recent developments in high frequency financial ... - Index of

92
subtracting the deviations from equilibrium depths at the kth quote at time t+1 and t
and taking expectations which yields
Eð pþk;tþ1 ðQþk;tþ1Qþk;tÞ mtÞ
¼ 0
Eð p k;tþ1 þ ðQk;tþ1Qk;tÞ mtÞ
¼ 0
k ¼ 1; 2;:::
k ¼ 1; 2;:::;
where Δpj,t+1= p j,t+1−p j,t. We refer to the equations in Eq. (5) as ‘marginal update
conditions’. They relate the expected changes in the order book to the market order
flow. An obvious additional moment condition to identify the expected market
order size is given by
(5)
EðjXtj Þ ¼ 0: (6)
Moment conditions Eqs. (4), (5) and (6) can conveniently be exploited for
GMM estimation a la Hansen (1982).
Såndas (2001) derives the moment conditions from the basic model setup
outlined by Glosten (1994). Both Glosten's framework and Såndas' empirical
implementation entail a set of potentially restrictive assumptions that may be
problematic when confronting the model with real world data. Maybe the most
crucial assumption of the Glosten framework is that limit order traders are assumed
to be uninformed and that private information is only revealed through the arrival
of market orders. Recent literature, however, suggests that limit orders may also be
information-motivated (Seppi (1997); Kaniel and Liu (2001); Cheung et al. (2003)).
Bloomfield et al. (2005) observe in an experimental limit order market that
informed traders use more limit orders than liquidity traders. Since both break even
and update conditions are derived from the assumption of uninformed limit order
traders, the rejection of the model when confronted with real world data might be a
result from a violation of this fundamental assumption. 10 Another important consideration
is the number of active liquidity providers. Glosten (1994) assumes perfect
competition. Biais et al. (2000) propose solutions for oligopolistic competition.
The following section proposes a revised set of moment conditions which are
derived from a relaxation of the expected marginal profit condition and the
parametric assumption of the market order distribution. However, we leave the
basic assumption of uninformed limit order traders intact. Its relaxation would
entail a fundamental revision of the theoretical base model. This is left for further
research.
3.2 Revised moment conditions
3.2.1 Alternatives to the distributional assumption on market order sizes
S. Frey, J. Grammig
Reviewing the Såndas/Glosten framework Hasbrouck (2004) conjectures that the
parametric specification for the market order size distribution Eq. (2) may be
incorrect. 11 Indeed, the plot of the empirical market order distribution against the
fitted exponential densities depicted in Figure 3 in Såndas (2001) sheds some doubt
10 We are grateful to a referee for pointing this out.
11 It should be noted that the exponential assumption in DeJong et al.’s (1996) implementation of
the Glosten model did not seem to be a restrictive assumption.