Gaussian and Quasi-Maximum-Likelihood GARCH Estimation

Why Model Conditional Variance?

In many economic time series, large movements tend to cluster: a period of unusually large shocks is often followed by further large shocks, while quieter periods tend to persist. A constant-variance model cannot represent that changing uncertainty. The GARCH(1,1) model describes how the variance conditional on past information evolves over time, using both the previous squared shock and the previous conditional variance.

The earlier material on ARCH effects motivates testing whether past squared residuals help explain current variation. GARCH extends that idea by allowing the conditional variance itself to carry forward. This tutorial focuses on specifying and estimating the resulting model, not on choosing among more elaborate volatility specifications.

Definition: Let $y_t$ have a specified conditional mean $m_t$, and define its innovation as $\epsilon_t=y_t-m_t$. The GARCH(1,1) model is $$ \epsilon_t=\sqrt{h_t}z_t,\qquad h_t=\omega+\alpha\epsilon_{t-1}^2+\beta h_{t-1}, $$ where $h_t$ is the conditional variance of $\epsilon_t$ given information available through time $t-1$. The usual positivity restrictions are $\omega>0$, $\alpha\geq 0$, and $\beta\geq 0$. The standardized innovation $z_t$ has conditional mean zero and conditional variance one.

The conditional mean may be a constant, a regression function, or a dynamic mean model. It must be specified well enough to define $\epsilon_t$; the volatility recursion then models its conditional variance. For Gaussian maximum likelihood, the additional distributional assumption is $z_t\mid\mathcal{F}_{t-1}\sim N(0,1)$, where $\mathcal{F}_{t-1}$ denotes information available through time $t-1$. That assumption is not required merely to define the GARCH variance recursion.

Conditional Likelihood

Under the conditional Gaussian assumption, $\epsilon_t\mid\mathcal{F}_{t-1}\sim N(0,h_t)$. Its conditional density is therefore the normal density with mean zero and variance $h_t$. For observations $t=1,\ldots,n$, the conditional likelihood multiplies these densities, treating the initial variance or the presample values needed to compute it as given.

f(\epsilon_t\mid\mathcal{F}_{t-1}) =\frac{1}{\sqrt{2\pi h_t}} \exp\left(-\frac{\epsilon_t^2}{2h_t}\right), \qquad \ell(\theta) =-\frac{1}{2}\sum_{t=1}^{n} \left[\log(2\pi)+\log(h_t)+\frac{\epsilon_t^2}{h_t}\right].

Here $\theta$ collects the parameters in the mean and variance equations, including $\omega,\alpha,\beta$, and each $h_t$ is calculated recursively from those parameters and the observed residuals. The log-likelihood rewards a variance that is neither too small nor needlessly large: the term $\log(h_t)$ penalizes large variances, while $\epsilon_t^2/h_t$ penalizes variances too small to account for the observed shock.

Worked Example: Calculating Conditional Likelihood Contributions

Suppose, hypothetically, that the conditional mean is zero and three observed residuals are $0.4,-0.8,1.2$, in the series’ measurement units. Set $\omega=0.10$, $\alpha=0.20$, $\beta=0.70$, and condition on $h_1=0.50$. These values satisfy the positivity restrictions. Compute the next variances:

h_2=0.10+0.20(0.4^2)+0.70(0.50)=0.482, \qquad h_3=0.10+0.20((-0.8)^2)+0.70(0.482)=0.5654.

Using the Gaussian log-likelihood contribution at each observation gives

\ell_t=-\frac{1}{2} \left[\log(2\pi)+\log(h_t)+\frac{\epsilon_t^2}{h_t}\right].

For the first observation, $\ell_1=-\tfrac12[1.8379+\log(0.50)+0.4^2/0.50]\approx-0.7324$. For the second, $\ell_2=-\tfrac12[1.8379+\log(0.482)+0.8^2/0.482]\approx-1.2179$. For the third, $\ell_3=-\tfrac12[1.8379+\log(0.5654)+1.2^2/0.5654]\approx-1.9073$. The conditional log-likelihood for these three observations is their sum, approximately $-3.8576$. This is the criterion evaluated at the specified parameters, not an estimate obtained by maximizing it.

