Information Loss in Volatility Measurement with Flat Price Trading 1

Transcription

1 Inforation Loss in Volatility Measureent with Flat Price Trading Peter C. B. Phillips Yale University, University of Auckland, University of York & Singapore Manageent University Jun Yu Singapore Manageent University May 7, 8 Our thanks go to Neil Shephard, Jean Jacod and Sungbae An for helpful coents on aspects of this work. Phillips acknowledges NSF support under Grant Nos. SES and SES Yu acknowledges support fro the Ministry of Education AcRF Tier fund under Grant No. T6B43-RS. Peter Phillips, Cowles Foundation, Yale University, Box 88, Yale Station, New Haven, Connecticut Eail: peter.phillips@yale.edu. Jun Yu, School of Econoics, Singapore Manageent University, 9 Staford Road, Singapore Eail: yujun@su.edu.sg.

2 Abstract A odel of nancial asset price deterination is proposed that incorporates at trading features into an e cient price process. The odel involves the superposition of a Brownian seiartingale process for the e cient price and a Bernoulli process that deterines the extent of at price trading. The approach is related to sticky price odeling and the Calvo pricing echanis in acroeconoic dynaics. A liit theory for the conventional realized volatility (RV) easure of integrated volatility is developed. The results show that RV is still consistent but has an in ated asyptotic variance that depends on the probability of at trading. Estiated quarticity is siilarly a ected, so that both the feasible central liit theore and the inferential fraework suggested in Barndor -Nielson and Shephard () reain valid under at price trading even though there is inforation loss due to at trading e ects. The results are related to work by Jacod (993) and Mykland and Zhang (6) on realized volatility easures with rando and interittent sapling, and to ACD odels for irregularly spaced transactions data. Extensions are given to include odels with icrostructure noise. Soe siulation results are reported. Epirical evaluations with tick-by-tick data indicate that the e ect of at trading on the liit theory under icrostructure noise is likely to be inor in ost cases, thereby a ring the relevance of existing approaches. Keywords: Bernoulli process, Brownian seiartingale, Calvo pricing, Flat trading, Microstructure noise, Quarticity function, Realized volatility, Stopping ties. JEL classi cation: C5, G

3 . Introduction The expression at trading refers to situations in arket trading where consecutively sapled prices in calendar tie take on the sae value. The phenoenon of at pricing is extreely coon in stock arket trading, a ecting alost all traded stocks, especially (but not exclusively) over sall tie intervals. An iediate iplication of the phenoenon is that both returns and volatility are zero over the at price subinterval, an outcoe that has null probability of occurrence in any odel where price behaves like a continuous Brownian seiartingale. This characteristic of the realized data inevitably has iplications for the econoetric easureent of volatility. The present paper seeks to explore soe of these iplications in the context of the use of realized variance (RV) estiates of integrated variance (IV). Part of the task is to develop a odel that copounds the presued seiartingale behavior of underlying e cient arket prices with a echanis that produces periods of at prices in practical trading. Flat trading is a regular feature of any nancial arkets, especially for stock price data that is sapled at odest to high frequencies, where it ay be regarded as a arket icrostructure phenoenon arising fro discrete trading practices, inforation arrival in discrete packets, and trading volue e ects. Without developing a full icrostructure theory, we posit a stochastic echanis that accords a constant probability of the occurrence of a trading at over each given subinterval. The forulation leads to the copounding of the e cient price Brownian seiartingale with a Bernoulli process that deterines the tiing and length of the at trading periods. The approach is related to sticky price odeling in acroeconoic dynaics and new Keynesian Phillips curve odels. In these odels, Calvo (983) pricing is frequently used in which only a xed share of rs are able to optiize price each period, leading to price stickiness and soe tie duration between price changes. Under the at trading odel in this paper, we develop a liit theory for standard econoetric estiates of volatility by nonparaetric RV easures. It turns out that when we allow for at trading RV is still consistent, converges to IV, and follows a ixed Gaussian liit theory under standard regularity conditions corresponding to those used in the original work of Barndor -Nielson and Shephard (, BNS hereafter). These results generalize the standard theory on epirical quadratic variation estiates. Notably, however, there is soe inforation loss when using RV to do inference about IV due to the presence of at price e ects. This loss takes the for of an increase in the asyptotic variance. The e ects are of a agnitude to be very signi cant in practical applications. For exaple, if the RV estiate is constructed fro 5-inute returns for Alcoa (AA) stock prices on April 5, 995, the proportion of at pricing on this day aounts to soe 6% of the saple and our results iply that the correct variance quadruples that of the variance obtained fro a seiartingale process without at pricing. As with uch other recent research on volatility, our interest in the use of RV easures is otivated by the availability of ultra-high frequency data which has ade it feasible to easure volatility accurately in a direct nonparaetric way. The idea is well explained in earlier work and siply involves the calculation of the su of squared intra-day returns obtained fro observed intra-day prices. The theoretical justi cation for easuring volatility in this way relies on standard properties of the epirical quadratic variation process for seiartingales

