Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) is a kind of ARMA (p,q) model to represent volatility:
$$ \sigma_t^2=\alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \cdots + \alpha_q \epsilon_{t-q}^2 + \beta_1 \sigma_{t-1}^2 + \cdots + \beta_p\sigma_{t-p}^2 = \alpha_0 + \sum_{i=1}^q \alpha_i \epsilon_{t-i}^2 + \sum_{i=1}^p \beta_i \sigma_{t-i}^2 $$
where $\epsilon$ is a i.i.d random variable generally from a normal N(0,1) or student distribution and $\alpha_0 >0 , \alpha_t ≥ 0 , \beta_t ≥0, \alpha_t + \beta_t <1 $. In practice, generally low order of GARCH models are used in many application such as GARCH(1,1), GARCH (1,2), GARCH (2,1).
By utilising GARCH (1,1), to forecaste variance in
k step ahead, following formula can be used:
$$ \sigma_t^2 (k)=\frac{\alpha_0}{1-\alpha_1 - \beta_1} ; k \to \infty$$
Lets have a look how to apply GARCH over monthly return of SP 500 from 1926 within R.
>library("fGarch") ## Library for GARCH
# get data
>data=read.table("http://www.mif.vu.lt/~rlapinskas/DUOMENYS/Tsay_fts3/sp500.dat",header=F)
> dim(data)
[1] 792 1
## First step is to model monthly return.
>data=read.table("http://www.mif.vu.lt/~rlapinskas/DUOMENYS/Tsay_fts3/m-intc7308.txt",header=T)
# lets find out lag
>ret=pacf(data)
> which.max(abs(ret$acf))
[1] 3
## Lets use ARMA(3,0) and GARCH(1,1) to model return
> m1=garchFit(~arma(3,0)+garch(1,1),data=data,trace=F)
> summary(m1)
Title:
GARCH Modelling
Call:
garchFit(formula = ~arma(3, 0) + garch(1, 1), data = data, trace = F)
Mean and Variance Equation:
data ~ arma(3, 0) + garch(1, 1)
[data = data]
Conditional Distribution:
norm
Coefficient(s):
mu ar1 ar2 ar3 omega alpha1
7.7077e-03 3.1968e-02 -3.0261e-02 -1.0649e-02 7.9746e-05 1.2425e-01
beta1
8.5302e-01
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 7.708e-03 1.607e-03 4.798 1.61e-06 ***
ar1 3.197e-02 3.837e-02 0.833 0.40473
ar2 -3.026e-02 3.841e-02 -0.788 0.43076
ar3 -1.065e-02 3.756e-02 -0.284 0.77677
omega 7.975e-05 2.810e-05 2.838 0.00454 **
alpha1 1.242e-01 2.247e-02 5.529 3.22e-08 ***
beta1 8.530e-01 2.183e-02 39.075 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
So, monthly return can be modeled as:
$$ r_t= 0.0078+0.032r_{t-1}-0.03r_{t-2}-0.01r_{t-3} + a_t$$
$$ \sigma_t^2=0.000080 + 0.12\alpha_{t-1}+0.85(\sigma_{t-1})^2$$
But since as t values of ARMA models are insignificant, we can use directly GARCH(1,1) to model it
> m2=garchFit(~garch(1,1),data=data,trace=F)
> summary(m2)
Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(1, 1), data = data, trace = F)
Mean and Variance Equation:
data ~ garch(1, 1)
[data = data]
Conditional Distribution:
norm
Coefficient(s):
mu omega alpha1 beta1
7.4497e-03 8.0615e-05 1.2198e-01 8.5436e-01
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 7.450e-03 1.538e-03 4.845 1.27e-06 ***
omega 8.061e-05 2.833e-05 2.845 0.00444 **
alpha1 1.220e-01 2.202e-02 5.540 3.02e-08 ***
beta1 8.544e-01 2.175e-02 39.276 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
So we can reduce model to more simpler form such as:
$$ r_t= 0.0074+a_t$$
$$ \sigma_t^2=0.000080 + 0.12a{t-1}+0.85(\sigma_{t-1})^2$$
Hence, variance of $a_t$ is:
$$ \frac{0.000080}{1-0.12 - 0.85} = 0.0027$$
Lets analyse if residuals are serially independent.
>stress=residuals(m2,standardize=T)
> Box.test(stress,12,type='Ljung')
Box-Ljung test
data: stress
X-squared = 11.9994, df = 12, p-value = 0.4457
So, above test indicates that there is not significant serial correlation in residuals, so we can conclude GARCH(1,1) is a good model for SP 500. We can also predict next 5 volatility of monthly return of SP 500 as:
> predict(m2,5)
meanForecast meanError standardDeviation
1 0.007449721 0.05377242 0.05377242
2 0.007449721 0.05388567 0.05388567
3 0.007449721 0.05399601 0.05399601
4 0.007449721 0.05410353 0.05410353
5 0.007449721 0.05420829 0.05420829
