>. Market Миroslructiire

1.....N.

(3.l.li)

Whereas (3.1.-I) shows lhal in die presence of nonlrading the observed returns process is a (stochastic) function of all past returns, the equivalent relation (3.1.0) reveals that i, may also be viewed as a random sum with a random number ol terms.

Л third and perhaps most natural way to view observed returns is the following:

0 with probability я,

,/ with probability (1-я,)-

in + i with probability (1-я,)-я,

>,i + -i + г/, ч with probability (1-л,)-л}

r +

+ i t with probability (1- я,)-я*

(3.1.7)

Expressed in this way, il is apparent that nontrading can induce spm ions serial correlation in observed returns because each r° contains within it the sum of past h consecutive virtual returns for every к with some positive probability (1 - я,)-я*.

Го see how the nontrading probability я, is related to the duration of nontrading, consider the mean and variance of k,:

\k,\ =

I - я,

Var[A,] =

(1 - jt,)-

(3.1.8)

If я,= then security / goes without Hading for one period at a time on average; if jt,= -[ then ihe average number of consecutive periods of nontrading

Hits is similar inspirit in the Sr holes ami Williams (ll.)77) subordinated stochastic process representation ol observed returns, although we do not restrict the trading limes lo take values in a lixed tiliilc interval. With suitable iiormali/ations il may be shown dial our nontrading

.....del converges weaklvlo ihcennliiiiioiis-iinic Ioisson process of .Scholes and Williams (1(177).

from (3.1.-1) die observed u-turns piocess may also be considered an inlinite-oidcr moving average ol virtual returns where the МЛ < oellii ienls are stochastic. This is in contrast lo Cohen, Maier. Si hwan/. and Whin oinh (I Ж(>, (:haplei li) in which observed reliiinsare assumed lo be a liliile-oidei МЛ pmrt-ss willi nonsloi liastie coeflicienls. Although our llontrading pro< ess is more general, theii observed returns piocess includes a hid-ask spread с oiiiponclll; ours does not.

3.I. Nonsynchionous Dmling

is three. As expected, if the security trades every period so that jt, = 0, both the mean and variance of к, are zero.

Implications for Individual Security Returns

To see how nontrading can affect the time-series properties of the observed returns of individual securities, consider the moments of r°, which, in turn, depend on the moments of Хц{к).ю For the nontrading process (3.1.2)-

(3.1.3), the observed returns processes (r°) (i = 1.....N) are covariance-

stationary with the following first and second moments:

E[r ] = Hi (3.1.9)

VarK] = af + --nf (3.1.10)

1 - n,

CovtC r°l+ll]

-nfjr for i = j, n > 0

frPjofn; for #;-.п>о

(3.1.11)

CorrlCJ = n > 0, (3.1.12)

where af = Var[r,(] and aj = Var[yi].

From (3.1.9) and (3.1.10) it is clear that nontrading does not affect the mean of observed returns but does increase their variance if the security has a nonzero expected return. Moreover, (3.1.12) shows that having a nonzero expected return induces negative serial correlation in individual security returns at all leads and lags which decays geometrically. The intuition for this phenomenon follows from the fact that during nontrading periods the observed return is zero and during trading periods the observed return reverts back to its cumulated mean return, and this mean reversion yields negative serial correlation. When /с,-0, there is no mean reversion hqnee no negative serial correlation in this case.

Maximal Spurious Autocorrelation These moments also allow us to calculate the maximal negative autocorrelation attributable to nontrading in individual security returns. Since the autocorrelation of observed returns (3.1.12) is a nonpositive continuous function of я, that is zero at я;=0 and approaches zero as яг,- approaches unity, it must attain a minimum for some я, in [0,1). Determining this lower bound is a straightforward exercise in calculus, and hence we calculate it only for the first-order autocorrelation and leave the higher-order cases to the reader.

To conserve space, we summarize die results here and refer readers to Ix> and MacKinlay (l1.Юа. Itl.IOc) lor further details.

