Temporal Aggregation of a Strong PGARCH(1,1) Process

Bollerslev’s (1986) standard GARCH(1,1) model has been successful in the literature of volatility modelling and forecasting in the past two decades. Many of its extensions are contributed to examine the stylized features often observed with financial asset data. One of the distinct success is Bollerslev and Ghysels’ (1996) periodic GARCH model, which takes into account periodic variation in the volatility of the underlying process. However, Drost and Nijman (1993) find that the conventional GARCH formulation works for only one sampling interval arbitrarily decided for the data in hand. This formulation does not apply to any other time intervals due to the assumption of an i.i.d. probability assumption for the underlying data. One of the problems caused by this will be that we cannot use ML method to estimate the GARCH model if the model is not for the original data set, but rather, for its temporally aggregated or dis-aggregated counterpart. Dorst and Nijman (1993) introduce the so-called weak GARCH formulation to tackle this problem and find this form of GARCH models apply to all sampling intervals for any given set of data. However, the ML method does not apply to the weak form of GARCH models since this formulation does not assume any probability distribution for the underlying standardised innovations. They thus propose a set of formulae to map the parameters of a weak GARCH(1,1) process sampled at one time interval to those of the same process but sampled at any other time interval. However, there is hitherto no analytical results for a weak PGARCH process. It is the main purpose of the paper to investigate the relationship amongst the parameters of a weak GARCH process before and after temporal aggregation. Our simulation results tend to suggest that a two-stage PGARCH process will aggregate into a weak GARCH process. Some analytical results about the aggregated process are introduced too.