On the use of R 2 as a measure of stock price efficiency

Transcription

1 On the use of R as a measure of stock prce effcency Govann Petrella Catholc nversty Largo Gemell Mlano (Italy) Emal govann.petrella@uncatt.t Dego Zappa Catholc nversty Largo Gemell Mlano (Italy) Emal dego.zappa@uncatt.t Abstract An actve debate s currently takng place on the use of R as a measure of stock prce nformatonal effcency. Some scholars argue that R s a measure of effcency (.e., the larger the R, the hgher s the level of effcency), some other researchers argue that R s a measure of neffcency (.e., the larger the R, the lower s the level of effcency). In ths paper, frst, we model the relatonshp between the market model R and the delay n the prce dscovery process and, then, we propose a research strategy for the nterpretaton of R as a measure of prce effcency based on the correlaton among R and delay. Two man fndngs arse from our emprcal analyss. Frst, the R s strctly ncreasng n frm sze. Ths pattern s consstent across all markets n our sample and casts some doubts on the nterpretaton of R as a prce neffcency measure. Second, the correlaton among R and delay s consstently negatve and statstcally sgnfcant from 001 onward for NYSE, NASDAQ and Chna. Our paper shows that prevous fndngs on stock prce synchroncty that lead to the use of R as prce neffcency measure were strongly sample-specfc and that usng the correlaton among R and delay helps n the nterpretaton of the R as a measure of prce effcency (or neffcency). Keywords: R ; Informatonal Effcency; Delay JEL Classfcaton: G14; C1 Ths draft: January 15, 010

2 On the use of R as a measure of stock prce effcency: I. INTRODCTION An actve debate s currently takng place on the use of R as a measure of stock prce nformatonal effcency. Some scholars argue that R s a measure of prce neffcency (.e., the larger the R, the lower s the level of effcency). The basc ntuton behnd ths hypothess s that a lower R s caused by greater ncorporaton of frm-specfc nformaton. Studes that ether assume or conclude that R s a measure of prce neffcency are, respectvely, Morck et al.'s (000) analyss at the country level and Durnev et al.'s (003) analyss at the frm level. Followng these papers, a number of studes nterpret a low R as evdence of prce neffcency. However, other studes (Ashbaugh et al., 006; Grffn et al., 006; Kelly, 007; Dasgupta et al., 008) document emprcal fndngs that strongly challenge the nterpretaton of R as a measure of prce neffcency. In fact, some scholars argue that R s a drect measure of prce effcency (.e., the larger the R, the hgher s the level of effcency). Pagano and Schwartz (003), n ther study on the effects of the ntroducton of a closng call aucton at Euronext Pars, nfer market qualty from the synchroncty of prce changes based upon the fact that a hgher R s consstent wth serally uncorrelated returns. In ths paper, frst, we model the relatonshp between the market model R and the delay n the prce dscovery process and, then, we propose a research strategy for the nterpretaton of R as a measure of prce effcency. In the last part of the paper we conduct an emprcal analyss to assess the valdty of our conjectures. Despte the conflctng evdence on the relatonshp between R and prce effcency, ths metrc s largely employed n fnance and accountng studes 1. Our paper s motvated by the mperatve need to better understand the nterpretaton of ths measure n order to approprately use t n emprcal analyses. 1 A non exhaustve lst of studes nclude Morck et al. (000), Wurgler (000), Durnev et al. (003), Pagano and Schwartz (003), Potrosk and Roulstone (004), Barbers et al. (005), Chan and Hameed (006), Jn and Myers (006). 1

3 The rest of ths paper s organzed as follows. In Secton II we present the current state of the debate on the use of R as a measure of prce (n)effcency. In Secton III, n order to assess whether a larger R s an ndcator of hgher or lower prce effcency, we model the relatonshp between R and delay. In partcular we show that, f effcency exsts, when delay s low, R should be hgh. In Secton IV we present the results of our emprcal analyss and n Secton V we provde summary and conclusons. II. THE R DEBATE Several studes use the R statstc of a market model as a measure of stock prce effcency. In ths Secton we brefly revew some hghly-cted papers that have been used as a reference n later studes to justfy the use of R as a measure of stock prce effcency or, dependng on the study, neffcency. Morck et al. (000) use b-weekly stock returns n the year 1995 for a sample of 15,90 frms spannng 40 countres to nvestgate stock returns synchroncty n emergng and developed economes. They defne the country-level R for country j as : R j = R, j SST, j SST, j [1] where R, j measures the percent of the varaton n the b-weekly returns of stock n country j explaned by varatons n country j's market return and SST, s the j sum of squared total varatons for stock n country j. Morck et al. (000) consder the country-level R as a measure of stock prce synchroncty and fnd that stock prces n economes wth hgh per capta gross domestc product (GDP) move n a relatvely unsynchronzed manner. By contrast, stock prces n low per capta GDP economes tend to move up or down together. To control for the sze of the country and other explanatory varables, they regress a Appendx A presents an applcaton of the Chsn approach to nterpret the country-level R as a prce effcency measure.