Under (3.1.2)-(3.1.3) the minimum first-order autocorrelation of the observed returns process [rtf with respect to nontrading probabilities n, is given by

Min Corr [ *, *,.,]

( ы у

where = д,/сг and the minimum is attained at

я, =

1 + V2~£,l

Over all values of л, € [0, 1) and £ e (-co, +oo), we have

Inf Corr[r° r

(3.1.13)

(3.1.14)

(3.1.15)

whicjt is the limit of (3.1.13) as increases without bound, but is never attained by finite f

Although the lower bound оГ - j seems quite significant, it is virtually unattainable for any empirically plausible parameter values. For example, if we consider a period to be one trading day, typical values for ji, and o , arc .05% and 2.5%, respectively, implying a typical value of 0.02 for According to (3.1.13), this would induce a spurious autocorrelation of al most -0.037% in individual security returns and would require a nontrading probsbility of 97.2% to attain, which corresponds to an average nontrading durat on of 35.4 days!

Tiese results also imply that nontrading-induced autocorrelation is magn Red by taking longer sampling intervals since under the hypothesized virtual returns process, doubling ihc holding period doubles ц., but only multiplies ст, by a factor of Therefore more extreme negative autocorrelations arc feasible for longer-horizon individual returns. However, this isjnot of direct empirical relevance since the effects of time aggregation have been ignored. To sec how, observe that the nontrading process (3.1.2)-(3.1.3) is not independent of the sampling interval but changes in a nonlinear fashion. For example, if a period is taken to be one week, the possibility of daily nontrading and all its concomitant effects on weekly observed returns is eliminated by assumption. A proper comparison of ob-scrved returns across distinct sampling intervals must allow for nontrading at the finest time increment, after which the implications for coarser-sampled returns may be developed. We shall postpone further discussion of this and other issues of time aggregation until later in this section.

Asymmetric Cross-Autocovariances

Several other important empirical implications of this nontrading model m arc captured by (3.1.11). In particular, the sign of the cross-autocovariances

&

is determined by the sign of (>,[>r Also, the expression is not symmetric with respect to i and /. If я, = 0 and л, / 0, then there is spurious cross-aittocovariancc between r° and r fll but no cross-autocovariance between r t and r l+ll for any n > 0. The intuition for this result is simple: When security j exhibits nontrading, the returns lo a constantly trading security i can forecast j due to the common factor f, present in both returns. Thai j exhibits nontrading implies that future observed returns rjl+n will be a weighted average of all past virtual returns r(<+ -/, (with the X;(+ (/[)s as random weights), of which one term will be the current virtual return rt,. Since the contemporaneous virtual returns i;, and rlt are correlated (because of the common factor), r/j can forecast rjl+li. However, r( is itself unforccastablc because r tl = r for all i (since л, = 0) and r is llf) by assumption, thus rf is uncorrelated with r l+ll for any n > 0.

The asymmetry of (3.1.11) yields an empirically testable restriction on the cross-autocovariances of returns. Since the only source of asymmetry in (3.1.11) is cross-sectional differences in the probabilities of nontrading, information regarding these probabilities may be extracted from sample moments. Specifically, denote by r the vector [ rj( rJJ( ] of observed reaims of the N securities and define the autocovariance matrix Г as

r = K[(r;-/x)(r;+ -/t)i, i = ею. (3.i.ui)

Denoting the (/, j)lh element of Г by yy(n), we have by definition

(1 - л,)(1 - я.) .,

Yl](n) =---J--(l.fc, n . (3.1.17)

1 - n.Jtj 1 1

If the nontrading probabilities я,- differ across securities, Г is asymmetric. From (3.1.17) it is evident that

Yij(n)

(3.1.18)

я,

Yjin)

Therefore relative nontrading probabilities may be estimated directly using sample autocovarianccsFn. To derive estimates of the probabilities я, themselves wc need only estimate one such probability, say л\, and the remaining probabilities maybe obtained from the ratios (3.1.18). A consistent estimator of Я is readily constructed with sample means and autocovariances via (3.1.11).

11 Ли alternative interpret л (ion of this asymmetry may he found in the lime-series I Herat иге concerning GrangeKausality (see Granger 11**(>*.*). in which t ( is said lo (hungrr-ctiusr r if the return to / predicts the return to j. In the above example, security i ( rtingrr-cnurs .security j when j is subject to nontrading but (is not. Since our nontiading process may be viewed as a torn, of measurement error, the latl thai the returns to one seemity may be exogenous with respect to the. returns of another has been pmposed under a difiereni guise in Sims (Н>74, 1077).