4 (e.g., Protter, 4), a set up which is coonly assued for nancial asset prices in the literature (see, for exaple, Andersen, Bollerslev, Diebold and Labys ( hereafter ABDL). The ain object of interest in this research is the value of IV over a speci c tie period such as a day. This approach to easuring volatility has attracted a great deal of attention recently and has led to nuerous successful applications see, for exaple, ABDL (, 3), Andersen, Bollerslev, Diebold and Ebens (, hereafter ABDE), Andersen, Bollerslev, Diebold and Wu (5), Andersen, Bollerslev and Meddahi (5), Bandi and Russell (8) and Fleing, Kirby and Ostdiek (3). For overviews of the literature, see Andersen, Bollerslev, Diebold (5) and BNS (7). Direct application of epirical quadratic variation liit theory requires that e cient or equilibriu prices be observed. This requireent appears too strong at ultra-high frequencies, such as the tick-by-tick frequency, because of the presence of various arket icrostructure e ects. These arket icrostructure e ects ay be regarded as containating the e cient price process and ay be, albeit soewhat crudely, odeled as noise. Ignoring these e ects produces bias and inconsistency in realized volatility estiates. Other institutional eleents of iportance arise fro epirical realities such as thin trading - the fact that the population of arket participants is large but nevertheless nite so that the usual (central liit theory) ingredients that underly the deterination of in nitesial Brownian increents is typically absent. Thin trading inevitably arises in situations where trading takes tie and there are nite nubers of traders. Accordingly, we ust expect to observe soe ats in the real data as we ove the sapling frequency to zero. This feature in cobination with icrostructure noise arising fro sources such as bid/ask bouncing and isrecording eans that the e cient price process is a latent variable that is observed interittently and with soe degree of error. While aintaining the assuption of artingale-like behavior for e cient prices, the literature has produced three di erent strands of research on how to deal with icrostructure noise in realized volatility calculations with intra-day data. One strand of research is to use all available tick-by-tick data and seek to explicitly odel icrostructure noise in this ne-grain sapling context. Assuptions about the properties of the icrostructure noise are typically ade for analytic convenience and include both iid and stationarity conditions. Iportant contributions to this literature include Zhang, Mykland and Aït-Sahalia (5, hereafter ZMA), Aït-Sahalia, Mykland and Zhang (5b), and Barndor -Nielson, Hansen, Lunde and Shephard (8). A second strand of research in the literature is to saple sparsely relative to the available sapling frequency, usually at odest frequencies, of 5 or inute intervals. This approach is otivated by the fact that any sources of icrostructure noise (such as bid/ask bounce), which occur in ultra-high frequency data, are itigated when prices are sapled at these odest frequencies. Correspondingly, it has been argued that these ore sparsely sapled prices better approxiate the e cient price process, and therefore standard seiartingale theory can be invoked. Under such seiartingale conditions, the consistency of RV was used in ABDL () and the asyptotic distribution of RV was developed in Jacod (993) and BNS (). In the third strand of the literature, researchers have focused on the nite saple properties Our copy of this iportant paper is undated and we have therefore used here the 993 date given in citation of the paper by Delattre and Jacod (997).

5 Figure : Tie series plots of transaction prices for AA on April 5, at three di erent frequencies. The horizontal axis is the tie stap (in seconds) since the arket opening at 9:3a. The rst panel is based on tick-by-tick observations. The second panel is based on data that are sapled every inute. The third panel is based on data that are sapled every 5 inutes. The prices at the - and 5-inute frequencies are obtained using the previous tick ethod. See Hansen and Lunde (6) for a detailed discussion of the di erent sapling schees. of the RV estiates. Here it is argued that the choice of sapling frequency e ectively trades o estiation variance against bias. When icrostructure noise is explicitly odelled, an optial sapling frequency, which iniizes the ean squared error of the RV estiate, ay be calculated. Studies following this approach include Zhou (996), Hansen and Lunde (6) and Bandi and Russell (8). None of the above analyses explicitly odels or allows for at trading in observed prices even though at trading is a salient feature in actual stock data at ost of the frequencies that have been used in this literature, fro tick-by-tick data through to 5-inute trading data. Flat trading is a characteristic of both actively traded and inactively traded stocks. To illustrate the forer, Fig. plots transaction prices for the stock AA fro the Dow Jones Industrial Average (DJIA) on the New York Stock Exchange (NYSE) at three di erent frequencies: tickby-tick, -inute, and 5-inute frequencies on April 5,. Flat trading is obvious at all three frequencies and it becoes a doinant feature in the tick-by-tick data. Table reports the proportions of at transaction prices for AA when sapling is perfored at ve di erent frequencies (-, -, 3-, 4-, 5-inute intervals) on the rst Wednesday in April fro 993 to 4. Although at pricing e ects are less pronounced after the decialization of trading in 3

6 January, they reain a non-negligible feature of these data. Flat pricing also takes place in tick sapling and in quote data; see, for exaple, Table in Hansen and Lunde (6) for the percentages of at quote prices at the tick-by-tick level for 3 DJIA stocks. Note that AA is a DJIA stock and DJIA stocks are aong the ost actively traded equities. Flat trading is naturally even ore of an issue for less liquid stocks. This feature of trading data deserves attention both in nancial odeling and econoetric volatility estiation with high frequency data. Table : Proportion of at trading in AA stock prices Date # of ticks Proportion of at trading -in -in 3-in 4-in 5-in April 7, April 6, April 5, April 3, April, April, April 7, April 5, April 4, April 3, April, April 7, The contribution of the present paper to these issues relates to both the second strand of the literature on odest frequency sapling and to studies on arket icrostructure noise. First, the odel introduced here extends the odels used in ABDL () and BNS () by gaining soe additional realis in its allowance for at trading saple paths. Second, we extend the liit theory of RV to the new odel, showing that while RV still consistently estiates IV and asyptotically follows a ixed Gaussian law in the presence of at trading, the asyptotic variance of the RV estiate is in ated, thereby revealing the loss of a substantial aount of inforation about underlying e cient price volatility in at trading. Third, we show that the estiated variation of RV based on epirical quarticity is siilarly a ected by the occurence of trading ats. In consequence, and iportantly for epirical research, both the feasible central liit theore and the inferential fraework developed in BNS () reain valid under at price trading. A further contribution of the paper is to relate the speci c odel used here and associated liit theory to odels with sapling at rando stopping ties considered in recent work by Mykland and Zhang (6) and in an earlier fundaental paper by Jacod (993). Soe new explicit asyptotic results for cases with rando sapling are given which show the e ects of rando duration ties between trades on the liit theory for integrated variance and quarticity. A nal contribution is to study the e ects of rando duration ties between trades on the liit theory in the presence of icrostructure noise of the type studied in ZMA (5). Iportantly, 4

