An Econometric Analysis of Volatility Discovery

Abstract We investigate information processing in the stochastic process driving stock’s volatility (volatility discovery). We apply fractionally cointegration techniques to decompose the estimates of the market-specific integrated variances into an estimate of the common integrated variance of the efficient price and a transitory component. The market weights on the common integrated variance of the efficient price are the volatility discovery measures. We relate the volatility discovery measure to the price discovery framework and formally show their roles on the identification of the integrated variance of the efficient price. We establish the limiting distribution of the volatility discovery measures by resorting to both long span and in-fill asymptotics. The empirical application is in line with our theoretical results, as it reveals that trading venues incorporate new information into the stochastic volatility process in an individual manner and that the volatility discovery analysis identifies a distinct information process than that based on the price discovery analysis.


Introduction
The proliferation of trading venues within many developed security markets has been a new trend in the financial equity market.The U.S., for instance, features 11 exchanges, 85 alternative trading systems (ATS) and more than 200 dealers. 1he market fragmentation phenomenon experienced in the U.S. highlights the importance of the microstructure of trading venues, as traditional listing exchanges have created markets within markets, changed markets' pricing structures, and set up specialized microstructures to attract specific trading clienteles (Menkveld, 2014;O'Hara, 2015;Menkveld, 2016).Retail brokers have responded to this new market setting by routing their orders to multiple trading platforms (Battalio, Corwin, and Jennings, 2016).Consequently, the way traders and markets learn and disseminate information is also dispersed across markets (O'Hara, 2015).
In this context of a decentralized system in which there are multiple prices for one homogeneous security, the study of information processing among trading venues becomes highly complex and, at the same time, of great significance for both market participants and regulators.In this article, we take a different angle to the price discovery literature and investigate information processing in the stochastic process driving stock's volatility. 2 The economic reasoning of the volatility discovery framework hinges on the fact that economic agents do not process volatility related news instantaneously and on the relevance of the informational content driving the volatility process.As for the former, recent evidence in Lochstoer and Muir (2022) shows that financial agents have sticky expectations about volatility that generates initial underreaction and delayed overreaction to volatility news.As for the volatility informational content, French and Roll (1986) document that volatility is mainly caused by private information which affects prices when informed investors trade.Differently from lagged returns, the volatility process contains important information that produces some degree of predictability in future returns (see, for instance, Ng and Ludvigson 2007;Bollerslev, Tauchen, and Zhou 2009).More specifically, Bollerslev et al. (2013) find that the implied and realized variation fractionally cointegrate and this long-run equilibrium relationship predicts returns over interdaily horizons.Finally, the information contained in the volatility process has compelled traders to use derivative markets to trade volatility (Ni, Pan, and Poteshman, 2008).All in all, the volatility process reveals information that is not present in lagged returns and hence it is not captured by the price discovery framework.This means that the volatility discovery analysis can be a valuable tool to identify the source (what market) of this information.
The price discovery literature has identified the unobserved efficient price by extracting commonalities within the transaction prices of a homogeneous security.In this context, prices are cointegrated, and the vector error correction (VEC) model has become the workhorse of price discovery analyses.We rely on a similar intuition and exploit the evidence that estimates of the integrated variance (realized measures) are long memory processes and cointegrated in order to identify the latent integrated variance of the efficient price using the fractionally cointegrated vector autoregressive (FCVAR) model of Johansen and Nielsen (2012). 3We then investigate how different markets impound information into the integrated variance of the efficient price.
We make the following methodological and theoretical contributions: First, we build upon the continuous-time theory and characterize the volatility discovery phenomenon, leading to the derivation of the volatility discovery measures.Second, we show that recovering the efficient price by means of a permanent and transitory decomposition (building block of any price discovery measure) generally does not suffice to identify the integrated variance of the efficient price.Third, we resolve the issue of the generated regressor problem in the FCVAR framework when using realized measures.We employ double asymptotics to establish the limiting distribution of the volatility discovery measures, enabling correct inferences on the volatility discovery process.
Our empirical application investigates the volatility discovery mechanism for 30 of the most actively traded stocks in the U.S over a large sample period of 7 years.The tick-by-tick quotes consists of firms from different industries that are traded in three trading venues.We document that markets indeed trade toward the efficient integrated variance, that is, market specific integrated variances are fractionally cointegrated.Our results reveal that trading venues incorporate new information into the stochastic volatility process in an individual manner.In line with our theoretical results, the price discovery analysis identifies distinct market leaders than those based on the volatility discovery measures.
Concurrently with this work, Dimpfl and Elshiaty (2021) reformulate our model to discrete time and study volatility discovery in cryptocurrencies markets.More specifically, they employ a stochastic volatility model fitted to daily returns to obtain daily volatility estimates and compute the volatility discovery measures using the FCVAR model.Forte and Lovreta (2019) also apply FCVAR model.They find that the CDS implied equity volatility and a equity options based index (VSTOXX) are fractionally cointegrated.Finally, Baule, Frijns, and Tieves (2017) deviate from our approach and adapt Hasbrouck's (1995) information share measure to stationary processes to study volatility discovery for options and bank-issued options (warrants).
The remainder of the article proceeds as follows.Section 2 introduces the volatility discovery framework.Next, Section 3 shows how to identify the integrated variance of the efficient price as a function of both price and volatility discovery measures.Section 4 derives the asymptotic properties of the volatility discovery measures and Section 5 presents an empirical application.Section 6 offers concluding remarks.The supplementary material presents the proofs, a simulation exercise, and additional empirical results.

