MICHAEL BARCLAY [*]
FRANK C. TORCHIO [**]
For approximately two decades, the General Trading Model ("GTM") has been used in securities litigation to estimate the number of shares damaged by alleged fraudulent misrepresentations by defendants. The GTM estimates the fraction of in-and-out trading volume and the fraction of retained volume. "In-and-out volume" refers to shares bought and sold within the class period; "retained volume" refers to shares purchased and held through the final disclosure that reveals the fraud. This is typically the last day of the class period. Estimates of the number of damaged shares from the GTM have been used in conjunction with a theory of true value (or conversely, artificial inflation) for the security to estimate aggregate monetary damages. 
Over the years, variations of the GTM predicated on different assumptions and/or parameters have been developed. The variations include single-trader models, such as the proportional and accelerated trading models, and multi-trader models.  This article compares the results of these models and critically evaluates the conclusions reached in previously published research.
This article demonstrates that results from the proportional single-trader model, GTM (1x), are consistent with the results of multi-trader GTMsfc when appropriate assumptions and parameters are used. No evidence was found to reject the GTM (1x) as a scientific method to estimate the number of damaged shares in securities litigation.
In securities litigation, damages arise when defendants make false or misleading statements that artificially inflate the stock price.  If an investor purchases the stock at this artificially inflated price, and the price later declines when the fraud is revealed, the investor will suffer damages from paying too much for the stock. In general, damages per share are calculated as the artificial inflation when the shares were purchased minus the artificial inflation when the shares were sold. For example, shares purchased when the stock price was artificially inflated and held through a disclosure that reveals the fraud typically are considered to be damaged. Shares purchased and then sold before any revelation of the fraud, however, are typically not considered to be damaged because these shares were passed on before any deflation in value.
Experts on damages in securities class actions generally do not have access to the trading records of individual class members. Consequently, the number of damaged shares is commonly estimated from a security's reported daily trading volume. Although the reported trading volume is quite reliable, the number of damaged shares is generally less than reported volume for several reasons.
First, reported volume may overstate the trading volume by the plaintiff class because it includes trades by specialists on the New York Stock Exchange ("NYSE") or market makers on the National Association of Securities Dealers Automated Quotation system ("NASDAQ") who buy from one investor and sell to another. One must adjust the reported volume to remove these double-counted trades. Recently published research suggests that a suitable correction is obtained by reducing NYSE reported volume by approximately ten percent and reducing NASDAQ volume by approximately fifty-eight percent. 
A second adjustment to volume is necessary to eliminate shares that were purchased during the class period and sold before the revelation of the alleged fraud. In many cases, these in-and-out shares have no associated damages because they were purchased and sold at prices with the same artificial inflation.  Historically, it has been common practice among economic experts for both plaintiffs and defendants to adjust volume for non-damaged, in-and-out volume using a statistical trading model.  The trading model is a mathematical model that estimates, on each day of the class period, the fraction of volume that is in-and-out volume and the fraction that is retained volume. …