10.2 Bollerslev’s GARCH Model

An important extension of the ARCH(p) model proposed by Bollerslev (1986) replaces the AR$(p)$ representation of $\sigma_{t}^{2}$ in (10.10) with an ARMA($p,q$) formulation giving the GARCH(p,q) model: $$\begin{align} \sigma_{t}^{2} & =\omega+\sum_{i=1}^{p}\alpha_{i}\varepsilon_{t-i}^{2}+\sum_{j=1}^{q}\beta_{j}\sigma_{t-j}^{2}\tag{10.14} \end{align}$$ In order for $\sigma_{t}^{2}>0,$ it is assumed that $\omega>0$ and the coefficients $\alpha_{i}$ $(i=0,\cdots,p)$ and $\beta_{j}$ $(j=1,\cdots,q)$ are all non-negative.⁵⁵ When $q=0$, the GARCH(p,q) model (10.14) reduces to the ARCH(p) model. In general, the GARCH($p,q)$ model can be shown to be equivalent to a particular ARCH($\infty)$ model.

Usually the GARCH(1,1) model, $$\begin{equation} \sigma_{t}^{2}=\omega+\alpha_{1}\varepsilon_{t-1}^{2}+\beta_{1}\sigma_{t-1}^{2},\tag{10.15} \end{equation}$$ with only three parameters in the conditional variance equation is adequate to obtain a good model fit for daily asset returns. Indeed, Hansen and Lund (2004) provided compelling evidence that is difficult to find a volatility model that outperforms the simple GARCH(1,1). Hence, for many purposes the GARCH(1,1) model is the de facto volatility model of choice for daily returns.

10.2.1 Statistical Properties of the GARCH(1,1) Model

The statistical properties of the GARCH(1,1) model are derived in the same way as the properties of the ARCH(1) model and are summarized below:

1. $\{R_{t}\}$ is a covariance stationary and ergodic process provided $\alpha_{1}+\beta_{1}<1$.
2. $\mathrm{v\mathrm{ar}}(R_{t})=E(\epsilon_{t}^{2})=E(\sigma_{t}^{2})=\omega/(1-\alpha_{1}-\beta_{1})=\bar{\sigma}^{2}.$
3. The distribution of $R_{t}$ conditional on $I_{t-1}$ is normal with mean $\mu$ and variance $\sigma_{t}^{2}$.
4. The unconditional (marginal) distribution of $R_{t}$ is not normal and $$\mathrm{kurt}(R_{t})=\frac{3(1+\alpha_{1}+\beta_{1})(1-\alpha_{1}-\beta_{1})}{1-2\alpha_{1}\beta_{1}-3\alpha_{1}^{2}-\beta_{1}^{2}}\geq3.$$
5. $\{R_{t}^{2}\}$ and $\{\varepsilon_{t}^{2}\}$ have an ARMA(1,1) representation $$\varepsilon_{t}^{2}=\omega+(\alpha_{1}+\beta_{1})\varepsilon_{t-1}^{2}+u_{t}-\beta_{1}u_{t-1},$$ with $u_{t}=\varepsilon_{t}^{2}-\sigma_{t}^{2}$. The persistence of the autocorrelations is given by $\alpha_{1}+\beta_{1}.$
6. $\sigma_{t}^{2}$ has an AR($\infty)$ representation $$\sigma_{t}^{2}=\frac{\omega}{1-\beta_{1}}+\alpha_{1}\sum_{j=0}^{\infty}\beta_{1}^{j}\varepsilon_{t-j-1}^{2}.$$

The derivations of these properties are left as end-of-chapter exercise.

• Need some comments here: more flexible autocorrelations structure for $R_{t}^{2}$ than ARCH(p) and with fewer parameters.

Example 10.1 (Simulate GARCH(1,1) model using rugarch)

Consider simulating $T=1000$ observations from the GARCH(1,1) model $$\begin{align*} R_{t} & =\varepsilon_{t}=\sigma_{t}z_{t},\text{ }z_{t}\sim iid\text{ }N(0,1)\\ \sigma_{t}^{2} & =0.01+0.07R_{t-1}^{2}+0.92\sigma_{t-1}^{2} \end{align*}$$ This process is covariance stationary since $\alpha_{1}+\beta_{1}=0.07+0.92=0.9<1,$ and $\bar{\sigma}^{2}=\omega/(1-\alpha_{1}-\beta_{1})=0.01/0.01=1.$ This GARCH(1,1) model has the same unconditional variance as the ARCH(5) model from the previous example but has much higher persistence. This model can be specified using the rugarch ugarchspec() function as follows:

garch11.spec = ugarchspec(variance.model = list(garchOrder=c(1,1)), 
                          mean.model = list(armaOrder=c(0,0)),
                          fixed.pars=list(mu = 0, omega=0.1, 
                                          alpha1=0.1, beta1 = 0.8))

This model has the same unconditional variance and persistence as the ARCH(5) model in the previous example. Simulated values for $R_{t}$ and $\sigma_{t},$ using the same random number seed as the ARCH(5) simulations, are created using

set.seed(123)
garch11.sim = ugarchpath(garch11.spec, n.sim=1000)

and are displayed in Figure xxx. The simulated GARCH(1,1) values of $R_{t}$ and $\sigma_{t}$ are quite different from the simulated ARCH(1) values. In particular, the simulated returns $R_{t}$ show fewer extreme values than the ARCH(1) returns and the simulated volatilities $\sigma_{t}$ values appear to show more persistence than the ARCH(1) volatilities. The sample autocorrelations in Figure xxx show the interesting result that $R_{t}^{2}$ exhibits little autocorrelation whereas $\sigma_{t}^{2}$ exhibits substantial positive autocorrelation.

sd(garch11.sim@path$seriesSim)

## [1] 0.9628771

kurtosis(garch11.sim@path$seriesSim)

## [1] 0.04431907

$\blacksquare$

References

Conrad, C., and B. R. Haag. 2006. Inequality Constraints in the Fractionally Integrated Garch Model. Journal of Financial Econometrics. Vol. 4.

Nelson, D. B., and Charles Q. Cao. 1992. Inequality Constraints in the Univariate Garch Model. Journal of Business & Economic Statistics. Vol. 10.

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55. Positive coefficients are sufficient but not necessary conditions for the positivity of conditional variance. See (Nelson and Cao 1992) and (Conrad and Haag 2006) for more general conditions.↩︎