Table 3.8b. Estimates nfmileml probit dope coefficients.

Ш Ш Ж

3, IU 8!;f ста .,! , . Mar.,i (:, )() i(m IftM- -m , 1 Qu. mm, cwa, гроши (t:Ul 26.027 na.lr,). УШпу,

1С w , > aKi A r U.lm r and TcleKrapli Cominuv IT

180.726 , ), for Oh- period fromj.iinurv 4, 1988 , lumber 30. 1988. (

aries are not evenly spaced, e.g., \a-\-vt = 1.765, wliereas a.,-ar> = 2.070 (it can be shown thai these two values arc statistically different). One implication is that the eighths-barrier model of discrete prices, e.g., that ol Marsh and Rosenfeld (1980), is not consistent with these transactions data. Another implication is that the estimated conditional probabilities of price changes need not look normal, but may (and do) display a clustering phenomenon similar to the clustering of the unconditional distribution of price changes on even eighths.

Table 3.8b shows that the conditional means of the Yk \s for all six stocks are only marginally affected by Д Moreover, the z-statistics are minuscule, especially in light of the large sample sizes. However, At does enter into the a; expression significantly-in fact, since all the parameters for crj; are significant, homoskedasticily may be rejected-and hence clock-time is important for the conditional variances, but not for the conditional means of Yk. Note that this does not necessarily imply the same for the conditional distribution ol the fVs, which is nonliiieaily related to the conditional distribution of the Yk s. For example, the conditional mean of the Yks may well depend on the conditional variance of the Fs, so that clock-time can still affect the conditional mean of obsei ved price changes even though il does not affect the conditional mean of Yk*.

Order Flow, Discreteness, and Price Impact

More striking is the significance and sign of the lagged price change coefficients P2, P:\, and Ai, which are negative for all stocks, implying a tendency towards price reversals. For example, if the past three price changes were each one lick, the conditional mean of 1* changes by Az+A:i4 At- However, if the sequence of price changes was 1/-I/1, then the effect on the conditional mean is py-ps+Pi, a quantity closer to zero for each of the securitys parameter estimates.

Note that these coefficients measure reversal tendencies beyond that induced by the presence of a constant bid-ask spread as in Roll (1984). The effect of bid-ask bounce on the conditional mean should be captured by the indicator variables IBS*. IBSj 2, and IBS :i. In the absence of all other information (such as market movements or past price changes), these variables pick up any price effects that buys and sells might have on the conditional mean. As expected, the estimated coefficients are generally negative, indicating the presence of reversals due to movements from bid to ask or ask to bid prices, llausman, l.o, and MacKinlay (1992) compare their magnitudes formally and conclude that the conditional mean of price-changes is path-dependent with respect lo past price changes-the sequence of price changes or order Jlow matters.

Using these parameter estimates, llausman, l.o, and MacKinlay (1992) are also able ;<> address the second two questions they put forward. Price

impel-the cflcri ol a trade on ilie market price-can be quantified with relatively high precision, il does increase with trade size although not linearly, and it (Idlers from slock lo stock. The more liquid stocks such as IBM lend lo have relatively Hal price-impact (unctions, whereas less liquid slocks such as 1 INI I are more sensitive to trade size (see, in particular, I lausman, I.o, and MacKinlay I 1*102, Figure 4).

Also, discreteness does mallei, in die sense that the conditional distribution of price changes implied by the ordered probit specification can capture certain nonlineai ilics-pt ice-clustei ing on even eighths versus odd eighths, for example-that other techniques such as ordinary least squares cannot.

While il is still too early to say whether the ordered probit model will have broader applications in market microstructure studies, it is currently the only model that can capture discreteness, irregular trade intervals, and ihe clfccis of economic variables on transaction prices in a relatively parsimonious fashion.

3.5 Conclusion

There are many on islanding economic and econometric issues that can now be resolved in the market mil rostnn lure literature thanks to ihe plethora of newly available transactions databases, lit this chapter we have touched on only three of the issues that are part of the burgeoning markei microstructure literaiure: nonsynchronous trading, the bid-ask spread, and modeling transactions data. 1 lowcver, ihe combination of transactions databases and ever-increasing computing power is sure to create many new directions of research. For example, the measurement and control of trading costs has been of primary concern to large institutional investors, but there has b*en relatively little academic research devoted to this important topic because die necessary data were unavailable unlit recently. Similarly, measures of market transparency, liquidity, and competitiveness all figure prominently in recent theoretical models of security prices, but it has been virtually impossible to implement any of these theories until recently because of a lack of data. The experimental markets literature has also contributed many insights into market microsmtt lure issues but its enormous potential is only beginning lo be realized, (liven the growing interest in market microstructure by academics, investment professionals and, most recently, policvmak-ers involved in rewriting securities markets regulations, the next few years are sure lo be an extremely exciting and fertile period for this area.