7 the ZMA two tie scale estiator is shown to retain its asyptotic properties in the presence of at trading with only a inor scale change to the asyptotic variance. We proceed as follows. After brie y reviewing the literature, we introduce the new odel and develop the corresponding liit theory in Section. Section 3 extends the odel and relates the liit theory to the work of Jacod (993) and Mykland and Zhang (6) where interittent sapling is eployed. Section 4 considers the case of icrostructure noise e ects. Section 5 reports the results of soe Monte Carlo experients to assess the accuracy of the liit theory in nite saples and soe epirical evaluations with tick-by-tick data are perfored to assess the e ect of at trading on the liit theory. Section 6 concludes. Proofs are in the Appendix.. A Flat Trading Model and Liit Theory Let p (t) be the logarith of the e cient price and assue p (t) evolves according to a Brownian seiartingale process on a ltered probability space (; F t ; P ). This assuption is justi ed by Back (99) in a frictionless, arbitrage-free econoy. As it is typical in the high frequency volatility literature, we further assue that p (t) follows the (driftless) di usion dp (t) (t)db(t); () where B(t) is a standard Brownian otion and (t) is an F t - easurable càdlàg volatility process. The quantity of interest is IV R (t)dt; the IV of p (t) over a certain unit tie period, say a day. The integral ay be de ned as the liit of the epirical quadratic variation IV pli h! i [p i; p i ;] ; () where p i; p (t i; ), t ; < t ; < < t ; is a sequence of deterinistic partitions of [; ], and h ; sup i jt i; t i ; j is the grid size. Since we are interested in interittent sapling of the process, it is also useful to de ne notation for interittent grid sizes h`; sup i jt i; t i `; j for ` : Soeties, it is convenient to assue that the partition involves a siple grid of equi-spaced points {t i; i : i ; :::; g and then h ; ; and h`; ` : The theory of quadratic variation allows the tie spacing in () to be stochastic and depend on the ltration, thereby allowing the ties of trade to have soe dependence on equilibriu prices. The liiting value IV in () is a (unit tie period) segent of the quadratic variation process of p. The saple counterpart is the epirical quadratic variation [p i; p i ;] : RV () (p ); i which is now coonly referred to as RV in nancial econoics. Since RV () (p ) p! IV as h ;! (e.g., Protter, 4), RV is a natural candidate for estiating IV, otivating the recent interest in this approach to volatility easureent. To 5

8 quantify the statistical di erence between RV and IV, BNS () used the liit theory p h RV () (p ) i d! IV MN ; Z 4 (t)dt ; (3) where M N signi es ixed norality. A feasible version of this liit involves the estiation of the quarticity functional R 4 (t)dt using epirical quarticity. BNS obtained the following result RV () (p ) IV d P! N (; ) ; (4) i [p i; p i ; ]4 q 3 which is convenient for use in inference. These asyptotic results all require knowledge of the log-e cient price, p i;. At ultra high frequencies arket icrostructure e ects challenge this requireent, containating observations with icrostructure noise so that the actual price data p i; p(t i; ) di ers fro p i; and RV () (p) 6 RV () (p ). To itigate such arket icrostructure e ects, ABDL (), ABDE () and BNS () suggested sapling sparsely, say at ve inute intervals, so that the accuulative e ects of noise are less iportant and p i; is treated the sae as p i;. ABDL justi ed the choice of ve inute intervals using the signature plot, a graphical device used to assess the degree of bias caused by arket icrostructure e ects at di erent sapling frequencies. Signature plots typically suggest that RV is ore severely biased when the sapling frequency increases but stabilizes at odest frequencies. This observation has propted researchers to view the observed price as a good approxiation to the e cient price and has the sae seiartingale characteristics at these odest frequencies. The ipact of arket icrostructure noise has also been exained in the ore speci c analytic fraework p(t) p (t) + u(t); (5) where u(t) is icrostructure noise. Most studies assue that the noise process u(t) and price process p (t) are independent. However, there are any di erent proposals in the literature about how to odel the noise process and how to treat the presence of noise in the estiation of IV. Soe studies (e.g., Zhou, 996, Bandi and Russell, 8, ZMA, 5, Sun, 6) assue a pure noise structure for u(t). Soe other studies (e.g. Hansen and Lunde, 6 and Aït- Sahalia, Mykland and Zhang, 5b) assue u(t) is covariance stationary. Neither pure noise nor covariance stationary icrostructure e ects explain at trading. In fact, when the e cient price follows a Brownian seiartingale as in (), then during periods of at trading prices the icrostructure noise e ect copletely o sets the e cient price uctuations to produce a sustained at transactions price. The noise process therefore inherits the sae local artingale-like behavior of the e cient price process over this subinterval. Inspection of trading data such as that shown in Fig. shows that while sapling at odest frequencies reduces the e ects of at trading it too does not copletely resolve the proble. Accordingly, we propose to build a odel that directly incorporates at trading features, so that the e ects of at pricing on RV asyptotics can be assessed. We follow the existing literature and assue that the e cient price process p (t) follows (). This speci cation iplies that, for any t i [; ]; p i; has the local artingale structure 6