4 logstc transformaton of the country-level R on country-related structural varables and a good government ndex. The authors fnd that the synchroncty of stock returns s hgher n countres where property rghts are not well protected. Morck et al. (000) conjecture that weaker property rghts protecton dscourage arbtrage actvty and make stock prces to be drven more by market-wde poltcal events than by frm-specfc nformaton. Ths evdence motvates the nterpretaton of R as a measure of prce neffcency. Durnev et al. (003) emprcally address the queston of how to nterpret R statstcs, as evdence of prce effcency or neffcency, by examnng a sample made of all S lsted companes avalable n the CRSP/Compustat database from 1983 to They estmate a regresson model to explan ndvdual frm return varaton wth both market and ndustry returns, and defne as frm-specfc prce varaton the porton of a frm's stock return varaton unexplaned by market and ndustry returns, whch s equal to (1 R ). To dscern whether hgher frm-specfc prce varaton mples more or less prce effcency, Durnev et al. (003) nvestgate the relatonshp between current prces and future earnngs as evdence of prce nformatveness. They fnd that frms and ndustres wth lower R exhbt hgher stock prce nformatveness, ndcatng more nformaton about future earnngs mpounded n current stock prces. Ther results also supports the nterpretaton of R as a measure of prce neffcency. Pagano and Schwartz (003), by contrast, take the opposte stance and assume that hgher R s assocated wth hgher prce effcency. They study the effects of ntroducng a closng call aucton on stock market qualty. In May 1996 and June 1998 Euronext Pars ntroduced a closng call aucton for, respectvely, less-lqud and more lqud stocks. Pagano and Schwartz (003) use data from Euronext Pars to compare market qualty ndcators before and after the ntroducton of the closng call n order to assess the effects of ths nnovaton. They nfer market qualty from the synchroncty of ndvdual stock returns wth respect to market returns (.e., from the market model R ) and take an ncrease n R as evdence of ncreased prce effcency. 3

5 Pagano and Schwartz (003)'s nterpretaton s based on the argument that tradng frctons (e.g., transacton costs) generate lead/lag relatonshps between ndvdual stock returns and market returns, whch n turn affect both estmated beta and market model R. In fact, f stock returns are serally uncorrelated, both stock beta and market model R wll be ndependent of the dfferencng nterval (.e., the return measurement nterval). By contrast, ntertemporal correlaton n stock returns whch s evdence of prce neffcency ntroduces ntervallng-effect bases: beta estmates obtaned from short perod returns are based downward for stocks that lag the market and upward for stocks that lead the market, and the market model R decreases the lower s the measurement nterval for stock returns. E.g., f stock returns are serally correlated, the R estmated from daly data s lower than the R estmated from monthly returns. The ntervallng-effect bases s caused by delays n the prce dscovery process (Schwartz and Whtcomb, 1977; Hawawn, 1980). Specfcally, delayed adjustments of stock prces to news smultaneously cause negatve autocorrelaton n ndvdual stocks resduals and postve autocorrelaton n market returns 3. These effects mply that, as the measurement nterval ncreases, the decrease n the varance of the resduals (deflated by the negatve autocorrelaton) s larger than the ncrease n the varance of the market returns (nflated by the postve autocorrelaton). Consequently, the presence of delay n the prce dscovery process causes R to decrease wth the measurement nterval and ths n turn mples that as evdence of serally uncorrelated returns a larger R means a more effcent prcng. Grffn et al. (006) study the degree of prce effcency n 56 (33 emergng and 3 developed) stock markets around the world. They collect daly data from 1994 through 005 and use varous effcency measures: the delay n the prce dscovery process, the autocorrelaton of returns, the post-earnngs announcement drft, the market model R. Grffn et al. (006) examne market model R wthn country 3 Suppose that a news that would rase all stock prces occurs on day t. Suppose also that the prce of some stocks reacts on the same day (actve stocks), whereas the prce of others reacts n the followng tradng day (nactve stocks). The market return wll be postve on day t because of the actve stocks prce movement and on day t+1 because of the nactve stocks prce movement. Ths generates 4

6 .e., at the frm level and fnd that R s are nearly monotoncally ncreasng wth frm sze and state that "there are several puzzlng fndngs for the average R s, but the most puzzlng s stll our frst fndng, that small cap stocks have lower and not hgher R s" (page 4). In fact, f the nterpretaton of Morck et al. (000) s correct (.e., that R ncreases wth neffcency), ths would mply that large cap stocks are less nformatonally effcent than small cap stocks. However, ths nterpretaton of R s also n contrast wth other effcency measures that Grffn et al. (006) compute (delay, autocorrelaton and post-earnngs drft), all clearly ndcatng that small cap stocks exhbt slower nformaton ncorporaton than large cap stocks. Studes usng market model's R as a measure of prce neffcency post that prvate nformaton s the prmary source of the poor market model ft. However, Roll (1988) suggests an alternatve hypothess: lower R s may also depend on a "frenzy unrelated to concrete nformaton" (page 566). Kelly (007) uses data for NYSElsted stocks from 1993 to 00 to explore ths possblty. Frst, he nvestgates the nformaton envronment surroundng stocks to examne whether mpedments to nformed tradng, whch obvously hurt prce effcency, are related to market model R. Kelly (007) consders nformaton costs (proxed by analyst coverage, sze and age), tradng costs and lqudty to characterze the mpedments to nformed tradng and fnd that greater nformaton costs, greater tradng costs, and lower lqudty are consstently assocated wth low market model R s. Ths evdence suggests that stocks wth low market model R may be those wth the greatest possblty of msprcng and ths mples that a low market model R s a sgn of relatvely less nformatonally effcent prcng. Ashbaugh et al. (006) conduct fve tests, usng data from sx large equty markets (Australa, France, Germany, Japan, K and S), to assess the valdty of the R as a measure of frm specfc nformaton mpounded nto share prces. Frst, they look at the relatonshp between current prces and future earnngs n order to evaluate whether low R s are assocated wth prces that are more nformatve regardng future earnngs. They actually fnd that hgher R stocks exhbt more nformatve prces n Germany and the S, and no statstcally sgnfcant relatonshp between postvely autocorrelated market returns. Negatvely autocorrelated resduals arse snce actve (nactve) stocks resduals wll be postve (negatve) on day t and negatve (postve) on day t+1. 5

