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Order aggressiveness and order book dynamics 139
P
i 1 1fti 0jF t ; k ¼ 1; ...; K; (1)
denotes the conditional intensity function associated with the counting process
N k (t), given the information setF t consisting of the history of the complete order
k and trading process up to t. In this framework ðt;FtÞ corresponds to the
instantaneous arrival rate of an aggressive order or cancellation, and thus is a
natural continuous-time measure for the degree of order aggressiveness at each
instant.
Russell (1999) proposes parameterizing kðt;FtÞ in terms of a proportional
intensity structure
k
ðt; F tÞ
¼ k Mt ðÞ k 0 t ðÞskðÞ; t k ¼ 1; ...; K; (2)
where Ψi k is a function capturing the dynamics of the k-type process,l0 k (t) denotes
a k-type baseline intensity component that specifies the deterministic evolution of
the intensity until the next event and s k (t) isak-type seasonality component that
may be specified using a spline function. The basic idea of the ACI model is to
specify the dynamic component Ψi k in terms of an autoregressive process. Assume
that Ψi k is specified in log-linear form, i.e.
k
i ¼ exp e k i þ z0i 1 k
; (3)
where z i denotes the vector of explanatory variables capturing the state of the
market at arrival time t i and γ k the corresponding parameter vector associated with
process k. Then, the ACI(1,1) model is obtained by parameterizing the (K×1)
vector e i ¼ e 1 i ; e 2 i ; ...; e K i in terms of a VARMA type specification,
e i ¼ XK
k¼1
A k ei 1 þ Be i 1 y k i 1 ; (4)
where A k ={αk j } denotes a (K×1) innovation parameter vector and B={β ij }isa
(K×K) matrix of persistence parameters. Moreover, yi k defines an indicator variable
that takes the value 1 if the i-th point of the pooled process is of type k.