returns a column vector that is a stack of the columns of x, an m by n matrix.
middle = kronecker(H,diag(n))+kronecker(diag(n),H)
vecS = solve(middle) %*% vec(StS)
S = matrix(vecS,nrow=n,byrow=FALSE)
eigens = eigen(S)$values
lambda = sqrt(max(eigens[eigens>0]))
######################################################
###### final ######
######################################################
#mufinal = mu[1,]
##sigmafinal = vecS[1,]
sigmavec[i] = lambda
}
################################################################################################
################################################################################################
################################################################################################
###### divide by 10^10 and center it ######
################################################################################################
################################################################################################
################################################################################################
################################################
###### read the data ######
################################################
################################################
################################################
data = read.csv("cover1.csv",header=TRUE)
data = data[,c(2,3,5,6,7)]
data[,1] = as.Date(data[,1])
total_raw = data[,3] + data[,4] + data[,5]
new = data.frame(data[,1],total_raw)
colnames(new) = c("date","total_raw")
####remove "Sandy"
sandy = which(new[,2]==0)
new = new[-206,]
#### divide by 10^10
#new[,2] = new[,2]/(10^10)
#plot(new,type="l",main="true data",ylab="RLN in 10 billion")
##### center
new[,2] = as.vector(scale(new[,2],center=TRUE,scale=FALSE))/(10^10)
plot(new,type="l",main="true data",ylab="centered RLN")
################################################
###### seperate seasonal component ######
################################################
newts = ts(new[,2], frequency = 21)
#dsimu = decompose(simu, "additive")
dnew = stl(newts, "periodic")
seasonal <- dnew$time.series[,1]
trend <- dnew$time.series[,2]
random <- dnew$time.series[,3]
#plot(as.ts(dsimu$seasonal))
#plot(as.ts(dsimu$trend))
#plot(as.ts(dsimu$random))
#plot(dsimu)
plot(seasonal)
plot(trend)
plot(random)
#seasonal = as.ts(dsimu$seasonal)
#trend = as.ts(dsimu$trend)
#random = as.ts(dsimu$random)
dseason = newts-seasonal
plot(dseason,type="l")
dseason2 = ts(dseason,frequency=1)
plot(dseason2,type="l")
######################################################
###### represent into AR infinity model ######
######################################################
#armafit <- arma(dseason2, order = c(0,3))
armafit = arima(dseason2, order=c(1,1,3))
#########Method1: directly use the estimated arma coefficients
arorder = armafit$coef[1]
maorder = armafit$coef[2:4]
#########Method2: use the values from acf and pacf (not correct)
#Switch the phi and the theta in the ARMAtoMA module to get the PIs of the infinite AR process
n = 10
coeff = ARMAtoMA(ar = c(maorder), ma = c(arorder), lag.max=n)
######################################################
###### estimate epsilon t ######
######################################################
x = new[,2]
ct = seasonal[length(seasonal)]
pastn = as.matrix(x[(length(seasonal)-n):(length(seasonal)-1)])
et = x[length(x)]-t(as.matrix(coeff))%*%pastn-ct
######################################################
###### construct vector and matrix ######
######################################################
cvec = as.matrix(c(ct,rep(0,(n-1))))
C = rbind(coeff,cbind(diag((n-1)),0))
evec = as.matrix(c(et,rep(0,(n-1))))
######################################################
###### estimation ######
######################################################
H = -logm(C,method="Eigen")
#H = -logm(C,method="Higham08")
mu = solve(diag(n)-C) %*% cvec
StS = C %*% cov(evec %*% t(evec)) %*% t(C)
###vec function returns a column vector that is a stack of the columns of x, an m by n matrix.
