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 Asymmetric ILLIQ model assumes that the ILLIQ series follows an Asymmetric Multiplicative Error Model (Asy-MEM). More specifically, we postulate that ${\mathrm{ILLIQ}}_t={\mu }_t{\epsilon }_t$, 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, ${\mu }_t={𝔼}_{t-1}\text{[}{\mathrm{ILLIQ}}_t\text{]}$ 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 Asy-MEM specification for this dependence is: $${\mu }_t=\omega +\left(\alpha +\gamma I_{t-1}\right){\mathrm{ILLIQ}}_{t-1}+\beta {\mu }_{t-1}$$and: $$I_{t-1}=\left\{\begin{array}{ll}0& if{r}_{t-1}\ge 0\\ 1& if{r}_{t-1}<0\end{array}\right.$$where $r_t$ is the return at time $t$. So, the effective coefficient associated with a negative shock is $\alpha +\gamma$. If $\gamma$ is positive, a negative shock at time $t-1$ has a stronger impact in the illiquidity at time $t$, a positive shock.

Estimation

V-Lab applies the Quasi-Maximum Likelihood (QML) procedure to estimate the Asymmetric ILLIQ model, as it does to estimate the Asymmetric MEM model for a variance proxy (See Asy-MEM documentation).

Prediction

Let ${\mathrm{ILLIQ}}_t$ be the last observation in the sample, and let $\hat{\omega },\hat{\alpha },\hat{\gamma }$ and $\hat{\beta }$ be the QML estimates of parameters $\omega ,\alpha ,\gamma$ and $\beta$. The Asymmetric MEM specification implies that the forecast of the conditional expectation of ILLIQ at time $T+h$ is: $${\hat{\mu }}_{T+h}=\hat{\omega }+\left(\hat{\alpha }+\frac{\hat{\gamma }}{2}+\hat{\beta }\right){\hat{\mu }}_{T+h-1}$$where the $\frac{1}{2}$ multiplying $\gamma$ comes from the assumption of symmetric conditional distribution for the returns, which is the maintained assumption in various volatility models used by V-Lab. By applying the above formula iteravtively, we can forecast the conditional expectation of ILLIQ at time $T+h$ for any horizon $h$.

Amihud, Y. 2002. Illiquidity and stock returns: cross-section and time-series effects. Journal of Financial Markets 5:31 – 56. URL http://www.sciencedirect.com/science/article/pii/S1386418101000246.