7 R and prce effcency n Australa, France, Japan and K. Second, they nvestgate whether R s assocated wth analyst forecast errors. If low R s are assocated wth greater amounts of frm-specfc nformaton mpounded nto stock prces, then a postve relatonshp between analysts' forecast errors and R should exst. However, Aushbaugh et al. (006) fnd such a postve assocaton only n Japan, whereas they document that frms wth larger R have smaller analysts' forecast errors n Australa, France, Germany, K and S. Thrd, they nvestgate whether there s a change n prce effcency surroundng frms' cross-lstngs n the S. Followng ths event, frms usually dsclose more nformaton than those requred n the home markets. If lower R represent relatvely more frm-specfc nformaton, one should expect a declne n R values followng frms' cross-lstngs. However, Aushbaugh et al. (006) fnd no evdence of R declne for Australan, German, Japanese and S stocks, and an ncrease n the R for French and K frms followng ther cross-lstng n the S. Fourth, Aushbaugh et al. (006) test the assocaton between R and proxes for the quantty and qualty of frms' nformaton flows (e.g., analyst coverage). They document nconsstent relatonshps snce they fnd postve relatons between the R and nformaton proxes n some countres and negatve relatons n others. Ffth, smlarly to Barbers et al. (005), Aushbaugh et al. (006) examne the assocaton between R and the addton to a German stock ndex that requres frms to provde addtonal dsclosures. They fnd that ndex membershp s sgnfcantly assocated wth hgher R s. Ths fndng s n contrast wth the nterpretaton of R as a measure of prce effcency. Collectvely, Aushbaugh et al.'s (006) results ndcate that hgher R s do not consstently mply lower prce effcency. Dasgupta et al. (008) adopt a dynamc settng and argue that more nformatve prces today should be assocated wth less frm-specfc varaton (.e., hgher return synchroncty) n the future. In fact, an effcent stock prce should already factor n the lkelhood of occurrence of future events. Consequently, when the events actually happen n the future, the prce reacton n an effcent market should be lower, ceters parbus, than the reacton n an neffcent market. Specfcally, they propose a theoretcal model where a more transparent nformaton envronment leads to hgher, rather than lower, stock return synchroncty. The basc ntuton of 6

8 ther model s that for a more transparent frm the surprse effect assocated wth future events (.e., the dosyncratc volatlty) s reduced. They test ther model's predctons on a sample of stock returns for S frms durng the perod Frst, they fnd that returns synchroncty s hgher as a frm becomes older. Ths fndng s related to the greater ablty of market partcpants to learn about stock fundamentals for older frms. Second, they fnd that returns synchroncty decreases pror to seasoned equty ssues and ADR lstngs and ncreases afterward. Ths result also shows that returns synchroncty, when an nformaton event occurs, frst decreases to ncorporate the frm specfc varaton and subsequently ncreases snce relevant nformaton s already mpounded nto stock prces. Collectvely, the results reported n ths Secton strongly challenge any uncontroversal nterpretaton of the R as prce effcency measure. III. R AND PRICE EFFICIENCY Two basc nterpretatons of the R statstc arse from the dscusson n the prevous Secton: "R as prce effcency measure" (nterpretaton # 1) vs. "R as prce neffcency measure" (nterpretaton # ). The nterpretaton of a larger R as evdence of hgher prce effcency subsumes the dea that an neffcent stock s ncorporatng relevant nformaton wth a delay (e.g., after one or two days). In fact, as shown n the prevous Secton, a delay n the ncorporaton of nformaton causes ntertemporal correlaton n stock returns whch, n turn, mples lower R. Specfcally, the delay refers to the senstvty of current returns to past market-wde nformaton. Several papers use the delay as a measure of stock prce effcency (Mech, 1993; Brennan, Jegadeesh and Swamnathan, 1993; Chorda and Swamnathan, 000; Hou and Moskowtz, 005). Dfferently from the R, the nterpretaton of the delay as a measure of prce effcency s uncontroversal. For ths reason we use the delay measure to nterpret the results of the R as a measure of prce effcency. In order to assess whether a larger R s an ndcator of hgher or lower prce effcency, we propose to look at the relatonshp between the delay measure and the 7

9 R. If the correlaton between delay and R s found to be negatve, ths would mply that nterpretaton #1 s correct and that hgher R s are evdence of prce effcency 4. By contrast, f the correlaton between delay and R s found to be postve, ths would mply that stocks exhbtng a low R also dsplay a low delay n ncorporatng nformaton. Ths would support the nterpretaton # that R s a measure of prce neffcency. We measure the delay as the dfference between the R of an unrestrcted model (that consders both current and past nformaton) and the R of a restrcted model (that consders current nformaton only). Let the restrcted model be the followng: where tme t. r = α + β r + ε [] R R t, mt, t, r, t s the return for stock at tme t; Let the unrestrcted model be: t,,0 mt,,1 mt, 1, mt, t, r m, t s the return for the stock market at r = α + β r + β r + β r + η [3] where r t, s the return for stock at tme t; r mt, s the return for the stock market at tme t. The unnormalzed defnton of delay s: * * delay δ = R R =1,...,k [4], where δ s the delay estmated for stock ; R s the R of the restrcted model R, estmated for stock ; R, s the R of the unrestrcted model estmated for stock. In what follows we assume that the market s made of k stocks and for each of them the R and, R have been estmated. For the sake of smplcty, n the rest of the paper we use R to ndcate the R from the restrcted model estmated for stock (.e., R R R, ). 4 If the nterpretaton #1 holds true, a stock wth a low R should have a hgh delay and vce versa (.e., hgh R should be assocated to low delay). 8

