The paper titled, Temporal Aggregation of GARCH processes, by Drost and Njiman is a classic paper that introduces three forms of GARCH processes: Strong form of GARCH, Semi-strong form of GARCH and Weak form of GARCH. Only the __Weak form of GARCH__ is appropriate for connecting volatility estimates and parameters of models built at various frequencies. In most of the literature on volatility estimation from high frequency data, the authors assume Weak form of GARCH.

In this post, I will briefly summarize the main points of the paper :

__Introduction __

Many financial time series exhibit conditional heteroskedasticity. ARCH, GARCH models are the usual goto models. If you are building an ARIMA model with high frequency data, then the aggregation properties of a low frequency ARIMA model is a well researched topic. However little is known about the aggregation properties of these models. This paper attempts to fill that void. The only known result before this paper appeared is that as the sampling frequency increases, the conditional heteroskedastic pattern disappears. One of the questions that this paper tries to answer is :

Is a GARCH model that is built using high frequency data consistent with GARCH model built using low frequency data ?

What’s the purpose of the paper ?

- This paper shows that the classic GARCH assumptions are not robust to the specification of sampling interval. Independent daily rescaled innovations, imply dependent rescaled innovations at the weekly frequency
- Introduce three types of GARCH forms - Strong GARCH, Semi-strong GARCH and weak GARCH
- To show that classical(semi-) strong GARCH assumptions on the available data are arbitrary. Generally a strong GARCH aggregates to a weak GARCH
- Assumption of symmetric weak GARCH models are internally consistent. Every ARMA model with a symmetric weak GARCH errors, aggregates to a model in the same class
- Strongly consistent estimators of low frequency parameters is possible with low frequency data sets
- High frequency parameters can be identified from the corresponding low frequency ones

The central message of the paper is

The class of ARMA models with weak GARCH errors is closed under temporal aggregation

__Definitions and Notation __

The basic model for conditional variance is given in this section. The three forms of GARCH are defined. Strong GARCH assumes rescaled innovations are iid, Semi-Strong form of GARCH assumes rescaled innovations are uncorrelated and Weak form of GARCH has a constraint only on projections of the innovations.

__Aggregation of GARCH(1,1) __

This section has four examples that illustrate the following points :

- The class of symmetric weak GARCH(1,1) processes with stock variables is closed under temporal aggregation.
- The class of symmetric weak GARCH(1,1) processes with flow variables is closed under temporal aggregation.
- The class of strong GARCH models is not closed under temporal aggregation.
- The class of semi-strong GARCH models is not closed under temporal aggregation.

Aggregation of ARMA - GARCH

This section shows that the class of ARMA models with symmetric GARCH errors are closed under temporal aggregation. There is a stylized fact mentioned in Rama Cont’s article - The coarse grained volatility serves as an estimate for fine grained volatility, than the other way around. In the GARCH case too, the high frequency parameters can be identified from low frequency parameters. The aliasing problem that infinite high frequency models can be identifies for a given low frequency model is not present in GARCH. The authors use data from some of the previous papers to reemphasize many aspects of this paper.

__Concluding remarks__

This paper derives a low frequency model that is implied by a assumed high frequency GARCH model and shows that iid assumptions on the rescaled innovations at the data frequency which one happens to have available is arbitrary. The low frequency variance parameters generally depend on mean, variance and kurtosis parameters of the high frequency model. Identification of the parameters in the high frequency strong GARCH model from low frequency data is often possible. Also, high frequency observations can be used to obtain estimates of low frequency variance parameters which are likely to be better than direct estimates of low frequency data .

This paper introduces three forms of GARCH models, the strong, semi-strong and weak form. It also derives relations connecting model parameters of a GARCH process and a temporally aggregated GARCH. The class of models containing symmetric weak GARCH is closed under temporal aggregation. The class of models containing strong GARCH or semi-strong GARCH is not closed under temporal aggregation. So, if there is some seasonality that is present in the intraday data, then temporal aggregation breaks down. Hence besides the usual testing that can be done by looking for autocorrelations, another way test for seasonality is to relate the GARCH parameters of low frequency models and high frequency models,and check whether the relations derived in this paper hold good or not.