The Volatility Discovery Framework
Asset price theory postulates that the efficient log-price of a financial security at time t must follow a semimartingale process defined in some filtered probability space ( , F, P) (see, among others, Andersen et al. 2003;Barndorff-Nielsen et al. 2011).Additionally, econometrics and finance literature have widely considered the volatility process of the efficient price a stochastic process, such that the efficient price can be written as where a is a predictable locally bounded drift, σ m is a càdlàg volatility process denoted as the efficient stochastic volatility process, and, without loss of generality, W is a Brownian motion.
It is important to note that m t is driven by two separate stochastic processes: W and σ m , which reflect new information arrival to the price and volatility processes, respectively.4Furthermore, the efficient price is considered a latent process not only due to trading frictions but also as a result of market fragmentation.
Our framework consists of investigating a homogeneous asset traded in S different markets.In general terms, the S-dimensional log-price process is modeled as a Brownian semimartingale defined in some filtered probability space ( , F, P).The S-dimensional daily log-price process then reads where a is a predictable locally bounded drift, is a càdlàg volatility matrix with = denoting the càdlàg spot covariance matrix, and W is a S-dimensional vector of Brownian motions.The quadratic variation of P is defined as where P t , P t t−1 : t = plim n−→∞ n i=1 P t i − P t i−1 P t i − P t i−1 with n denoting the number of intraday observations such that t − 1 = t 0 < t 1 . . .< t n = t and T being the total number of trading days, such that t = 1, . . ., T. Because the price process in (2) does not consider jumps, the diagonal elements of P t , P t t−1 : t are the market-specific integrated variances (IV).The IV is central to the pricing of financial instruments, portfolio allocation, and risk management.In a fragmented market context, not only market prices but also their stochastic volatilities are expected to gather information at different speeds, following market characteristics such as market design, trading costs, trading clientele, liquidity, and the presence of informed traders.
The price discovery literature typically adopts the VEC model to approximate the dynamics of (2), identify the single common stochastic trend seen as the efficient price, and obtain consistent estimates of the price discovery measures.Among the most popular price discovery measures are Booth, So, and Tseh's (1999) component share measure and variants of Hasbrouck's (1995) information share (IS) measure. 5o investigate the dynamics of the market-specific IVs, we first acknowledge the fact that these IVs are tied to a longrun equilibrium and track a common latent stochastic process seen as the integrated variance of the efficient price.Second, we need to define feasible estimates of market-specific IVs originated from (3).We generically denote these estimates as realized measures (RM), which are consistent estimators computed with ultra-high-frequency data.There are several consistent estimators for the IV (see Mcaleer and Medeiros (2008) for a comprehensive survey of realized measures estimators), thus, at this point, we do not distinguish among these different estimators, apart from requiring them to be consistent estimates of the market-specific IVs.Third, as opposed to security prices, which are known to be integrated of order one, I (1), processes, there is well-documented evidence that RMs are persistent and characterized by a long memory.The long memory feature of RMs follows from early work on the conditional volatility of daily financial returns (Baillie, Bollerslev, and Mikkelsen 1996) and on modeling RMs (Andersen and Bollerslev 1997;Andersen et al. 2003;Corsi 2009).In particular, Andersen et al. (2003) note that The slow hyperbolic autocorrelation decay symptomatic of long memory is evident ....This stylized fact of RMs implies that these series are fractionally integrated of some order d.A process z t is said to be fractionally integrated of order d if (1 − L) d z t = η t , where η t is an integrated of order zero (stationary) process.It follows that d := (1 − L) d is the fractional difference operator defined in terms of the binomial expansion One way to accommodate these characteristics is to approximate the daily dynamics of RMs using the fractionally cointegrated vector autoregressive (FCVAR) model of Johansen (2008) and Johansen and Nielsen (2012), ) is a S × 1 vector that concatenates the element-wise logarithmic of the daily marketspecific RMs, R t is a S × S consistent estimator of b is the usual lag operator for fractional processes, d and b are the fractional order of the RMs and the degree of fractional cointegration, respectively, and ε t is iid with a mean zero and positive-definite covariance matrix .The parameters in the FCVAR framework have the same common interpretation as the parameters in the standard VEC model.In that, α corresponds to the speed of adjustment and β is the cointegrating vector so that β R t is integrated of order d − b and represents the long-run equilibrium relation.It is relevant to note two important aspects of the approximation in (4).First, because the RMs are not expected to diverge over time, the rank of αβ should equal S − 1 in that the RM's share a single common fractional stochastic trend.We see this stochastic trend as an estimate of the logarithm of the IV of the efficient price.Second, we expect deviations between the RMs in the different markets to be transient, that is, a short memory (covariance stationary) process driven by the trading-venue-specific characteristics, such as cost structure, different degrees of transparency, the speed of order execution and different trader groups, among other features.This stylized fact is accommodated by assuming d = b in (4), which implies that β R t is an I (0) process.Finally, we allow the RMs to present a fractional order in the range of 0.5 ≤ d < 1, implying that these measures may be nonstationary but mean reverting. 6It is possible to extend the FCVAR approximation in (4) to the estimates of the market-specific spot volatilities at a cost of a larger measurement error on their estimates.Nevertheless, our simulation study shows that fitting a FCVAR model to the daily RMs still allows us to capture the partial adjustment dynamics that occurs on the spot volatilities. 7ultivariate random walk decompositions are central to constructing any price discovery measure, as they decompose observed prices into two components: the I (1) efficient pricethe stochastic trend-and an I (0) process that is associated with the portion of information that has no permanent impact on prices.In the same way that the random walk decomposition constitutes the basic building block of price discovery analysis, the fractional counterpart of the Granger representation theorem serves as a building block for volatility discovery analysis.By decomposing the RMs into a long memory term common to all markets and market-specific I (0) terms, we can identify the efficient price's IV and investigate how different venues incorporate information and adjust to the long-run equilibrium.Johansen (2008) and Johansen and Nielsen (2012) provide the fractional counterpart of the Granger representation theorem and decompose the process R t into I (d) and I (0) components: where + is a truncated version of the fractional difference operator of order d; X t is an I (0) stochastic term; and α ⊥ and β ⊥ are the orthogonal complements of α and β, respectively, with α α ⊥ = 0, α ⊥ ι S = 1, β β ⊥ = 0, and ι S denoting a S × 1 vector of ones.Similarly to the price discovery case, if β = (I S−1 , −ι S−1 ) and rk αβ = S − 1, the matrix has common rows.Section 5.1 provides strong empirical support for the cointegrating vector akin to β = (I S−1 , −ι S−1 ) in the FCVAR setup.8Hence, a similar interpretation as that for the price discovery analysis holds in the volatility discovery setup, implying that R t shares a single fractional stochastic trend that is integrated of order d, given by (6) where ψ accounts for the common row of .Notably, R m,t possesses the long-term persistence and slow hyperbolic decay discussed in Andersen et al. (2003), and it can be seen as the natural estimate of the logarithm of the efficient price's IV.Given that R m,t can be recovered by ψ −d + ε t , we can investigate how it is tied to innovations to the RMs of the different markets.