middle = kronecker(H,diag(n))+kronecker(diag(n),H)
vecS = solve(middle) %*% vec(StS)
S = matrix(vecS,nrow=n,byrow=FALSE)
eigens = eigen(S)$values
lambda = sqrt(max(eigens[eigens>0]))
######################################################
###### final ######
######################################################
mufinal = mu[1,]
##sigmafinal = vecS[1,]
sigma = lambda
muvec = rep(mufinal,length(seasonal))
#####plot:
plot(new[,2],type="l",main="True data with 95%",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(muvec,col="red",type="l")
lines(muvec+qnorm(0.975)*sigma,col="blue",type="l")
lines(muvec-qnorm(0.975)*sigma,col="blue",type="l")
###95% confidence interval
mufinal = mu[1,]
muvec = rep(mufinal,length(seasonal))
plot(new[,1:2],type="l",main="True data with 95% Confidence Interval (max error)",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(cbind(new[,1],muvec),col="red",type="l")
lines(cbind(new[,1],muvec+qnorm(0.975)*max(sigmavec)),col="blue",type="l")
lines(cbind(new[,1],muvec-qnorm(0.975)*max(sigmavec)),col="blue",type="l")
###95% confidence interval
mufinal = mu[1,]
muvec = rep(mufinal,length(seasonal))
plot(new[,1:2],type="l",main="True data with 95% Confidence Interval (95% Percentile)",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(cbind(new[,1],muvec),col="red",type="l")
lines(cbind(new[,1],muvec+qnorm(0.975)*as.numeric(quantile(sigmavec, 0.95))),col="blue",type="l")
lines(cbind(new[,1],muvec-qnorm(0.975)*as.numeric(quantile(sigmavec, 0.95))),col="blue",type="l")
try1 = c(rep(0,499),sigmavec)
plot(new[,2],type="l",main="True data with 95%",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(muvec,col="red",type="l")
lines(muvec+qnorm(0.975)*try1,col="blue",type="l")
lines(muvec-qnorm(0.975)*try1,col="blue",type="l")
############################################################################################################
###### backtest ############################################################
############################################################################################################
new = data.frame(new)
new$year = as.numeric(format(new$date,"%y"))
#table(new$year)
sigvecmax95 = rep(0,5)
sigvecmax70 = rep(0,5)
sigvecmax = rep(0,5)
subindex = c("12","13","14","15","16")
for (k in 1:(length(subindex))){
if (k==length(subindex)){
sub1 = subset(new,(year=="16")|(year=="17"))
} else {
yearn = subindex[k]
sub1 = subset(new,year==yearn)
}
first = 220
total = nrow(sub1)-first+1
sigmavec = rep(0,total)
#for (i in 1:length(sigmavec)){
# print(i)
# nstart = first+(i-1)
# newcut = ts(new[1:nstart,2],frequency = 21)
for (i in 1:total){
# print(i)
nstart = first+(i-1)
subcut = ts(sub1[i:nstart,2],frequency = 21)
################################################
###### seperate seasonal component ######
################################################
dnew = stl(subcut, "periodic")
seasonal <- dnew$time.series[,1]
trend <- dnew$time.series[,2]
random <- dnew$time.series[,3]
dseason = subcut-seasonal
#plot(dseason,type="l")
dseason2 = ts(dseason,frequency=1)
#plot(dseason2,type="l")
######################################################
###### represent into AR infinity model ######
######################################################
#armafit <- arma(dseason2, order = c(1, 1))
armafit = arima(dseason2, order=c(1,1,3))
#########Method1: directly use the estimated arma coefficients
arorder = armafit$coef[1]
maorder = armafit$coef[2:4]
#########Method2: use the values from acf and pacf (not correct)
#Switch the phi and the theta in the ARMAtoMA module to get the PIs of the infinite AR process
n = 10
coeff = ARMAtoMA(ar = c(maorder), ma = c(arorder), lag.max=n)
######################################################
###### estimate epsilon t ######
######################################################
x = sub1[i:nstart,2]
ct = seasonal[length(seasonal)]
pastn = as.matrix(x[(length(seasonal)-n):(length(seasonal)-1)])
et = x[length(x)]-t(as.matrix(coeff))%*%pastn-ct
######################################################
###### construct vector and matrix ######
######################################################
cvec = as.matrix(c(ct,rep(0,(n-1))))
C = rbind(coeff,cbind(diag((n-1)),0))
evec = as.matrix(c(et,rep(0,(n-1))))
######################################################
###### estimation ######
######################################################
#H = -logm(C,method="Higham08")
H = -logm(C,method="Eigen")
mu = solve(diag(n)-C) %*% cvec
StS = C %*% cov(evec %*% t(evec)) %*% t(C)
###vec function returns a column vector that is a stack of the columns of x, an m by n matrix.
