Abstract

Modeling high frequency returns under a spatial representation with possible spatial long memory in volatility is a current research topic of great interest. For univariate return series, different fractionally integrated (FI) volatility models with long memory, such as the FIGARCH, FIEGARCH and FIAPARCH are proposed. Most recently, a FILog-GARCH model, which is a long memory extension of the Log-GARCH, and an adjusted modulus asymmetric FILog-GARCH (MAFILog-GARCH), have been introduced. Both models share some common advantages over the existing long memory volatility models, as their stationary solutions have simple closed-form formulas and explicit conditions for the existence of finite moments until any given order can be easily obtained. The most important feature of the FILog-GARCH is that it is a log-linear process of the squared returns. However, the performance of this model is strongly affected by exact or nearly zero returns. Moreover, possible asymmetric volatility effect is also not considered in this model. Additionally, in the FILog-GARCH model, negative coefficients are permitted but only to some extent. The MAFILog-GARCH is proposed to overcome all of those disadvantages, where the log-volatility process is defined as a FARIMA model of an asymmetric function of the log-modulus transformation of the innovations in the past.

In this paper, the spatial extensions of the FILog-GARCH and MAFILog-GARCH, called SFILog-GARCH and SMAFILog-GARCH, respectively, are introduced. Like in the univariate case, the SFILog-GARCH is a log-linear spatial FARIMA (SFARIMA) of the squared-returns. While for the SMAFILog-GARCH only the log-volatility process is linear, but the squared returns are no longer log-linear. Stationary solutions, conditions for the existence of finite moments until a given order as well as closed-form formulas of the acf of the log-volatility processes and the squared returns are obtained. The SFILog-GARCH can be estimated by the least squared estimator (LSE) of Beran et al, (2009) with the same asymptotic properties. For practical implementation, the SFILog-GARCH can be fitted with the “sfarima.est” function in the “DCSmooth” R package, where a quick algorithm for calculating the spatial residuals was employed so that the running time in case with large sample sizes in both dimensions is still acceptable. Following Zaffaroni (2009), it is shown that both proposed models are members of the spatial exponential volatility family. Hence, the log-process of the squared returns by a SMAFILog-GARCH exhibits a spatial linear signal-plus-noise representation. A LSE for fitting this model is proposed based on this factor and the above-mentioned quick algorithm. Application to high-frequency returns of a few German firms showed that the proposed models are very useful in practice.

References:
Beran, J., Ghosh, S., and Schell, D. (2009). On least squares estimation for long-memory lattice processes. J. Multivar. Anal., 100, 2178–94.

Zaffaroni, P. (2009). Whittle estimation of EGARCH and other exponential volatility models. J. of Econometrics, 151, 190–200.