We develop a behavioral model for liquidity and volatility based on empirical regularities in trading order flow in the London Stock Exchange. This can be viewed as a very simple agentbased model in which all components of the model are validated against real data. Our empirical studies of order flow uncover several interesting regularities in the way trading orders are placed and cancelled. The resulting simple model of order flow is used to simulate price formation under a continuous double auction, and the statistical properties of the resulting simulated sequence of prices are compared to those of real data. The model is constructed using one stock (AZN) and tested on 24 other stocks. For low volatility, small tick size stocks (called Group I) the predictions are very good, but for stocks outside Group I they are not good. For Group I, the model predicts the correct magnitude and functional form of the distribution of the volatility and the bid-ask spread, without adjusting any parameters based on prices. This suggests that at least for Group I stocks, the volatility and heavy tails of prices are related to market microstructure effects, and supports the hypothesis that, at least on short time scales, the large fluctuations of absolute returns jrj are well described by a power law of the form Pðjrj4RÞ$R Àar , with a value of a r that varies from stock to stock. r 2007 Published by Elsevier B.V.JEL classification: G10

In this comment we discuss the problem of reconciling the linear efficiency of price returns with the long-memory of supply and demand. We present new evidence that shows that efficiency is maintained by a liquidity imbalance that co-moves with the imbalance of buyer vs. seller initiated transactions. For example, during a period where there is an excess of buyer initiated transactions, there is also more liquidity for buy orders than sell orders, so that buy orders generate smaller and less frequent price responses than sell orders. At the moment a buy order is placed the transaction sign imbalance tends to dominate, generating a price impact. However, the liquidity imbalance rapidly increases with time, so that after a small number of time steps it cancels all the inefficiency caused by the transaction sign imbalance, bounding the price impact. While the view presented by Bouchaud et al. of a fixed and temporary bare price impact is self-consistent and formally correct, we argue that viewing this in terms of a variable but permanent price impact provides a simpler and more natural view. This is in the spirit of the original conjecture of Lillo and Farmer, but generalized to allow for finite time lags in the build up of the liquidity imbalance after a transaction. We discuss the possible strategic motivations that give rise to the liquidity imbalance and offer an alternative hypothesis. We also present some results that call into question the statistical significance of large swings in expected price impact at long times. Contents

We show how this can be caused by delays in market clearing. Under the common practice of order splitting, large orders are broken up into pieces and executed incrementally. If the size of such large orders is power-law distributed, this gives rise to power-law decaying autocorrelations in the signs of executed orders. More specifically, we show that if the cumulative distribution of large orders of volume v is proportional to v −␣ and the size of executed orders is constant, the autocorrelation of order signs as a function of the lag is asymptotically proportional to −͑␣−1͒. This is a long-memory process when ␣ Ͻ 2. With a few caveats, this gives a good match to the data. A version of the model also shows long-memory fluctuations in order execution rates, which may be relevant for explaining the long memory of price diffusion rates.