middle = kronecker(H,diag(n))+kronecker(diag(n),H)
vecS = solve(middle) %*% vec(StS)
S = matrix(vecS,nrow=n,byrow=FALSE)
eigens = eigen(S)$values
lambda = sqrt(max(eigens[eigens>0]))
######################################################
###### final ######
######################################################
#mufinal = mu[1,]
##sigmafinal = vecS[1,]
sigmavec[i] = lambda
}
sigvecmax95[k] = as.numeric(quantile(sigmavec, 0.95))
sigvecmax70[k] = as.numeric(quantile(sigmavec, 0.7))
sigvecmax[k] = max(sigmavec)
#print(sigvecmax)
}
#c(as.numeric(table(new$year)))
try1 = rep(sigvecmax95,c(247,250,250,250,336))
try2 = rep(sigvecmax70,c(247,250,250,250,336))
plot(new[,1:2],type="l",main="True data with 95% (In-sample)",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(cbind(new[,1],muvec),col="red",type="l")
lines(cbind(new[,1],muvec+qnorm(0.975)*try1),col="blue",type="l")
lines(cbind(new[,1],muvec-qnorm(0.975)*try1),col="blue",type="l")
lines(cbind(new[,1],muvec+qnorm(0.975)*try2),col="green",type="l")
lines(cbind(new[,1],muvec-qnorm(0.975)*try2),col="green",type="l")
legend( x="bottomleft",
legend=c("95 perc","70 perc"),
col=c("blue","green"), lwd=1, lty=c(1,1), merge=FALSE)
############################################################################################################
###### table ############################################################
############################################################################################################
data = read.csv("cover1.csv",header=TRUE)
data = data[,c(2,3,5,6,7)]
data[,1] = as.Date(data[,1])
total_raw = data[,3] + data[,4] + data[,5]
new = data.frame(data[,1],total_raw)
colnames(new) = c("date","total_raw")
####remove "Sandy"
sandy = which(new[,2]==0)
new = new[-206,]
(mufinal+qnorm(0.975)*sigvecmax95)*(10^10)+mean(new[,2])
# ############################################################################################################
# ###### volatility ############################################################
# ############################################################################################################
#
# new = data.frame(new)
# new$year = as.numeric(format(new$date,"%y"))
#
# volatility = rep(0,nrow(new))
#
#
# subindex = c("12","13","14","15","16")
#
# for (k in 1:(length(subindex))){
# print(k)
# if (k==length(subindex)){
# sub1 = subset(new,(year=="16")|(year=="17"))
# index = which((new$year=="16")|(new$year=="17"))
# } else {
# yearn = subindex[k]
# sub1 = subset(new,year==yearn)
# index = which(new$year==yearn)
# }
#
# for (i in 31:length(index)){
# volatility[index[i]] = sd(sub1[1:i,2])
# }
# }
#
# volatility[which(volatility==0)]="NA"
#
# index = which(volatility !="NA")
#
# plot(volatility,col="red",type="l",main="Volatility in each year")
#
# plot(new[index,1],volatility[index],type="l",main="True data with 95%",ylab="RLN in 10 billion")
#
#
# plot(new[,1:2],type="l",main="True data with 95% (Out-of-Sample)",ylab="RLN in 10 billion",ylim=c(-3,3))
# lines(cbind(new[,1],muvec),col="red",type="l")
# lines(cbind(new[248:nrow(new),1],muvec[248:nrow(new)]+qnorm(0.975)*try1),col="blue",type="l")
############################################################################################################
###### out-of-sample ############################################################
############################################################################################################
new = data.frame(new)
new$year = as.numeric(format(new$date,"%y"))
#table(new$year)
sigvecmax95 = rep(0,5)
sigvecmax70 = rep(0,5)
subindex = c("12","13","14","15","16")
for (k in 1:(length(subindex))){
if (k==length(subindex)){
sub1 = subset(new,(year=="16")|(year=="17"))
} else {
yearn = subindex[k]
sub1 = subset(new,year==yearn)
}
first = 220
total = nrow(sub1)-first+1
sigmavec = rep(0,total)
#for (i in 1:length(sigmavec)){
# print(i)
# nstart = first+(i-1)
# newcut = ts(new[1:nstart,2],frequency = 21)
for (i in 1:total){
# print(i)
nstart = first+(i-1)
subcut = ts(sub1[i:nstart,2],frequency = 21)
################################################
###### seperate seasonal component ######
################################################
dnew = stl(subcut, "periodic")
seasonal <- dnew$time.series[,1]
trend <- dnew$time.series[,2]
random <- dnew$time.series[,3]
dseason = subcut-seasonal
#plot(dseason,type="l")
