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functions.R
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functions.R
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# CONTAINS CONSTRUCTED FUNCTIONS THROUGHOUT THE THESIS. NOT COMMENTED YET.
#
#
#
# DOES NOT CONTAIN CODE FROM LAST YEARS PROJECT.
#
#
#
#############################################################################
########################################################################################################################
#
# RISKMETRICS FILTERS
#
########################################################################################################################
ewma.filter <- function(x, half.life, correlation = F, realized = F, lambda = NULL){
iT <- nrow(x)
covar <- array(0L, dim = c(ncol(x),ncol(x), iT))
#unconditional covariance for first month.
if(realized){
covar[,,1] <- realCov(x[1:100, ])
}
covar[,,1] <- cov(x[1:100, ])
#calculation for half-life: lmd^t = 0.5 (half-life def) <=> t = ln(0.5)/ln(lmd) <=> lmd = exp(ln(0.5)/t)
if(is.null(lambda)){
lambda <- exp(log(0.5)/half.life)
}
print(sprintf("Lambda: %s", lambda))
half.life <- log(0.5)/log(lambda)
print(sprintf("half life: %s", half.life))
for(i in 2:iT){
covar[,,i] <- lambda * covar[,,i-1] + (1-lambda) * t(x[i-1,])%*%x[i-1,]
}
if(correlation){
corr <- array(0L, dim = c(ncol(x),ncol(x), iT))
#corr[,,1] <- cor(x[1:100, ])
for(i in 1:iT){
d <- sqrt(diag(covar[,,i]))
d <- diag(d, ncol(covar), ncol(covar))
corr[,,i] <- inv(d) %*% covar[,,i] %*% inv(d)
}
return(corr)
}
return(covar)
}
ewma.filter2006 <- function(data, correlation = F, realized = FALSE, tau1 = NULL, rho = NULL){
#Reference values found in RiskMetrics 2006:
#Rho and tau1 controls the half-life construction in the 1993 model.
#set Rho =1 and vary tau1 to vary the half-life.
if(is.null(tau1)){
tau1 <- 4
}
if(is.null(rho)){
rho <- sqrt(2)
}
tau0 <- 1560
taumax <- 512
if(is.null(tau1) && is.null(rho)){
kmax <- round(log(taumax/tau1)/log(rho))
}
else{kmax <- 14 }
#The one business day horizon at which
#the data are available is denoted by dt.
dt <- 1
tau <- numeric(kmax)
mu <- numeric(kmax)
iT <- nrow(data)
vol <- array(0L, dim = c(ncol(data), ncol(data), iT))
volk <- list()
weights <- numeric(kmax)
if(realized){
vol[,,1] <- realCov(data[1:100, ])
}
vol[,,1] <- cov(data[1:100, ])
for(k in 1:kmax){
#now it is initialized using sample covariance
tau[k] <- tau1 * rho^(k-1)
mu[k] <- exp(-dt/tau[k])
weights[k] <- 1-log(tau[k]/tau0)
#Calculate recursion of kth EWMA.
#need to initialize the vol formula. He initializes it using a backcast??
for(t in 2:iT){
vol[,,t] <- mu[k] * vol[,,t-1] + (1 - mu[k]) * t(data[t-1,])%*%data[t-1, ]
}
#saves kth recursion in list.
volk[[k]] <- vol
}
#we pre-calculated the weights and made them sum to 1 by normalizing by the sum of the weights:
weigths <- weights / sum(weights)
#efficient vol is now a weighted sum of the k EWMA:
effvol <- array(0L, dim=c(ncol(data), ncol(data), iT))
for(k in 1:kmax){
effvol <- effvol + weights[k] * volk[[k]]
}
if(correlation){
corr <- array(0L, dim = c(ncol(data),ncol(data), iT))
#corr[,,1] <- cor(x[1:100, ])
for(i in 1:iT){
d <- sqrt(diag(effvol[,,i]))
d <- diag(d, ncol(effvol), ncol(effvol))
corr[,,i] <- inv(d) %*% effvol[,,i] %*% inv(d)
}
return(corr)
}
return(effvol)
}
ewma.filter2006.realized <- function(covariance, correlation = F, tau1 = NULL, rho = NULL){
#Reference values found in RiskMetrics 2006:
#Rho and tau1 controls the half-life construction in the 1993 model.