In view of the Granger representation in (6), is essentially a normalization factor, meaning that α ⊥ plays the key role on identifying the fractional stochastic trend.In turn, we define α ⊥ as a measure of volatility discovery.The elements of α ⊥ show how innovations from the logarithm of the market-specific RMs contribute to R m,t .As the element of α ⊥ for a given market increases, the importance of this market for the volatility discovery process increases.
The economic interpretation of α ⊥ is therefore given by its ability to identify where (in what markets) the informational content driving the efficient volatility is revealed.In the context of highly fragmented markets, α ⊥ has the potential to guide what markets should be used when computing the realized measures, meaning that the markets with the highest volatility discovery measures should be preferred to markets with low importance to the volatility discovery mechanism.Moreover, similar to the price discovery analysis, the volatility discovery framework can be a valuable component of a comprehensive toolbox for assessing market quality.By combining the insights from volatility discovery measures with other relevant indicators, we can gain a deeper understanding of market dynamics, information efficiency, and overall market quality.Because the information content of the volatility process has predictive power for future returns and volatility (see, for instance, Patton and Sheppard 2015; Baltussen, van Bekkum, and van der Grient 2018), knowing the source of this information has the potential to improve derivatives pricing and out-of-the sample forecasts.

Identification of the Efficient Price's IV
The discussion posed in the previous section relies on discretetime approximations to introduce the volatility discovery framework.This section assumes a parametric structure to (2) and to the daily dynamics of its quadratic variation, such that we can work out the identification of the IV of the efficient price.Next, we provide a theoretical bridge between the price and volatility discovery measures and disentangle the role played by them on impounding information to the IV of the efficient price.The proofs for the theoretical results are available on the supplementary material.
Our first goal is to write the IV of the efficient price as a function of both price and volatility discovery measures.To this extent, we need to specify a data-generating process for the general market-specific price process in (2) and the daily dynamics for the market-specific IVs.The price process is such that the drift parameter has reduced rank, that is, prices cointegrate, and the diffusion matrix is stochastic.Assumption 1 provides the technical conditions.
Assumption 1.The log-prices on any given day of an asset that trades on S multiple venues follow the S-dimensional Brownian semimartingale where γ and δ have dimensions 1 with all eigenvalues having negative real parts, meaning that the eigenvalues of γ δ lie in the left open half-plane of the complex plane and at zero, and t = t t is a càdlàg process with [ t ] s,s > 0 for all s, s = 1, . . .S, where [.] s,s denotes the (s, s )th element of a matrix.
The process in ( 7) is Markovian and the discrete-time parameters obtained from its discretization coincide with those from the reduced-rank Gaussian Ornstein-Uhlenbeck (OU) process in Kessler and Rahbek (2004).In that, Kessler and Rahbek's (2004) Theorem 1 holds automatically, meaning that the subspace in R S−1 spanned by the columns of the discrete-time speed-of-adjustment parameters is the same of those spanned by γ .
Next, we establish a multivariate random walk decomposition of the process in (7).To decompose (7) into longand short-run components, we employ the identity where γ ⊥ and δ ⊥ are the orthogonal complements of γ and δ, respectively.The first and second terms in this identity are orthogonal and hence satisfy the Gonzalo and Granger's (1995) definition of permanent and transitory components.More specifically, γ ⊥ δ ⊥ −1 γ ⊥ P t returns the stochastic trend common to all markets, that is, a common Brownian semimartingale with initial value given by γ ⊥ δ −1 γ ⊥ P 0 , whereas δ P t is an I(0) component that admits a continuous-time Wold representation.Proposition 1 formalizes the results.
Proposition 1.Let Assumption 1 holds.The permanent and transitory (P-T) decomposition then reads where f t = γ ⊥ t t 0 u dW u + P 0 is the stochastic trend and δ P t = Z t .
The P-T decomposition generalizes the results in Kessler and Rahbek (2001) to the case of a stochastic covariance matrix.The common stochastic trend, f t , driving the price process is a Brownian semi-martingale and γ ⊥ is used as a price discovery measure (the component share measure).Furthermore, according to Proposition 1, the IV of f t is given by Next, we investigate the relationship between the quadratic variation of P t and IV f ,t .To this extent, we start by writing the daily quadratic variation of P t in (7) as9 In the fragmented markets setting and in the absence of jumps in (7), the diagonal elements of ( 9) are the marketspecific IVs, which we denote as IV t = (IV 1,t , . . ., IV S,t ) = diag( P t , P t t−1:t ).The empirical estimators of the quadratic variation usually adopted in the realized measures literature, for example, the realized variance estimator, are consistent estimators of P t , P t t−1:t .Therefore, in the context of an homogeneous asset traded on multiple markets, the RMs estimate the market-specific IVs (and covariances) rather than the IV of the efficient price.The price discovery literature advocates using f t from the P-T decomposition to identify the IV of the efficient price.However, this identification strategy has an important caveat: IV f ,t only takes into consideration the weights given by γ ⊥ and hence provides no information about any potential learning and temporal feedback mechanisms embedded into t in Assumption 1.More concretely, in order to identify the partial adjustments dynamics in the market-specific IVs, it is necessary to specify a parametric data-generating process for these quantities.