dseason2 = ts(dseason,frequency=1)
#plot(dseason2,type="l")
######################################################
###### represent into AR infinity model ######
######################################################
#armafit <- arma(dseason2, order = c(1, 1))
armafit = arima(dseason2, order=c(1,1,3))
#########Method1: directly use the estimated arma coefficients
arorder = armafit$coef[1]
maorder = armafit$coef[2:4]
#########Method2: use the values from acf and pacf (not correct)
#Switch the phi and the theta in the ARMAtoMA module to get the PIs of the infinite AR process
n = 10
coeff = ARMAtoMA(ar = c(maorder), ma = c(arorder), lag.max=n)
######################################################
###### estimate epsilon t ######
######################################################
x = sub1[i:nstart,2]
ct = seasonal[length(seasonal)]
pastn = as.matrix(x[(length(seasonal)-n):(length(seasonal)-1)])
et = x[length(x)]-t(as.matrix(coeff))%*%pastn-ct
######################################################
###### construct vector and matrix ######
######################################################
cvec = as.matrix(c(ct,rep(0,(n-1))))
C = rbind(coeff,cbind(diag((n-1)),0))
evec = as.matrix(c(et,rep(0,(n-1))))
######################################################
###### estimation ######
######################################################
#H = -logm(C,method="Higham08")
H = -logm(C,method="Eigen")
mu = solve(diag(n)-C) %*% cvec
StS = C %*% cov(evec %*% t(evec)) %*% t(C)
###vec function returns a column vector that is a stack of the columns of x, an m by n matrix.
middle = kronecker(H,diag(n))+kronecker(diag(n),H)
vecS = solve(middle) %*% vec(StS)
S = matrix(vecS,nrow=n,byrow=FALSE)
eigens = eigen(S)$values
lambda = sqrt(max(eigens[eigens>0]))
######################################################
###### final ######
######################################################
#mufinal = mu[1,]
##sigmafinal = vecS[1,]
sigmavec[i] = lambda
}
sigvecmax95[k] = as.numeric(quantile(sigmavec, 0.95))
sigvecmax70[k] = as.numeric(quantile(sigmavec, 0.7))
#print(sigvecmax)
}
#c(as.numeric(table(new$year)))
try1 = rep(sigvecmax95[1:4],c(250,250,250,336))
#try2 = rep(sigvecmax70[1:4],c(250,250,250,336))
plot(new[,1:2],type="l",main="True data with 95% (Out-of-Sample)",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(cbind(new[,1],muvec),col="red",type="l")
lines(cbind(new[248:nrow(new),1],muvec[248:nrow(new)]+qnorm(0.975)*try1),col="blue",type="l")
lines(cbind(new[248:nrow(new),1],muvec[248:nrow(new)]-qnorm(0.975)*try1),col="blue",type="l")
#lines(cbind(new[248:nrow(new),1],muvec[248:nrow(new)]+qnorm(0.975)*try2),col="green",type="l")
#lines(cbind(new[248:nrow(new),1],muvec[248:nrow(new)]-qnorm(0.975)*try2),col="green",type="l")
legend( x="bottomleft",
legend=c("95 perc","70 perc"),
col=c("blue","green"), lwd=1, lty=c(1,1), merge=FALSE)
############################################################################################################
###### in-sample and out-of-sample for only 2012 to 2015 ############################################################
############################################################################################################
try1 = rep(sigvecmax95[1:4],c(247,250,250,250))
try2 = rep(sigvecmax[1:4],c(247,250,250,250))
index = which(new$year<=15)
plot(new[index,1:2],type="l",main="True data with 95% (In-sample)",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(cbind(new[index,1],muvec[index]),col="red",type="l")
lines(cbind(new[index,1],muvec[index]+qnorm(0.975)*try1),col="blue",type="l")
lines(cbind(new[index,1],muvec[index]-qnorm(0.975)*try1),col="blue",type="l")
lines(cbind(new[index,1],muvec[index]+qnorm(0.975)*try2),col="green",type="l")
lines(cbind(new[index,1],muvec[index]-qnorm(0.975)*try2),col="green",type="l")
legend( x="bottomleft",
legend=c("95 perc","max"),
col=c("blue","green"), lwd=1, lty=c(1,1), merge=FALSE)
try1 = rep(sigvecmax95[1:3],c(250,250,250))
try2 = rep(sigvecmax[1:3],c(250,250,250))
index1 = which((new$year<=15))
index = which((new$year<=15)&(new$year>=13))
plot(new[index1,1:2],type="l",main="True data with 95% (Out-of-sample)",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(cbind(new[index1,1],muvec[index1]),col="red",type="l")
lines(cbind(new[index,1],muvec[index]+qnorm(0.975)*try1),col="blue",type="l")
