A GARCH model is a conditional-variance model in which current variance depends on past shocks and past conditional variance. Tim Bollerslev introduced Generalized Autoregressive Conditional Heteroskedasticity as an extension of Robert Engle's ARCH framework.
The common GARCH(1,1) equation
For a return innovation εt, a standard GARCH(1,1) variance equation is σt2 = ω + αεt-12 + βσt-12.
Here ω is a variance intercept, α controls the response to the most recent squared innovation, and β controls persistence from the previous conditional variance estimate.
The model forecasts variance, not price direction
GARCH describes the conditional scale of uncertainty around the mean equation. A larger forecast variance does not specify whether the next return should be positive or negative.
The mean model and variance model are separate components even when they are estimated together.
Why squared innovations appear
Positive and negative innovations both contribute positively when squared, so the basic symmetric GARCH model responds to shock magnitude. This helps represent volatility clustering, where large movements tend to occur near other large movements.
It also means standard GARCH(1,1) does not by itself model an asymmetric response to positive versus negative shocks. Other variants exist for that purpose.
Coefficient restrictions depend on the specification
For a conventional GARCH(1,1) with positive conditional variance, restrictions commonly include ω > 0 and nonnegative α and β. A frequently used covariance-stationarity condition is α + β < 1 under the standard specification.
These conditions should not be copied mechanically to every GARCH-family model. Different distributions, variance equations and integrated or asymmetric variants have different properties.
Persistence needs careful interpretation
When α + β is close to one in the standard GARCH(1,1) setting, volatility shocks decay slowly in the fitted model. That can describe persistent conditional variance, but a high estimated persistence parameter is not proof that the same persistence will continue through future structural changes.
The innovation distribution is a modeling choice
Conditional normality is not the only option. Heavy-tailed distributions such as Student-t are often considered because financial return residuals can contain more extreme observations than a normal model would imply.
Choosing a distribution after repeatedly inspecting the same test period can, however, become another form of model selection bias.
Fit is not enough
Researchers should inspect residuals and standardized residuals, remaining autocorrelation, parameter stability and forecast performance on later data. A model can fit the in-sample variance pattern well while producing weak out-of-sample risk estimates.
Time-Series Cross-Validation and Walk-Forward Analysis provide the broader evaluation context.
GARCH is one volatility model, not the definition of volatility
Volatility Modelling includes rolling estimates, exponentially weighted approaches, realized measures and other model families. GARCH should be chosen because its conditional-variance structure fits the research problem, not because it is mathematically familiar.
Risk applications inherit model risk
GARCH forecasts can feed position sizing, VaR or simulation, but downstream calculations inherit the assumptions and forecast errors of the variance model. A precise numerical output should not be confused with a precise future variance.
GARCH is valuable because it makes changing conditional variance explicit. Its usefulness depends on specification, diagnostics, stability and out-of-sample performance rather than on the model name alone.