As briefly discussed in Section 2, much evidence from both the empirical and theoretical realized measures literature point toward the importance of taking into account the high degree of persistence and the long memory property in the integrated variance and realized measures (see, among others, Andersen et al. 2003;Corsi 2009;Bollerslev, Patton, and Quaedvlieg 2016;Shi and Yu 2022).When fitting ARFIMA models to the RMs, statistically significant long memory parameter estimates typically fall within the range of 0 < d < 1.Consequently, RMs can exhibit either stationary or mean-reverting nonstationary processes (Rossi andde Magistris, 2014 andShi andYu, 2022).Alternatively, the long memory feature of RMs can be addressed using Corsi's (2009) heterogeneous autoregressive (HAR) model.The HAR family of models are based on a restricted autoregressive (AR) model that captures the persistence of the RMs by considering the RM in the previous day, its average over the previous week, and its average over 22 days.Baillie et al. (2019) reconcile both approaches and estimate an ARFIMA model with the restrictions implied by the HAR model.Their findings demonstrate the benefits of incorporating the long memory parameter into the HAR model to explain higher-order dynamics.To capture the long memory feature of the IV of the efficient price as well as the fact that market-specific IVs do not diverge and are tied to a long-run equilibrium, we assume the dynamics of the daily market-specific logarithmic IVs follow a FCVAR process.Assumption 2 formalizes their data-generating process.
Assumption 2. 1.The daily logarithmic IVs of a given asset that trades on multiple venues follow a FCVAR model as where 0 Assumption 2 ensures that the market-specific IVs cointegrate and share a single common fractional stochastic trend with long memory order d.This common factor corresponds to the logarithm of the IV of the efficient price and reads Alternatively, we could reformulate Assumption (2) in terms of primitive assumptions regarding both the efficient spot volatility and market-specific and efficient price IVs.Specifically, the fractional Ornstein-Uhlenbeck process of Comte and Renault (1998) models the logarithmic of the spot volatility as a long memory process of order d.Consequently, the logarithmic of the spot volatility associated with the efficient price reads where is the drift parameter, ϕ is the volatility (volatility of volatility) parameter, and W d m is a fractional Brownian motion (fBm). 11It is important to note that the fBm process in (12) does not imply an arbitrage opportunity because the price process in Assumption 1 is a semi-martingale.The logarithmic of the market-specific spot volatility in (12) inherits the long memory property from the fBm process, as its (pseudo) spectral density has a pole in zero that is a function of d: f σ m (x) ∼ cx −2d as x → 0, where c ∈ R + and 0 ≤ d < 1. Importantly, Proposition 2.2 in Comte and Renault (1998) show that log(σ m ), W d m , and σ m share the same long memory order d.Additionally, Proposition 1 in Rossi and de Magistris (2014) establishes that the integrated variance also possesses the same long memory degree du is a long memory process of order d.This long memory feature of t t−1 σ 2 m,u du is equivalent to the fractionally integrated property of the IV of the efficient price implied by Assumption 2.Moreover, because the market-specific integrated variances are tied to a long run equilibrium they can then be expressed as a function of IV m,t and an I(0) zero-mean covariance stationary process X t , such that IV s,t = IV m,t + X s,t with s = 1, . . .S. This specification establishes a connection with two important conditions implied by Assumption 2. First, it satisfies the definition of fractional cointegration with β = (I S−1 , −ι S−1 ), that is, IV s,t and IV s ,t , s, s = 1, . . ., S and s = s , are I(d) processes and IV s,t − IV s ,t is an I(0) process.Second, it is consistent with Johansen and Nielsen's (2012) Granger representation theorem as implied by Assumption 2, as IV s,t and IV s ,t share IV m,t as the common fractional stochastic trend.Finally, the main advantage of employing Assumption 2 over the set of primitive assumptions discussed earlier is that it allows for the identification of IV m,t using d and α (and consequently α ⊥ ).Additionally, it does not require any constraints on the parameters to ensure that estimates of IV m,t are positive, as the FCVAR is defined in terms of the logarithm of the market-specific IVs.Next, we characterize the IV of the efficient price as a function of the price and volatility discovery measures, that is γ ⊥ and α ⊥ , respectively.
11 The fBm is defined as 0 , where is the Gamma function.The fBm process can also be written in terms of the Hurst index H rather than d, such that d = H − 1/2.Comte, Coutin, and Renault (2012) proposes an affine class of long memory volatility processes that involve fractional integration of the square root of the spot volatility process, rather than its logarithm as in (12).Additionally, the fractional OU in ( 12) can be written in terms of log(σ 2 m,t ) (Rossi and de Magistris 2014).