lines(cbind(new[index,1],muvec[index]-qnorm(0.975)*try1),col="blue",type="l")
lines(cbind(new[index,1],muvec[index]+qnorm(0.975)*try2),col="green",type="l")
lines(cbind(new[index,1],muvec[index]-qnorm(0.975)*try2),col="green",type="l")
legend( x="bottomleft",
legend=c("95 perc","max"),
col=c("blue","green"), lwd=1, lty=c(1,1), merge=FALSE)
############################################################################################################
###### get the values roberto - start here to the end #############################################
############################################################################################################
###### improve the model ###################################################################################
## https://math.stackexchange.com/questions/89030/expectation-of-the-maximum-of-gaussian-random-variables ##
## https://www.gwern.net/docs/conscientiousness/2008-nadarajah.pdf #########################################
######## in particular the function to compute the pdf of the max ##########################################
############################################################################################################
data = read.csv("cover1.csv",header=TRUE)
data = data[,c(2,3,5,6,7)]
data[,1] = as.Date(data[,1])
total_raw = data[,3] + data[,4] + data[,5]
new = data.frame(data[,1],total_raw)
colnames(new) = c("date","total_raw")
####remove "Sandy"
sandy = which(new[,2]==0)
new = new[-206,]
new[,2] = as.vector(scale(new[,2],center=TRUE,scale=FALSE))/(10^10)
meanrln = mean(new[,2])
##in-sample 95%:
(mufinal + qnorm(0.975)*sigvecmax95)*(10^10)+meanrln
##out-of-sample 95%:
(mufinal + qnorm(0.975)*sigvecmax95[1:4])*(10^10)+meanrln
####only 2012 to 2015
##in-sample 95%:
(mufinal + qnorm(0.975)*sigvecmax95[1:4])*(10^10)+meanrln
##in-sample max:
(mufinal + qnorm(0.975)*sigvecmax[1:4])*(10^10)+meanrln
############################################################################################################
###### filtered EWMA ############################################################
############################################################################################################
new = data.frame(new)
new$year = as.numeric(format(new$date,"%y"))
first = 500
total = nrow(new)-first+1
###using first 150 as average
sigmavecori = rep(0,total)
sigmavecewma = rep(0,total)
lambdaewma = 0.97
avernum = 250
for (i in 1:total){
# print(i)
nstart = first+(i-1)
subcut = ts(new[i:nstart,2],frequency = 21)
################################################
###### seperate seasonal component ######
################################################
dnew = stl(subcut, "periodic")
seasonal <- dnew$time.series[,1]
trend <- dnew$time.series[,2]
random <- dnew$time.series[,3]
dseason = subcut-seasonal
#plot(dseason,type="l")
dseason2 = ts(dseason,frequency=1)
#plot(dseason2,type="l")
######################################################
###### represent into AR infinity model ######
######################################################
#armafit <- arma(dseason2, order = c(1, 1))
armafit = arima(dseason2, order=c(1,1,3))
#########Method1: directly use the estimated arma coefficients
arorder = armafit$coef[1]
maorder = armafit$coef[2:4]
#########Method2: use the values from acf and pacf (not correct)
#Switch the phi and the theta in the ARMAtoMA module to get the PIs of the infinite AR process
n = 10
coeff = ARMAtoMA(ar = c(maorder), ma = c(arorder), lag.max=n)
######################################################
###### estimate epsilon t ######
######################################################
x = new[i:nstart,2]
ct = seasonal[length(seasonal)]
pastn = as.matrix(x[(length(seasonal)-n):(length(seasonal)-1)])
et = x[length(x)]-t(as.matrix(coeff))%*%pastn-ct
######################################################
###### construct vector and matrix ######
######################################################
cvec = as.matrix(c(ct,rep(0,(n-1))))
C = rbind(coeff,cbind(diag((n-1)),0))
evec = as.matrix(c(et,rep(0,(n-1))))
######################################################
###### estimation ######
######################################################
#H = -logm(C,method="Higham08")
H = -logm(C,method="Eigen")
mu = solve(diag(n)-C) %*% cvec
StS = C %*% cov(evec %*% t(evec)) %*% t(C)
###vec function returns a column vector that is a stack of the columns of x, an m by n matrix.