#set Rho =1 and vary tau1 to vary the half-life.
if(is.null(tau1)){
tau1 <- 4
}
if(is.null(rho)){
rho <- sqrt(2)
}
tau0 <- 1560
taumax <- 512
if(is.null(tau1) && is.null(rho)){
kmax <- round(log(taumax/tau1)/log(rho))
}
else{kmax <- 14 }
#The one business day horizon at which
#the data are available is denoted by dt.
dt <- 1
tau <- numeric(kmax)
mu <- numeric(kmax)
iT <- length(covariance[1,1,])
vol <- array(0L, dim = c(2, 2, iT))
volk <- list()
weights <- numeric(kmax)
sampleR <- matrix(
c(mean(covariance[1,1,1:100], na.rm = T),
mean(covariance[2,1,1:100], na.rm = T),
mean(covariance[2,1,1:100], na.rm = T),
mean(covariance[2,2,1:100], na.rm = T)),
ncol=2, nrow=2)
#initializing by sample mean of realized covariance
vol[,,1] <- sampleR
for(k in 1:kmax){
#now it is initialized using sample covariance
tau[k] <- tau1 * rho^(k-1)
mu[k] <- exp(-dt/tau[k])
weights[k] <- 1-log(tau[k]/tau0)
#Calculate recursion of kth EWMA.
#need to initialize the vol formula. He initializes it using a backcast??
for(t in 2:iT){
vol[,,t] <- mu[k] * vol[,,t-1] + (1 - mu[k]) * covariance[,,t-1]
}
#saves kth recursion in list.
volk[[k]] <- vol
}
#we pre-calculated the weights and made them sum to 1 by normalizing by the sum of the weights:
weigths <- weights / sum(weights)
#efficient vol is now a weighted sum of the k EWMA:
effvol <- array(0L, dim=c(2, 2, iT))
for(k in 1:kmax){
effvol <- effvol + weights[k] * volk[[k]]
}
if(correlation){
corr <- array(0L, dim = c(ncol(data),ncol(data), iT))
#corr[,,1] <- cor(x[1:100, ])
for(i in 1:iT){
d <- sqrt(diag(effvol[,,i]))
d <- diag(d, ncol(effvol), ncol(effvol))
corr[,,i] <- inv(d) %*% effvol[,,i] %*% inv(d)
}
return(corr)
}
return(effvol)
}
ewma.filter.realized <- function(cov, half.life, correlation = F, lambda = NULL, trace){
iT <- length(cov[1,1,])
covar <- array(0L, dim = c(2,2, iT))
#unconditional covariance for first month.
sampleR <- matrix(
c(mean(cov[1,1,1:100], na.rm = T),
mean(cov[2,1,1:100], na.rm = T),
mean(cov[2,1,1:100], na.rm = T),
mean(cov[2,2,1:100], na.rm = T)),
ncol=2, nrow=2)
covar[,,1] <- sampleR
#calculation for half-life: lmd^t = 0.5 (half-life def) <=> t = ln(0.5)/ln(lmd) <=> lmd = exp(ln(0.5)/t)
if(is.null(lambda)){
lambda <- exp(log(0.5)/half.life)
}
half.life <- log(0.5)/log(lambda)
if(trace == 1){
print(sprintf("Lambda: %s", lambda))
print(sprintf("half life: %s", half.life))
}
for(i in 2:iT){
covar[,,i] <- lambda * covar[,,i-1] + (1-lambda) * cov[,,i-1]
}
if(correlation){
corr <- array(0L, dim = c(2,2, iT))
#corr[,,1] <- cor(x[1:100, ])
for(i in 1:iT){
d <- sqrt(diag(covar[,,i]))
d <- diag(d, ncol(covar), ncol(covar))
corr[,,i] <- inv(d) %*% covar[,,i] %*% inv(d)
}
return(corr)
}
return(covar)
}
min.qlike.riskmetrics <- function(cov, half.life, proxy, correlation = F, lambda){
filter <- ewma.filter.realized(cov, half.life,F, lambda)
qlikes.riskmetrics <- numeric()
for(i in 1:length(filter[1,1,])){
qlikes.riskmetrics[i] <- QLIKE(filter[,,i], proxy, 2)
}
return(mean(qlikes.riskmetrics))
}
########################################################################################################################
#
# REALIZED AND INTERNAL FUNCTIONS FOR REALIZED MEASURES
#
########################################################################################################################
