Optim function in R

Question

my colleagues and I have an assignment to submit with the following directions:

• Without using any built-in R functions, write a general function that estimates a GARCH(1,1) for a time series, and that returns the parameters, standard errors and the filtered variance process.

• Download 15 years of daily SP500 data and estimate the GARCH model.

• Use the estimated parameters and the filtered volatility to simulate a 95% confidence interval for a 30-days prediction period. Do this for every day in your sample.

• Verify how often the realizations 30 days ahead violate the confidence interval. Makeanice plot.

The main issue in our code is that it appears that there is an error in the optim function and we can't figure out how to fix it. Due to this problem, we can't find the standard error (since it would derive from the hessian in the optim function). We hope you'll be able to help us. Thank you. Note that with "h" we are referring to the variance. And with "ster" we mean "standard error".

Please, find attach the code below:

#R PROJECT GARCH(1,1) MODEL written by Alice, Christian and Diana



#S&P 500 data from 01.01.2005 to 31.12.2021
library(xts)
library(ggplot2)
library(readr)
library(quantmod)
library(cvar)



SPX1 <- getSymbols("^GSPC",auto.assign = FALSE, from = "2005-01-01", to="2021-12-31")
SPX_Adj <- SPX1$GSPC.Adjusted



#SP500 <- read_csv("SP500.csv", col_types = cols(Close = col_number(), Date = col_character()))
#View(SP500)



#BETTER FITTING THE DATA FOR OUR WORK
SP500log <- as.data.frame(log10(SPX_Adj[,1]))
N <- length(SP500log[,1])
SP500_ret <- ((SP500log[2:N,]-SP500log[1:N-1,]))
x <- SP500_ret



#WRITING THE GARCH (1,1) FUNCTION



GARCH <-function(para,x){
# Volatility and loglik initialisation
mu <- mean(x)
para <- c(0.2,0.8,0.15)
Beta0 <- para[1]
Beta1 <- para[2]
Beta2 <- para[3]
e <- x - mean(x)
e2 <- e^2
h <- c(0)
loglik=c(0)
# Start of the loop
vol=c()
for (t in 2:length(x)){
h[t]=Beta0+Beta1*e2[t-1]+Beta2*h[t-1]
ster <- sqrt(h[t]/N)
loglik[t]=loglik[t-1]+log(dnorm(x[t],mean(x[1:t]),sqrt(h[t])))
}
Ster <<- ster
H <<- h
return(tail(loglik,1))
}



paraoptim <- optim(para, GARCH, gr=NULL, method=c("BFGS"),lower = 0, upper = Inf, control = list(), hessian = FALSE, SP500_ret)



paraoptim$par
Beta0 <- paraoptim$par[1]
Beta1 <- paraoptim$par[2]
Beta2 <- paraoptim$par[3]



GARCH(para,SP500_ret)



#CONFIDENCE INTERVAL



data <- diff(h, lag=29)
ConfInt <- function(data, alpha=0.05) {
x <- x[!is.na(data)]
m <- mean(x)
z <- qnorm(1-alpha/2)
s <- sd(x)/sqrt(length(x))
CONFIDENCE <- c("inf"=m-z*s, "sup"=m+z*s)
return(CONFIDENCE)
}



ConfInt(data)
bounds <- ConfInt(data)

#
ValueAtRisk <- function(x){
VaR = c()
for (t in 1:length(x)) {
VaR[t] <- as.numeric(VaR(H[1:t], p=0.95, method=c("historical")))
}
return(VaR)
}



ValueAtRisk(SP500_ret)
VaRline <- (h/ValueAtRisk(SP500_ret))/1000
#



#PLOT
plot(SP500_ret, type = "l", col="blue", ylim = c(-0.08,0.05))
lines(VaRline)

Answer 1

I you replace

paraoptim <- optim(para, GARCH, gr=NULL, method=c("BFGS"),lower = 0, upper = Inf, control = list(), hessian = FALSE, SP500_ret)

with

para <- c(0.2,0.8,0.15)
paraoptim <- optim(para, GARCH, gr=NULL, method=c("BFGS"),lower = 0, upper = Inf, control = list(), hessian = FALSE, SP500_ret)
h<-H

your code will run.

If believe your 2 issues are

1. As pointed out by @G.Grothendieck, you are defining the para parameter within the GARCH function. Therefore its not global, and the garch call cannot see para.
2. GARCH create H, but your subsequent calls use h.