Function umemfit implements the estimation and evaluation of the MEM, MEM-MIDAS MEM-X and MEM-MIDAS-X models,
with and without the asymmetric term linked to negative lagged daily returns. The general framework assumes that:
$$x_{i,t}= \mu_{i,t}\epsilon_{i,t} = \tau_{t} \xi_{i,t} \epsilon_{i,t},$$
where
\(x_{i,t}\) is a time series coming from a non-negative discrete time process for the \(i\)-th day (\(i = 1, \ldots, N_t\))
of the period \(t\) (for example, a week, a month or a quarter; \(t = 1 , \ldots, T\));
\(\tau_{t}\) is the long-run component, determining the average level of the conditional mean, varying each period \(t\);
\(\xi_{i,t}\) is a factor centered around one, labelled as the short--run term, which plays the role of dumping or amplifying \(\tau_{i,t}\);
\(\epsilon_{i,t}\) is an \(iid\) error term which, conditionally on the information set, has a unit mean, an unknown variance, and a probability density function
defined over a non-negative support.
The short--run component of the MEM-MIDAS-X is:
$$\xi_{i,t}=(1-\alpha-\gamma/2-\beta) + \left(\alpha + \gamma \cdot {I}_{\left(r_{i-1,t} < 0 \right)}\right) \frac{x_{i-1,t}}{\tau_t} + \beta \xi_{i-1,t} + \delta \left(Z_{i-1,t}-E(Z)\right),$$
where \(I_{(\cdot)}\) is an indicator function, \(r_{i,t}\) is the daily return of the day \(i\) of the period \(t\) and \(Z\) is
an additional X term (for instance, the VIX). When the X part is absent, then the parameter \(\delta\) cancels.
The long-run component of the MEM-MIDAS and MEM-MIDAS-X is:
$$\tau_{t} = \exp \left\{ m + \theta \sum_{k=1}^K \delta_{k}(\omega) X_{t-k}\right\},$$
where \(X_{t}\) is the MIDAS term. When the "skew" parameter is set to "NO", \(\gamma\) disappears.
The MEM and MEM-X models do not have the long- and short-run components. Therefore, they directly evolve according to \(\mu_{i,t}\).
When the "skew" and X parameters are present, the MEM-X is:
$$\mu_{i,t}= \left(1-\alpha - \gamma / 2 - \beta \right)\mu + (\alpha + \gamma I_{\left(r_{i-1,t} < 0 \right)}) x_{i-1,t} + \beta \mu_{i-1,t}+\delta \left(Z_{i-1,t}-E(Z)\right),$$
where \(\mu=E(x_{i,t})\). When the "skew" parameter is set to "NO", in the previous equation \(\gamma\) cancels.
Finally, when the additional X part is not present, then we have the MEM model, where \(\delta\) disappears.