3 Univariate GARCH models

• Univariate GARCH models (Generalized Autoregressive Conditional Heteroskedasticity) are econometric models used to model, estimate, and forecast the conditional volatility (conditional variance of returns) of a single asset, and represent a parsimonious extension of ARCH models

• The presence of ARCH effects in financial time–series, confirmed by significant autocorrelation of absoulte or squared returns up to and including lag $p$, indicates that the variance of returns in not constant over time (term heteroscedasticity)

• Since the time–varying variance of returns is conditioned on its past history we use the term conditional heteroscedasticity

Conditional heteroscedasticity is as a simple autoregression AR($p$) of the squared innovations, from which the ARCH($p$) model, proposed by Engle, was established: $$\begin{equation} \begin{aligned} r_t&=\mu+u_t,~~~~~~~~~~~~~~~~u_t\sim i.i.d.~(0,~\sigma^2_t) \\ \\ \sigma_t^2&=\omega+\alpha_1 u_{t-1}^2+ \alpha_2 u_{t-2}^2 + \cdots +\alpha_p u_{t-p}^2 \end{aligned} \tag{3.1} \end{equation}$$

• The first equation in (3.1) is the conditional mean equation of returns which includes only intercept $\mu$ (mean), while standardized errors terms (called innovations) follow a white noise process, typically with zero mean and unit variance, i.e. $\dfrac{r_t-\mu}{\sigma_t}=\dfrac{u_t}{\sigma_t}=z_t \sim i.i.d.(0,~1)$

• The second equation in (3.1) is the conditional variance equation

In practice it is found that a large number of lags $p$ is required to obtain a good fit, and therefore a more parsimonious model proposed by Bollerslev replaces the ARCH($p$) with GARCH($1,1$) $$\begin{equation} \begin{aligned} r_t&=\mu+u_t,~~~~~~~~u_t=z_t\sigma_t \\ \\ \sigma_t^2&=\omega+\alpha_1 u_{t-1}^2+ \beta_1 \sigma_{t-1}^2 \end{aligned} \tag{3.2} \end{equation}$$ Note: GARCH($1,1$) model consists of two equations: (a) the conditional mean equation and (b) the conditional variance equation, which uses lagged squared innovations from the first equation and lagged variance of returns

• Although the specification of a GARCH($p,q$) model depends on time lags $p$ and $q$, typically one or two time lags are sufficient for a good fit

• GRACH($1,0$) is a special case of ARCH($1$) when lag $q=0$

• By incorporating additional variables into the conditional mean equation and/or the conditional variance equation, various types of GARCH($p,q$) models can be obtained

• The type of GARCH($p,q$) model also depends on the assumed distribution of innovations (e.g., a standard normal distribution, a Student’s t–distribution, or other distribution), which can be chosen to better align with the empirical properties of the returns

• Assumed distribution of innovations is crucial in specifying the likelihood function for estimation purposes, i.e. parameters $\mu$, $\omega$, $\alpha_1$ and $\beta_1$ are being estimated using maximum likelihood method – MLE

• Parameters of any GARCH type model should meet certain conditions to ensure: (a) the positivity of the conditional variance $\sigma_t^2$, and (b) the convergence to the long–run (unconditional) variance

• Standard GARCH($1,1$) model conditions are: $\omega \gt0$, $\alpha_1 \ge0$, $\beta_1 \ge 0$ and $(\alpha_1 + \beta_1) \lt 1$

• The sum $(\alpha_1 + \beta_1)$ is called volatility persistence, which is typically close to $1$, indicating a slow mean reverting to the long–run variance

$$\begin{equation} \lim_{t \to \infty} \sigma_t^2 =\dfrac{\omega}{1-(\alpha_1+\beta_1)} \tag{3.3} \end{equation}$$

• Related to the persistence parameter, not only long–run variance can be calculated, but also the hife–life

$$\begin{equation} hl=\dfrac{\ln(0.5)}{\ln(\alpha_1+\beta_1)} \tag{3.4} \end{equation}$$

• Half–life is defined as the number of days it takes for the effect of a shock to volatility to be reduced by half

• In many applications when modeling the volatility of exchange rates, it has been observed that persistence is equal to $1$. This means that exchange rates volatility follows a random walk, making the GARCH model integrated (IGARCH). However, IGARCH models are not popular because the $\beta_1$ is never estimated but instead calculated by enforcing the sum of the ARCH and GARCH parameters to be $1$, and thus unconditional variance and half –life can not be determined

• The IGARCH($p,q$) model without a constant term, which is equivalent to the EWMA model (Exponential Weighted Moving Average), is one of those belonging to the class of long–memory models. Additionally, the class of long–memory models includes FIGARCH($p,q$) models, i.e., fractional integrated GARCH($p,q$) models, which are not considered here

• Furthermore, if the mean return is influenced by the level of volatility (higher volatility leads to higher expected returns), you might use a GARCH–in–Mean (GARCH–M model). GARCH-in-Mean is an extension of standard GARCH model, where the conditional mean of returns depends on the conditional variance (in the first equation)

$$\begin{equation} \begin{aligned} r_t&=\mu+\delta\sigma^2_t+u_t \\ \\ \sigma_t^2&=\omega+\alpha_1 u_{t-1}^2+ \beta_1 \sigma_{t-1}^2 \end{aligned} \tag{3.5} \end{equation}$$

• In above GARCH–M($1,1$) model parameter $\delta$ represents the risk premium (assuming that investors should be “rewarded” for taking additional risk by investing in stocks with higher returns)

• In the conditional mean equation, variance can appear in its logarithmic form $\ln(\sigma^2_t)$ or as the standard deviation $\sqrt{\sigma^2_t}$. Nevertheless, it is expected positive value of $\delta$ when returns and risk are positively correlated