Abstract: In the paste decades, much effort has been devoted to exploring the relation between stock price movements and trading volume to gain a better understanding of an issue that is the law of financial market price changes. Trading volume, however, only measures the absolute quantity of trading activity, but ignores the important information that this trading is buyer-initiated or seller-initiated. Order imbalances can provide additional power beyond trading activity measures such as volume in explaining stock return volatilities. In fact, order imbalance can reflect the information buyer-initiated or seller-initiated. In addition, order imbalance can signal excessive investor interest in a stock, and if this interest is auto-correlated, then order imbalance could be related to future returns.In this paper the relationship between order imbalances and daytime stock returns is investigated to obtain more detailed results. We often confront with two main difficulties in the study. The first one is the heterogeneous effects of the former on the latter under different market conditions. Second, it always involves large-scale data processing. To this end, quantile regression approach is used for large-scale data to reveal heterogeneous effects across different quantiles and hope to obtain more reliable results. Quantile regression approach for large-scale data consists of three steps:(1) computing a well-conditioned basis via QR factorization, (2) computing a sampling matrix to reduce the number of observations, and (3) using standard quantile regression for the reduced subset to compute a high-precision approximate solution. Compared to standard quantile regression, memory requirement and CPU time are reduced obviously by the proposed approach. For empirical illustration, first, Shanghai and Shenzhen stock markets are selected to test the effectiveness o quantile regression approach for large-scale data. Second, two lags of order imbalance are used to study the relationship between lagged order imbalances and daytime stock returns. Third, the contemporaneous imbalance is controlled and two lags of order imbalance are used to study the indirect effects of lagged order imbalances to the returns. Finally, the conditional density of response is predicted through estimated conditional quantiles. The empirical results show that one period lagged order imbalance has positive effects with increasing trend on stock returns at higher quantiles while has negative effects at lower quantiles. Furthermore, the lagged order imbalance has negative effects on stock returns when the current order imbalance is controlled, and the negative effect presents a downward trend with the increasing of quantiles. This implies that the order imbalance has good qualities of explanation and prediction for stock returns.

Key words: order imbalance, stock returns, large scale data, quantile regression