middle = kronecker(H,diag(n))+kronecker(diag(n),H)
vecS = solve(middle) %*% vec(StS)
S = matrix(vecS,nrow=n,byrow=FALSE)
eigens = eigen(S)$values
lambda = sqrt(max(eigens[eigens>0]))
######################################################
###### final ######
######################################################
#mufinal = mu[1,]
##sigmafinal = vecS[1,]
sigmavecori[i] = lambda
}
sigmavecewma[avernum] = quantile(sigmavecori[1:avernum],0.95)
for (i in (avernum+1):length(sigmavecewma)){
sigmavecewma[i] = max(sigmavecewma[avernum],lambdaewma*sigmavecori[i]+(1-lambdaewma)*sigmavecewma[i-1])
}
sigmavecewma2 = rep(0,total)
sigmavecewma2[avernum] = quantile(sigmavecori[1:avernum],0.95)
for (i in (avernum+1):length(sigmavecewma2)){
sigmavecewma2[i] = max(sigmavecewma2[avernum],0.93*sigmavecori[i]+(1-0.93)*sigmavecewma2[i-1])
}
plot(new[(first+avernum-1):nrow(new),1:2],type="l",main="True data with 95% Confidence Interval (EWMA-0.97)",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]),col="red",type="l")
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]+qnorm(0.975)*sigmavecewma[avernum:length(sigmavecewma)]),col="blue",type="l")
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]-qnorm(0.975)*sigmavecewma[avernum:length(sigmavecewma)]),col="blue",type="l")
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]+qnorm(0.975)*as.numeric(quantile(sigmavecori, 0.95))),col="orange",type="l")
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]-qnorm(0.975)*as.numeric(quantile(sigmavecori, 0.95))),col="orange",type="l")
legend( x="bottomleft",
legend=c("EWMA","Asymptotic Mean","Rolling"),
col=c("blue","red","orange"), lwd=1, lty=c(1,1,1), merge=FALSE,cex=0.75)
plot(new[(first+avernum-1):nrow(new),1:2],type="l",main="True data with 95% Confidence Interval (EWMA-0.93)",ylab="RLN in 10 billion",ylim=c(-3,3))
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]),col="red",type="l")
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]+qnorm(0.975)*sigmavecewma2[avernum:length(sigmavecewma2)]),col="blue",type="l")
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]-qnorm(0.975)*sigmavecewma2[avernum:length(sigmavecewma2)]),col="blue",type="l")
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]+qnorm(0.975)*as.numeric(quantile(sigmavecori, 0.95))),col="orange",type="l")
lines(cbind(new[(first+avernum-1):nrow(new),1],muvec[(first+avernum-1):nrow(new)]-qnorm(0.975)*as.numeric(quantile(sigmavecori, 0.95))),col="orange",type="l")
legend( x="bottomleft",
legend=c("EWMA","Asymptotic Mean","Rolling"),
col=c("blue","red","orange"), lwd=1, lty=c(1,1,1), merge=FALSE,cex=0.75)
# ###############no 2016 considered
#
# newcut = subset(new,year<16)
#
# first = 200
# total = nrow(newcut)-first+1
# ###using first 150 as average
# sigmavecori = rep(0,total)
# sigmavecewma = rep(0,total)
# lambdaewma = 0.97
# avernum = 250
#
# for (i in 1:total){
# # print(i)
# nstart = first+(i-1)
# subcut = ts(newcut[i:nstart,2],frequency = 21)
# ################################################
# ###### seperate seasonal component ######
# ################################################
# dnew = stl(subcut, "periodic")
# seasonal <- dnew$time.series[,1]
# trend <- dnew$time.series[,2]
# random <- dnew$time.series[,3]
#
# dseason = subcut-seasonal
