Application of GARCH models in R – Part I



You probably heard about the GRACH models and its application to forecast volatility for financial stock prices. In this post I don’t want to go deeply about the concept of the GARCH models, instead I want to show you an application of using these models to forecast volatility.
This is the first post of a GARCH application series, for the future we explain more about the more advanced application and how to interpret the result.
The general model of GARCH (Generalized Autoregressive Conditional Heteroskedasticity) is GARCH(p, q), which p and q are the number of lag variances and the number of lag residual errors respectively.
If we define p=0 i.e. GARCH(0, q) is equivalent to an ARCH(q) model. In the ARCH(q) process the conditional variance is specified as a linear function of past sample variances only, whereas the GARCH(p, q) process allows lagged conditional variances to enter as well. This corresponds to some sort of adaptive learning mechanism.
GARCH(1,1):

and
ARCH(1):
If we need to have higher level lags in the model for example GARCH(4,1) the model will defined as follows:

In these models 𝟂 is constant and we need to estimate 𝟂, 𝞪 and 𝞫.

In the following code I used fGarch package for estimating the model coefficients for SP500 daily returns for the period between "2009-01-01" and most up to date price data.
library(quantmod)
library(fGarch)
Symb = '^GSPC'
# download the SP500 adjusted close price from yahoo finance
P = getSymbols(Symbols = Symb, src = "yahoo", from =c("2009-01-01") ,
              to = Sys.Date(), auto.assign = F,periodicity = "daily")[,6]
log_return = diff(log(P))*100
Zero_mean_return= log_return - mean(as.vector(log_return),na.rm = TRUE)
# Garch modeling: ARCH(1)
garch01=garchFit(formula = ~garch(1,0), data =na.omit(Zero_mean_return),
                               include.mean = FALSE)
names(garch01@fit)
 [1] "par"         "objective"   "convergence" "iterations"  "evaluations" "message"   
 [7] "value"       "coef"        "llh"         "hessian"     "ics"         "series"    
[13] "params"      "cvar"        "se.coef"     "tval"        "matcoef"   

Now we can find the estimated the coefficients by this command:
garch01@fit$coef
omega    alpha1 
0.7990782 0.2747121 


Remember the ARCH(1) model was.


And we could successfully estimate the 𝟂 and 𝞪 in R.
Now let us plot results.
By plotting the estimated model you will have 13 different plot options that you can use for your analysis.
> plot(garch01)
Make a plot selection (or 0 to exit): 

1:   Time Series   
SP500 Returns 2009-01-01 to 2019-11-01  
         
                   
                       

 2:   Conditional SD 
                                       

 3:   Series with 2 Conditional SD Superimposed 

 

4:   ACF of Observations       
                        

 5:   ACF of Squared Observations     
                     

 6:   Cross Correlation
    

 7:   Residuals
                                                                          
8:   Conditional SDs

                                                

 9:   Standardized Residuals
                                         

 10:   ACF of Standardized Residuals 
                      

 11:   ACF of Squared Standardized Residuals      

 12:   Cross Correlation between r^2 and r      

 13:   QQ-Plot of Standardized Residuals 

 Now lets model the volatility with more complex model i.e. GARCH(1,1) with skew student-t conditional distribution.

garch11_s_t=garchFit(
formula = ~garch(1,1),data=na.omit(Zero_mean_return),
                 include.mean = FALSE, cond.dist = "sstd",trace= F)
garch11_s_t@fit$coef
garch11_s_t@fit$llh
plot(garch11_s_t)



According to the GARCH(1,1) we just need the last volatility and return in order to forecast volatility for the next period. You can see the estimated coefficients for GARCH(1,1) with skew student_t conditionla distribution as follows.
omega = 0.01883026   
alpha1 = 0.14946961   
beta1    = 0.84392795   
skew     = 0.8884789
shape    = 5.32634804




























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