9 p i; P i j " j;, where " j; R t j; t j ; (s)db(s). The new odel adds a siple Bernoulli process to () to deterine the trading price p i; p i; if i p i ; if i ; (6) where i is a Bernoulli sequence independent of p with E( i ) ; and p ; p ; O p (). Thus, while p i; follows an underlying local artingale in the background, the observed price copounds this e cient process with an independent Bernoulli sequence that deterines whether at trading occurs in the price realization. Whenever p i; 6 p i ;, the realization follows the e cient price and we observe p i;. Otherwise, at trading occurs. In that event, the icrostructure noise e ect copletely o sets the e cient price oveent over the subinterval in which at trading occurs. We can think of p as a tie changed process arising fro the equilibriu price p where the tie change is rando and discontinuous, depending on the realizations of the Bernoulli sequence. The odel ay therefore be linked to other literatures, particularly the iportant paper by Jacod (993) which allows for rando stopping ties, the work of Mykland and Zhang (6) which allows for deterinistic interittent sapling, and the work by Engle and Russell (998) on ACD odeling of duration. Soe of the connections with this literature are explored in Section 3, where the odel (6) is generalized to allow for rando stopping ties which ay depend on past prices. In other related work, Delattre and Jacod (997) introduced round-o to the easureent of a stochastic process and exained the ipact of this type of easureent error on the lit theory for certain functionals of the process. In such cases, the e cient price p (t) is e ectively rounded o to a grid of values deterined by a paraeter ; say, so that at t i we observe p (t i ) [p (t i ) ] where [] signi es the integer part of the arguent. Thus, p (t i ) is easured in increents of : Iportantly, this odel incorporates the institutional feature of the arket that trading takes place at well de ned increents. In practice, of course, these increents (like /8 cent or decialization) change over tie, which is a further institutional coplication. Delattre and Jacod (997) show that the e ects of this type of roundo on the liit theory are subtle and depend on the rate at which! as! ; inconsistencies arising in the estiation of volatilities, for exaple, when the roundo paraeter! too slowly. Round-o e ects of this type will induce soe at trading in the observed process when the e cient price wanders around a particular level over successive observations. Duration of ats is then copletely deterined by the e cient price process and the echanics of the roundo process. In the odel (6), the duration of at trading is deterined by the nuber of successive zero draws in the Bernoulli sequence i : In units of the sapling grid, the induced duration K is therefore rando and unbounded. In particular, it is known (e.g. Schilling, 99) that the axiu run tie, K, for a sequence of identical Bernoulli draws in a saple of size has ean E K O log f ( )g O log f ( )g log and variance Var( K ) 6 log ( ) : It follows that K O p (log ) : So the induced duration K of trading ats in (6) is unbounded but at ost O p (log ) as! : In cases where the sapling grid 7

10 is equispaced with t i; i ; it follows that the calendar duration tie is K and therefore o p () as! : In the case of non-equispaced grid ft i; g ; the axiu calendar duration tie on the grid is h K; sup i jt i; t i K; j: The odel (6) ay be construed as producing sticky price e ects analogous to those in Calvo (983) pricing schees. In those schees, onopolistic power enables soe fraction (the Calvo share) of rs to set prices, so that at any point in tie there is a probability of sticky pricing and a corresponding probability of a price change. Siilar echaniss appear in other pricing odels. For instance, inforation ay reach only a fraction of traders as in the Mankiw and Ries () wholesale pricing odel or there ay be rational inattention in the sense that certain traders ay update their inforation irregularly due to various costs of inforation gathering or other frictions, as in the pricing odel of Reis (6). In a siilar fashion, the present odel (6) allows for at trading with a constant probability of ; so that there is a positive probability of at trading at each point on the teporal grid when [; ): When, p i; p i; alost surely and the odel reduces to the earlier odel of ABDL (), ABDE () and BNS (). If p i ; p i ; and p i; p i ;, then p i; p i; p i ; p i; " i;. So the new odel allows for noise in the observed price and the noise depends on the e cient price. The noise can be interpreted as a discrete price e ect, according to which the realized price changes only when the inforation content is strong enough. Eventually, of course, the observed price will change and follow the e cient price provided > : One consequence of the speci cation is that when noise occurs in the odel it takes the for p i; p i; " i; R t i; t i ; (s)db(s) and is therefore negatively correlated with the e cient price process. Negative correlation between icrostructure noise and the e cient price has been epirically docuented in Hansen and Lunde (6, page 3). However, since p i; p i; when p i; 6 p i ;, the present odel eliinates noise e ects when the price changes. Thus, this particular odel ay be ore appropriate at odest frequencies rather than at ultra-high frequencies. The odel is extended in Sections 3 and 4 to allow for ore general sapling schees and icrostructure noise, aking it better suited to data at very high frequencies. We reark that the paraeter is assued to reain xed as! ; so that the no atter how nely the data are observed there is always a positive probability of a at when < : This assuption ensures that, although there is a positive probability of at trading, the length of the ats will inevitably shorten as increases because the ats are easured in increents of for an equi-spaced grid. In this respect, the odel clearly di ers fro data such as that observed in Fig. where no atter how frequently we ake observations the ats reain xed in size. Thus, the odel abstracts fro the reality in the data, and this abstraction fro the data applies in the sae way for all the other approaches discussed above because the data is typically of this for. We further reark that if the odel were to accoodate xed periods of at trading as the sapling frequency! then it would not be possible to consistently estiate the integrated volatility R (t)dt of p (t) because there would be xed subperiods of the interval [; ] in which the integrated volatility is inconsistently set to zero to accord with trading ats in the observed price p (t). Siilar probles would arise with all other approaches in this case because integrated volatility is not identi ed over these subperiods. 8

