ILLIQ-MFMEM

Definition

Amihud (2002) develops a measure of illiquidity of a security at time $t$ as$${\mathrm{ILLIQ}}_t=\frac{|R_t|}{{\mathrm{VOLD}}_t}$$Where $R_t$ is the stock return and ${\mathrm{VOLD}}_t$ is volume in 100 million dollars (V-Lab chooses the scaling of the dollar volume to enhance readability). This measure captures the intuition that a security is less liquid if a given trading volume generates a greater move in its price. It only requires daily data on prices and trading volumes, yet it is highly correlated with other illiquidity measures that require more data inputs.The Multiplicative Factor Multi Frequency ILLIQ model assumes that the ILLIQ series follows a Multifactor Multiplicative Error Model (MF-MEM). More specifically, we postulate that ${\mathrm{ILLIQ}}_t={\mu }_t{\tau }_t{\epsilon }_t$, where ${\epsilon }_t~\text{D(}1,{{\sigma }_{\epsilon }}^2\text{)}$ and $\text{D}$ is a distribution with non-negative support with unit mean and variance ${{\sigma }_{\epsilon }}^2$. So, ${\mu }_t{\tau }_t={𝔼}_{t-1}\text{[}{\mathrm{ILLIQ}}_t\text{]}$ is the conditional mean of ${\mathrm{ILLIQ}}_t$. Despite of having serially uncorrelated error terms, the ILLIQ series does not need to be serially independent. For instance, its conditional mean may depend on past information. The MF-MEM specification for this dependence is: $${\mu }_t=\left(1-\alpha -\gamma /2-\beta \right)+\left(\alpha +\gamma I_{t-1}\right)\frac{{\mathrm{ILLIQ}}_{t-1}}{{\tau }_{t-1}}+\beta {\mu }_{t-1}$$is a usual MEM specification and:$${\tau }_t={\lambda }_1+{\lambda }_2{V}_{t-1}^{\left(m\right)}+{\lambda }_3{\tau }_{t-1}$$represents the low-frequency movements of ILLIQ. The idea of adding this ${\tau }_t$ is to capture lower frequency variations on the illiquidity, like seasonalities and trends. The variable, $m$, is chosen by minimizing the Bayesian Information Criterion (BIC), also known as Schwarz Information Criterion (SIC).

Estimation

The family of MEMs is isomorphic to the family of ARCH and GARCH models. Hence, this model can be estimated using the Quasi-Maximum Likelihood (QML) procedure for MF2-GARCH estimation by making $\sqrt{{\mathrm{ILLIQ}}_t}$ the dependent variable, specifying it to have zero mean and an error process assumed MF2-GARCH with exogenous variables $\epsilon$ (See MF2-GARCH documentation). The only difference is the distributional assumption for $D$. V-Lab assumes that the distribution of the fundamental innovation ${\epsilon }_t$ is Chi-Square, i.e. $D={{\chi }_{(1)}}^2$, as it does in the MEM estimation for a variance proxy or ILLIQ (For example, see MEM documentation).