  While the long-ranged correlation of market orders and their impact on prices
has been relatively well studied in the literature, the corresponding studies
of limit orders and cancellations are scarce. We provide here an empirical
study of the cross-correlation between all these different events, and their
respective impact on future price changes. We define and extract from the data
the "bare" impact these events would have, if they were to happen in isolation.
For large tick stocks, we show that a model where the bare impact of all events
is permanent and non-fluctuating is in good agreement with the data. For small
tick stocks, however, bare impacts must contain a history dependent part,
reflecting the internal fluctuations of the order book. We show that this
effect can be accurately described by an autoregressive model on the past order
flow. This framework allows us to decompose the impact of an event into three
parts: an instantaneous jump component, the modification of the future rates of
the different events, and the modification of the future gaps behind the best
quotes. We compare in detail the present formalism with the temporary impact
model that was proposed earlier to describe the impact of market orders when
other types of events are not observed. Finally, we extend the model to
describe the dynamics of the bid-ask spread.