11 The following result con rs that the copound odel preserves the artingale property for trading prices. THEOREM. (Martingale Property): If p (t) follows () with E R r (t)dt < for all r [; ); and the trading price p(t) follows (6) with (; ]; then fp i; g is a artingale with E(p i; jf i ; ) p i ; and the natural ltration F i; (p i; ; p i ; ; ). The rst results of the paper now follow. Under at trading, RV consistently estiates IV as in the standard theory of epirical quadratic variation (e.g., Jacod and Shiryaev, 987; ABDL, ). In fact, the result is covered by standard liit theory because the axiu duration of at trading is h K; ; which for an equi-spaced grid is of order O p (log ) and therefore tends to zero. Theore.3 derives the corresponding central liit theore (CLT) for RV and Theore.4 provides a feasible version of the CLT for inference about IV using an epirical quarticity estiate. For the CLT results it is convenient to assue that the discrete sapling grid is equi-spaced, so that ft i; i : i ; :::; g: This requireent ts in with earlier conditions used in BNS on RV liit theory without at trading, and is relaxed in Section 3. THEOREM. (Consistency): If (; ] and if, for soe > ; h ;! as! ; then RV () (p)! p IV: THEOREM.3 (Infeasible CLT): Assue the observation grid is equi-spaced with ft i; i : i ; :::; g: If (; ]; then as! p h i RV () d! (p) IV MN ; 4 Z 4 (t)dt ; (7) where M N signi es ixed noral. When :5; soe 5% of the data involves at trading and the asyptotic variance in (7) is three ties as large as when : This agnitude sees to be in line with what has been docuented epirically in Hansen and Lunde (6, page 37). Table shows the ratio of the asyptotic variance to the case where there is no at trading for various values of and Fig. plots this nonlinear relationship. As becoes sall, the ratio blows up rapidly. Table : Ratio of asyptotic variance with ats to that without at Result (7) holds even when ats are reoved fro the saple. This is because the epirical quadratic variation is una ected by the presence of at trading periods. Hence reoving at prices fro data does not reduce the asyptotic variance or change the liit theory. In e ect, the liit result shows that, when trading which does not re ect the true e cient price occurs, the asyptotic variance of the RV estiate increases proportionately. That is, when there is at price trading there is less inforation about the e cient price p (t), and the asyptotic theory re ects this reduction in inforation by an in ation of the variance. In e ect, there is a reduced saple size due to at trading. 9

12 Figure : Ratio of the asyptotic variance under at trading to that with no at trading ( ). To use (7) in practice the asyptotic variance ust be estiated, which involves estiating the integrated quarticity functional R 4 (t)dt: Following BNS, integrated quarticity can be estiated consistently and used in a feasible CLT that is suitable for inference about IV. Lea.4: Under the conditions of Theore.3, as! When, Z [p i; p i ; ] 4 p! 4 (t)dt: (8) i and result (8) is identical to that of BNS. THEOREM.5 (Feasible CLT): Under the conditions of Theore.3, as! r 3 (RV () (p) IV ) d q P! N (; ) : (9) i [p(t i;) p(t i ; )] 4 Interestingly, the standardization in the feasible CLT (9) does not depend on and the feasible CLT result is therefore the sae as that given in BNS for the case where there is no at trading ( ). In e ect, the quantity involving appears as a factor in the asyptotic variance and the estiated quarticity functional and is therefore scaled out in the feasible CLT. Nonetheless, the e ects of at trading are iplicitly ebodied in the feasible CLT since they are carried in the epirical easure P i [p(t i;) p(t i ; )] 4 ; which is correspondingly reduced by periods of at pricing. Thus, the asyptotic inferential apparatus of BNS continues to hold under the present odel where at trading is anifest. 3. Stopping Tie Models In odel (6) it is assued that p (t) is interittently observed according to the realizations of a Bernoulli sequence. More generally, we ay assue that p is generated by () and observed

13 at rando ties f j : j ; :::; J g deterined according to a schee in which the increents take the for j j D j ; ; () where D j is a strictly stationary and ergodic sequence of nonnegative rando variables for which ED j D > and EDj! D : The sequence D j easures the duration between observations (in units of the interval ) and ay be (partly) dependent on past prices. In the odel (6), the integer process D j + L j is deterined by the outcoe of a sequence of independent Bernoulli draws i at each of the points t i i of the original grid, those draws representing whether the e cient price process p is observed ( i ) or not ( i ) at this point. The integer L j is then the nuber of successive zero draws in the sequence prior to j. The terinal quantity J is such that p ( J ) is the last p observed on the grid fp (t i; ) : i ; :::; g : In order to develop an asyptotic theory for this ore general case, we need ore speci c conditions on the sequence D j in (). We also want to consider the ore general case where the duration sequence ay for an array and depend on ; in which case we use the notation fd ;j g in place of fd j g : Assuption D (i) fd j g j is a strictly stationary and ergodic sequence of nonnegative rando variables with nite ean ED j D > and second oent EDj! D : Partial sus of the centred values D j D satisfy the functional law fd j D g ) V () j[] for soe Brownian otion V with variance D : (ii) ax j D j o p ( p ) as! : Assuption D (i) fd ;j g j is an array of nonnegative rando variables whose conditional rst and second oents satisfy E D ;[s] jf [s]! p D (s) ; E D;[s] jf [s]! p! D (s) ; () as! ; where D (s) and! D (s) are nonnegative càdlàg processes for s [; ]. Partial sus of the centred values D ;j E D ;j jf j satisfy the functional law D;j E D ;j jf j ) V () j[] for soe Brownian otion V with variance D :

