Modeling Volatility and GARCH Family › Parameter Estimation

GARCH parameters are typically estimated via Maximum Likelihood Estimation (MLE). As discussed in the statistical inference chapter (Part I, Chapter 4), MLE finds parameters that maximize the probability of observing the data. For GARCH models, this means finding the variance equation parameters that make our observed return sequence most probable.

The likelihood function for GARCH builds on the assumption that standardized residuals are i.i.d. with a known distribution. Given parameters, we can compute the conditional variance at each point in time. We then evaluate how likely each observed return is given that conditional variance. Returns that are small relative to their conditional variance contribute more to the likelihood; returns that are large relative to their variance are penalized.

For a sample of returns $\{r_1, r_2, \ldots, r_T\}$, assuming conditional normality, the log-likelihood is:

$$\mathcal{L}(\theta) = -\frac{1}{2} \sum_{t=1}^{T} \left[ \ln(2\pi) + \ln(\sigma_t^2) + \frac{(r_t - \mu)^2}{\sigma_t^2} \right]$$

where:

• $\mathcal{L}(\theta)$: log-likelihood of the sample
• $\ln(\sigma_t^2)$: penalty term for higher variance (uncertainty)
• $\frac{(r_t - \mu)^2}{\sigma_t^2}$: penalty term for poor fit (squared standardized residual)
• $\theta$: vector of parameters $(\mu, \omega, \alpha, \beta)$
• $T$: sample size
• $\sigma_t^2$: conditional variance at time $t$
• $r_t$: return at time $t$

This expression has an intuitive structure. The first term inside the brackets is a constant that normalizes the density. The second term $\ln(\sigma_t^2)$ penalizes high variance. All else equal, the model prefers parameters that produce lower variance because this implies more precision in the forecasts. The third term $(r_t - \mu)^2 / \sigma_t^2$ penalizes returns that are large relative to the predicted variance. The model prefers parameters that make large returns occur when variance is high and small returns occur when variance is low.

The estimation proceeds as follows:

1. Initialize: Set starting values for parameters and initial variance (often the sample variance)
2. Recursion: For each $t$, compute $\sigma_t^2$ using the GARCH equation
3. Evaluate: Calculate the log-likelihood contribution from each observation
4. Optimize: Use numerical optimization to find parameters maximizing total log-likelihood

The recursive structure of GARCH means that the conditional variance at time $t$ depends on the conditional variance at time $t-1$. We must specify an initial variance to start the recursion, typically set to the unconditional variance implied by preliminary parameter estimates or simply the sample variance. As the sample grows, the influence of this initialization diminishes, but for short samples it can matter.

While conditional normality is convenient, you know from our stylized facts chapter that return distributions have fatter tails than normal. Even after accounting for time-varying variance, standardized residuals often exhibit excess kurtosis. This suggests that the normal distribution may not adequately capture the probability of extreme returns. Two common alternatives address this limitation:

• Student-t distribution: Adds a degrees-of-freedom parameter $\nu$ to capture excess kurtosis, where lower $\nu$ means fatter tails. As $\nu$ approaches infinity, the Student-t converges to the normal distribution. Typical estimates for financial data fall between 4 and 10, indicating substantial tail risk.
• Generalized Error Distribution (GED): Flexible family that nests the normal as a special case. A shape parameter controls the tail thickness, with values less than 2 producing fatter tails than normal and values greater than 2 producing thinner tails.

Using fat-tailed distributions often improves model fit and produces more accurate tail risk estimates. For risk management applications where extreme events matter most, the choice of error distribution can substantially affect VaR and expected shortfall calculations.