Theorem 1.Let Assumptions 1 and 2 hold.Let = γ δ γ −1 δ , k(s) be a S × 1 vector that has its sth row equal to one and all the remaining entries equal to zero, K = (vech (K 1 ) , . . ., vech (K S )) be a S × 1 2 S(S + 1) matrix with K s denoting a S × S matrix that has the (s, s)th entry equal to one and all the remaining entries equal to zero, L be a 1 2 S(S + 1) × S 2 elimination matrix such that, for any symmetric matrix A, vech (A) = Lvec (A), and log(.)be the element-wise logarithm.Then, Theorem 1.1 carries two important implications.First, it shows how the price and volatility discovery mechanisms emerge as different sources on impounding information to the IV of the efficient price.Unlike IV f ,t , IV m,t fully encompasses the informational content and error correction mechanisms presented in t through α ⊥ .Consistent with empirical evidence that demonstrate investor trade based on volatility information (e.g., Ni, Pan, and Poteshman 2008), incorporating IV m,t instead of IV f ,t is expected to enhance trading decisions, while leveraging α ⊥ identifies what are the sources of information impacting IV m,t .Second, the identification of the IV of the efficient price using f t critically hinges on the condition γ ⊥ = α ⊥ = k(s) with s ∈ [1, . . ., S] (Theorem 1.1).Failure to satisfy this condition renders any price discovery measure relying on estimates of IV f ,t inconsistent.This has significant implications for any variation of Hasbrouck's (1995) IS measure, as such measures give the relative contribution of each market to the IV of the efficient price.
To further appreciate the limitation of using IV f ,t instead of IV m,t , we shall discuss the case in which the restriction γ ⊥ = α ⊥ = k(s) or γ ⊥ = α ⊥ with s ∈ [1, . . ., S] with s ∈ [1, . . ., S] holds, that is, Theorem 1.1.In the very special case of α ⊥ = γ ⊥ = k(s) with s ∈ [1, . . ., S], a single market is fully responsible for impounding information to both P t and IV m,t , meaning that the remaining exchanges are irrelevant with respect to information processing.Such scenario appears to be rather unlikely to hold in practise, as trading platforms fiercely compete for order flow and high-frequency traders operate across markets (see, for instance, O'Hara 2015).Second, the two last terms on the righthand side of ( 13) would collapse to zero if the diffusion matrix in ( 7) is constant, which rules out the presence of stochastic volatility altogether.In such case, there is neither intra-nor interdaily learning with respect to the volatility processes across the different trading platforms.
Another possible scenario consists of examining the case where price and volatility discovery processes do not happen exclusively in one market, but γ ⊥ α = 0 holds.In such case, Theorem 1.1 shows that log IV m,t remains a function of both price and volatility discovery measures.More specifically, the cross-quadratic variation is zero and α ⊥ in (15) shows how log IV m,t is related to elements of the daily quadratic variation of the transitory component in Proposition 1.

Estimation and Inference
We now turn our attention to the estimation and inference tools for the volatility discovery measures.We extend the Johansen and Nielsen's (2012) results on consistency and the asymptotic normality of the maximum likelihood (MLE) estimator to the context of double asymptotics (mixed case), that is, where both long span and infill asymptotics hold.This approach is necessary because we need to replace the latent market-specific IVs by their consistent intraday-based estimates in the FCVAR models.Because consistency of these realized measures are established for n → ∞, consistency and asymptotic normality of the FCVAR-based volatility discovery measures require both n, T → ∞.This approach is consistent with previous work on estimating diffusion processes based on discretely observed data in which the asymptotic properties of the estimators depend on both n, T → ∞ (see, for instance, Prakasa Rao 1983;Tang and Chen 2009).Specifically, in the context of extremum estimators similar to ours, Phillips and Yu (2009) use the feasible central limit theorem for realized variances in Barndorff- Nielsen and Shephard (2002) as the first step in their two-stage estimation procedure to estimate the parameters in the drift function of a diffusion process.
To obtain the log-likelihood function of the FCVAR model in (10), first note that Assumption 2.2 does not imply that the errors are Gaussian.This means that the estimation strategy based on the Gaussian likelihood in Johansen and Nielsen (2012) can be interpreted in the context of quasi MLE estimation.In addition to Assumption 2, we need Assumption 3 to ensure the limiting behavior of the likelihood function for the FCVAR model.12Assumption 3. The initial values IV −t , are uniformly bounded with IV −t = 0 for 0 ≤ t ≤ T 0 and IV −t = 0 for t > T 0 .
Assumption 3 concerns the initial values in which the likelihood function is conditioned on and essentially determines that d IV t is well-defined for any 0 ≤ d ≤ 1.Given Assumptions 2.1, 2.2, and 3, we follow Johansen and Nielsen (2012) and define the Gaussian likelihood function conditional on IV −t , 0 ≤ t ≤ T 0 and T 0 < T: where λ = ( d, vec( α) , vec( β) , vech( ) ) is a 1 + 2S(S − 1) + S(S+1)/2×1 vector of free parameters and ε t = d log (IV t )− α β L d log (IV t ) are the corresponding residuals. 13he latent nature of the IVs means that we cannot compute the MLE estimator directly from ( 16).Therefore, we replace log(IV t ) in ( 16) by their consistent high-frequency based estimates R t .Assume prices are observed at regular spaced intervals, such that there is n intraday prices in a given day.Define R t = n−1 i=1 P t i+1 − P t i P t i+1 − P t i as the realized covariance estimator of ) and log(.)denotes the element-wise logarithm.The feasible conditional log-likelihood function then reads where Note that the notation in (17) includes the number of intraday observations n, which serves to highlight the fact that R t are consistent estimates of the logarithm of the IVs as n → ∞.As a result, n,T ( λ) also depends on n.More precisely, we impose the following assumptions on the RMs and feasible conditional log-likelihood function.
where t is a positive-definite matrix (Barndorff-Nielsen and Shephard 2004).

sup λ∈ ∂ n,T λ ∂ λ
= O p (1), with λ ∈ and .denoting the Euclidean norm, and is a compact set.3. The initial values R −t are uniformly bounded with R −t = 0 for 0 ≤ t ≤ T 0 and R −t = 0 for t > T 0 .