14 (ii) ax j D ;j o p ( p ) as! : D(i) involves standard stationarity, oent and partial su functional law conditions on the sequence D j. In D(i) the convergence of the conditional oents in () is satis ed for certain autoregressive conditional duration (ACD) odels, as we discuss below. Condition (ii) involves the relative stability of the axia of a stationary sequence and is quite weak. For instance, for Gaussian sequences when the th autocovariance decays according to the rate o p then it is known (Beran, 96) that ax j D ;j O p log : Priitive conditions for (ii) are also available in the literature (e.g. Naveau, 3). THEOREM 3. Suppose p is observed at rando ties j deterined as in () with D j satisfying Assuption D. Then as! and J j Z [p ( j ) p ( j )]! p (t)dt; () 8 9 p < J Z [p ( j ) p ( j )] (t)dt : ; ) MN j ;! D D Z 4 (t)dt If the D j satisfying Assuption D, then () continues to hold as! and (3) is replaced by 8 9 p < J Z Z [p ( j ) p ( j )] (t)dt : ; ) MN ; 4 (t)! D (t) D (t) dt (4) j The corresponding self noralized quantity has the following liit under D as! (3) r P J 3 [p ( j j ) p ( j )] R (t)dt n PJ o ) N (; ) : (5) j [p ( j ) p ( j )] 4 Exaple : Mykland and Zhang (6) As an exaple, suppose the sequence D j is non rando and the increents j j D j are such that Dj!! D; D j! D ; j j and J log D j + O ; j

15 so that there are J increents over the interval [; ] ; as earlier. This type of interittent deterinistic sapling is covered in recent work by Mykland and Zhang (6). Set j j j and, using the notation of Mykland and Zhang (6) for the average interval length, we nd that and () P J j D j J j: j t D j j D + o () ; D j f P j i D jtg Dj f jed i tg + o () j Dj + o () jt D ED i t D + o () ()!! D D t: Then, again in the notation of Mykland and Zhang (6), we have Pj: H () (t) j t ( j) P j: j t D j D!! D + o () t : H(t): D Under these conditions of deterinistic sapling and noting that H (t)! D, Proposition D of Mykland and Zhang (6) gives the following liit theory () 8 < : J j 9 Z [p ( j ) p ( j )] (t)dt ; ) MN ;! D D Z 4 (t)dt ; which, since () K + o () ; confors to result (3) above. However, Mykland and Zhang (6) ipose the following additional condition (Assuption A(i) in their paper) on the interittent sapling sequence O () ; () where D j ax j : (6) Allowing for rando interittent sapling with duration D j between observations, condition (6) is equivalent to ax j D j ED j O p () ; Mykland and Zhang assue the sapling points j to be deterinistic but note later in their paper that the schee covers the case where the sapling points are rando but independent of the observed process. The stopping tie schee () is rando and allows for dependence on past prices. 3

16 which does not hold in any interesting cases. For exaple, in the odel (6) where sapling is deterined by a siple Bernoulli schee, we have ax j D j ED j O p (log ) : Thus, Mykland and Zhang s Assuption A(i) holds only for deterinistic sequences or bounded rando sequences fd j g and excludes the odel (6), although as shown above in Theore 3. the liit theory does extend to this case so condition (6) is, in fact, not necessary. Observe that for the odel (6) we have 3 D E ( + L i ) and! D E ( + L i) so that! D D 4 corresponding to the scale factor in the asyptotic variance in (7). ; ; (7) Exaple : Jacod (993) As a second exaple, we note that the sapling schee () ts into the sae stopping tie fraework as that used in Jacod (993). The associated epirical easures that appear in Jacod s paper can be worked out as follows using Jacod notation: [; t] : f j <tg j j j f P`j D`<tg f P`j ED`+O p( )<tg + o p ()! p t D : [; t] ; say (8) j[t] D p D [; t] : f j < tg! p t p D : [; t] ; say. j[t] Note that ( [; t] t) ( [; t] t) ; thereby satisfying result (3.) of Jacod (993). More generally under D, the liit easure 4 is [; t] R t D (s) ds; so that (dt) dt D (t) : The liit theory in Section 6 of Jacod (993) applies in the current context but to standardized price increents. In particular, equations (3.8), (6.5) and Theore 6. of Jacod (993) lead to the following liit theory for the epirical quadratic variation of the standardized increents fp ( j ) p ( j )g ( j j ) J j fp ( j ) p ( j )g j j! p Z (t) (dt) : (9) 3 Note that the values L i ; ; ; :::; correspond to realizations i ; i ; i ; i ; i ; i with n respective probabilities o ; ( ) ; ( ) ; ::::: 4 In this case we have P`j D ;` < t ; which under D is asyptotically equivalent to n R o j D (s) ds < t : The easure [; t] is then given by [; t] r (t) where R r D (s) ds t so that D (r) dr dt: 4

17 Correspondingly, D J j fp ( j ) p ( j )g j j! p Z (t)dt In the case of D, the liit is R D (t)dt and under D the liit is R (t) dt D (t): A proof is given in the Appendix. For the liit distribution, we nd that 8 9 p < J fp ( j ) p ( j )g Z Z! (t) (dt) ) MN ; 4 (t)! D (t) + D (t) : ( j j ) ; D (t) (dt) ; j which after restandardization in the case of D becoes 8 9 p < J D fp ( j ) p ( j )g Z (t)dt ; : j j ; ) MN! D + D j D Z 4 (t)dt Result (9) is copatible with (). The di erent weight factors re ect the use of standardized price increents in the realized variance. In this case, the epirical quadratic variation converges to a weighted version of integrated volatility where the weight function depends on the ean duration process of ats D (t). The central liit theory () is siilar to (4) but has a di erent weighting function in the conditional variance. As shown in the Appendix, the liit distribution stes fro a artingale ter that involves both the nuerator and the denoinator of the standardized squared price increents. Interestingly, because of the standardization of the price increents the weight function in the conditional variance of () is () ()! D (t) + D (t) D (t)! D (t) D (t)! D (t) D (t) D (t) <! D (t) D (t) ; showing that the variance in the liit distribution () is less than that of (4). In the special case where D applies, we ay restandardize by D ; as shown in (), to produce a consistent estiator of IV. In the case of odel (6) we have the decoposition! D + D 4 D 3 ; () showing the variance in ation fro at trading on the estiation of IV using standardized squared increents is less than it is in the case of squared increents in the odel (6). Of course, in () the IV estiator depends on scaling by D : Further, since ean duration can be consistently estiated by ^ D J P J j D j ; we ay construct the feasible estiator ^ D J j fp ( j ) p ( j )g j j! p Z (t)dt: (3) Fro Assuption D(i), we have p (^ D D ) ) N (; D ) where the variance depends on the sapling properties of the array D j including autocorrelation and heterogeneity. The 5

