Analysis of Financial Time Series, Third Edition

3.6 The Integrated GARCH Model

If the AR polynomial of the GARCH representation in Eq. (3.15) has a unit root, then we have an IGARCH model. Thus, IGARCH models are unit-root GARCH models. Similar to ARIMA models, a key feature of IGARCH models is that the impact of past squared shocks ηt−i = for i > 0 on is persistent.

An IGARCH(1,1) model can be written as

where {ϵt} is defined as before and 1 > β1 > 0. For the monthly excess returns of the S&P 500 index, an estimated IGARCH(1,1) model is

where the standard errors of the estimates in the volatility equation are 0.0017, 0.000013, and 0.0144, respectively. The parameter estimates are close to those of the GARCH(1,1) model shown before, but there is a major difference between the two models. The unconditional variance of at, hence that of rt, is not defined under the above IGARCH(1,1) model. This seems hard to justify for an excess return series. From a theoretical point of view, the IGARCH phenomenon might be caused by occasional level shifts in volatility. The actual cause of persistence in volatility deserves a careful investigation. ...