realsemicov <- function(matrix,type, correlation = FALSE){
# Realized SemiCovariance estimator for 1 day.
#
# Args:
#
# matrix: A matrix of returns. Nxk
#
# N: number of intraday returns, k: number of asset dimension, T: number of days.
realcovariance <- matrix()
positive <- matrix * (matrix >= 0)
negative <- matrix * (matrix <= 0)
if(type =='P'){
realcovariance <- t(positive) %*% (positive)
}
if(type == 'N'){
realcovariance <- t(negative) %*% (negative)
}
if(type == 'M'){
realcovariance <- t(positive) %*% negative + t(negative) %*% positive
}
if(type == 'hM'){
realcovariance <- t(positive) %*% negative
}
if(correlation){
d <- sqrt(diag(realcovariance))
d <- diag(d, ncol(realcovariance), ncol(realcovariance))
corr <- inv(d) %*% realcovariance %*% inv(d)
return(corr)
}
return( realcovariance )
}
bandwidthH <- function(list, sparsedata){
n <- as.vector(sapply(list,length))
c <- ((12)^2/0.269)^(1/5)
w <- numeric()
IV <- numeric()
for(i in 1:length(list)){
w[i] <- 1/(2*n[i]) * realCov(list[[i]])
IV[i] <- realCov(sparsedata[[i]])
}
e <- w/(IV/n)
H <- c * e^(4/5) * n^(3/5)
return(H)
}
########################################################################################################################
#
# GENERAL FUNCTIONS
#
########################################################################################################################
riskparity_2dim <- function(matrix, risktarget, rt = F){
#
#
#
#NOTE TO YOURSELF: Palomar uses non-sqrt portrisk, which gives
#reasonable risk for the unlevered risk-parity portfolio. Using sqrt
#
#REMEMBER TO USE COV(),SD() etc. FOR CLOSE-TO-CLOSE RETURNS, SINCE REALCOV OVERESTIMATES
#THE COVARIANCE FOR CLOSE-TO-CLOSE RETURNS
bonds <- matrix[1,1]
stocks <- matrix[2,2]
w_1 <- sqrt(bonds)^-1 / (sqrt(stocks)^-1 + sqrt(bonds)^-1)
w_2 <- sqrt(stocks)^-1 / (sqrt(stocks)^-1 + sqrt(bonds)^-1)
w <- matrix(c(w_1, w_2), ncol=1, nrow=2) #dimnames = list(c(), c("Bond", "Stock"))
#Palomar uses portfolio variance as portfolio risk, thus no sqrt.
portrisk <- as.numeric(sqrt(t(w) %*% (matrix) %*% w))
riskcont <- w * (matrix %*% w)/portrisk
relativeriskcont <- (w * (matrix %*% w)) / portrisk^2
if(rt){
alpha <- risktarget / portrisk
w_new <- w %*% alpha
w_new <- matrix(c(w_new[1], w_new[2]), ncol=1, nrow=2)
#here is sqrt, while palomar uses variance, no sqrt.
portrisk <- as.numeric(sqrt((t(w_new) %*% matrix %*% w_new))) #gives marginal risk for each asset.