18 liit distribution of the feasible estiator (3) therefore involves additional ters arising fro the use of the tted ean ^ D : Finally, as shown in the Appendix, under Assuption D we have the following liit theory for epirical quarticity in the case of standardized increents J j fp ( j ) p ( j )g 4 ( j j )! p 3 which reduces in the case Assuption D to D J j Z fp ( j ) p ( j )g 4 ( j j )! p 3 4 (t) dt; (4) D (t) Z 4 (t)dt: Exaple 3: Relation to the ACD Model Engle and Russell (998) introduced autoregressive conditional duration (ACD) forulations to odel the duration between transaction ties. In such odels the conditional expectation D;j E (D ;j D ;j ) jf j of the intra trade tie is a easurable function of past durations. In a GARCH(,) fraework, we have D;j + D ;j + D;j ; (5) where we allow the coe cients (! ; ; ) to depend on because of the array structure of D ;j : Models of this type have been studied by Ghysels and Jasiak (997) and Engle () aong any others. Nelson (99) showed that under certain conditions on the coe cients in (5) GARCH odels of the for (5) satisfy in the liit as! a di usion equation d D (s) ( D (s)) ds + dw (s) ; (6) where W is a standard Brownian otion. In particular, in an appropriately de ned and enlarged probability space and for the paraeterization ; p ; and we have the convergence D[s]! p D (s) ; where the liit is a stochastic process satisfying (6). This type of liit theory has also been studied by Corradi () and Boswijk (, 5). Thus, the stopping tie odel with duration array D ;j satisfying condition D subsues certain ACD speci cations and condition () ay be regarded as version of the associated convergence of the discrete conditional expectation to a continuous process. 4. Microstructure Noise E ects Microstructure noise in the for of additive errors ay be incorporated into the fraework (6) by setting p p i; i; + u i; if i p i ; if i ; (7) 6

19 so that when prices do change a noise coponent u i; perturbs the observed price. Siilarly, adding noise to the ore general forulation in Section 3 leads to a process p that is observed at stopping ties j according to the schee p ( j ) p ( j ) + u j ; j j D ;j ; ; where the durations D ;j satisfy Assuption D. Various assuptions about the noise process are possible, but for the purposes here it will be su cient to assue that u j is iid with zero ean, variance u and nite fourth oent. The e ects of this noise ay be taken into account using ethods such as the use of di erent tie scales (ZMA, 5) or kernel soothing techniques (Hansen and Lunde, 6; Aït-Sahalia, Mykland and Zhang, 5b, Barndor -Nielson, et al, 8). We will illustrate by considering the two tie scale approach developed in ZMA. Our approach di ers fro the existing literature in that all ats are retained whereas ats are reoved in other analysis. The otivation arises fro the desire to utilize all available high frequency data as suggested in Aït-Sahalia, Mykland and Zhang (5a). Let the epirical quadratic variation of a process Y t on the full grid G f j : j ; :::J g be denoted [Y ] G t Y j Y j ; with [Y ] G [Y ] G ; j G ; j t with siilar de nitions for the epirical quadratic covariation between Y t and another process t ; i.e., [Y; ] G t Y j Y j j j : Then i G ; j t [p] G [p ] G + [p ; u] G + [u] G [p ] G + J u + O p J : (8) Following ZMA, we construct a two tie scale estiator of IV using [p] G estiator using K nonoverlapping grids and an average G k f k ; k +K ; k +K ; :::; k +k Kg involving every K th observation in G for k ; :::; K, and where k is the integer for which k +k K is the last eleent in G: k The average estiator has the explicit for [p] (avg) K K k [p] Gk ; (9) where j [p] Gk [p ( k +sk ) p k +(s )K ] ; k ; :::; K: (3) s 7

20 De ne K K k k J K + : (3) K As in ZMA, we require K! and K! as! : It follows that KJ! : We also need to odify D(ii) to take account of the tie scale induced by the new grids G k : Assuption D (ii) ax j D ;j o p 3 : The two tie scale estiator of ZMA is d[p ] [p] (avg) J [p] G (3) and uses the two di erent tie scales in coputation, one based on the subgrids G k and the other on the full grid G : The liit theory for d [p ] follows. THEOREM 4. Suppose p is observed at rando ties j deterined as in () with D ;j satisfying Assuption D (i) and (ii). Let K c 3 for soe constant c. Then as! Z d 8 6 [p ] (t)dt ) c 4 u + c N (; ) ; (33) where n R o 4 ( s) D ( s) ds Z R 4 (t) dt: (34) D (s) ds This result corresponds with the liit theory in ZMA and the only di erence occurs in the de nition of, which directly involves the ean duration e ects arising fro the process D (s) : In particular, the e ect of at trading (or rando durations in interittent sapling) on the liit theory of the two tie scale estiator of ZMA arises through the conditional ean functional of duration, D (s) ; in the liiting variance. Interestingly, as copared with (4), when there is no icrostructure noise, the conditional second oent of duration,! D (s) ; does not a ect the liit distribution. When D (s) D a:s: so that conditional ean duration is constant throughout the period of observation, (34) reduces to 4 Z ( s) ds Z 4 (t) dt 4 3 Z 4 (t) dt; (35) corresponding to equation (5) in ZMA, as obtained for the case of equispaced observations. This shows that the liit theory of ZMA reains correct in the presence of at trading of the Bernoulli type. 8