Assumption 4.1 provides information on the rate of convergence of the RMs estimates, which is relevant for establishing the limiting distribution of α ⊥ in the mixed asymptotics case.The limiting result in Barndorff- Nielsen and Shephard (2004) is obtained under the assumption of "no-leverage, " which corresponds to the case where t is random but independent of the Brownian motion in (7) and achieves the optimal convergence rate.Assumption 4.2 imposes a Lipschitz type smoothness condition to the feasible conditional log-likelihood function.Assumption 4.3 relates to the initial values and it is analogous to Assumption 3.
The estimator we consider is in the class of extremum estimators and hence it is necessary to show that feasible and infeasible conditional log-likelihood functions and their gradients converge uniformly in probability as n, T → ∞.It then follows that, for a fixed d, the conditional MLE estimator based on (17) reduces to a reduced rank regression similar to the standard VEC case.The likelihood function can be simplified by concentrating out the set of parameters vec(α) , vec(β) , vech( ) ) .This allows us to estimate d through the numerical optimization of the profile likelihood function.As a result, λ is the MLE estimator of λ.The limiting distribution of the MLE estimator for λ determines the asymptotic distribution underlining the LR-based cointegration rank test, which is used to determine the number of cointegrating vectors in the FCVAR model (see Johansen and Nielsen, 2012 for more details).Finally, we could use the MLE estimates of α to compute the estimates of the volatility discovery measures: α ⊥ .Theorem 2 formalizes the limiting distribution of the volatility discovery measures.To show that, define α ⊥ as α ⊥ = that ι S α ⊥ = 1 and α α ⊥ = 0 hold.Next, note that α ⊥ is a vector-valued continuously differentiable function and define ∂ λ λ= λ be the first-order derivative of n,T ( λ 1 , λ 2 ) and T ( λ 1 , λ 2 ), respectively, with respect to λ and evaluated at a vector λ that is between λ 1 and λ 2 , J = (0 S(S−1)×1 , I S(S−1) , 0 S(S−1)×S(S−1) , 0 S(S−1)×S(S+1)/2 ) be a S(S − 1) × 1 + 2S(S − 1) + S(S + 1)/2 deterministic matrix such that vec(α) = Jλ, and Theorem 2. Let Assumptions 1, 2.1, 2.2, 3, and 4 hold and suppose The limiting distribution in Theorem 2 allows us to construct single hypothesis tests on the elements of α ⊥ and to formulate a Wald test for joint zero constraints on the α ⊥ .Specifically, the Wald statistic has the usual limiting chi-squared distribution under the null hypothesis.
In the two-market setting, testing single zero restrictions on the elements of α ⊥ is equivalent to testing for weak exogeneity of one of the IVs with respect to the cointegrating vector β.This follows because the null hypothesis α ⊥,1 = 0 implies α 2 = 0 and the volatility discovery only occurs in market 2. Similarly, the null hypothesis α ⊥,2 = 0 is equivalent to test the null hypothesis α 1 = 0 and implies that volatility discovery happens exclusively in market 1.The rejection of both individual null hypotheses (α ⊥,1 = 0 and α ⊥,2 = 0) suggests that volatility discovery occurs in the two markets.Finally, because Assumption 2 implies a FCVAR with zero autoregressive components, testing for weak exogeneity implies testing for Granger causality (strong exogeneity).
In the general S-dimensional market setting, testing multiple joint S − 1 zero constraints on the elements of α ⊥ allows us to examine whether a single market is solely responsible for impounding information into the efficient IV.For instance, if S = 3, rejecting both null hypotheses α ⊥,1 = α ⊥,2 = 0 and α ⊥,1 = α ⊥,3 = 0, while failing to reject the null hypothesis α ⊥,2 = α ⊥,3 = 0 imply finding statistical evidence that market 1 is the single driver for the volatility discovery process.However, this procedure neglects the multiple testing issue to correct for Type I error.We address this issue in our empirical application by applying the Holm-Bonferroni procedure.This procedure involves sorting the multiple tests p-values in ascending order and sequentially comparing them against the significant level scaled by a function of the number of tests.

Empirical Application
Our data consist of 30 of the most actively traded assets in the U.S. extracted from the TAQ database.We consider the two major listing exchanges in the U.S.-Nyse (N) and Nasdaq (T)and one of the most liquid trading venues, Arca (P).This sample selection reflects a balance between capturing relevant market information and avoiding estimation difficulties associated with a large number of markets.All stocks are simultaneously traded in at least two of the three markets covered in this analysis and represent a broad set of industries.We use tick-by-tick quotes from a sample period of 7 years, from January 2007 to December 2013, which captures the implementation of the National Market System regulation (Reg NMS) and the market fragmentation phenomenon.14Table S.3 in the supplementary material displays the details.We compute the daily RMs using Barndorff-Nielsen et al.'s (2011) realized kernel estimator because it provides consistent estimates of the market IV measures under time-dependent and endogenous market microstructure noise, irregularly spaced data, and jumps.These features ultimately allow us to take advantage of mid-quotes observed at a tick-bytick frequency.

