The generalized autoregressive conditional heteroscedastic (GARCH) model is an extension of Engle's ARCH model for variance heteroscedasticity [1]. If a series exhibits volatility clustering, this suggests that past variances might be predictive of the current variance.

The GARCH(P,Q) model is an autoregressive moving average model for conditional variances, with P GARCH coefficients associated with lagged variances, and Q ARCH coefficients associated with lagged squared innovations. The form of the GARCH(P,Q) model in Econometrics Toolbox™ is


whereεt=σtzt and


    Note:   The Constant property of a garch model corresponds to κ, and the Offset property corresponds to μ.

For stationarity and positivity, the GARCH model has the following constraints:

  • κ>0

  • γi0,αj0

  • i=1Pγi+j=1Qαj<1

To specify Engle's original ARCH(Q) model, use the equivalent GARCH(0,Q) specification.


[1] Engle, Robert F. "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation." Econometrica. Vol. 50, 1982, pp. 987–1007.

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