riskcont <- (w_new * (matrix %*% w_new)) / portrisk
w_riskfree <- uniroot(function(x) colSums(w_new)+x-1, interval = c(-100,100), extendInt="yes")$root
w_new <- matrix(c(w_new, w_riskfree), ncol=1, nrow=3)
rownames(w_new) <- c("TLT", "SPY", "riskfree")
lout <- list(w_new, portrisk, riskcont, alpha)
names(lout) <- c("w", "portrisk", "riskcont", "alpha")
return(lout)
}
w_riskfree <- uniroot(function(x) colSums(w)+x-1, interval = c(-100,100), extendInt="yes")$root
w <- matrix(c(w_1, w_2, w_riskfree), ncol = 1, nrow = 3)
lout <- list(w, portrisk, riskcont, relativeriskcont)
names(lout) <- c("w", "portrisk", "riskcont", "relativeriskcont")
return(lout)
}
minvar <- function(Covar){
ones <- matrix(rep(1, ncol(Covar)), ncol=1, nrow=ncol(Covar))
t <- try(inv(Covar), silent = F)
if("try-error" %in% class(t)){return(matrix(rep(NaN,ncol(Covar)), ncol=2, nrow =1))}
else{
w1 <- t %*% ones
w2 <- t(ones) %*% t %*% (ones)
w <- w1 %*% w2^-1
return(w)
}
}
#Kevin sheppard 2 sided hessian function:
hessian_2sided <- function(fun, theta, ...){
f <- fun(params = theta, ...)
h <- 1e-5 * abs(theta)
thetah <- theta + h
h <- thetah - theta
k <- length(theta)
h <- diag(h)
fp <- numeric(k)
fm <- numeric(k)
for (i in 1:k){
fp[i] <- - fun(params =theta+h[i, ], ...)$fulldLLK #negating it since sum is
fm[i] <- - fun(params =theta-h[i, ], ...)$fulldLLK
}
fpp <- matrix(0L, nrow = k, ncol = k)
fmm <- matrix(0L, nrow = k, ncol = k)
for(i in 1:k){
for(j in i:k){
fpp[i,j] <- fun(params = theta + h[i, ] + h[j, ], ...)$fulldLLK
fpp[j,i] <- fpp[i,j]
fmm[i,j] <- fun(params = theta - h[i, ] - h[j, ], ...)$fulldLLK
fmm[j,i] <- fmm[i,j]
}
}
hh <- diag(h)
hh <- matrix(hh, nrow = k, ncol = 1)
hh <- hh %*% t(hh)
H <- matrix(0L, nrow = k, ncol = k)
for(i in 1:k){
for(j in i:k){
H[i,j] <- (fpp[i,j] - fp[i] - fp[j] + f$fulldLLK + f$fulldLLK - fm[i] - fm[j] + fmm[i,j])/hh[i,j]/2
H[j,i] <- H[i,j]
}
}
return(H)
}
standardized.res <- function(returns, variances){
#variances should be a matrix of ncol = number of assets, nrow = days.
lT <- ncol(returns)
rT <- nrow(returns)
res <- matrix(0L, nrow = rT, ncol = lT)
for(i in 1:lT){
for(j in 1:rT){
res[j,i] <- returns[j,i] * (sqrt(variances[j,i]))^-1
}
}
return(res)
}
standardized.res.intraday <- function(returns, variances){
#variances should be a matrix of ncol = number of assets, nrow = days.
lT <- ncol(returns)
rT <- nrow(returns)
res <- matrix(0L, nrow = rT, ncol = lT)
variances <- matrix(variances, nrow = 1, ncol = lT)
variances <- matrix(rep(variances[1, ],rT), nrow =rT, ncol=lT, byrow = T)
for(i in 1:lT){
for(j in 1:rT){
res[j,i] <- returns[j,i] * (sqrt(variances[j]))^-1
}
}
return(res)
}
#UNIVARIATE INSANITY FILTER!
volatility.insanity.filter <- function(forecasts, lb, ub, adjusted_forecast){
ut1 <- forecasts
items <- sum(forecasts < lb) + sum( forecasts > ub)
ut1[forecasts < lb] = adjusted_forecast
ut1[forecasts > ub] = adjusted_forecast
lOut <- list()
lOut[["vol"]] <- ut1
lOut[["items"]] <- items
return(lOut)
}
RTQ <- function(rData) {
returns <- as.vector(as.numeric(rData))
n <- length(returns)
tq <- n * (n/(n-2)) *((2^(2/3) * gamma(7/6) * gamma(1/2)^(-1))^(-3)) * sum(abs(returns[1:(n - 2)])^(4/3) *
abs(returns[2:(n-1)])^(4/3) * abs(returns[3:n])^(4/3))
return(tq)
}
#for univariate HAR models.
