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A Proofs
Proof of Lemma 1
If it is optimal for a trader with valuation v to submit a buy order, then it is also optimal for traders
with private values v > v to submit buy orders. Let s be an arbitrary sell order, and suppose that
dbuy(v) = 1. Then,
b,t
buy buy buy buy buy buy
Èb,t v - pb,t + ¾b,t + V (1 - Èb,t ) - c > Èb,t (v - pb,t) + ¾b,t + V (1 - Èb,t ) - c
sell sell sell
e" Ès,t (ps,t - v) - ¾s,t + V (1 - Ès,t ) - c
sell sell sell
e" Ès,t ps,t - v - ¾s,t + V (1 - Ès,t ) - c. (A1)
The first line follows because v > v; the second line follows because it is optimal for a trader with
value v to submit a buy order at b; the third line follows because v > v.
Let pb ,t be the optimal buy submission for the trader with valuation v . By optimality,
buy buy buy buy buy buy
Èb,t (v - pb,t) + ¾b,t + V (1 - Èb,t ) - c, e" Èb ,t (v - pb ,t) + ¾b ,t + V (1 - Èb ,t ) - c (A2)
buy buy buy buy buy buy
Èb ,t (v - pb ,t) + ¾b ,t + V (1 - Èb ,t ) - c e" Èb,t (v - pb,t) + ¾b,t + V (1 - Èb,t ) - c. (A3)
Subtracting equation (A3) from equation (A2) and rearranging
buy buy
(v - v )(Èb,t - Èb ,t ) e" 0. (A4)
The result follows from equations (A4) and (16). Symmetric arguments hold on the sell side.
Proof of Proposition 1
The threshold characterization follows from the monotonicity in Lemma 1.
B The conditional log-likelihood function
Let ti denote the time of the ith order submission and I the total number of order submissions.
Conditioning on the common value, order size, xti and zti the conditional log-likelihood function is
I
sell
ds ln Gti ¸ti (0, 1) - yti »ti
0,ti
i=1
St
sell s
+ dsell ln Gti ¸ti (Marginal) - yti - Gti ¸ti(0, 1) - yti »ti
s,ti
s=1
buy
+ dbuy ln 1 - Gti ¸ti (0, 1) - yti »ti
0,ti
Bt
buy buy
+ dbuy ln Gti ¸ti (0, 1), -yti - Gti ¸ti (Marginal) - yti »ti
b,ti
b=1
ti
buy
sell
- Gt(¸t (Marginal) - yt) - Gt(¸t (Marginal) - yt) »tdt . (B1)
ti-1
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The first line is contribution from the instantaneous probability of a sell market order at time ti;
the second line is the contribution from the instantaneous probability of a sell limit order; the third
line is the contribution from the instantaneous probability of a buy market order; fourth line is
the contribution from the instantaneous probability of a buy limit order; and the final line is the
integrated hazard rate.
In our estimation, we assume that the common value yt only changes when an order is submitted
at ti.
C Execution probabilities in the Weibull competing risks model
Suppose a limit order is submitted at time ti. Let te be the hypothetical execution time for the
order and tc the hypothetical cancellation time for an order. Assume that the times are independent
Weibull random variables, with cdf s Fe and Fc:
Fe(Ä) = 1 - exp - exp(³exti)(Ä - ti)±e , (C1)
Fc(Ä) = 1 - exp - exp(³cxti)(Ä - ti)±c , (C2)
The execution probability is defined as the probability that the order executes between ti and ti+T ,
ti+T
Pr (Äe d" Äc, Äe d" ti + T ) = (1 - Fc(t - ti)) dFe(t - ti)
Ü Ü Ü
ti
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