GARCH, or Generalized Autoregressive Conditional Heteroskedasticity, is a statistical model used in finance and econometrics to analyze and model the volatility of financial time series data. The GARCH process is an extension of the ARCH (Autoregressive Conditional Heteroskedasticity) model, introduced by Robert Engle in 1982.

The primary objective of GARCH models is to capture the time-varying volatility or variance observed in financial returns. Volatility clustering, where periods of high volatility tend to be followed by periods of high volatility and vice versa, is a common characteristic of financial time series data.

The GARCH(p, q) model is characterized by two main components:

1. **Autoregressive (AR) Component (p):** Similar to the AR process in time series analysis, the GARCH model includes an autoregressive component that captures the dependence of the current conditional variance on past conditional variances.

\[ \sigma_t^2 = \alpha_0 + \sum_{i=1}^{p} \alpha_i \varepsilon_{t-i}^2 \]

Here, \(\sigma_t^2\) is the conditional variance at time \(t\), \(\alpha_0\) is the constant term, \(\alpha_i\) are the autoregressive parameters, and \(\varepsilon_{t-i}^2\) are the squared past innovations (residuals).

2. **Moving Average (MA) Component (q):** The GARCH model also includes a moving average component that captures the dependence of the current conditional variance on past conditional innovations.

\[ \sigma_t^2 = \alpha_0 + \sum_{i=1}^{q} \beta_i \sigma_{t-i}^2 \]

Here, \(\beta_i\) are the moving average parameters, and \(\sigma_{t-i}^2\) are the past conditional variances.

The GARCH process is useful for modeling financial time series data because it allows for the representation of changing volatility over time. Traders, risk managers, and researchers often use GARCH models to forecast future volatility and manage risk. The estimation of GARCH parameters is typically done using maximum likelihood estimation (MLE) or other statistical methods.

Extensions of the basic GARCH model include models that account for asymmetry in volatility (e.g., GJR-GARCH) or models that incorporate long-term components (e.g., integrated GARCH or IGARCH). These models provide flexibility in capturing various aspects of volatility dynamics in financial markets.