mlag <- function(x,n=1,init=0){
xlag <- matrix(0L, ncol = ncol(x)*n, nrow = nrow(x))
icnt <- 0
for(i in 1:ncol(x)){
for(j in 1:n){
xlag[(j+1):nrow(x), icnt+j] <- x[1:(nrow(x)-j),i]
}
icnt <- icnt+n
}
xlag[xlag == 0] <- init
return(xlag)
}
#requires pracma
#takes entire array
invsquarerootmat <- function(Covar){
iT <- length(Covar[1,1,])
d <- ncol(Covar[,,1])
isrmat <- array(0L, dim = c(d,d,iT))
for(i in 1:iT){
isrmat[,,i] <- solve(sqrtm(t(chol(Covar[,,i])))$B %*% sqrtm(chol(Covar[,,i]))$B)
}
return(isrmat)
}
turnover <- function(weights, portret, assetret){
assets <- ncol(weights)
init.weight <- rep(0, assets)
turnover <- numeric()
turnover[1] <- abs(sum(weights[1, ])) #only when initial weights are zero.
for(i in 2:(nrow(portret)-1)){
turnover[i] <- sum(abs( weights[i, ] - weights[i-1, ] * (1 + assetret[i-1, ]) * (1+portret[i-1, ])^-1 ))
}
return(turnover)
}
portconcentration <- function(weights){
concentration <- numeric()
for(i in 1:(nrow(weights)-1)){
concentration[i] <- sqrt(sum(weights[i, ]^2))
}
return(concentration)
}
########################################################################################################################
#
# DRD-HAR MODEL AND CORRELATION MODELS.
#
########################################################################################################################
minimizingfunc <- function(data, params){
dA <- params[1]
dB <- params[2]
dC <- params[3]
y <- as.numeric(data[[1]])
x1 <- as.numeric(data[[2]])
x2 <- as.numeric(data[[3]])
x3 <- as.numeric(data[[4]])
dint <- (1-dA-dB-dC) * mean(y)
mini <- (y - dint - dA * x1 - dB * x2 - dC * x3)^2
minis <- mini
mini <- sum(mini)
lOut <- list()
lOut[["mini"]] <- mini
lOut[["minis"]] <- minis
return(lOut)
}
ineqconstraint <- function(vPar, ...){
dA <- vPar[1]
dB <- vPar[2]
dC <- vPar[3]
return(dA+dB+dC)
}
min.RSS <- function(vPar, data){
rss <- minimizingfunc(data, vPar)$mini
return(rss)
}
EstimatecorrHAR <- function(variances, correlation = NULL, proxy, trace=1, Forecasts = F, ineqfun = ineqconstraint, ineqLB = 0.00, ineqUB = 0.9999){
#variances should be produced by the univariate models and not come from realized measures!
#correlation should be found via eg. five min samples but adhere to the same frequency as variances from HAR models.
dA = 0.01
dB = 0.2185
dC = 0.4
## vector of starting parameters
vPar = c(dA, dB, dC)
correlation <- as.matrix(correlation)
corrday <- correlation
corrweek <- rowMeans(cbind(correlation, mlag(correlation,4,mean(correlation))), na.rm = T)
corrweek[is.na(corrweek)] <- mean(corrweek, na.rm = T)
corrmonth <- rowMeans(cbind(correlation, mlag(correlation,21,mean(correlation))), na.rm = T)
corrmonth[is.na(corrmonth)] <- mean(corrmonth, na.rm = T)
end <- length(correlation)
data <- list(proxy[23:end], corrday[22:(end-1)], corrweek[22:(end-1)], corrmonth[22:(end-1)])
optimizer = solnp(vPar, fun = min.RSS, data = data,
ineqfun = ineqfun, #the inequality constraint
ineqLB = ineqLB, ## the inequality lower bound
ineqUB = ineqUB, ## the inequality lower bound, i.e. 0.0 <= a + b + c < 0.9999
## lower and upper bounds for all parameters
LB = c(0, 0, 0), UB = c(0.9999, 0.9999, 0.9999),
control = list(tol = 1e-22, outer.iter = 800, inner.iter = 1200, delta = 1e-8,
trace = trace))
params <- optimizer$pars
min <- tail(optimizer$values, 1)
hessian <- optimizer$hessian
#calculating covariances:
hhatcorrHAR <- cbind(corrday[22:(end-1)], corrweek[22:(end-1)], corrmonth[22:(end-1)]) %*% matrix(params)
if(Forecasts){ hhatcorrHAR <- cbind(corrday[end], corrweek[end], corrmonth[end]) %*% matrix(params) }
covs <- hhatcorrHAR * sqrt(variances[,1]) * sqrt(variances[,2])
vSigma2 <- array(0L, dim = c(2,2, (nrow(variances))))
for(i in 1:(nrow(variances))){
vSigma2[,,i] <- matrix(c(variances[i,1], covs[i], covs[i], variances[i,2]), ncol=2, nrow=2)
}
#R-squared
Rsquared <- 1-var(correlation[23:end] - hhatcorrHAR)/var(correlation[23:end])
lOut <- list()
lOut[["vPar"]] <- params
lOut[["vSigma2"]] <- vSigma2
lOut[["R2"]] <- Rsquared
lOut[["estcor"]] <- hhatcorrHAR
lOut[["MSE"]] <- min/end
lOut[["hessian"]] <- hessian
return(lOut)
}
EstimatecorrHAR2 <- function(returns, variances, proxy, trace=1, ineqfun = ineqconstraint, ineqLB = 0.00, ineqUB = 0.9999){
#mEta is standardized residuals from univariate models!
#variances should be produced by the univariate models and not come from realized measures!
#correlation should be found via eg. five min samples but adhere to the same frequency as variances from HAR models.
mEta <- numeric()
for(i in 1:nrow(variances)){
#returns should start at 22.
mEta2 <- standardized.res.intraday(returns[[i]], variances[i, ])
mEta[i] <- realCov(mEta2, T)[2,1]
}
dA = 0.45
dB = 0.09185
dC = 0.26
## vector of starting parameters
vPar = c(dA, dB, dC)
mEta <- as.matrix(mEta)
corrday <- mEta
corrweek <- rowMeans(cbind(mEta, mlag(mEta,4,mean(mEta))))
corrmonth <- rowMeans(cbind(mEta, mlag(mEta,21,mean(mEta))))
data <- list(proxy, corrday, corrweek, corrmonth)
optimizer = solnp(vPar, fun = min.RSS, data = data,
ineqfun = ineqfun, #the inequality constraint
ineqLB = ineqLB, ## the inequality lower bound
ineqUB = ineqUB, ## the inequality lower bound, i.e. 0.0 <= a + b + c < 0.9999
## lower and upper bounds for all parameters
LB = c(0, 0, 0), UB = c(0.9999, 0.9999, 0.9999),
control = list(tol = 1e-18, outer.iter = 800, inner.iter = 1200, delta = 1e-7,
trace = trace))
params <- optimizer$pars
min <- tail(optimizer$values, 1)
hessian <- optimizer$hessian
#calculating covariances:
hhatcorrHAR <- cbind(corrday, corrweek, corrmonth) %*% matrix(params)
covs <- hhatcorrHAR * sqrt(variances[,1]) * sqrt(variances[,2])
vSigma2 <- array(0L, dim = c(2,2, (nrow(variances))))
for(i in 1:(nrow(variances))){
vSigma2[,,i] <- matrix(c(variances[i,1], covs[i], covs[i], variances[i,2]), ncol=2, nrow=2)
}
lOut <- list()
lOut[["vPar"]] <- params
lOut[["vSigma2"]] <- vSigma2
lOut[["estcor"]] <- hhatcorrHAR
lOut[["MSE"]] <- min/2516
lOut[["hessian"]] <- hessian
lOut[["mEta"]] <- mEta
return(lOut)
}
########################################################################################################################
#
# REALIZED SCALAR-BASED BIVARIATE GARCH MODEL (rBG)
#
########################################################################################################################
BivarGARCHFilter <- function(dailyret, params, covariance, inference = F){
dAlpha <- params[1]
dBeta <- params[2]
samplecov <- cov(dailyret)
if(inference){
dAlpha <- params[1]
dBeta <- params[2]
samplecov <- params[3]
}
days <- length(covariance[1,1,])
d <- 2 #being bivariate
rcov <- array(covariance, dim = c(2,2,days))
samplercov <- matrix(
c(mean(rcov[1,1,]),
mean(rcov[2,1,]),
mean(rcov[2,1,]),
mean(rcov[2,2,])),
ncol=2, nrow=2)
mSigma <- array(0L, dim = c(2, 2, days))
#initializing with unconditional mean.