Model Specification
We use the log RMs to estimate bivariate and trivariate FCVAR models. 15We consider four market combinations: Nasdaq-Arca, Nasdaq-Nyse, Arca-Nyse, and Nasdaq-Arca-Nyse.This choice accommodates assets that are not simultaneously traded on the three exchanges and ultimately yields a richer and broader sample, which is particularly important for handling firms that are listed on Nasdaq but are not traded on Nyse.We choose the lag length, κ, as the minimum value that makes the LM test for serial correlation on the residuals at the first 10 lags nonsignificant at the 5% significance level. 16o evaluate whether the RMs at different trading venues are fractionally cointegrated, we implement the sequential likelihood ratio (LR) test for the cofractional rank of Johansen and Nielsen (2012).For each two-market combination, the RMs are fractionally cointegrated and share a single common fractional stochastic trend if rk αβ = 1.As for the three-market case, if rk αβ = 2, then the RMs share a single fractional stochastic trend.Table 1 presents the p-values associated with the LR cofractional rank test. 17The null is not rejected at the 5% significance level in all but five stocks, suggesting that the RMs are fractionally cointegrated and share the IV of the efficient price as a common fractional stochastic trend.As for the long memory feature of the RMs, we find that d is statistically significant at 1% significance level and usually greater than 0.5, confirming the RMs' long memory characteristic (nonstationary and mean reverting).In line with Assumption 2.3, Table 1 provides evidence that β = (1, −1) and β = (I 2 , −ι 2 ) hold for the two-and three-market combinations respectively, indicating that the market stochastic volatilities are expected to be equal in equilibrium.To appreciate that, recall that β is not uniquely defined, in that we normalize the left-hand S − 1 × S − 1 block of β to be the identity matrix and find that the S − 1 × 1 righthand block of β equals −ι S−1 .In summary, Table 1 provides the necessary preliminary results to ensure that Assumption 2 holds and both identification of the IV m and inference on the volatility discovery measures are valid.

Volatility Adjustment
We now turn our attention to the parameter estimates that determine the volatility discovery mechanism.We focus on the estimates of volatility discovery measures α ⊥ , which is normalized such that α ⊥ ι S = 1. 18Recall that the market associated with the highest element of α ⊥ leads the volatility discovery process.In view of the Granger representation in (6) and Assumption 2, the logarithm of the integrated variance of the efficient price is given by a weighted sum of the innovations in the different market-specific integrated variances, with weights given by the elements of α ⊥ .In that, it is natural to interpret these weights as nonnegative, such that 0 ≤ α ⊥,s ≤ 1 with s = 1, . . ., S and α ⊥ ι S = 1.In finite sample analysis, it is still possible for some elements of the estimates of α ⊥ to turn out to be negative.In these cases, we should not reject the null hypothesis that these volatility discovery measures are equal to zero.
Table 2 presents the estimates of α ⊥ for each market combination, along with the significance levels (one-sided test) of the null hypothesis that a single market does not contribute to the volatil- We report results for 30 assets considering the Nasdaq and Arca, Nasdaq and Nyse, Arca and Nyse, and Nasdaq, Arca and Nyse market combinations.For each twomarkets set of results, the first column (rk αβ ) displays the number of cointegrating relations used in the estimation of the FCVAR model.Considering the two-market setting, the first column has the p-value of the likelihood-ratio (LR) cofractional rank test when the null hypothesis is rk αβ ≤ 1.The second column displays the estimates of d and the last column presents the second element of the cointegrating vector, such that β = (1, β 2 ) .As for the three-markets set of results, the first column presents the p-value of the likelihood-ratio (LR) cofractional rank test when the null hypothesis is rk αβ ≤ 2. The second column displays the estimates of d.
ity discovery process, that is, α ⊥,s = 0 with s ∈ (T, P, N). 19 For the three-market setting, we provide the corresponding p-value for the null hypotheses that two volatility discovery measures are equal to zero.These p-values are denoted as Hs, s , where s, s ∈ (T, P, N) and s = s .For example, HT, P represents the p-value associated with the null hypothesis α ⊥,T = α ⊥,P = 0. Considering the first market combination, pointwise analyses show that α ⊥,P > α ⊥,T holds for 19 of the 28 assets.This suggests that Arca is more important than Nasdaq.A more precise analysis is obtained by examining the hypothesis tests derived from Theorem 2. At the 5% significance level, we find that the volatility discovery mechanism occurs exclusively at Arca for 12 of the 28 stocks.This means that we fail to reject the null hypothesis α ⊥,T = 0, while simultaneously rejecting the null hypothesis α ⊥,P = 0. Similar analysis offers statistical evidence that the Nasdaq is the single driver for 5 assets only.In order to address the issue of multiple testing, we apply the Holm-Bonferroni correction procedure at the 5% significance level.We find that that Arca and Nasdaq are the single drivers of the volatility discovery process for 9 and 4 stocks, respectively. 20 19 Because we find κ = 0 for nearly all stocks (see Table S.4 in the supplementary material), the hypothesis tests below can be most often interpreted as Granger causality tests. 20See Table S.7 in the supplementary material for the p-values of the volatility discovery measures.
Results from the second market combination also suggest that Nasdaq is less important.We find that α ⊥,N > α ⊥,T holds for 13 of the 19 stocks, indicating that Nyse leads the volatility discovery process.At the 5% significance level, we find statistical evidence that Nyse is the unique contributor for 9 of the 19 assets, which also implies that Nasdaq does not Granger-cause the Nyse for these stocks.Considering the same significance levels, we find that Nasdaq is the sole contributor for 6 stocks only.These findings remain consistent when applying the Holm-Bonferroni correction procedure.
Regarding the Arca-Nyse market combination Arca and Nyse lead in 11 and 10 of the 21 markets, respectively.When applying the hypothesis tests, we find that Arca and Nyse individually drive volatility discovery for 8 and 9 stocks, respectively, at the 5% significance level.Controlling for multiple tests does not alter the qualitative findings.
