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Documentation>Liquidity Analysis>Spline ILLIQ

Definition

Amihud (2002) develops a measure of illiquidity of a security at time t asILLIQt=|Rt|VOLDtWhere Rt is the stock return and VOLDt 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 Spline ILLIQ model assumes that the ILLIQ series follows an Spline Multiplicative Error Model (Spline-MEM). More specifically, we postulate that ILLIQt=μt τtεt, where εt~D(1,σε2) and D is a distribution with non-negative support with unit mean and variance σε2. So, μtτt=𝔼t-1[ILLIQt] is the conditional mean of ILLIQt. 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 Spline-MEM specification for this dependence is:μt=ω+αILLIQt-1+βμt-1is a usual MEM specification and: τt = exp i=1k φi t - ti 2 is the exponential of a quadratic Spline with k knots t1,t2, . . . ,tk. The idea of adding this τt is to capture lower frequency variations on the illiquidity, like seasonalities and trends. The number of knots, k, is chosen by minimizing the Bayesian Information Criterion (BIC), also known as Schwarz Information Criterion (SIC), while the knots are placed equidistantly.

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 Spline-GARCH estimation by making ILLIQt the dependent variable, specifying it to have zero mean and an error process assumed Spline-GARCH with exogenous variables ε (See Spline-GARCH documentation). The only difference is the distributional assumption for D. V-Lab assumes that the distribution of the fundamental innovation εt is Chi-Square, i.e. D=χ(1)2, as it does in the MEM estimation for a variance proxy or ILLIQ (For example, see MEM documentation).

Prediction

Let ILLIQt be the last observation in the sample, and let ω^,α^,β^ and φ^1,φ^2,...,φ^k be the QML estimates of parameters ω,α,β and φ1,φ2,...,φk, respectively. For out-of-sample forecasts, we need to extrapolate the spline. We have good results when the extrapolation simply uses the last in-sample value. That is, V-Lab assumes that, h0: τ^ T+h = τ^T = exp i=1 k φi T - ti 2 Hence, the Spline MEM specification implies that the forecast of the conditional expetation of ILLIQ at time T+h is: μ^ T+h τ^ T+h = ω^ + α^ + α^ μ^ T+h-1 exp i=1 k φi T - ti 2 By applying the above formula iteratively, we can forecast the conditional expectation of ILLIQ at time T+h for any horizon h.

References

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.