GARCHlike

GARCHlike refers to time-series models for conditional variance that resemble the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) framework. In these models, the conditional variance of a series at time t is modeled as a function of past squared innovations and past variances, enabling volatility clustering and time-varying risk estimates common in financial data.

The standard GARCH(p,q) specification is a core example: h_t = omega + sum_{i=1}^q alpha_i ε_{t-i}^2 + sum_{j=1}^p beta_j h_{t-j}, where ε_t is the innovation at time t. GARCHlike models maintain this autoregressive structure for conditional variance but may differ in features such as the distribution of errors, the inclusion of asymmetries, or nonlinear transformations.

Common GARCHlike variants include EGARCH, which models the log of the variance and can capture leverage effects; TGARCH or GJR-GARCH, which include asymmetric responses to negative shocks; APARCH, which generalizes the power term; FIGARCH and related long-memory variants; and stochastic volatility models that share similar goals though they differ in specification.

Estimation usually proceeds via maximum likelihood under an assumed error distribution (normal, Student-t, etc.), or via quasi-maximum likelihood, with stationarity requiring that the sum of the persistence parameters be less than one.

Applications span financial risk management, volatility forecasting, and option pricing. Limitations include sensitivity to distributional assumptions, potential misspecification, and challenges in crisis periods where volatility dynamics may depart from finite-parameter models. See also ARCH and stochastic volatility models.