The two-market setting shows Nasdaq as the least important exchange to the volatility discovery process, whereas Nyse appears as the most efficient one.To obtain a more precise answer to which market drives the volatility discovery process, the natural step forward in our analysis is to extend it to a threemarket setting.The final set of results in Table 2 displays the results.Pointwise analyses reveal a stronger role of Nyse when compared to the two-market setting.Nyse dominates in 12 out of the 19 stocks, while Arca and Nasdaq leads in only 5 and 2 stocks, We report the volatility discovery measures for 30 assets considering the Nasdaq and Arca, Nasdaq and Nyse, Arca and Nyse, and Nasdaq, Arca and Nyse market combinations.T, P, and N denote Nasdaq, Arca, and Nyse, respectively.The FCVAR parameters are computed using the MLE estimator of Johansen and Nielsen (2012) where rk αβ = 1 and rk αβ = 2 for the two-and three-market settings, respectively, κ is chosen according to Table S.4, and d = b.For each set of results, we report the elements of the orthogonal complement of α, where the subscript identifies the market, for example, α ⊥,T denotes the orthogonal complement associated with the Nasdaq market.As for the Nasdaq-Arca-Nyse set of results, the three last columns report the p-values associated with the null hypothesis that volatility discovery measures of two markets are equal to zero, for example, H T,P reports the p-value of the null given by α ⊥,T = α ⊥,P = 0. * , * * , and * * * denote significance at 1%, 5%, and 10% significance levels."-" implies that the stock is not traded in at least one of the two trading venues during the entire sample.
respectively.With joint hypothesis tests at the 5% significance level, we reject the null hypotheses α ⊥,T = α ⊥,N = 0 and α ⊥,P = α ⊥,N = 0 and fail to reject the null hypothesis α ⊥,T = α ⊥,P = 0 for 8 of the 19 stocks, indicating that Nyse is the sole contributor to the volatility discovery process.Similarly, our analysis shows that Arca and Nasdaq are the sole contributor for 1 and 0 stocks, respectively.The Holm-Bonferroni correction procedure confirms Nyse as the single driver in 7 stocks.Overall, our findings in Table 2 indicate that Nasdaq plays a less significant role compared to Nyse, which emerges as the leading exchange in impounding information for the efficient integrated variance.Using the limiting distribution established in Theorem 2, we find statistical evidence suggesting that a single market typically acts as the sole driver of the volatility discovery process.
We further expand our empirical application to investigate the implications of our theoretical framework, which highlights the presence of two distinct sources of information within the price and volatility discovery mechanisms.Having documented the secondary role of Nasdaq for the volatility discovery mechanism, we next examine whether the same feature holds in the price discovery analysis.We follow the price discovery literature and adopt the VEC model as a discrete approximation of the observed prices in the three market combinations discussed previously. 21Our choice of price discovery measure is the component share (CS), which is invariant to the sampling frequency, meaning that we can consistently estimate it from prices sampled at any sampling intervals (Dias, Fernandes, and Scherrer, 2020).As there is a tradeoff between the choice of sampling interval and the size of the standard errors of the CS measures, we use mid-quotes aggregated at 10 sec to balance out the negative effect of market microstructure noise and the positive effect of having information at the highest possible frequency.
In general, the price discovery results in Table 3 suggest that Nasdaq leads the price discovery mechanism when compared to both Arca and Nyse.We find that Nasdaq leads in 22 of the 28 stocks for the Nasdaq-Arca market combination and 14 of the 19 stocks for the Nasdaq-Nyse market combination.The Arca-Nyse combination shows generally more balanced results, with Arca leading in 13 of the 21 stocks.The results from the Nasdaq-Arca-Nyse market combination are robust to the twomarket setting, with Nasdaq leading in 12 of the 19 stocks.In contrast with the volatility discovery analysis, we find that all markets contribute to the price discovery process irrespectively of the market combination, that is, we reject the null hypotheses γ ⊥,s = 0 for all s = 1, . . ., S at the 5% significance level for all but We report price discovery results for 30 assets considering Nasdaq and Arca, Nasdaq and Nyse, Arca and Nyse, and Nasdaq, Arca, and Nyse market combinations.The VEC parameters are computed using the OLS estimator where rk γ δ = S − 1, δ = I S−1 , −ι S−1 with S denoting the number of markets, and the lag length is chosen as the minimum value which makes the LM test for serial correlation on the residuals at lag 15 to be insignificant at the 5% significance level.For the first three sets of results, the first and second column show the estimate of the CS measure associated with the first and second market, respectively.As for the Nasdaq-Arca-Nyse set of results, the first three columns display the price discovery measures for Nasdaq, Arca, and Nyse, respectively.The last three columns report the p-values associated with the null hypothesis that volatility discovery measures of two markets are equal to zero, for example, H T,P reports the p-value of the null given by γ ⊥,T = γ ⊥,P = 0. * , * * , and * * * denote significance at 1%, 5%, and 10% significance levels."-" implies that the stock is not traded in at least one of the two trading venues during the entire sample.
two stocks.This result holds when controlling for the multiple testing issue.Overall, our results suggest that the volatility and price discovery mechanisms do not necessarily occur in the same trading venue.This finding is in line with Theorem 1, which shows that the price and volatility discovery measures capture how markets incorporate news into two distinct efficient stochastic processes: the efficient price and the efficient stochastic volatility.

Conclusions
In the current context of market fragmentation, we develop a theoretical framework to investigate how distinct markets contribute to the efficient price's IV.The economic rationale supporting the volatility discovery analysis rests on the premise that the efficient volatility process is a separate stochastic process.Just as the price discovery measure, the volatility discovery analysis contributes to assess market efficiency and investor protectionreliable and transparent information when making investment decisions using volatility.
The volatility discovery framework builds on evidence that the RMs of a homogeneous asset are tied to a long run equilibrium and share a common factor: the efficient stochastic volatility.We exploit RMs' long memory feature and adopt the FCVAR model to identify and estimate how innovations to market volatilities contribute to the efficient volatility process.We show the role played by both price and volatility discovery measures on the identification of the integrated variance of the efficient price.Next, we establish the limiting distribution of the volatility discovery measures and discuss how to make inference on market leadership.Finally, our empirical application corroborate our theoretical findings: RMs are tied to a long run equilibrium; markets incorporate information distinctively; and volatility and price discovery do not necessarily occur in the same trading venue.

Table 1 .
Cofractional rank test and estimates of the cointegrating vector.