10 Let R, * R and δ be the sets of respectvely The covarance between * R and δ s: R,, * R and δ for =1,...,k. * Cov( R, δ )=Cov( R, R R )=E[ R ( R R )] E( R )E( R R ) =E[ R R ] E[ R ] E( R )E( R )+[E( R )] =Cov( R, R ) Var( R ) * It follows that Cov( R, δ ) 0 (.e., the nterpretaton # "R as prce neffcency measure" s correct) when Var( R ) that s when Corr( R, R ) Var( R ) Cov( R, R ) Var( R ) [5] 1/. Equaton [5] s equvalent to γ 1 1, where γ 1 s the estmated coeffcent of the model R =γ 0 +γ 1 R. Analogously, Cov( * R, δ ) 0 when γ 1 1. Ths result mples that the relatonshp between the R and the delay measures depends on γ 1, whch descrbes the relatonshp between the R s of the unrestrcted and restrcted models. By equatons [] and [3], t holds that R R. Consequently, the upper left trangle n Panel A of Fgure 1 represents the expected doman of the bvarate varable ( R, R ). If we assume that t s very large (.e., t ), the estmates of the pars ( R, R ) may be consdered akn to the generally unknown true populaton parameters. Furthermore, f we assume that k s very large (.e., k ), the least nformatve pror about the unknown relatonshp between R and R would suggest that ths doman s unformly flled. nder the three prevous hypotheses (t, k and unform dstrbuton), the dashed lne n Panel A of Fgure 1 represents the locus of E[ R R ] and t s straghtforward to observe γ 1 1 (.e., proportonally than R ncreases less than R ). Consequently, we expect to fnd on average a negatve 9

11 * relatonshp between δ and forward at the begnnng of ths Secton. R, whch s consstent wth the nterpretaton #1 put In the lterature several authors use the followng normalzed defnton of delay: delay R δ = 1 R R,, [6] Analogously to the prevous dervaton, the covarance between * R and δ s: R Cov( R,δ)=Cov R,1 R R = Cov R, R The trangle below the bsecton lne n Panel B of Fgure 1 s the expected doman of ( R, R / R ). It s agan straghtforward to observe that Cov( R, R / R ) s postve and consequently we agan expect to fnd on average a negatve * relatonshp between δ and R. The prevous conclusons hold under qute strong assumptons: a) t ; b) k ; c) unform dstrbuton of ( R, R ). In practce we generally observe the followng condtons: a*) t s fnte; b*) k may be small; c*) the dstrbuton of ( R, R ) over the whole doman s unknown. As a consequence of the prevous condtons we mght observe γ 1 1 or, whch s equvalent, Cov( R,δ ) 0. Ths evdence, whch would support nterpretaton # put forward at the begnnng of ths Secton, mght be drven by a sample of ( R, R ) pars that does not guarantee ether n the cross secton or n the tme seres dmenson consstent estmates of the R s. Addtonally, t should also be stressed that the coeffcent of correlaton s not a robust statstcs. Ths means that the presence of outlers strongly affects ts level 5. 5 The coeffcent of correlaton changes sgnfcantly n value and/or n sgn even f only one observaton of the sample set s far off the cloud descrbed by all the remanng observatons. 10

12 IV. EMPIRICAL ANALYSIS 6 A. Data We selected, accordng to the classfcaton by the Internatonal Fnance Corporaton (IFC) of the World Bank, one developed country (nted States) and two emergng countres (Chna and Poland). Ths choce was motvated by the fact that prevous studes lookng at R based on nternatonal samples (Morck et al., 000; Ashbaugh- Skafe et al., 006; Fernandes and Ferrera, 009; La et al., 009) consstently show that the S market dsplays the lowest level of stock prce synchroncty and, on the opposte sde of the rankng, Chna and Poland dsplay the hghest level of stock return synchroncty. We obtan from Thomson Reuters' Datastream (TRD) data for all the stocks traded on the NYSE and the NASDAQ for the nted States, the Shenzen Exchange and the Shangha Exchange for Chna, and the Warsaw Stock Exchange for Poland. Specfcally, we collected the daly total return ndex (RI s the varable name n TRD), the daly adjusted prce (P) and the market captalzaton (MV) for common stocks tradng n the company's home market. We also ncluded dead stocks n the sample. The data span from January 1995 through December 008. The sample perod was chosen to get results comparable wth prevous studes, especally wth Morck et al. (000) who use data from Table 1 reports the number of stocks and the frm average market captalzaton (n mllon SD) per year for both the full sample and 5 sze-sorted portfolos for S, Chna and Poland. For the nted States we separately report NYSE and NASDAQ data. The number of stocks vares across years. In 1995, the startng year for our analyss, the sample ncludes 6,941 stocks and 95% of the stocks are from the S market. In 008, the last year of our analyss, the sample ncludes 7,167 stocks and the percentage of stocks from S drops to 7% due to the growth n the Chnese (3%) and Polsh (5%) stock market. The average market captalzaton also vares across years. In 1995, the average market cap was about SD 1.46 bllon for NYSE stocks, SD 0.5 bllon for 6 The emprcal analyss was performed usng R. The software code s freely downloadable from the authors webstes. 11

13 NASDAQ stocks, SD 0.13 bllon for Chna, SD 0.08 for Poland. In 008, the average market cap was about SD 5.89 bllon for NYSE stocks, SD 1.11 bllon for NASDAQ stocks, SD 1.75 bllon for Chna, SD 0.7 for Poland. The relatve frm sze across countres changed over tme. The ncrease n the average market cap of Chnese stocks s partcularly remarkable. A very smlar pattern emerges for sze-sorted portfolos. The samplng frequency s weekly. We decded to use weekly data, as Hou and Moskowtz (005), as opposed to monthly or daly data because of the shortcomngs assocated wth lower and hgher frequences. At monthly frequency, delay measures exhbt lttle dsperson because most of the stocks respond to news wthn a month. Daly data, albet ntroducng more dsperson n the delay measures, may also ntroduce confoundng mcrostructure-related effects (e.g., the bd-ask bounce). We use the followng data flters. Frst, we exclude a stock f the proporton of zeroreturn days s larger than 75% n a gven perod. Ths screen s necessary snce TRD reports a constant stock prce after delstng. Second, we set the observaton to mssng f the absolute value of the return s larger than 1. B. Results Table 1 compares R and delay measures across countres and sze-sorted portfolos. For each stock n the sample and each year of weekly data we ran a market model regresson, ncludng both a country market return ndex and a S market return ndex adjusted for currency effects. Ths procedure produces yearly estmates for market model parameters and R. In Table 1 we report mean and medan of the yearly estmates, separately for each market, for both the full sample and 5 szesorted portfolos. The R s larger n emergng countres (Chna and Poland) than nted States. Ths fndng confrms the evdence orgnally presented by Morck et al. (000). Lookng at sze-sorted portfolos, the R s strctly ncreasng n frm sze. Ths pattern s consstent across all markets n our sample and casts some doubts on the nterpretaton of R as a prce neffcency measure. In fact, f ths nterpretaton 1