21 The rando coe cient in (34) depends on the average conditional ean duration which ay be estiated by D J J j D ;j J J j Z E D ;j jf j + op ()! p D (s) ds: In view of Assuption D we then have the consistent estiate n ^h 4 J R o j 4 ( s) D ( s) ds D ;J j! p R DJ J j : (36) D (s) ds Soe other aspects of the estiation of and the use of the result in epirical application are discussed in ZMA. In a siilar way, it is possible to exaine the e ects of at trading on other consistent estiators of integrated volatility in the presence of icrostructure noise. In particular, the ultiple tie scale estiator of Zhang (6) and the realized kernel ethod of Barndor - Nielsen et al. (8) ay be treated siilarly and have the advantage of the faster rate of convergence Siulations and Epirical Applications The siulation study in this section considers two odels the Brownian otion odel and Heston s stochastic volatility odel. Using the Brownian otion odel we check the accuracy of the CLT (7) and the CLT (). Using the stochastic volatility odel we check the accuracy of the CLT (9). Data are siulated over a day so that t ; and t ;. A day is assued to have 6.5 hours and 3,4 seconds. In our Monte Carlo design we chose 39, 78, 3, 95, and 39. These values correspond to frequencies of, 5, 3,, inutes, respectively. The epirical application assesses the practical ipact of at trading on the CLT (33) using two dataset discussed in the introduction and three other datasets. 5. The Brownian otion odel This subsection reports siulations fro a siple Brownian otion odel where volatility is a known constant ( (t) ) so that dp (t) db(t): (37) This forulation allows us to assess the accuracy of CLT (7) R 4 (t)dt and then the asyptotic variance in (7) is siply 4 : Table 3 shows both the asyptotic and nite saple siulated variances of the statistic p RV () (p) IV based on 5, replications for various cobinations of and. The asyptotic forula is clearly very accurate except for very sall values of : The e ect of at trading on the asyptotic variance is draatic, producing a three fold increase in variance when :5: 9

22 Table 3: Asyptotic and nite saple variances Asyp. variance Asyp variance Variance ( 39) Variance ( 78) Variance ( 3) Variance ( 95) Variance ( 39) We also exaine the accuracy of CLT () of the restandardized quantity. Bernoulli process odel, the asyptotic variance of the restandardized quantity is With the Z 4 (t)! D (t) + D (t) dt! D + D (t) 3 D (t) D : Table 4 shows both the asyptotic and nite saple siulated variances of the statistic 8 9 p < J : fp ( j ) p ( j )g Z D (t)dt j j ; j with D, based on 5, replications for various values of and 39. The asyptotic forula is clearly very accurate in all cases and even when is as sall as 39. Although not reported here, we have also found saller bias in the restandardized quantity. Table 4: Asyptotic and nite saple variances Asyp. variance Variance ( 39) A stochastic volatility odel In this subsection, price data is siulated fro Heston s stochastic volatility odel with volatility following a square root odel (Heston, 993): dp (t) (t)db (t); d (t) ( (38) (t))dt + (t)db (t): Feller (95) showed that the density of (t+h) conditional on (t) is ce u v (vu) q I q ((uv) ) and the arginal density of (t) is w w (w ) e w (w ), where c ( ( e h )); u c (t)e h ; v c (t + h); q ; w ; w, and I q () is the odi ed Bessel function of the rst kind of order q. The conditional density together with the arginal density are used for data siulation. The paraeters in the odel are set at :; and :5.

23 Table 5: Asyptotic and nite saple variances Asyptotic variance Variance ( 39) Variance ( 78) Variance ( 3) Variance ( 95) Variance ( 39) The ai of the experient is to assess the accuracy of the epirical quarticity forula in the feasible CLT (9). Table 5 gives q the Monte Carlo results. In particular, we report the 3 (RV variance of the standardized statistic () (p) IV ) p P fro, replications, for i [p(t 4 i;) p(t i ; )] various cobinations of and ; shown against the asyptotic variance of unity. Soe conclusions can be drawn fro the table. First, the asyptotic theory clearly works better for large and large. This is unsurprising because larger values of iply fewer at price trading periods and therefore larger e ective saple sizes. Second, the asyptotic theory does eventually work well even for sall, but needs larger values of to provide a good approxiation. The ain reason for these e ects is that it is ore di cult to estiate the integrated quarticity than the integrated volatility. This corroborates existing ndings in the literature on realized volatility without at pricing. Table 4 shows that these e ects are exacerbated when there is at trading, especially when is sall, because of the saller e ective saple size. 5.3 Epirical Evaluation Using the liit theory in Section 4 under icrostructure noise, we ay evaluate the epirical e ect of at trading in the application of the asyptotics. Here we consider the epirical ipact of at trading on (33). Using the tick-by-tick transaction data for AA on April, 997 and on April, 998, both discussed in the introduction, and for three other copanies traded in the NYSE (3M, GE and Boeing) on April, 997, we nuerically copare the coe cient given in (34) with that of (35). 5 This involves a siple coparison of a consistent estiate of 4 R ( s) D ( s) ds R D (s) ds with the quantity 43 that applies in the case of equispaced sapling and regular allocation to subgrids. Table 6 reports the consistent estiate ^h 4 P J DJ j paraeter 4 R ( s) D ( s) ds R j J D;J j of the duration D (s) ds together with the e ective saple sizes for several stocks, including the AA data set discussed in the introduction. It is interesting that while ats are a doinating feature of the tick data, the estiates ^h di er little fro the factor 4/3 in all these cases, suggesting that the iplied variances are very close to each other and that the ZMA liit theory should be a good approxiation. For these data, as we have seen, ost ticks do not involve price changes. So if the ats are reoved prior to econoetric 5 The datasets and dates are arbitrarily selected but are illustrative of heavily traded stocks.