Econometric Measures of Liquidity

Abstract

A security is liquid to the extent that an investor can trade significant quantities of the security quickly, at or near the current market price, and bearing low transaction costs. As such, liquidity is a multidimensional concept. In this chapter, I review several widely used econometrics or statistics-based measures that researchers have developed to capture one or more dimensions of a security’s liquidity (i.e., limited dependent variable model (Lesmond, D. A. et al. Review of Financial Studies, 12(5), 1113–1141, 1999) and autocovariance of price changes (Roll, R., Journal of Finance, 39, 1127–1139, 1984). These alternative proxies have been designed to be estimated using either low-frequency or high-frequency data, so I discuss four liquidity proxies that are estimated using low-frequency data and two proxies that require high-frequency data. Low-frequency measures permit the study of liquidity over relatively long time horizons; however, they do not reflect actual trading processes. To overcome this limitation, high-frequency liquidity proxies are often used as benchmarks to determine the best low-frequency proxy. In this chapter, I find that estimates from the effective tick measure perform best among the four low-frequency measures tested.

Appendix 1: Solution to LOT (1990) Model

To estimate transaction costs based on their model in Eq. 47.3, Lesmond et al. (1999) introduce the limited dependent variable regression model of Tobin (1958) and Rosett (1959). Tobin’s model specifies that data are available for the explanatory variable, x, for all the observation while data are only partly observable for the dependent variable, y, and for the other unobservable region, the information is given whether or not data are above a certain threshold.

Considering this aspect of Tobin’s model, the limited dependent variable model is an appropriate econometric method for the LOT model because a nonzero observed return occurs only when marginal profit exceeds marginal transaction costs.

Assuming that market model is correct in the presence of transaction costs, Lesmond et al. (1999) estimate transaction costs on the basis of Eqs. 47.1 and 47.3. The equation system is

$$ {\mathrm{R}}_{\mathrm{it}}^{*}={\upbeta}_{\mathrm{it}}{\mathrm{R}}_{\mathrm{mt}}+{\upvarepsilon}_{\mathrm{it}}, $$

where

$$ \begin{array}{ll}{\mathrm{R}}_{\mathrm{it}}={\mathrm{R}}_{\mathrm{it}}^{*}\hbox{--} {\upalpha}_{1\mathrm{i}}\hfill & \mathrm{if}\kern0.75em {\mathrm{R}}_{\mathrm{it}}^{*}<{\upalpha}_{1\mathrm{i}}\hfill \\ {}{\mathrm{R}}_{\mathrm{it}}=0\hfill & \mathrm{if}\kern0.75em {\upalpha}_{1\mathrm{i}}<{\mathrm{R}}_{\mathrm{it}}^{*}<{\upalpha}_{2\mathrm{i}}\hfill \\ {}{\mathrm{R}}_{\mathrm{it}}={\mathrm{R}}_{\mathrm{it}}^{*}\hbox{--} {\upalpha}_{2\mathrm{i}}\hfill & \mathrm{if}\kern0.75em {\mathrm{R}}_{\mathrm{it}}^{*}>{\upalpha}_{2\mathrm{i}}\hfill \end{array} $$

(47.4)

The solution to this limited dependent regression variable model requires a likelihood function to be maximized with respect to α_1i, α_2i, β_i, and σ_i.

$$ \begin{array}{l}\mathrm{L}\left({\upalpha}_{1\mathrm{i}},{\upalpha}_{2\mathrm{i}},{\upbeta}_{\mathrm{i}},{\upsigma}_{\mathrm{i}}/{\mathrm{R}}_{\mathrm{i}\mathrm{t}},{\mathrm{R}}_{\mathrm{mt}}\right)=\\ {}{\displaystyle \prod_1}\O \left(\frac{{\mathrm{R}}_{\mathrm{i}\mathrm{t}}+{\upalpha}_{1\mathrm{i}}-{\upbeta}_{\mathrm{i}}{\mathrm{R}}_{\mathrm{mt}}}{\upsigma_{\mathrm{i}}}\right){\displaystyle \prod_2}\left[\Phi \left(\frac{{\mathrm{R}}_{\mathrm{i}\mathrm{t}}+{\upalpha}_{2\mathrm{i}}-{\upbeta}_{\mathrm{i}}{\mathrm{R}}_{\mathrm{mt}}}{\upsigma_{\mathrm{i}}}\right)-\Phi \left(\frac{{\mathrm{R}}_{\mathrm{i}\mathrm{t}}+{\upalpha}_{1\mathrm{i}}-{\upbeta}_{\mathrm{i}}{\mathrm{R}}_{\mathrm{mt}}}{\upsigma_{\mathrm{i}}}\right)\right]{\displaystyle \prod_3}\O \left(\frac{{\mathrm{R}}_{\mathrm{i}\mathrm{t}}+{\upalpha}_{2\mathrm{i}}-{\upbeta}_{\mathrm{i}}{\mathrm{R}}_{\mathrm{mt}}}{\upsigma_{\mathrm{i}}}\right),\end{array} $$

(47.5)

where Ø refers to the standard normal density function and Φ refers to the cumulative normal distribution. The product is over the Region 1, 2, and 3 of observations for which R*_it < α_1i, α_1i < R*_it < α_2i, and R*_it > α_2i, respectively. The log likelihood function is

$$ \begin{array}{c} \log \kern0.5em \mathrm{L}={\Sigma}_1 \log \left[\frac{1}{\left(2{{\uppi \upsigma}_{\mathrm{i}}}^2\right)\frac{1}{2}}\right]-\frac{1}{2{\upsigma_{\mathrm{i}}}^2}{\Sigma}_1{\left({\mathrm{R}}_{\mathrm{i}\mathrm{t}}+{\upalpha}_{1\mathrm{i}}-{\upbeta}_{\mathrm{i}}{\mathrm{R}}_{\mathrm{mt}}\right)}^2+{\Sigma}_2 \log \left[{\Phi}_2-{\Phi}_1\right]\\ {}+{\Sigma}_3 \log \left[\frac{1}{{\left(2{{\uppi \upsigma}_{\mathrm{i}}}^2\right)}^{\frac{1}{2}}}\right]-\frac{1}{2{\upsigma_{\mathrm{i}}}^2}{\Sigma}_3{\left({\mathrm{R}}_{\mathrm{i}\mathrm{t}}+{\upalpha}_{2\mathrm{i}}-{\upbeta}_{\mathrm{i}}{\mathrm{R}}_{\mathrm{mt}}\right)}^2.\end{array} $$

(47.6)

Given Eq. 47.6, α_1i, α_2i, β_i, and σ_i can be estimated. The difference between α_2i and α_1i is the proxy of a round-trip transaction cost in the LOT model.