  We propose a microscopic model to describe the dynamics of the fundamental
events in the limit order book (LOB): order arrivals and cancellations. It is
based on an operator algebra for individual orders and describes their effect
on the LOB. The model inputs are arrival and cancellation rate distributions
that emerge from individual behavior of traders, and we show how prices and
liquidity arise from the LOB dynamics. In a simulation study we illustrate how
the model works and highlight its sensitivity with respect to assumptions
regarding the collective behavior of market participants. Empirically, we test
the model on a LOB snapshot of XETRA, estimate several linearized model
specifications, and conduct in- and out-of-sample forecasts.The in-sample
results based on contemporaneous information suggest that our model describes
returns very well, resulting in an adjusted $R^2$ of roughly 80%. In the more
realistic setting where only past information enters the model, we observe an
adjusted $R^2$ around 15%. The direction of the next return can be predicted
(out-of-sample) with an accuracy above 75% for time horizons below 10 minutes.
On average, we obtain an RMSPE that is 10 times lower than values documented in
the literature.