Gaussian maximum likelihood (Gaussian ML) chooses parameters that maximize this conditional log-likelihood, subject to the model’s restrictions. In applications the recursion is evaluated repeatedly at candidate parameter values, and the numerical maximization must preserve admissible values. The likelihood is conditional on the initialization described above; when the initial variance is fixed or initialized by a practical rule, this is commonly called a conditional likelihood. For a long series, the influence of a finite number of initial terms is typically small under regularity conditions, but initialization should still be reported.

Constraints, Persistence, and Long-Run Variance

The restrictions $\omega>0,\alpha\geq0,\beta\geq0$ ensure a positive conditional variance when the recursion starts from a positive variance. The coefficient $\alpha$ measures the response to the most recent squared innovation; $\beta$ measures the carryover of the previous conditional variance. Their sum, $\rho=\alpha+\beta$, is commonly used as a measure of volatility persistence.

If the innovations have conditional variance one, a finite unconditional variance requires the stronger condition $\alpha+\beta<1$. Taking expectations of the variance recursion in a covariance-stationary process gives $E(h_t)=\omega+\alpha E(\epsilon_{t-1}^2)+\beta E(h_{t-1})$. Since $E(\epsilon_{t-1}^2)=E(h_{t-1})$, the unconditional variance is $\omega/(1-\alpha-\beta)$. Thus nonnegative coefficients alone do not imply a finite long-run variance.

Key result: When $\omega>0$, $\alpha,\beta\geq0$, and $\rho=\alpha+\beta<1$, the unconditional variance is $$ E(\epsilon_t^2)=E(h_t)=\frac{\omega}{1-\rho}. $$ For forecasts, the conditional expectation of future variance deviations from this long-run level decays geometrically at rate $\rho$. A value of $\rho$ near one indicates highly persistent volatility.

Worked Example: Interpreting Estimated Persistence

Suppose a hypothetical fitted model for a return series, measured in percentage points, gives $\hat{\omega}=0.08$, $\hat{\alpha}=0.12$, and $\hat{\beta}=0.82$. The estimates satisfy positivity, and $\hat{\rho}=0.12+0.82=0.94<1$, so the model implies a finite unconditional variance.

The estimated long-run variance is $0.08/(1-0.94)=1.333$ squared percentage points. Its corresponding standard deviation is $\sqrt{1.333}=1.155$ percentage points. Because $\hat{\rho}$ is close to one, shocks to conditional variance fade slowly. The geometric half-life solves $0.94^j=0.5$, so

j=\frac{\log(0.5)}{\log(0.94)}\approx 11.20\text{ periods}.

This is an approximate half-life in the fitted variance-decay recursion; it does not mean that every observed shock’s realized effect will follow that exact path. The estimate $\hat{\rho}<1$ gives a finite-variance implication within the fitted model, not proof that the model is correctly specified.

Gaussian ML and Quasi-Maximum Likelihood

Gaussian ML treats conditional normality as part of the model. In economic data, conditional normality may be a convenient approximation rather than a credible description of every shock. Gaussian quasi-maximum likelihood (Gaussian QML) uses the same Gaussian log-likelihood as an objective function, but interprets it as a quasi-likelihood: the innovations need not be normally distributed. The important moment restrictions are that the conditional mean is specified correctly and that the conditional variance follows the proposed recursion, with standardized innovations having conditional mean zero and variance one.

Under suitable regularity conditions—including identification, stationarity or ergodicity appropriate to the model, a correctly specified conditional mean and variance, and adequate finite moments—Gaussian QML can consistently estimate the variance parameters even when the conditional distribution is nonnormal. Finite fourth moments are commonly needed for conventional asymptotic variance estimation. These conditions matter: if the variance recursion is wrong, robustness to nonnormality does not make the resulting estimates valid for the intended model.

