2. The Predictability of Asset Returns

subperiods. Which of the skewness, kurtosis, and siudemizcd range estimates ape statistically different from the skewness, kurtosis, and studen-tized range of a normal random variable at the 5% level? For these twelve series, perform the same calculations using monthly data. What do you jmclu.de about the normality of these return series, and why?

Market Microstructure

Wllll.H it is ALWAYS the case that some features of the data will be lost in the process of modeling economic phenomena, determining which features to focus on requires some care and judgment. In exploring the dynamic properties of financial asset prices in Chapter 2, we have taken prices and returns as the principal objects of interest without explicit reference to the institutional structures in which they are determined. We have ignored the fact that security prices are generally denominated in fixed increments, typically eighths of a dollar or ticks for stock prices. Also, securities do not trade at evenly spaced intervals throughout the day, and on some days they do not trade at all. Indeed, the very process of trading can have an important impact on the statistical properties of financial asset prices: In markets with designated marketmakers, the existence of a spread between the price at which the marketmaker is willing to buy (the bid price) and the price at which the marketmaker is willing to sell (the offer, or ask price) can have a nontrivial impact on the serial correlation of price changes.

For some purposes, such aspects of the markets microstructure can be safely ignored, particularly when longer investment horizons are involved. For example, it is unlikely that bid-ask bounce (to be defined in Section 3.2) is responsible for the negative autocorrelation in the five-year returns of US stock indexes such as the Standard and Poors 500, even though the existence of a bid-ask spread does induce negative autocorrelation in returns (see Section 3.2.1).

However, for other purposes-the measurement of execution costs and market liquidity, the comparison of alternative marketmaking mechanisms, the impact of competition and the potential for collusion among market-makers-market microstructure is central. Indeed, market inicTosiriicturc is now one of the most active research areas in economics and finance, span-

lScc Section 2.5 in Chapter 2 and Section 7.2.1 in Chapter 7 lor further discussion of long-horizon returns.

ninjr many mai kcls and many models.- To test some of these models, and to determine the importance of market microstructure effects for other research areas, we require some empirical measures of market microstructure effects. We shall construct such measures in this chapter.

In Section 3. I, we present a simple model of the trading process to capture the elfects of nonsynchronous trading. In Section 3.2, we consider the effects of the bid-ask spread on the time-series properties of price changes, and in Section 3.3 we explore several techniques for modeling transactions data which pose several unique challenges including price discreteness and irregular sampling intervals.

3.1 Nonsynchronous Trading

I he nonsynchronous trading or nontrading effect arises when time series, usually asset prices, are taken to be recorded at lime intervals of one length when in fact they are recorded at time intervals of other, possibly irregular, lengths. For example, the daily prices of securities quoted in the financial press are usually closing prices, prices at which the last transaction in each of those securities occurred on the previous business day. These closing prices generally do not occur at the same lime each day, but by referring to them as daily juices, we have implicitly and incorrectly assumed ihat they are equally spaced al 21-hour intervals. As we shall see below, such an assumption ran create a false impression of predictability in price changes and returns even if true price changes or returns are statistically independent.

In particular, the nontrading effect induces potentially serious biases in the moments and co-moments ol asset returns such as their means, variances, covariances, betas, and autocorrelation and cross-autocorrelation coefficients. For example, suppose that die returns to stocks A and В are temporally independent but A trades less frequently than B. If news affecting the aggregate stock market arrives near the close of the market on one day, it is more likely that B-s end-of-day price will reflect this information than As, simply because A may not trade after the news arrives. Of course, A will respond to this information eventually but the fact that it responds with a lag induces spurious cross-autocorrelation between the daily returns of A and В when calculated with closing prices. This lagged response will

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j.l. jvuusyiunwnous Hading

also induce spurious own-autocorrelation in the daily returns of A: During periods of nontrading, As observed return is zero and when A does trade, its observed return reverts to the cumulated mean return, and this mean-reversion creates negative serial correlation in As returns. These effects have obvious implications for tests of predictability and nonlinearity in asset returns (see Chapters 2 and 12), as well as for quantifying the trade-ofTs between risk and expected return (see Chapters 4-6).

Perhaps the first to recognize the importance of nonsynchronous prices was Fisher (1966). More recently, explicit models of nontrading have been developed by Atchison, Butler, and Simonds (1987), Cohen, Maier, Schwartz, and Whitcomb (1978,1979), Cohen, Hawawini, Maier, Schwartz, and Whit-comb (1983b), Dimson (1979), Lo and MacKinlay (1988, 1990a, 1990c), and Scholes and Williams (1977). Whereas earlier studies considered the effects of nontrading on empirical applications of the Capital Asset Pricing Model and the Arbitrage Pricing Theory,3 more recent attention has been focused on spurious autocorrelations induced by nonsynchronous trading.4 Although the various models of nontrading may differ in their specifics, they all have the common theme of modeling the behavior of asset returns that are mistakenly assumed to be measured at evenly spaced time intervals when in fact they are not.

3.1.1 A Model of Nonsynchronous Trading

returns

Since most empirical investigations of stock price behavior focus on i or price changes, we take as primitive the (unobservable) return-generating process of a collection of N securities. To capture the effects of nontrading, we shall follow the nonsynchronous trading model of Lo and MacKinlay (f990a) which associates with each security i in each period / an unobserved or virtual continuously compounded return r . These virtual returns represent changes in the underlying value of the security in the absence of any trading frictions or other institutional rigidities. They reflect both cornpany-specific information and economy-wide effects, and in a frictionless market these returns would be identical to the observed returns of the security.

