Estimating Volatility using GARCH vs Standard Deviation

Load the EuStockMarkets (DAX Index) data

library(fGarch)
library(tseries)

data(EuStockMarkets) #load data

Looking at the DAX Index - Closing Price & Returns

plot( EuStockMarkets[,"DAX"] , ylab = "Price", main = "Time Series of Closing Price of DAX")

#first differencing on the closing price to get returns
dax <- diff(log(EuStockMarkets))[,"DAX"] 

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

##  [1]  0.0037622676 -0.0003217412 -0.0167942368 -0.0061509827 -0.0005362285
##  [6] -0.0313150592  0.0022470827 -0.0066312038  0.0132238036 -0.0076720025
## [11] -0.0149217633 -0.0096893845 -0.0183408812 -0.0155528317  0.0126187588
## [16] -0.0249390115 -0.0325073453  0.0189573098 -0.0059411996  0.0219221523

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

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: 0x00000000176aa3a0>
 [data = dax]

Conditional Distribution:
 norm 

Coefficient(s):
        mu       omega      alpha1       beta1  
6.5351e-04  4.7543e-06  6.8417e-02  8.8761e-01  

Std. Errors:
 based on Hessian 

Error Analysis:
        Estimate  Std. Error  t value Pr(>|t|)    
mu     6.535e-04   2.158e-04    3.029  0.00245 ** 
omega  4.754e-06   1.264e-06    3.760  0.00017 ***
alpha1 6.842e-02   1.478e-02    4.630 3.66e-06 ***
beta1  8.876e-01   2.356e-02   37.677  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log Likelihood:
 5966.214    normalized:  3.209368 

Description:
 Fri Apr 14 23:51:32 2017 by user: User 


Standardised Residuals Tests:
                                Statistic p-Value  
 Jarque-Bera Test   R    Chi^2  13380.69  0        
 Shapiro-Wilk Test  R    W      0.9477474 0        
 Ljung-Box Test     R    Q(10)  3.195816  0.9764329
 Ljung-Box Test     R    Q(15)  10.13427  0.8112099
 Ljung-Box Test     R    Q(20)  12.80196  0.8857182
 Ljung-Box Test     R^2  Q(10)  0.893265  0.9998977
 Ljung-Box Test     R^2  Q(15)  1.329651  0.9999981
 Ljung-Box Test     R^2  Q(20)  1.756904  1        
 LM Arch Test       R    TR^2   1.08588   0.9999776

Information Criterion Statistics:
      AIC       BIC       SIC      HQIC 
-6.414432 -6.402538 -6.414441 -6.410049 

Predict 30-days ahead returns and volatilities

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

#1-day ahead volatility forecast
sd1 <- predict(fit, n.ahead = 30,plot = T)[1,3] 

print(sd1)

## [1] 0.01526939

Therefore, the 1-day ahead forecast volatility is 0.0152694.

Calculating volatility using sample standard deviation

sd2 <- sd(dax)
print(sd2)

## [1] 0.01030084

The sample standard deviation of returns is 0.0103008, which is lower than the 1-day ahead forecast volatility, 0.0152694.