# Documentation

## GARCH Model

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

${y}_{t}=\mu +{\epsilon }_{t},$

where${\epsilon }_{t}={\sigma }_{t}{z}_{t}$ and

${\sigma }_{t}^{2}=\kappa +{\gamma }_{1}{\sigma }_{t-1}^{2}+\dots +{\gamma }_{P}{\sigma }_{t-P}^{2}+{\alpha }_{1}{\epsilon }_{t-1}^{2}+\dots +{\alpha }_{Q}{\epsilon }_{t-Q}^{2}.$

 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:

• $\kappa >0$

• ${\gamma }_{i}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha }_{j}\ge 0$

• ${\sum }_{i=1}^{P}{\gamma }_{i}+{\sum }_{j=1}^{Q}{\alpha }_{j}<1$

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

## References

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