# One Day in the Life of a Very Common Stock

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14 The Review of Financial Studies /v 10 n means that this probability is the product D Pr{(B d, S d, N d ) D d=1 α, δ, µ, ε} = Pr{(B d, S d, N d ) α, δ, µ, ε} (14) d=1 where (B d, S d, N d ) is the outcome on day d, d = 1,...,D. The likelihood function, as usual, is an efficient description of all of the data information about the parameters. The form of this likelihood function has a number of implications for our estimation. First, using only one day s data, it is clear that α and δ are not identified. The likelihood in Equation (13) is bilinear in α and δ, so the maximum likelihood estimators will be zeros or ones. This situation is analogous to estimating a Bernoulli probability from one trial. For any given day, our sufficient statistic is at best three-dimensional, so it is possible to estimate at most three parameters. But if B + S + N is approximately a constant (as may be the case for many stocks) or is ancillary (as is the case if we are considering fixed time intervals such as seconds or minutes), then our statistic is actually two-dimensional, limiting our estimation ability accordingly. In any case, our statistic is sufficient to allow estimation of the two parameters µ and ε from daily data. Second, in our model, information on µ and ε accumulates at a rate approximately equal to the square root of the number of trade outcomes, while information on α and δ accumulates at a rate approximately equal to the square root of the number of days. Hence, using many days of data greatly enhances our ability to estimate more parameters. For our estimation problem, this means that using a multiday sample can provide sufficient information to estimate α and δ. What is also apparent, however, is that while it may be sensible to use large sample methods to estimate µ and ε, it is less so for α and δ. The presumed stationarity of information is unlikely to be true over a long sample period, dictating a natural limit to the number of days we can sensibly employ. The difference in information accumulation rates also dictates that the precision of our µ and ε estimates will exceed the precision of our α and δ estimates. Of course, this is reflected in the standard errors of the estimators. Having defined the likelihood function for our model, we can now calculate the parameter values of α, δ, µ, and ε that maximize this function for a given stock. Roughly, what our procedure does is to classify days into buy-led high-volume days, sell-led high-volume days, and low-volume days (as in Figure 1). In our structural interpretation, these are good-event days, bad-event days, and no-event days, respectively. The likelihood function given in Equation (13) is a mixture of trinomials, with specific terms reflecting the numbers of buys, sells, and no-trades and with restrictions across the compo- 818

15 One Day in the Life of a Very Common Stock nents of the mixture. For tractability in the estimation, we use a log transformation of our likelihood function, which after simplifying and dropping a constant term, is given by [ D ( log α(1 δ) 1+ µ ) B+αδ ( 1+ µ ) ( ) ] S+(1 α) 1 S+B+N x x 1 µ d=1 + D log[((1 µ)(1 ε)) N x S+B ], (15) d=1 where x = (1 µ) 1 2 (ε). Note that the weights on the trinomial components reflect the information event parameters α and δ. 8 Hence, a simple test of the information model is to compare the goodness-of-fit of the model s likelihood function with that of the simpler trinomial in which α and δ do not appear. In this specification, each day is the same as far as information is concerned. As will be apparent, maximizing the likelihood function of Equation (15) provides parameters yielding a direct measure of the effect of information on trades, quotes, and prices. 3. The Data and Maximum Likelihood Estimation The estimation described in the last section requires trade outcome data for a specific stock over some sample period. As our focus here is on the estimation approach rather than on any specific performance measure, which particular stock is analyzed is not important, nor is the sample period of any particular concern. Of course, since we intend our methods to be quite generally applicable, we do not wish to select a bizarre stock or unusual period for illustration. For our analysis, we selected Ashland Oil to be our sample stock. This selection was dictated both by convenience, and by the relatively active trading found in the stock. This latter characteristic is important given that it is the information contained in trade data that is the focus of our work. Trade data for Ashland Oil were taken from the ISSM transactions database for the period October 1, 1990, to November 9, A30 trading-day window was chosen to allow sufficient trade observations for our estimation procedure. The ISSM data provide a complete listing 8 This mixture of trinomials is reminiscent of the work of Clark (1973), who postulated that the distribution of security prices could be represented by such mixtures. Here our focus is on the trade process, but this in turn affects the price process. 9 Over this period there was an earnings announcement in day 10 of our sample, but the data suggest little impact of this on the stock. 819

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