We propose a model similar to Biais, Hillion, and Spatt (1999). We assume that the overnight change in the valueρtN is on average incorporated fully into the stock price after 10 minutes of continuous trading:
rOt =ρNt +ωOt , (4.1)
where rOt is the return of the stock price (at 9:10) on the previous day closing price, andωOt is an error term uncorrelated with the overnight returnρNt . The indext distin-guishes observations for different days or stocks.
The opening auction allows market participants to discover the overnight change in the fundamental value of the traded asset without the risk of immediate execution. The indicative price of the auction at each timeican partially or fully reflect that overnight change:
rit = 1
βρtN+ωti. (4.2)
The parameter β1 measures the amount of price discovery. The parameterωit captures microstructure effects and is uncorrelated to the pricing error after 10 minutes of trad-ing. Prices are fully informative if β is equal to one. Partial informativeness would 110
s2i = 1
β2σN2 +σi2, (4.3)
whereσN2 and σi2are the variances of the overnight return ρtN and the microstructure noisewit respectively. The indicative price variance can therefore be decomposed into the part stemming from changes in the value and the part originating in microstructure effects. The latter might stem e.g. from short-term mispricing, price manipulation, or from submitted orders that aim at scanning the book and are cancelled before the auction is called. Both components of the indicative price variance are unobserved.
From equation (4.2), the estimated equation reads:
rOt =brti+et. (4.4)
Our main deviation from the model in Biais, Hillion, and Spatt (1999) is that we consider a different proxy for the value of the asset. Information spills over the market during the trading day, due to, e.g., informed trades or the opening of foreign markets (mostly the American ones). Therefore, the close-to-close return, while a good proxy for the change in the value during that day, may depart dramatically from the actual change in the value overnight. Moreover, the opening price might not incorporate all the overnight information if the auction fails at its goal. This consideration leads us to pick the price at 9:10 a.m. instead of the closing price. In the following, the overnight return refers to the return of the price at 9:10 a.m. on the previous day closing price.
Biais, Hillion, and Spatt (1999) estimate equation (4.4) by means of standard OLS.
However, as pointed out in Barclay and Hendershott (2003), the observed returnsrOt andrit are equal to the true change in the value plus some noise. Measurement errors in the variables imply that the OLS estimate ˆbOLS is a biased estimate of the parameter β.
There exists a variety of approaches to diminish or circumvent the effects of mea-surement errors. A list, by no means exhaustive, starts with weighted regression,
in-4. AUCTION DESIGN IN ORDER BOOK MARKETS
provide a comprehensive review of the methods at hand. We use an adjustment of the OLS regression that can be seen as a special case of the consistent adjusted least squares (CALS) estimator. The main advantage of CALS lies in its ability to estimate the amount of measurement error σi2. We simply need to provide reasonable identifi-cation assumptions. The CALS estimator is described in the appendix.
In our framework, we need to identify value from microstructure noise. In our model, the covariancesEOC,Cof the return of the close-to-9 a.m. indicative price return, and the close-to-close return is equal to the variance of the overnight return adjusted by the parameter of the amount of price discovery:
sEOC,C = 1
βσ2N. (4.5)
All remaining components of the two prices vanish as they are assumed to be mutu-ally uncorrelated to the others. If we additionmutu-ally assume that the magnitude of price discovery is stable throughout the auction, then we are able to identify the two com-ponents in the indicative price variance s2i. This turns out to be sufficient to obtain unbiased, microstructure-free estimates of the βparameter.
The CALS estimator then reads (see the appendix for the exact derivation):
bˆCALS = si,O
where si,O is the covariance of the return of the indicative price and the return until 9:10 a.m.
By comparison, the OLS estimator forβreads:
bˆOLS = si,O
s2i . (4.8)
Both estimators for βshare the same numerator, but the denominator is different.
Instead of the variance of the regressor s2i, which includes microstructure noise, the CALS estimator denominator is the covariance of the closing return and the auction re-turn at the end of the opening, which measures the amount of price discovery. The dif-ference between the two estimators corresponds to the amount of measurement error (microstructure noise) in the observed returns. The asymptotic variance/covariance-matrix for the estimated parameter vector(bˆCALS, ˆσe,CALS2 ) is derived in the appendix.
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mate the model every 5 minutes between 7:30 and 8:50 a.m., every 15 seconds between 8:50 and 8:59 a.m., and finally every second until 9:00:30 a.m.6 For reporting purposes, and to improve the quality of the estimation, we adopt a classification of the thirty DAX stocks in four groups based on trading activity.7 The first group is then com-posed of the seven most frequently traded stocks whereas the eight least frequently traded stocks are in group four. The second and third group contain respectively eight and seven stocks each (see Table 4.6). Results based on a single stock or on the pooled regression (one regression for each quartile) lead to similar results. In the following we present results based on the pooled regressions.
Figure 4.6 reports the estimates for thebcoefficient with both OLS and CALS esti-mation methods. The OLS estimate starts from zero at the beginning of the pre-trading phase (7.30 a.m.), and linearly increases to reach the value of one during the random phase. The CALS estimate displays a markedly different pattern: according to CALS, prices are already fully informative (β = 1) around 8:50, which is the launch of the opening call phase. Interestingly, although the estimated value reaches the value of one early, the variance of the estimator decreases sharply as we get closer to the open-ing. Our interpretation is that the indicative price incorporates most of the overnight information quite early, but that the quality of the indicative price improves as we head toward the end of the call phase. This is confirmed by the size of the microstructure noise (measurement error) reported in Figure 4.6. Microstructure noise is very large at the beginning of the call phase, and decreases towards zero during the auction. Since the OLS estimate does incorporate microstructure effects, it reaches the value of one only at the very end of the auction (around 9:00). Therefore, contrary to Biais, Hillion, and Spatt (1999), we conclude that the indicative price during the auction becomes in-formative very early, but is then very noisy due to the importance of microstructure effects.
4. AUCTION DESIGN IN ORDER BOOK MARKETS