Model risk is the risk of using the wrong model

GARCH Models in R

Kris Boudt

Professor of finance and econometrics

Sources of model risk and solutions

Sources:

modeling choices
starting values in the optimization
outliers in the return series

Solution: Protect yourself through a robust approach

model-averaging: averaging the predictions of multiple models
trying several starting values and choosing the one that leads to the highest likelihood
cleaning the data

Model averaging

variance.models <- c("sGARCH", "gjrGARCH")
distribution.models <- c("norm", "std", "sstd")
c <- 1
for (var.model in variance.models) {
    for (dist.model in distribution.models) {
        garchspec <- ugarchspec(mean.model = list(armaOrder = c(0, 0)),
         variance.model = list(model = var.model), distribution.model = dist.model)
        garchfit <- ugarchfit(data = msftret, spec = garchspec)
        if (c==1) { msigma <- sigma(garchfit)
        } else { msigma <- merge(msigma, sigma(garchfit))} 
        c <- c + 1 }
}

The average vol prediction

avesigma <- xts(rowMeans(msigma), order.by = time(msigma))

Definition of beta

Robustness to starting values

coef(garchfit)

          mu        omega       alpha1        beta1         skew        shape 
5.669200e-04 6.281258e-07 7.462984e-02 9.223701e-01 9.436331e-01 6.318621e+00

Those estimates are the result of a complex optimization of the likelihood function
Optimization is numeric and iterative: step by step improvement, which can be sensitive to starting values
rugarch has a default approach in getting sensible starting values
You can specify your own starting values by applying the setstart() method to your ugarchspec() GARCH model specification

Estimation with default starting values

garchspec <- ugarchspec(mean.model = list(armaOrder = c(0,0)),
                        variance.model = list(model = "sGARCH"),
                        distribution.model = "sstd")

garchfit <- ugarchfit(data = sp500ret, spec = garchspec)

coef(garchfit)

          mu        omega       alpha1        beta1         skew        shape 
5.669200e-04 6.281258e-07 7.462984e-02 9.223701e-01 9.436331e-01 6.318621e+00

likelihood(garchfit)

24280.33

Estimation with modified starting values

garchspec <- ugarchspec(mean.model = list(armaOrder = c(0, 0)),
  variance.model = list(model = "sGARCH"), distribution.model = "sstd")
setstart(garchspec) <- list(alpha1 = 0.05, beta1 = 0.9, shape = 8)
garchfit <- ugarchfit(data = sp500ret, spec = garchspec)

coef(garchfit)

          mu        omega       alpha1        beta1         skew        shape 
5.638002e-04 6.303949e-07 7.466503e-02 9.224117e-01 9.438978e-01 6.309185e+00

likelihood(garchfit) # returns 24280.33

Cleaning the data

Avoid that outliers distort the volatility predictions
How? Through winsorization: reduce the magnitude of the return to an acceptable level using the function Return.clean() in the package PerformanceAnalytics with method = "boudt":

# Clean the return series
library(PerformanceAnalytics)
clmsftret <- Return.clean(msftret, method = "boudt")
# Plot them on top of each other
plotret <- plot(msftret, col = "red")
plotret <- addSeries(clmsftret, col = "blue", on = 1)

Definition of beta

Impact of cleaning on volatility prediction

Make the volatility predictions using raw and cleaned Microsoft returns

garchspec <- ugarchspec(mean.model = list(armaOrder = c(1, 0)),
     variance.model = list(model = "gjrGARCH"), distribution.model = "sstd")
garchfit <- ugarchfit(data = msftret, spec = garchspec)
clgarchfit <- ugarchfit(data = clmsftret, spec = garchspec)

Compare them in a time series plot

plotvol <- plot(abs(msftret), col = "gray")
plotvol <- addSeries(sigma(garchfit), col = "red", on = 1)
plotvol <- addSeries(sigma(clgarchfit), col = "blue", on = 1)
plotvol

Definition of beta

Be a robustnik: it is better to be roughly right than exactly wrong

GARCH Models in R