14 holds, we could nfer that the stock market s less nformatonally effcent for large cap stocks. However, ths concluson s somewhat counter-ntutve gven the fact that large frms are usually more actvely traded, attract more attenton by meda, and are followed by more analysts than small frms. Table also reports mean and medan values for the delay n the prce dscovery process as well as the correlaton among R and delay. The delay measure decreases wth frm sze. Ths mples that large cap stocks as expected exhbt lower delay n the prce dscovery process. However, takng together the R and delay fndngs, the followng contrasted pcture arses: large (small) cap show the lowest (hghest) delay n the prce dscovery process and, f we nterpret the R as a prce neffcency measure, also the lowest (hghest) level of prce effcency. The correlaton among R and delay s negatve and stronger as the market captalzaton ncreases. Ths pattern s consstent across all markets n our sample. Ths fndng confrms the predctons developed n Secton III and supports the nterpretaton of the R as a prce effcency measure. Our sample spans across 14 years. Such a long perod mght mask dfferent patterns n the data. To nvestgate ths possblty, Table 3 reports mean and medan values of R and delay, as well as the correlaton among R and delay, by market and by year. Three clear patterns emerge from the S markets. Frst, the average (medan) R ncreases from 7% (5%) for NYSE and 6% (4%) for NASDAQ n 1995 to, respectvely, 35% (36%) and 3% (0%) n 008. Second, the average (medan) delay n the prce dscovery process slghtly decreases from 9% (7%) for NYSE and 9% (8%) for NASDAQ n 1995 to, respectvely, 8% (6%) and 8% (7%) n 008. Thrd, the correlaton among R and delay decreases from -0.0 for NYSE and 0.00 for NASDAQ n 1995 to, respectvely, and -0.9 n 008. For Chna and Poland, by contrast, no clear pattern emerges. Panel A to D of Fgure provde a graphcal representaton of the tme seres dynamcs of the R s over the entre sample perod for the 5 sze-sorted portfolos for the four markets separately and Panel A to D of Fgure 3 do the same for the delay measures. Fgures and 3 confrm the evdence reported n Tables and 3. Frst, the R (delay) ncreases (decreases) wth market captalzaton. Second, the R 13

15 (delay) ncreases (decreases) over tme n our sample perod. These fndngs hold for the S markets, the evdence s mxed for Chna and Poland. Addtonally, Fgures and 3 provde an addtonal fndng: the dfference n R s and delays across sze-sorted portfolos ncreases over tme n our sample perod. Gven the fact that R and delay change over tme, the relatonshp among them may also change over tme. To nvestgate ths possblty, for each stock n the sample and each observaton, we estmate both a restrcted and an unrestrcted market model regresson on the prevous 5 weeks of data. Ths procedure produces rollng estmates for the market models parameters, the R restrcted and the R unrestrcted. Fgure 4 shows the tme seres dynamcs of 5-week rollng correlaton among R and delay as well as the bootstrapped.5% lower- and 97.5% upper-lmt confdence nterval. From 001 onward the correlaton s consstently negatve and statstcally sgnfcant for NYSE, NASDAQ and Chna. For Poland the correlaton among R and delay s also negatve, but not always statstcally sgnfcant. Ths evdence mght be explaned by the low market captalzaton of Polsh stocks. Fgure 5 shows the correlaton among R and delay for three sze-sorted portfolos: the larger s the market captalzaton, the stronger s the (negatve) correlaton. V. SMMING P In ths paper we contrbute to the debate on the use of R as a measure of stock prce nformatonal effcency. Some scholars argue that R s a measure of effcency (.e., the larger the R, the hgher s the level of effcency), some other researchers argue that R s a measure of neffcency (.e., the larger the R, the lower s the level of effcency). In ths paper, frst, we model the relatonshp between the market model R and the delay n the prce dscovery process and, then, we propose a research strategy for the nterpretaton of R as a measure of prce effcency based on the correlaton among R and delay. Three fndngs arse from our emprcal analyss. Frst, the R s strctly ncreasng n frm sze. Ths pattern s consstent across all markets n our sample and casts some doubts on the nterpretaton of R as 14

16 a prce neffcency measure. Second, the R (delay) ncreases (decreases) over tme n our sample perod. Ths fndng holds for the S markets, the evdence s mxed for Chna and Poland. Thrd, the correlaton among R and delay s consstently negatve and statstcally sgnfcant from 001 onward for NYSE, NASDAQ and Chna. Our paper shows that prevous fndngs on stock prce synchroncty that lead to the use of R as prce neffcency measure were strongly sample-specfc. The R s a very ntutve and easy to use ndcator, however users need to know ts lmts. Our paper shows that usng the correlaton among R and delay helps n the nterpretaton of the R as a measure of prce effcency (or neffcency). 15