Problems-Chapter 3

3.1 Derive the mean, variance, autocovariance, and autocorrelation functions (3.1 .*.!)-( 3.1 12) of the observed returns process r( ) for the nontrading model of Section 3.1. Hint: Ise the representation (3.1.4).

t lUUWtltS

3.2 Under the nontrading process defined by (3.1.2)-(3.1.3), and assuming lhal virtual returns have a linear one-factor structure (3.1.1), showjhow nontrading affects the estimated beta of a typical security. Recall that a securitys beta is defined as the slope coefficient of a regression of the securitys returns on the return of the market portfolio. ;

3.3 Suppose that the trading process [S ) defined in (3.1.2) were not IID, but followed a two-state Markov chain instead, with transition probabil ties given by S:,

0 / Jf, I -тгД

3.3.1 Derive the unconditional mean, variance, first-order autocovjari-ance, and steady-state distribution of Sit as functions of я, and ttj.

3.3.2 Calculate the mean, variance, and autocorrelation function of the observed returns process r° under (3.5.1). How does serial correlation in Su affect the moments of observed returns? - !

3.3.3 Using daily returns for any individual security, estimate the parameters тг, and or- assuming that the virtual returns process is IID. Are the estimates empirically plausible?

3.4 F.xtend the Roll (1984) model to allow for a serially correlated order-type indicator variable. In particular, let /, be a two-state Markov with -1 and 1 as the two states, and derive expressions for the moments of tsP, in terms of j and the transition probabilities of How do these results differ from the IID case? How would you reinterpret Rolls (1984) findings in light of this more general model of bid-ask bounce?

3.5 How does price discreteness affect the sampling properties of the mean, standard deviation, and first-order autocorrelation estimators, if at all? Hint: Simulate continuous-state prices with various starting price levels, round to the nearest eighth, calculate the statistics of interest, and tabulate the relevant sampling distributions.

3.6 The following questions refer to an extract of the NYSEs TAQDatabase which consists of all transactions for IBM stock that occurred on January 4th and 5th, 1988 (2,748 trades).

3.6.1 Construct a histogram for IBMs stock price. Do you see any evidence of price clustering? Construct a histogram for IBMs stock price changes. Is there any price-change clustering? Construct the following two histograms and compare and contrast: the histogram of price changes conditional on prices fallingon an even eighth, and the histogram of price changes conditional on prices falling on an odd eighth. Using these his-

tograms, comment on the importance or unimportance of discrete prices for statistical inference.

3.6.2 What is the average time between trades for IBM? Construct a 95% confidence interval about this average. Using these quantities and the central limit theorem, what is the probability that IBM does not trade in any given one-minute interval? Divide the trading day into one-minute intervals, and estimate directly the unconditional and conditional probabilities of nontrading, where the conditional probabilities arc conditioned on whether a trade occurred during the previous minute (hint: think about Markov chains). Is the nontrading process independent?

3.6.3 Plot price and volume on the same graph, with tiinc-of-day as the horizontal axis. Are there any discernible patterns? Propose and perform statistical tests of such patterns and other patterns that might not be visible to the naked eye but are motivated by economic considerations; e.g., block trades are followed by larger price changes than nonblock trades, etc.,r

3.6.4 Devise and estimate a model that measures price impact, i.e., the

1actual cost of trading n shares of IBM. Feel free to use any statistical methods at your disposal-there is no single right answer (in particular, ordered probit is not necessarily the best way to do this). Think carefully i about the underlying economic motivation for measuring price impact.

3.7 The following questions refer to an extract of the NYSEs TAQDalabase which consists of bid-ask quote revisions and depths for IBM stock that were displayed duringjanuary 4th and 5th, 1988 (1,327 quote revisions).

3.7.1 Construct a histogram for IBMs bid-ask spread. Can you conclude from this that the dynamics of the bid-ask spread arc unimportant? Why or why not? You may wish to construct various conditional histograms to properly answer this question.

$.7.2 Are there any discernible relations between revisions in the bid-ask motes and transactions? That is, do revisions in bid-ask quotes cause rades to occur, or do trades motivate revisions in the quotes? Propose and estimate a model to answer this question.

3.7.3 How are changes in the bid and ask prices related to volume, if at all? For example, do quote revisions cause trades to occur, or do trades ihotivatc revisions in the quotes? Propose and estimate a model to answer this question.

3.7.4 Consider an asset allocation rule in which an investor invests fully in stocks until experiencing a sequence of three consecutive declines, after

wThe NYSE defines a block Irade as any irade consisting of 10.000 shares or more.

which he will switch completely into bonds until experiencing a sequence ol six consecutive advances. Implement this rule for an initial investment of $100 000 with the transactions .lata, but do it iwo ways: (1) use the average of the bid-ask spread for purchases or sales; (2) use the ask price lor purchases and the bid price for sales. I low much do you have left at the end of two days of trading? You may assume a zero r.sklrec rate lor this exercise.