id <- matrix(c(1,0,0,1), ncol = 2, nrow = 2)
omega <- samplecov - dAlpha * samplercov %*% id - dBeta * samplecov # samplecov^(-1)
mSigma[,,1] <- samplecov
#compute first observation of log-likelihood (we are minizing neg-log-likelihood):
dLLK <- log(det(mSigma[,,1])) + (dailyret[1, , drop = F]) %*% solve(mSigma[,,1]) %*% t(dailyret[1, , drop = F])
dLLKs <- numeric()
dLLKs[1] <- dLLK
for(i in 2:days){
mSigma[,,i] <- omega + dBeta * mSigma[,,i-1] + dAlpha * rcov[,,i-1]
#neglog collection for score calculation
dLLKs[i] <- (log(det(mSigma[,,i])) +
dailyret[i, , drop = F] %*% solve(mSigma[,,i]) %*% t(dailyret[i, , drop = F]))
}
fulldLLK <- - 0.5 *(d* log(2*pi) * days + sum(dLLKs)) #loglikelihood
lOut <- list()
lOut[["fulldLLK"]] <- fulldLLK
lOut[["mSigma"]] <- mSigma
lOut[["cov"]] <- omega
lOut[["dLLKs"]] <- dLLKs
lOut[["sampleH"]] <- samplecov
return(lOut)
}
#objective function specific for scalar Bivariate GARCH
ObjFBivarGARCH <- function(vPar, dailyret, covariance) {
dAlpha = vPar[1]
dBeta = vPar[2]
dLLK = BivarGARCHFilter(dailyret, c(dAlpha, dBeta), covariance)$fulldLLK
return(-as.numeric(dLLK))
}
ineqfun_GARCH_BIVAR <- function(vPar, ...) {
dAlpha = vPar[1]
dBeta = vPar[2]
return(dAlpha + dBeta)
}
EstimateBivarGARCH <- function(dailyret, covariance, bootstrap = FALSE, vPar=NULL, ineqfun_GARCH = ineqfun_GARCH_BIVAR, ineqLB = 0.00, ineqUB = 0.9999, tol = 1e-7){
# We set starting value for alpha equal to 0.05, dBeta = 0.94, and chose omega to target
# the empirical variance by targeting the unconditional variance of the
# GARCH model
dAlpha = 0.20
dBeta = 0.50
## vector of starting parameters
if(is.null(vPar)){
vPar = c(dAlpha, dBeta)
}
# have a look at help(solnp) ObjFBivarGARCH
##optimization step
optimizer = solnp(vPar, fun = ObjFBivarGARCH, dailyret = dailyret, covariance = covariance,
ineqfun = ineqfun_GARCH, #the inequality constraint
ineqLB = ineqLB, ## the inequality lower bound
ineqUB = ineqUB, ## the inequality lower bound, i.e. 0.0 < alpha + beta < 0.9999
## lower and upper bounds for all parameters
LB = c(0.0001, 0.0001), UB = c(0.999, 0.999),
control = list(tol = tol, outer.iter = 800, inner.iter = 1200, delta = 1e-7)
)
## extract estimated parameters
vPar = optimizer$pars
## extract the likelihood computed at its maximum
dLLK = -tail(optimizer$values, 1)
if(!bootstrap){
#CALCULATING ROBUST STD. ERRORS WHEN NO COVARIANCE TARGETING:
rse <- NULL
se <- NULL
scores <- NULL
if(is.null(vPar)){
scores <- matrix(0L, nrow=length(dailyret[,1]), ncol = length(vPar))
step <- 1e-5 * vPar
# This follows directly from Kevin Sheppard who uses percentage returns, Moreover scale open-to-close with 24/6.5.
for(i in 1:length(step)){
h <- step[i]