17 APPENDIX: CONTRY-LEVEL R, THE CHISINI MEAN AND THE CORRESPONDING MARKET MODEL As dscussed n Secton II, another ssue strctly related to the use of R as a prce effcency measure s how to nterpret the country-level R defned n [1]. In ths Appendx we contrbute to the R debate as follows. [1] s a weghted average of the ndvdual stocks R s. Then t must be consdered tself as an R. However, the weghts n [1] are nether frequences nor probabltes. Consequently, the nterpretaton of [1] as a coeffcent of determnaton (.e., as a measure of the ft of a model) must be approprately justfed. The queston s: does t exsts a model whose R s equal to [1]? The prevous queston s mportant both to nterpret [1] n an approprate way and to specfy under whch condtons [1] may be consdered as an average of the R s of j, the stocks traded n country j. For the sake of smplcty, snce n ths Appendx we only need to refer to a sngle country, we wll drop the subscrpt j. Let us rewrte [1] as follows R σ [A.1] σ = R = σ σ We propose an nterpretaton of [A.1] usng the Chsn s approach to compute a mean (see, e.g., Grazan and Veronese, 009). The ntuton behnd ths approach may be explaned as follows. Consder the varables Y, X 1, X,, X h,, X k and the correspondng sample set y, x 1, x,, x h,, x k for =1,,,n. Suppose that a functon f( ) exsts such that Y = f(x 1, X,, X h,, X k ). Because of these assumptons we may wrte K K (,,,,, ) y = f x x x x 1 h k 16

18 Consder the varable X h, for h=1,,k and let x h be a scalar whch maps {x 1h,x h,,x nh } R 1. For example, but ths not the only opton, x h may be the average of X h. If (,, K,, K, ) = (,, K,, K, ) f x x x x f x x x x 1 h k 1 h k [A.] holds, then x h s called the Chsn mean of the varable X h and t has the usual propertes of an average operator. Equaton [A.] s known as the nvarance requrement, because xh keeps nvarant the quantty y. Turnng back to [A.1],.e. on the R nterpreted as a prce effcency measure, the latter may be consdered as the soluton of a problem lke [A.], by supposng to keep nvarant ether a) the sum of the varances explaned by the market model estmated for all the or stocks n the country (.e. σ ), b) the sum of the overall stock return varances (.e. σ ). sng the Chsn approach, f a) holds, and observng that wrte σ = σ R, we may R Apply Chsn σ = σ R σ R = σ R R = σ σ [A.3] σ If b) holds and lettng σ =, then R σ = = = Apply Chsn σ σ σ σ R R R R σ R [A.4] whch s the harmonc mean of the R s wth weghts gven by the varances explaned by the market model 7. 7 We may equvalently refer to both market model [] or [3]. In what follows, for the sake of smplcty, we wll refer to model [] only. 17

19 Accordng to [A.3] (.e., f a) holds), a straghtforward nterpretaton of [A.1] s that we place greater weght on those R s that are assocated wth hghly volatle stocks, havng fxed the overall amount of varablty explaned by the market. By contrast, accordng to [A.4] (.e., f b) holds), we keep nvarant the overall varance of returns, weghtng ndvdual stocks on the bass of ther proporton of systematc rsk. Both approaches smplfes nto [A.1]. The man dfference s that [A.4] enttles to search for a model that, havng fxed the nformaton to be explaned,.e., σ, and keepng constant the overall varablty of the stock returns, has explaned varance equal to σ and, consequently, R equal to [A.1]. We use the Seemngly nrelated Regresson (SR) framework to fnd a model that satsfes the prevous two requrements, stressng that n the next paragraphs our am wll not be to make nference on the model parameters or on the model tself, whch are topcs already well addressed n the fnancal econometrcs lterature, but only to study under whch condtons a multvarate response model satsfes [A.1]. We wll make use of the followng notaton: a. I k s the dentty matrx of dmenson kxk; b. 1 k s the row vector wth k ones; c. r m s the (Tx1) vector of market returns (for the sake of smplcty and wthout loss of generalty, centred on zero) where r m,t for t=1,,...,t s the market return at tme t; d. r for =1,,...k s the (Tx1) vector of returns of the -th stock, where r,t s the return at tme t, for t=1,,...,t; e. R P s the Txk matrx wth the -th column equal to r ; f. μ and μ P are, respectvely, the vectors of the mean of the stock returns and the mean of the market; g. β s a (kx1) vector of parameters; h. vec( ) and are, respectvely, the vec and the Kronecker matrx operators. 18

20 Start assumng that both the varables nvolved n the computaton of the denomnator and the k models used at the numerator of [A.1] are at least crosssectonally uncorrelated. We wll come back to ths assumpton later. Condtonally to the market returns, r m, the unvarate restrcted model [] has the followng multvarate representaton Let Vec(R P ) = ( I k r m ) β + vec(e P ) [A.5] SST R P be the sum of squared of total varatons for Vec(R P ). To show that [A.5] s the model whose R s [A.1], we need to check whether T σ. Let us wrte SST R P = dev(vec(r P )) Tkμ P where, for a generc vector Q, dev(q)= Q T Q. In the country-level R the denomnator s = dev(vec(r P )) T ( μ T μ ) T σ From [A.6] and [A.7] t follows that SST R P wll equal T σ f SST R P s equal to [A.6] [A.7] T Tkμ P = ( ) T μ μ. Ths condton holds when μ =μ P =1,,...k and s usually not satsfed. For ths reason we propose to center returns on zero 8. We now need to verfy whether the varance explaned by [A.5] s equal to Let $ β be an estmate of β. The devance explaned by [A.5] s T σ. dev(( I k r m ) $ β ) = $ β T ( I k r m ) T ( I k r m ) $ β = = $ β T ( I k I k T rm r m ) β $ = T σ $ m β T β $ whch s equal to T σ only f the -th term n β $ corresponds to the one estmated on the -th stock, for =1,,,k, usng []. 8 The de-meanng procedure does not bas the estmate of R. 19

