Definition

Consider a return time series $r_t=\mu +{ϵ}_t$, where $\mu$ is the expected return and ${ϵ}_t$ is a zero-mean white noise. Despite of being serially uncorrelated, the series ${ϵ}_t$ does not need to be serially independent. For instance, it can present conditional heteroskedasticity. The Multiplicative Factor Multi Frequency GARCH (MF2-GARCH) model assumes a specific parametric form for this conditional heteroskedasticity. More specifically, we say that ${ϵ}_t~\mathrm{MF2-GARCH}$ if we can write ${ϵ}_t=\sqrt{{\sigma }_t^2{\tau }_t}{z}_t$, where $z_t$ is standard Gaussian:

$${\sigma }_t^2=\left(1-\alpha -\gamma /2-\beta \right)+\left(\alpha +\gamma I_{t-1}\right)\frac{{\epsilon }_{t-1}^2}{{\tau }_{t-1}}+\beta {\sigma }_{t-1}^2$$with$$I_{t-1}≔\{\begin{array}{lr}0& \text{if }{r}_{t-1}\ge \mu \\ 1& \text{if }{r}_{t-1}<\mu \end{array}$$is a GJR-GARCH component, and:$${\tau }_t={\lambda }_1+{\lambda }_2{V}_{t-1}^{\left(m\right)}+{\lambda }_3{\tau }_{t-1}$$where$$V_{t-1}^{\left(m\right)}=\frac{1}{m}\sum _{j=1}^m{V}_{t-j}$$

with $V_{t-j}={\epsilon }_{t-j}^2/{\sigma }_{t-j}^2$ denoting the standardized forecast error of the GARCH component at time $t-j$. The idea of adding this ${\tau }_t$ is to capture lower frequency variations in the volatility. The specification of the long-term component, ${\tau }_t$, is motivated by the empirical observation that the daily standardized forecast errors of one-component GARCH models are predictable when averaged at a lower frequency. We consider $V_{t-1}^{\left(m\right)}$ as a rolling window measure of the average forecast performance of the GARCH component over the previous $m$ days. Thus, ${\tau }_t$ scales the conditional volatility up/down if the GARCH component has under-/overestimated volatility in the recent past. The number $m$ is chosen by minimizing the Bayesian Information Criterion (BIC), also known as Schwarz Information Criterion (SIC).

Estimation

V-Lab estimates all the parameters $\left(\alpha ,\gamma ,\beta ,{\lambda }_1,{\lambda }_2,{\lambda }_3\right)$ simultaneously, by maximizing the log likelihood. The assumption that $z_t$ is Gaussian does not imply that the returns are Gaussian. Even though their conditional distribution is Gaussian, it can be proved that their unconditional distribution presents excess kurtosis (fat tails). In fact, assuming that the conditional distribution is Gaussian is not as restrictive as it seems: even if the true distribution is different, the so-called Quasi-Maximum Likelihood (QML) estimator is still consistent, under fairly mild regularity conditions.

Besides leptokurtic returns, the MF2-GARCH model, as the GJR-GARCH model, captures other stylized facts in financial time series, like volatility clustering and asymmetry. The GJR-GARCH model is in fact a restricted version of the MF2-GARCH, with ${\lambda }_2={\lambda }_3=0$.

Usual restrictions on the parameters are $\alpha >0$, $\beta >0$, $\alpha +\beta <1$, $\alpha +\gamma >0$, ${\lambda }_1>0$, ${\lambda }_2>0$, ${\lambda }_3>0$ and ${\lambda }_2+{\lambda }_3<1$.

Prediction

Let $r_T$ be the last observation in the sample, and let $\hat{\alpha }$, $\hat{\gamma }$, $\hat{\beta }$, ${\hat{\lambda }}_1$, ${\hat{\lambda }}_2$, ${\hat{\lambda }}_3$ be the QML estimators of the parameters $\alpha$, $\gamma$, $\beta$, ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, respectively. The MF2-GARCH model implies that the forecast of the conditional variance at time $T+1$ is:

$${\hat{\sigma }}_{T+1}^2{\hat{\tau }}_{T+1}.$$

Due to the dependence between the short-term and long-term component, the forecast of the conditional variance at time $T+i$ is not given by ${\hat{\sigma }}_{T+i}^2{\hat{\tau }}_{T+i}$ but, instead, needs to be computed recursively as described in Conrad and Engle (2021). Then, the forecast of the compound volatility at time $T+h$ is

$$\sqrt{\sum _{i=1}^h{𝔼}_T\widehat{\left[{\sigma }_{T+i}^2{\tau }_{T+i}\right]}}$$

Link to MF2-GARCH Toolbox for MATLAB: https://github.com/christian-conrad/mf2garch