Implications for Portfolio Returns

Suppose securities are grouped by their nontrading probabilities and equal-weighted portfolios are formed based on this grouping so that portfolio A contains N securities with identical nontrading probability я , and similarly for portfolio II. Denote by r tl and the observed time-/ returns on these two portfolios respectively, which are approximately averages of the individu; I returns:

* ТгЕ-e K = (3M9>

it I.

where the summation is over all securities i in the set of indices IK which comprise portfolio к. The reason (VI. 19) is not exact is that both observed and virtual returns are assumed to be continuously compounded, and the logarithm of a sum is not the sum of the logarithms.12 However, if r , lakes on small values and is not loo volatile-plausible assumptions lor the shoi i return intervals that nonsynchronous trading models typically focus on- the approximation error in (3.1.19) is negligible.

The time-series properties of (3.1.19) may be derived from a simple asymptotic approximation that exploits the cross-sectional independence of the disturbances c . Similar asymptotic arguments can be found in the Arbitrage Pricing Theory (APT) literature (see Chapter (i); hence our assumption of independence may be relaxed to the same extent that it may he relaxed in studies of the APT in which portfolios are required lo be well-diversified. 11 In such cases, as the number of securities in portfolios A and И (denoted by N and N/ respectively) increases without bound, the following equalities obtain almost surely:

= /< + ( l - л.)л, ]Г *,*/, *, (з. i .20)

*=(>

where

-A precise tiitci pii-i.tttihi ol is the return to a pordolio whose value is calculated as in unweighted цеотеи average ol the component securities prices. Пи* expected return ol such A portlnlio will be lower than that ol ;\n equal-weighted portfolio whose returns .tie calculated .is the arithmetic means ol ihe simple returns ol the compimeiit securities. This issue is examined in greater detail hy Modest and Sundaresan (I983) and Kytan and Ilarpav (l.)H(i) in the context ol ihe Value l.iue Index which was an unweighted geometric average until

l:1See, lor example. Clumbeilain (\\№л), (Ihamberhiin and Kolhs< hild (НШ). and Wang < HI.W). Ihe essence ol these weakei ( ondiiions is simply to allow a l iw of Large Numbers to he applied lo ihe averse ol ihe dislinb.mi es, so ih.it idiosyncratic risk vanishes almost surely as the cross set lion glows.

for к = a,b. The first and second moments of ihe portfolios returns are

then given by

E[r;,] = Ц. = E[rrl] (3.1.22)

Var[r;,] fi(Lz£)aj (3.1.23)

Cov[r; r;(+ ] 4 fifLzJy-aj, >0 (j.1.24)

corr[r; r j = n > о (s.

rw,-° r° l ° (I -я )(1 -жь) 2

Cov[ra(, rw+ ] = ---paf}b0fxb, (Л

1 - TXaTCb

where the symbol = indicates that the equality obtains only asymptotically.

same

From (3.1.22) we see that observed portfolio returns have the mean as the corresponding virtual returns. In contrast to observed individual returns, the variance of r°, is lower asymptotically than the variance of its virtual counterpart r , since

Na ,n. Na .6/.

= lla + Ы (f 1-28)

1.25) 1.26)

where (3.1.28) follows from the law of large numbers applied to trie last

term in (3.1.27). Thus Var[r ,] = $\ai, which is greater than or equal to

Var[C).

Since the nontrading-induced autocorrelation (3.1.25) declines geometrically, observed portfolio returns follow a first-order autoregressive process with autoregressive coefficient equal to the nontrading probability. In contrast to expression (3.1.11) for individual securities, the autocorrelations of observed portfolio returns do not depend explicitly on the expected return of the portfolio, yielding a much simpler estimator for n : the nth root of the nth order autocorrelation coefficient. Therefore, we may easily estimate all nontrading probabilities by using only the sample first-order own-autocorrelation coefficients for the portfolio returns.

Comparing (3.1.26) to (3.1.11) shows that the cross-autocovariance between observed portfolio returns takes the same form as that of observed individual returns. If there are differences across portfolios in the nontrading probabilities, the autocovariance matrix for observed portfolio returns will be asymmetric. This may give rise to the types of lead-lag relations empirically documented by \jq and MacKinlay (1988) in size-sorted portfo-