**6. Conclusions**

PHARCH models are good models for the analysis of high frequency data, since the financial market agents behave differently, incorporating heterogeneous information to the market microstructure. Nevertheless, their use still depends on solving some issues, mainly computational ones.

One big challenge in the analysis of high frequency data is dealing with large amounts of observations, and complex models bring computational difficulties, even with the recent technological breakthroughs in computing technology. Therefore, the first issue here is to develop techniques that help us to improve the computational algorithms. Maximum likelihood estimation may collapse, as we have described earlier. Techniques such as genetic algorithms and neural networks are viable optimization alternatives.

Another possibility is to use Bayesian techniques, such as the Griddy-Gibbs samples that we have used. The disadvantage of the Griddy-Gibbs sampler lies in its high computational load. From another viewpoint, more sophisticated volatility models might be developed, taking into account the arrival of information, for example, a stochastic volatility model or stochastic duration model; or we could adapt existing models such as CHARN (conditional heteroskedasticity nonlinear autoregressive) models to heterogeneity of information characteristics. Finally, extensions similar to those proposed to GARCH models could be studied for HARCH models.

A feature of the HARCH models is that the market components are chosen in a subjective way. In the analysis of the Euro-Dollar series, we considered five components, with different aggregations. A different number of components could be proposed, depending of the degree of information one has. This is clearly a matter for further studies.

One last remark is that the performance of the different estimation methods should be evaluated. This evaluation could be done using prediction capabilities, for example. Other possibility is calculating some measure of risk. Volatility models are often established with the purpose of computing the VaR (value at risk) or other risk measure or for establishing trading strategies. In this context, an evaluation of the performance of the proposed model and several estimation procedures should be interesting. A comparison of returns of different trading systems that use a proposed model will be of fundamental importance. Further details on these aspects can be found in Acar and Satchell (2002), Dunis et al. (2003), Ghysels and Jasiak (1994) and Park and Irwin (2005).

Other models for high frequency data use the realized volatility as a basis, instead of models such as ours and models of the ARCH family, which assume that volatility is a letent variable. Among the former, we mention the autoregressive fractionally integrated moving average (ARFIMA) models, the heterogeneous autoregressive model of realized volatilidade (HAR–RV) of Corsi (2009) and the mixed data sampling regression (MIDAS) proposed by Ghysels Santa-Clara and Valkanov (2002). A comparison of the PHARCH models with HAR and MIDAS would be useful, but due to the length of the present paper, this will be the object of future research.

**Author Contributions:** The authors study some theoretical properties of the PHARCH models and illustrate the theory with an application to real data. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially funded by Fapesp gran<sup>t</sup> 2013/00506-1.

**Conflicts of Interest:** The authors declare no conflict of interest.