The usual inverse-Hessian covariance formula for Gaussian ML relies on the information equality, which can fail under nonnormality. QML inference therefore uses a sandwich covariance estimator. In generic notation, let $s_t(\theta)$ be the score contribution and let $A$ be the expected negative derivative of the score, while $B$ is the variance of the score. The asymptotic covariance of $\sqrt{n}(\hat{\theta}-\theta)$ is $A^{-1}BA^{-1}$; its sample counterpart replaces these quantities by estimates. When correctly specified conditional moments make scores serially uncorrelated, $B$ is estimated from their outer products. If score contributions remain serially dependent, a long-run covariance estimate is needed instead.

The robust covariance does not change the Gaussian QML point estimate. It changes estimated uncertainty, and therefore standard errors, confidence intervals, and tests. For a derived quantity such as persistence $\rho=\alpha+\beta$, uncertainty must account for the covariance between $\hat{\alpha}$ and $\hat{\beta}$, not just their separate standard errors.

Worked Example: Robust Uncertainty for Persistence

For illustration, suppose $\omega$ is fixed and QML estimates $(\alpha,\beta)$ from $n=100$ hypothetical observations, giving $(\hat{\alpha},\hat{\beta})=(0.15,0.75)$. Suppose the estimated per-observation matrices for the sandwich calculation are

\hat A= \begin{pmatrix}4&0\\0&1\end{pmatrix}, \qquad \hat B= \begin{pmatrix}4&1\\1&2\end{pmatrix}.

Then $\hat A^{-1}=\begin{pmatrix}0.25&0\\0&1\end{pmatrix}$, so the estimated covariance of $\sqrt{n}(\hat{\alpha},\hat{\beta})$ is

\hat A^{-1}\hat B\hat A^{-1} = \begin{pmatrix}0.25&0.25\\0.25&2\end{pmatrix}.

Dividing by $n=100$ gives the estimated covariance of the parameter estimates, $\begin{pmatrix}0.0025&0.0025\\0.0025&0.0200\end{pmatrix}$. The estimated persistence is $0.15+0.75=0.90$, and its estimated variance is $0.0025+0.0200+2(0.0025)=0.0275$, with standard error $\sqrt{0.0275}=0.1658$. A rough normal-approximation interval is $0.90\pm1.96(0.1658)$, or approximately $(0.575,1.225)$. Since this interval crosses one, it does not establish that persistence is below one. Moreover, inference near the restriction $\alpha+\beta=1$ requires particular care because boundary behavior can make ordinary normal approximations unreliable.

Common Pitfall

A common mistake is to treat a Gaussian QML fit as evidence that the shocks are normally distributed, or to report ordinary inverse-Hessian standard errors without checking whether the Gaussian information equality is justified. Maximizing a Gaussian criterion does not itself establish conditional normality. The correction is to distinguish the estimation objective from the distributional assumption: call the procedure Gaussian ML when conditional normality is assumed, and Gaussian QML when the Gaussian criterion is used under weaker conditional-moment assumptions. For QML inference, use an appropriate sandwich covariance estimate and state the assumptions that support it.

A second interpretive error is to call a large $\hat{\alpha}+\hat{\beta}$ “proof” of permanent volatility. It is an estimate of high persistence under the fitted recursion. Whether the process has a finite unconditional variance depends on the relevant parameter restrictions, and uncertainty in the persistence estimate should be considered.

Key takeaway: GARCH(1,1) estimation combines a recursive conditional variance with a likelihood built from its conditional density. Positivity restrictions keep the recursion admissible; $\alpha+\beta$ summarizes persistence and determines the finite unconditional variance condition. Gaussian QML can support inference without conditional normality only when its moment and regularity assumptions hold, and robust standard errors are then essential.

Check Your Understanding

Answer each question by referring to the model equations and assumptions.

Why do the restrictions $\omega>0,\alpha\geq0,\beta\geq0$ help ensure positive conditional variances?
What additional assumption distinguishes Gaussian ML from Gaussian QML?
If $\omega=0.06$, $\alpha=0.10$, and $\beta=0.85$, calculate the persistence and, if finite, the unconditional variance.
Why does the sandwich covariance for $\hat{\alpha}$ and $\hat{\beta}$ need their covariance when calculating uncertainty for $\hat{\alpha}+\hat{\beta}$?
What does a persistence estimate close to one say about the fitted model, and what does it not establish?