GARCH Parameters and Estimation
You're building a volatility model for an equity return series and decide to use a GARCH(1,1) specification.
- Write down the GARCH(1,1) model and identify all the parameters. What constraints must they satisfy for the model to be stationary?
- Explain how you would estimate the parameters using maximum likelihood. What is the log-likelihood function, and how do you handle the recursion for the conditional variance?
- What is the long-run (unconditional) variance implied by the model, and how does it relate to the parameters?
Hints
- Start by writing the variance equation as a weighted combination of a constant, yesterday's squared innovation, and yesterday's variance. What do each of these terms capture economically?
- For stationarity, think about what happens to the variance recursion when $\alpha + \beta \geq 1$ -- does it converge to a finite long-run level?
- The log-likelihood under Gaussian innovations involves both $\ln(\sigma_t^2)$ and $\varepsilon_t^2 / \sigma_t^2$. You need to recursively build the entire sequence of $\sigma_t^2$ before you can evaluate it.
Worked Solution
How to Think About It: GARCH is the workhorse volatility model on every trading desk. The basic idea is simple: tomorrow's variance is a weighted average of three things -- a long-run baseline, today's squared return (the "news" component), and today's variance (the "persistence" component). If someone asks you about GARCH in an interview, they want to see that you understand both the model mechanics and the practical estimation, not just the formula.
Key Insight: The model has only three free parameters ($\omega$, $\alpha$, $\beta$), and they must satisfy $\alpha + \beta < 1$ for stationarity. The sum $\alpha + \beta$ controls how persistent volatility shocks are -- typical equity values have $\alpha + \beta \approx 0.95$, meaning shocks to volatility die out slowly.
The Model:
Returns follow $r_t = \mu + \varepsilon_t$ where $\varepsilon_t = \sigma_t z_t$ and $z_t \sim N(0,1)$. The conditional variance evolves as:
$\sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2$
Parameters:
- $\omega > 0$: the constant term, anchoring variance to a long-run level
- $\alpha \geq 0$: the ARCH coefficient -- how much the model reacts to yesterday's squared shock
- $\beta \geq 0$: the GARCH coefficient -- how much yesterday's variance persists into today
- Stationarity requires $\alpha + \beta < 1$
Long-run variance:
$\bar{\sigma}^2 = \frac{\omega}{1 - \alpha - \beta}$
This is the level to which volatility mean-reverts. The speed of mean reversion is governed by
Estimation via MLE:
Assuming Gaussian innovations, the log-likelihood is:
$\ell(\omega, \alpha, \beta) = -\frac{1}{2} \sum_{t=1}^{T} \left( \ln(2\pi) + \ln(\sigma_t^2) + \frac{\varepsilon_t^2}{\sigma_t^2} \right)$
The estimation procedure:
- Initialize $\sigma_0^2$ to the sample variance of the return series
- For each $t = 1, \ldots, T$, recursively compute $\sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2$
- Evaluate the log-likelihood given the full sequence of $\sigma_t^2$
- Use a numerical optimizer (BFGS or Nelder-Mead) to maximize $\ell$ over $(\omega, \alpha, \beta)$, subject to the positivity and stationarity constraints
Practical Considerations:
- Typical equity estimates: $\alpha \approx 0.05$-$0.10$ and $\beta \approx 0.85$-$0.95$
- Quasi-MLE (QMLE) provides consistent estimates even when the true innovation distribution is non-Gaussian
- The initialization of $\sigma_0^2$ matters less as $T$ grows, but for short series, a poor initialization can distort estimates
- Extensions like EGARCH or GJR-GARCH add asymmetry to capture the leverage effect (volatility rises more after negative returns)
Answer: GARCH(1,1) has three parameters: $\omega$ (baseline variance), $\alpha$ (shock sensitivity), and $\beta$ (persistence). They are estimated by maximizing the Gaussian log-likelihood via numerical optimization, with constraints $\omega > 0$, $\alpha \geq 0$, $\beta \geq 0$, and $\alpha + \beta < 1$. The long-run variance is $\omega / (1 - \alpha - \beta)$.
Intuition
GARCH captures the most important empirical feature of financial returns: volatility clustering. Big moves tend to follow big moves, and calm periods tend to follow calm periods. The model formalizes this by making tomorrow's variance a function of today's shock and today's variance. The parameter $\alpha$ controls how reactive the model is to new information (a large $\alpha$ means a single big return spikes the forecast), while $\beta$ controls how sticky the forecast is (a large $\beta$ means volatility stays elevated for a long time after a shock). In practice, $\beta$ dominates -- markets have persistent volatility regimes, and most of the day-to-day variance forecast comes from the previous day's forecast, not from the latest return.
The estimation via MLE is conceptually straightforward but numerically tricky. The log-likelihood is not globally concave, so starting values matter. A common trick is to initialize with rough moment-based estimates: set $\alpha + \beta$ close to 0.95, then let the optimizer refine. The key interview takeaway is that GARCH is a filter -- it takes the observable return series and produces an unobservable variance series, and MLE is the principled way to calibrate that filter.