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Asymmetries in bid and ask responses to innovations in the trading process 57
The terns f i B (MCt, D t) and f i S (MCt, D t), i2{a,b}, are functional forms of two
vectors of variables. The first vector (MC t) includes exogenous variables that
characterize the trade and the market environment. The second vector (D t) control
for trading-time regularities. The particular functional form considered is given in
Eq. (3.6). We impose linearity for simplicity reasons. The price impact of a given
trade is conditioned on these set of exogenous and deterministic variables, that we
will specify latter on.
f j
i MCt; ð DtÞ
¼ 1 þ Xn
k¼1
i;j
k MCk t
þ Xn0
h¼1
i; j
h Dh t
; i 2 fa; bg;
j 2 fB; Sg
(3.6)
From Eqs. (3.1) to (3.3), a t and b t are nonstationary, integrated of order one,
processes. Nonstationarity comes from the common long-run component (m t),
implying that the time series a t and b t must be co-integrated. 9 Our application has
the unusual advantage that the co-integration relationship has a known cointegration
vector (1,−1). The co-integration relationship is, therefore, a t−b t, the
bid–ask spread (henceforth, s t).
An increase in s t represents a departure from the long-run equilibrium
relationship between a t and b t. The error correction mechanism produces simultaneous
revisions in both ask and bid quotes that correct such deviations. For this
reason, we incorporate s t into Eqs. (3.2)–(3.3) as a determinant of the transitory
components of a t and b t. The coefficients α a EC and αb EC show how quickly do at and
b t revert to their common long-run equilibrium value.
3.2 The empirical model
The most common efficient parameterization of a vector autoregressive (VAR)
model with co-integrated variables is, from Granger’s representation theorem in
Engle and Granger (1987), a vector error correction (VEC) model. In the Appendix I,
we give an explicit derivation of the VEC model in Eq. (3.7) from the structural
model in the previous subsection,
0
10
1
1 0 AaB;t * AaB;t * at
B 0 1 AbB;t * AbS;t * CB
bt C
B
@
0 0 1 0
0 0 0 1
CB
A@
¼
0
B
@
x B t
x S t
C
A
1
ðLÞ ðLÞ C
ð Þ A
ð Þ
st
0
B
1 þ AtðLÞB @
EC
a
EC
b
B L
S L
at 1
bt 1
x B t 1
x S t 1
1
0
ua t
ub t
C
A þ
uB t
uS B
@
t
1
C
A ; (3.7)
9 Engle and Granger (1987), Stock and Watson (1988), Johansen (1991), and Escribano and Peña
(1994), among others, provide formal derivations of this result.