Estimating Volatility using GARCH vs Standard Deviation

Load the textbook_example data

library(fGarch)
library(tseries)

setwd("C:/Users/User/Google Drive/P_Project/volatility_garch_sd")
df <- read.fwf("textbook_example.txt", widths = 8, header = T)[,2] #load data
df <- ts(df)

Looking at the data - Closing Price & Returns

plot( df , ylab = "Price", main = "Time Series of Closing Price")

#first differencing on the closing price to get returns
dax <- diff(log(df))

#preview latest 20 days returns
print( tail(dax, 20) )

##  [1]  0.004987542 -0.010000083  0.005012542  0.024692613 -0.012270093
##  [6]  0.031594365  0.000000000  0.000000000 -0.007202912  0.000000000
## [11]  0.011976191  0.004750603 -0.009523882  0.000000000  0.016607736
## [16]  0.007034027  0.000000000 -0.007034027  0.023256862  0.011428696

#plot the returns time series
plot(dax, xlab = "Time", main = "Time Series of Returns")

Fit a GARCH(1,1) model on the returns

log <- capture.output({  #ignore this function (To suppress unnecessary message output ##
      
   fit <- garchFit(~garch(1, 1), data = dax) 
      
}) 

Print summary of the fitted model

summary(fit)

Title:
 GARCH Modelling 

Call:
 garchFit(formula = ~garch(1, 1), data = dax) 

Mean and Variance Equation:
 data ~ garch(1, 1)
<environment: 0x000000001673dc58>
 [data = dax]

Conditional Distribution:
 norm 

Coefficient(s):
        mu       omega      alpha1       beta1  
4.8475e-03  1.4785e-10  1.0000e-08  9.8923e-01  

Std. Errors:
 based on Hessian 

Error Analysis:
        Estimate  Std. Error  t value Pr(>|t|)    
mu     4.847e-03   2.648e-03    1.831   0.0671 .  
omega  1.478e-10   3.177e-05    0.000   1.0000    
alpha1 1.000e-08          NA       NA       NA    
beta1  9.892e-01   8.236e-02   12.011   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log Likelihood:
 60.40333    normalized:  3.020166 

Description:
 Wed Apr 26 00:01:31 2017 by user: User 


Standardised Residuals Tests:
                                Statistic p-Value  
 Jarque-Bera Test   R    Chi^2  1.443836  0.4858196
 Shapiro-Wilk Test  R    W      0.9341599 0.185607 
 Ljung-Box Test     R    Q(10)  6.627275  0.7601013
 Ljung-Box Test     R    Q(15)  11.36599  0.7262447
 Ljung-Box Test     R    Q(20)  NA        NA       
 Ljung-Box Test     R^2  Q(10)  4.982482  0.8923452
 Ljung-Box Test     R^2  Q(15)  11.12652  0.7435748
 Ljung-Box Test     R^2  Q(20)  NA        NA       
 LM Arch Test       R    TR^2   8         0.7851304

Information Criterion Statistics:
      AIC       BIC       SIC      HQIC 
-5.640333 -5.441186 -5.703860 -5.601457 

Predict 30-days ahead returns and volatilities

# predict(fit, n.ahead = 10,plot = T)

#1-day ahead volatility forecast
sd1 <- predict(fit, n.ahead = 15,plot = T)[1,3] 

print(sd1)

## [1] 0.01057778

Therefore, the 1-day ahead forecast volatility is 0.0105778.

Calculating volatility using sample standard deviation

sd2 <- sd(dax)
print(sd2)

## [1] 0.01215933

The sample standard deviation of returns is 0.0121593, which is higher than the 1-day ahead forecast volatility, 0.0105778.