6.6 GARCH(p,q) Model

The ARCH model is based on an autoregressive representation of the conditional variance. One may also add a moving average part. The GARCH($ p$,$ q$) process (Generalised AutoRegressive Conditionally Heteroscedastic) is thus obtained. The model is defined by


$\displaystyle y_{t}$ $\displaystyle =$ $\displaystyle u_{t}\cr
u_{t}$ (6.27)

where $ \alpha_{i}\geq 0,\beta_{i}\geq 0,\alpha_{0}>0$ are imposed to ensure that the conditional variance is strictly positive.

The conditional variance can be expressed as

$\displaystyle \sigma_{t}^{2}=
\alpha_{0}+\alpha(B)u_{t}^{2}+\beta(B)\sigma_{t}^{2}$

where $ \alpha(B)=\alpha_{1}B+...+\alpha_{q}B^{q}$ and $ \beta(B)=\beta_{1}B+...+\beta_{p}B^{p}$ are polynomials in the backshift operator B. If the roots of $ 1-\beta(Z)$ lie outside the unit circle, we can rewrite the conditional variance as

$\displaystyle \sigma_{t}^{2}=\frac{\alpha_{0}}{1-\beta(1)}+\frac{\alpha(B)}
{1-\beta(B)}u_{t}^{2}.$

Hence, this expression reveals that a GARCH($ p$,$ q$) process can be viewed as an ARCH($ \infty$) with a rational lag structure imposed on the coefficients.

The GARCH($ p$,$ q$) model may be rewritten in an alternative form, as an ARMA model on squared perturbations.

For this purpose, let us introduce $ \upsilon_{t}=u_{t}^{2}-\sigma_{t}^{2}$. Replacing $ \sigma_{t}^{2}$ by $ u_{t}^{2}-\upsilon_{t}$ in the GARCH representation yields

$\displaystyle u_{t}^{2}-\upsilon_{t}=\alpha_{0}+
\sum_{i=1}^{q}\alpha_{i}u_{t-i}^{2}+\sum_{j=1}^{p}\beta_{j}(u_{t-j}^{2}-\upsilon_{t-j})$

that is to say

$\displaystyle u_{t}^{2}=\alpha_{0}+
\sum_{i=1}^{\max(p,q)}(\alpha_{i}+\beta_{i})u_{t-i}^{2}+\upsilon_{t}-
\sum_{j=1}^{p}\beta_{j}\upsilon_{t-j}$

with $ \alpha_{i}=0$ for $ i>q$ and $ \beta_{i}=0$ for $ i>p$.

It is an ARMA ($ \max(p,q)$,$ p$) representation for the process $ u_{t}^{2}$ but with an error term, which is a white noise process that does not necessarily have a constant variance.


6.6.1 GARCH(1,1) Model

The most used heteroscedastic model in financial time series is a GARCH(1,1), (see Bera and Higgins (1993) for a very complete revision).

This particular model parameterises the conditional variance as

$\displaystyle \sigma_t^2 = \alpha_0 +\alpha_1 u^2_{t-1} + \beta_1 \sigma^2_{t-1}$ (6.28)

Using the law of iterated expectations

$\displaystyle \textrm{E}(u^2_t) = \textrm{E}(\textrm{E}(u^2_{t}\vert I_{t-1}))$ $\displaystyle =$ $\displaystyle \textrm{E}(\sigma_t^2) =
\alpha_0 + \alpha_1 \textrm{E}(u_{t-1}^2) + \beta_1 \textrm{E}(\sigma_{t-1}^2) \cr$ (6.29)

Assuming the process began infinitely far in the past with a finite initial variance, the sequence of the variances converge to a constant

$\displaystyle \sigma^2 = \frac{\alpha_0}{1-\alpha_1-\beta_1},$   if $\displaystyle \alpha_1+\beta_1 < 1
$

therefore, the GARCH process is unconditionally homoscedastic. The $ \alpha_1$ parameter indicates the contributions to conditional variance of the most recent news, and the $ \beta_1 $ parameter corresponds to the moving average part in the conditional variance, that is to say, the recent level of volatility. In this model, it could be convenient to define a measure, in the forecasting context, about the impact of present news in the future volatility. To carry out a study of this impact, we calculate the expected volatility $ k$-steps ahead, that is

$\displaystyle \textrm{E}(\sigma^2_{t+k}\vert
\sigma^2_t)=(\alpha_1+\beta_1)^k\sigma^2_t+
\alpha_0(\sum_{i=0}^{k-1}(\alpha_1+\beta_1)^i).$

Therefore, the persistence depends on the $ \alpha _1+\beta _1$ sum. If $ \alpha_1+\beta_1<1$, the shocks have a decaying impact on future volatility.

6.6.1.0.1 Example

In this example we generate several time series following a GARCH(1,1) model, with the same value of $ \alpha _1+\beta _1$, but different values of each parameter to reflect the impact of $ \beta_1 $ in the model.


Table 6.2: Estimates and t-ratios of both models with the same $ \alpha _1+\beta _1$ value
  $ \alpha_0$ $ \alpha_1$ $ \beta_1 $
Model A 0.05 0.85 0.10
estimates 0.0411 0.8572 0.0930
t-ratio (1.708) (18.341) (10.534)
Model B 0.05 0.10 0.85
estimates 0.0653 0.1005 0.8480
t-ratio 1.8913 3.5159 18.439


Figure 6.9: Simulated GARCH(1,1) data with $ \alpha _1=0.85$ and $ \beta _1=0.10$.
\includegraphics[width=0.59\defpicwidth]{sgarch1.ps}

Figure 6.10: Estimated volatility of the simulated GARCH(1,1) data with $ \alpha _1=0.85$ and $ \beta _1=0.10$ .
\includegraphics[width=0.59\defpicwidth]{svgarch1.ps}

Figure 6.11: Simulated GARCH(1,1) data with $ \alpha _1=0.10$ and $ \beta _1=0.85$.
\includegraphics[width=0.59\defpicwidth]{sgarch2.ps}

Figure 6.12: Estimated volatility of the simulated GARCH(1,1) data $ \alpha _1=0.10$ and $ \beta _1=0.85$.
\includegraphics[width=0.59\defpicwidth]{svgarch2.ps}

All estimates are significant in both models. The estimated parameters $ \hat\alpha_1$ and $ \hat\beta_1$ indicate that the conditional variance is time-varying and strongly persistent $ (\hat\alpha_1+\hat\beta_1\approx 0.95)$. But the conditional variance is very different in both models as we can see in figures 6.10 and 6.12.