21 The parameters n [] are usually estmated by OLS. Ths mples that the varance of the resduals are supposed to be homoschedastc,.e., E[vec(E P )vec(e P ) T ] = E[ E E P ] I k := dag( Σ ) I T. If we assume, more generally, that T P E[vec(E P )vec(e P ) T ] := V [A.8] where V s of full rank, then a GLS procedure should be used (see, e.g., Greene, 003). It s well known that n GLS P $β = (( I k r m ) T (V 1 )( I k r m ) ) 1 ( I k r m ) T (V 1 )vec(r P ) whch, for a general V, does not equal the one obtaned usng OLS. Consequently, under assumpton [A.8], the R wll not equal [A.1]. Let us assume the exstence of dependence of the resduals across stocks and the absence of heteroschedastcty wthn stocks. Then, n [A.8] where V = Σ P I T Σ P s not necessarly a dagonal matrx. Estmatng β by GLS, we get [A.9] $β = (( =( T I k T I k 1 Σ I k P T 1 r m )( Σ P I T )( I k r m ) ) 1 T T 1 ( I k r m )( Σ P I T )vec(r P ) r I T r m ) 1 1 ( Σ r I T ) vec(r P ) T m T I k = ( I k ( r T r m m ) 1 T r m ) vec(r P ) = ( σ m ) 1 Cov(R P,r m ) P T m where Cov( ) s the column vector of covarances across the k stocks and the market. The devance explaned (when estmatng [11] by GLS) s then dev(( I k r m ) β $ ) = T ( σ m ) 1 Cov(R P,r m ) T Cov(R P,r m ) = 0

22 = T Cov σ ( r, r ) m m = T σ where n the last equalty we have used the fact the -th term of $ β equals to the one estmated usng [] for each stock. Summng up, under the assumptons μ = μ P and [A.9], the R of the market model [A.5] s equal to [A.1]. Lastly, the mplcaton of [A.9] manly concerns the varance-covarance of $ β, whch s equal to Var( β $ ) = (( I k r m ) T ( Σ P I T ) 1 ( I k r m ) ) 1 = Σ P ( r T r m m ) 1 Any test of the knd C $ β =h, where C s a desgn matrx wth rank(c)>1, and h s a vector of constants, nvolves the correlaton across parameters. Snce the prevous consderatons also hold for the unrestrcted model [3], testng for the sgnfcance of the delay measure ( δ ) mples testng for the jont statstcal sgnfcance of the * lagged market returns parameters n [3] whch must take nto account the dependences across parameters and then across stocks. From an nferental pont of vew, ths approach may greatly mprove the precson of a test about the * relatonshp between ( R, δ ). 1

23 REFERENCES Alves, P., Peasnell, K., Taylor, P. (006), "The R Puzzle", Workng Paper, Lancaster nversty Ashbaugh-Skafe, H., Gassen, J., LaFond, R. (006), "Does Stock Prce Synchroncty Represent Frm-Specfc Informaton? The Internatonal Evdence", Workng Paper, nversty of Wsconsn at Madson Barbers, N., Shlefer, A., Wurgler, J. (005), "Comovement", Journal of Fnancal Economcs 75, Bernard V.L. (1987), "Cross-Sectonal Dependence and Problems n Inference n Market-Based Accountng Research", Journal of Accountng Research 5, 1-48 Brennan, M.J., Jegadeesh, N., Swamnathan, B. (1993), "Investment analyss and the adjustment of stock prces to common nformaton", Revew of Fnancal Studes 6, Chan, K., Hameed, A. (006), "Stock Prce Synchroncty and Analyst Coverage n Emergng Markets", Journal of Fnancal Economcs 80, Chorda, T., Swamnathan, B. (000), "Tradng volume and cross-autocorrelatons n stock returns", Journal of Fnance 55, Dasgupta, S., Gan, J., Gao, N. (008), "Does Lower R Mean More Informatve Stock Prces? Theory and Evdence", Journal of Fnancal and Quanttatve Analyss, Forthcomng Durnev, A., Morck, R., Yeung, B., Zarown, P. (003), "Does Greater Frm-Specfc Return Varaton Mean More or Less Informed Stock Prcng?", Journal of Accountng Research 41, Fernandes, N., Ferrera, M.A. (009), "Insder Tradng Laws and Stock Prce Informatveness", Revew of Fnancal Studes, Grazan, R., Veronese, P. (009), "How to Compute a Mean? The Chsn Approach and Its Applcaton", The Amercan Statstcan 63, Greene, W. (003), Econometrc Analyss, Prentce Hall, London Grffn, J., Kelly, P., Nardar, F. (006), Measurng Short-Term Internatonal Stock Market Effcency, Workng Paper, nversty of Texas at Austn Hawawn, G. (1980), "Intertemporal cross-dependence n securtes daly returns and the short-run ntervalng effect on systematc rsk", Journal of Fnancal and Quanttatve Analyss 15, Hou, K., Moskowtz, T. (005), "Market Frctons, Prce Delay, and the Cross- Secton of Expected Returns", Revew of Fnancal Studes 18, Jn, L., Myers, S. (006), "Return Synchroncty Around the World: New Theory and New Tests", Journal of Fnancal Economcs 79, 57-9

24 Kelly, P. (007), "Informaton Effcency and Frm-Specfc Return Varaton", nversty of South Florda La, S., Ng, L., Zhang, B. (009), "Informed Tradng Around The World", Sngapore Management nversty Mech, T.S. (1993), "Portfolo Return Autocorrelaton", Journal of Fnancal Economcs 34, Morck, R., Yeung, B., Yu, W. (000), "The nformaton content of stock markets: why do emergng markets have synchronous stock prce movements", Journal of Fnancal Economcs 58, Pagano, M., Schwartz, R. (003), "A closng call's mpact on market qualty at Euronext Pars", Journal of Fnancal Economcs 68, Potrosk, J., Roulstone, D. (004), "The Influence of Analysts, Insttutonal Investors and Insders on the Incorporaton of Market, Industry and Frm- Specfc Informaton nto Stock Prces", Accountng Revew 79, Roll, R. (1988), "R ", Journal of Fnance 43, Schwartz, R., Whtcomb, D. (1977), "The tme-varance relatonshp: Evdence on autocorrelaton n common stock returns", Journal of Fnance 3, Wurgler, J. (000), "Fnancal markets and the allocaton of captal", Journal of Fnancal Economcs 58,