To model the nontrading phenomenon as a purely spurious statistical artifact-not an economic phenomenon motivated by private information and strategic considerations-suppose in each period t there is some probability jt, that security i does not trade and whether the security tradesjor not is independent of the virtual returns (r ) (and all other random variables

See, for example, Cohen, Hawawini, Maier, Schwartz, and Whitcomb (1983a, b), Dimson (1У7У), Scholes and Williams (1977), and Shanken (1987b).

4See Atchison, Butler, and Simonds (1987), Cohen, Maier, Schwartz, and Whitcomb (1979, 1986), and U) and MacKinlay (1988,1988b, 1990a, 1990c).

in :his model).5 Therefore, this nontrading process can be viewed as an IID sequence of coin tosses, with different nontrading probabilities across secirities. By allowing cross-sectional differences in the random noun ing processes, we shall be able to capture the effects of nontrading < returns of portfolios of securities.

The observed return of security:, r-J, depends on whether security (trades in period /: If security j does not trade in period t, let its observed return be zero-if no trades occur, then the closing price is set to the previous periods closing price, and hence ru = log(/>,(/> -i) = log 1 = 0. If, on the other hand, security i docs trade in period /, let its observed return be the sum of thevirtual returns in period I and in all prior consecutive periods in which / did not trade.

For example, consider a sequence of five consecutive periods in which security i trades in periods 1,2, and 5, and does not trade in periods 3 and 4. The above nontrading mechanism implies that: the observed return in period 2 is simply the virtual return (r 2 = rl2); the observed returns in period 3 and 4 are both zero (r , = r°4 = 0); and the observed return in period 5 is the sum of the virtual returns from periods 3 lo 5 (r(° = rl5 -f rl4 + r,r,).7 This captures the essential feature of nontrading as a source of spurious autocorrelation: News affects those stocks that trade more frequently first and influences the returns of more thinly traded securities with a lag. In this framework the impact of news on returns is captured by the virtual returns process and the impact of the lag induced by nontrading is captured by the observed returns process r°.

To complete the specification of this nontrading model, suppose that virtual returns are governed by a one-factor linear model:

r,i = H. + Pift + d, г = 1.....N (3.1.1)

where / is some zero-mean common factor and e is zero-mean idiosyncratic noise that is temporally and cross-sectionally independent at all leads and lags. Since we wish to focus on nontrading as the sole source of autocorrelation, we also assume that the common factor / is IID and is independent of

The case where trading is correlated wills virtual returns is nut without interest, but it is inconsistent with the spirit of the nontrading as a kind of measurement error. In the >icscncc ol private information and strategic behavior, trading activity does typically depend on vii lual returns (suitably defined), and strategic trading can induce serial correlation in observed returns, bin such correlation can hardly be dismissed as spurious . See Section .4.1.2 lor further discussion.

This assumption may be relaxed lo allow for slate-dependent probabilities, i.e., auiot or-relatethionlrading; see the discussion in Section 3.1.2.

PdriiKl ls return obviously depends on bow many consecutive periods prior lo period I ttiat the} security did not trade. If it iraded in period 0, then the period-1 return is simply ettial to its virtual return; if it did not trade in period 0 but did trade in period -I, then petiotl Is observed return is the sum of petiod lis and period ls virtual returns; etc.

6 t for all ;, /, and к.н Each periods virtual return is random and captures movements caused by information arrival as well as idiosyncratic noise. The particular noun.tiling and return-cumulation process we assume captures the lag with which news and noise is incorporated into security prices due to infrequent trading. The dynamics ol such a stylized model are surprisingly rich, and they yield several important empirical implications.

To derive an explicit expression for the observed returns process and to deduce ils time-series properties we introduce two related random variables:

j 1 (no trade) with probability я, (312)

~~ [0 (trade) with probability I - тг/

X {k) = (I-5,-f)5,-, ,5 2 I. > 0

!1 with probability (1 -л,)л* (3.1.3)

0 with probability 1 - (\-л,)л-

where A (0) = 1 - <5 , {<5U is assumed to be independent of {Sj,\ for i ф j and temporally IID for each i = 1,2, .... N.

The indicator variable 8jt takes on the value one when security г docs not trade in period ( and is zero otherwise. X (k) is also an indicator variable and lakes on the value one when security ; Hades in period / but has not traded in any of the к previous consecutive periods, and is zero otherwise. Since 7г, is within the unit interval, for large к the variable X (k) will be zero with high probability. This is not surprising since it is highly unlikely that security / should trade today but never in the past.

Having defined the A (A)sil is now a simple matter to derive an explicit expression for observed returns r( ;

ru = £ A (/() r t * = I.....N. (3.1.4)

If security /does not trade in period /, then o =l which implies that X (A)=0 for all /с, and thus r;;=(). If i does trade in period /, then its observed return is equal lo the sum of todays virtual return and its past /(, virtual returns, where the random variable к, is the number of past consecutive periods that i has not traded. We call this the duration of nontrading, which may be expressed as

(v I *

к< = E *=i i,=i

Although (3.1.4) will prove lo be more convenient for subsequent calculations, /(, may be used to give a somewhat more intuitive definition of the

(3.1.5)

these sluing assumptions are made piiin.u ily loi exposi initial i oineiiiini e and may be relaxed considerably. See Section .4.1.2 lor liinliei discussion.