# #plot(dseason,type="l")
#
# dseason2 = ts(dseason,frequency=1)
# #plot(dseason2,type="l")
#
# ######################################################
# ###### represent into AR infinity model ######
# ######################################################
# #armafit <- arma(dseason2, order = c(1, 1))
# armafit = arima(dseason2, order=c(1,1,3))
#
# #########Method1: directly use the estimated arma coefficients
#
# arorder = armafit$coef[1]
# maorder = armafit$coef[2:4]
#
# #########Method2: use the values from acf and pacf (not correct)
#
# #Switch the phi and the theta in the ARMAtoMA module to get the PIs of the infinite AR process
#
# n = 10
# coeff = ARMAtoMA(ar = c(maorder), ma = c(arorder), lag.max=n)
#
#
# ######################################################
# ###### estimate epsilon t ######
# ######################################################
#
# x = newcut[i:nstart,2]
#
# ct = seasonal[length(seasonal)]
# pastn = as.matrix(x[(length(seasonal)-n):(length(seasonal)-1)])
# et = x[length(x)]-t(as.matrix(coeff))%*%pastn-ct
#
#
# ######################################################
# ###### construct vector and matrix ######
# ######################################################
#
# cvec = as.matrix(c(ct,rep(0,(n-1))))
# C = rbind(coeff,cbind(diag((n-1)),0))
# evec = as.matrix(c(et,rep(0,(n-1))))
#
#
# ######################################################
# ###### estimation ######
# ######################################################
# #H = -logm(C,method="Higham08")
# H = -logm(C,method="Eigen")
# mu = solve(diag(n)-C) %*% cvec
# StS = C %*% cov(evec %*% t(evec)) %*% t(C)
#
# ###vec function returns a column vector that is a stack of the columns of x, an m by n matrix.
#
# middle = kronecker(H,diag(n))+kronecker(diag(n),H)
# vecS = solve(middle) %*% vec(StS)
#
# S = matrix(vecS,nrow=n,byrow=FALSE)
# eigens = eigen(S)$values
#
# lambda = sqrt(max(eigens[eigens>0]))
#
#
#
# ######################################################
# ###### final ######
# ######################################################
# #mufinal = mu[1,]
# ##sigmafinal = vecS[1,]
# sigmavecori[i] = lambda
#
# }
#
# sigmavecewma[avernum] = quantile(sigmavecori[1:avernum],0.95)
# for (i in (avernum+1):length(sigmavecewma)){
# sigmavecewma[i] = max(sigmavecewma[avernum],lambdaewma*sigmavecori[i]+(1-lambdaewma)*sigmavecewma[i-1])
# }
#
#
# plot(newcut[(first+avernum-1):nrow(newcut),1:2],type="l",main="True data with 95% Confidence Interval (EWMA)",ylab="RLN in 10 billion",ylim=c(-3,3))
# lines(cbind(newcut[(first+avernum-1):nrow(newcut),1],muvec[(first+avernum-1):nrow(newcut)]),col="red",type="l")
# lines(cbind(newcut[(first+avernum-1):nrow(newcut),1],muvec[(first+avernum-1):nrow(newcut)]+qnorm(0.975)*sigmavecewma[avernum:length(sigmavecewma)]),col="blue",type="l")
# lines(cbind(newcut[(first+avernum-1):nrow(newcut),1],muvec[(first+avernum-1):nrow(newcut)]-qnorm(0.975)*sigmavecewma[avernum:length(sigmavecewma)]),col="blue",type="l")
# lines(cbind(newcut[(first+avernum-1):nrow(newcut),1],sigmavecori[avernum:length(sigmavecori)]),col="green",type="l")
#
#
#
# plot(new[first:nrow(new),1:2],type="l",main="True data with 95% (In-sample)",ylab="RLN in 10 billion",ylim=c(-10,10))
# lines(cbind(new[first:nrow(new),1],sigmavec),col="green",type="l")