25 Fgure 1 The relatonshp between R restrcted ( R ) and unrestrcted ( R ) Panel A. nnormalzed ( y = R ) R R Panel B. Normalzed ( y = R ) R R / R R 4

26 Fgure R tme seres dynamcs by quntle of market captalzaton Panel A. NYSE R_Q1 R_Q R_Q3 R_Q4 R_Q5 Panel B. NASDAQ R_Q1 R_Q R_Q3 R_Q4 R_Q5 Panel C. Chna 5

27 R_Q1 R_Q R_Q3 R_Q4 R_Q5 Panel D. Poland R_Q1 R_Q R_Q3 R_Q4 R_Q5 6

28 Fgure 3 Delay tme seres dynamcs by quntle of market captalzaton 0.18 Panel A. NYSE Delay_Q1 Delay_Q Delay_Q3 Delay_Q4 Delay_Q5 Panel B. NASDAQ Delay_Q1 Delay_Q Delay_Q3 Delay_Q4 Delay_Q5 Panel C. Chna 7

29 Delay_Q1 Delay_Q Delay_Q3 Delay_Q4 Delay_Q5 Panel D. Poland Delay_Q1 Delay_Q Delay_Q3 Delay_Q4 Delay_Q5 8

30 Fgure 4 Tme seres dynamcs of 5-week rollng correlaton between R and delay Panel A. NYSE /9/1995 1/9/1997 1/9/1999 1/9/001 1/9/003 1/9/005 1/9/ Rollng Correlaton.5% Lower Lmt Confdence Interval 97.5% pper Lmt Confdence Interval Panel B. NASDAQ /9/1995 1/9/1997 1/9/1999 1/9/001 1/9/003 1/9/005 1/9/ Rollng Correlaton.5% Lower Lmt Confdence Interval 97.5% pper Lmt Confdence Interval 9

31 Panel C. Chna /9/1995 1/9/1997 1/9/1999 1/9/001 1/9/003 1/9/005 1/9/ Rollng Correlaton.5% Lower Lmt Confdence Interval 97.5% pper Lmt Confdence Interval Panel D. Poland /9/1995 1/9/1997 1/9/1999 1/9/001 1/9/003 1/9/005 1/9/ Rollng Correlaton.5% Lower Lmt Confdence Interval 97.5% pper Lmt Confdence Interval 30

32 Fgure 5 Tme seres dynamcs of 5-week rollng correlaton between R and delay for three sze-sorted portfolos Panel A. NYSE /9/1995 1/9/1997 1/9/1999 1/9/001 1/9/003 1/9/005 1/9/ Corr_Q1 Corr_Q3 Corr_Q5 Panel B. NASDAQ /9/1995 1/9/1997 1/9/1999 1/9/001 1/9/003 1/9/005 1/9/ Corr_Q1 Corr_Q3 Corr_Q5 31

33 Panel C. Chna /9/1995 1/9/1997 1/9/1999 1/9/001 1/9/003 1/9/005 1/9/ Corr_Q1 Corr_Q3 Corr_Q5 Panel D. Poland /9/1995 1/9/1997 1/9/1999 1/9/001 1/9/003 1/9/005 1/9/ Corr_Q1 Corr_Q3 Corr_Q5 3

34 Table 1 - Sample Descrptve Statstcs Ths table reports, for each market separately, the number of stocks and the average market captalzaton (n mllon SD) of the stocks ncluded n the full sample as well as n 5 sze-sorted portfolos. Sze-sorted portfolos have been constructed on the bass of quntles of market captalzaton. nted States - NYSE nted States - NASDAQ # of Full Sze-Sorted Portfolos # of Full Sze-Sorted Portfolos Year stocks Sample stocks Sample ,434 1, , , , ,074, , , , ,918, ,3.8 11, , , ,99 3, , , , , ,858 3, , , , , ,674 4, , , ,774 1, , ,55 4, ,63. 19,7.3 3, , ,543 3, , ,13.4 3, , ,610 3, , , , , ,571 4, , ,847. 3, ,5. 005,536 5, , ,71. 3,167 1, , ,443 5, ,166.4, , ,13 1, , ,317 6, , ,56.8 8,515. 3,084 1, , ,199 5, ,18.6, ,901.,931 1, ,

35 Table 1 (contnued) Chna # of Full Sze-Sorted Portfolos # of Full Sze-Sorted Portfolos Year stocks Sample stocks Sample , , , , , , , , , , , , , , , , , ,563 1, , , ,669 1, , ,339.7 Poland 34

36 Table - R and Delay by Market Captalzaton Ths table reports, for each market separately, mean (Mean ) and medan (Medan )valuesofr and delay for the stocks ncluded n the full sample as well as n 5 sze-sorted portfolos. Correlaton s the average correlaton across stocks estmated for the full sample and the 5 sze-sorted portfolos. Delay Correlaton Delay Mean Medan Mean Medan Mean Medan Mean Medan R nted States - NYSE nted States - NASDAQ R Correlaton Full Sample Sze-Sorted Portfolos 1 (Smallest) (Largest) Chna Poland R Delay Correlaton R Delay Mean Medan Mean Medan Mean Medan Mean Medan Correlaton Full Sample Sze-Sorted Portfolos 1 (Smallest) (Largest)

37 Table 3 - R and Delay by Year Ths table reports, for each market and each year separately, mean (Mean ) and medan (Medan )valuesofr and delay for the stocks ncluded n the full sample. Correlaton s the average correlaton across stocks estmated for the full sample. nted States - NYSE nted States - NASDAQ R Delay Correlaton R Delay Mean Medan Mean Medan Mean Medan Mean Medan Correlaton