1006 Autoregressive Conditional Duration Models
xi=ψi (^) i, (21.1)
where{ (^) i}is a sequence of independent and identically distributed (i.i.d.) random
variables withE(
i)=1 and positive support, andψisatisfies:
ψi=α 0 +
∑p
j= 1
αjxi−j+
∑q
v= 1
βvψi−v, (21.2)
wherepandqare non-negative integers andαjandβvare constant coefficients.
Sincexiis positive, it is common to assume thatα 0 >0,αj≥0 andβv≥0 for
j∈{1,...,p}andv∈{1,...,q}. Furthermore, the zeros of the polynomialα(L)=
1 −
∑g
j= 1 (αj+βj)L
jare outside the unit circle, whereLdenotes the lag operator,
g=max{p,q}, andαj=0 forj>pandβj=0 forj>q.
LetFhbe theσ-field generated by{ (^) h, (^) h− 1 ,...}. It is easy to see thatE(xi|Fi− 1 )=
ψiE(
i|Fi− 1 )=ψi. Thus,ψiis the conditional expected duration of the next trans-
action givenFi− 1. Since (^) ihas a positive support, it may assume the standard
exponential distribution. This results in an exponential ACD model. For ease of
reference, we shall refer to the model in equations (21.1)–(21.2) as an EACD(p,q)
model when (^) ifollows the standard exponential distribution.
21.2.1 Properties of the EACD model
We start with the simple EACD(1,1) model:
xi=ψi (^) i, ψi=α 0 +α 1 xi− 1 +β 1 ψi− 1. (21.3)
Taking the expectation of the model, we obtain:
E(xi)=E(ψi (^) i)=E[ψiE(
i|Fi− 1 )]=E(ψi),
E(ψi)=α 0 +α 1 E(xi− 1 )+β 1 E(ψi− 1 ).
Under the weak stationarity assumption,E(xi)=E(xi− 1 ), so that:
μx≡E(xi)=E(ψi)=
α 0
1 −α 1 −β 1
.
Consequently, 0≤α 1 +β 1 <1 for a weakly stationary process {xi}. Next, making
use of the fact thatE(
i)=1 andE(
i^2 )=2, we haveE(x^2 i)= 2 E(ψi^2 ). Again, under
weak stationarity:
E(ψi^2 ) =
μ^2 x[ 1 −(α 1 +β 1 )^2 ]
1 − 2 α^21 −β 12 − 2 α 1 β 1
, (21.4)
Var(xi) =
μ^2 x( 1 −β^21 − 2 α 1 β 1 )
1 − 2 α^21 −β 12 − 2 α 1 β 1
. (21.5)
From these results, for the EACD(1,1) model to have a finite variance, we need
1 > 2 α 12 +β 12 + 2 α 1 β 1. Similar results can be obtained for the general EACD(p,q)
model, but the algebra involved becomes tedious.