A Score-Driven Conditional Correlation Model for Noisy and Asynchronous Data: an Application to High-Frequency Covariance Dynamics
Giuseppe Buccheri, Giacomo Bormetti, Fulvio Corsi, Fabrizio Lillo
aa r X i v : . [ q -f i n . T R ] M a r A Score-Driven Conditional Correlation Model for Noisy andAsynchronous Data: an Application to High-Frequency CovarianceDynamics ∗ Giuseppe Buccheri , Giacomo Bormetti , Fulvio Corsi , and Fabrizio Lillo Scuola Normale Superiore, Italy University of Bologna, Italy University of Pisa, Italy City University of London, UK CADS, Human Technopole, Milan, Italy
First version: December, 2017This version: March, 2019
Abstract
The analysis of the intraday dynamics of correlations among high-frequency returns is challenging due tothe presence of asynchronous trading and market microstructure noise. Both effects may lead to significant datareduction and may severely underestimate correlations if traditional methods for low-frequency data are employed.We propose to model intraday log-prices through a multivariate local-level model with score-driven covariancematrices and to treat asynchronicity as a missing value problem. The main advantages of this approach are: (i)all available data are used when filtering correlations, (ii) market microstructure noise is taken into account, (iii)estimation is performed through standard maximum likelihood methods. Our empirical analysis, performed on1-second NYSE data, shows that opening hours are dominated by idiosyncratic risk and that a market factorprogressively emerges in the second part of the day. The method can be used as a nowcasting tool for high-frequency data, allowing to study the real-time response of covariances to macro-news announcements and tobuild intraday portfolios with very short optimization horizons.
Keywords : Intraday Correlations; Dynamic Dependencies; Asynchronicity; Microstructure Noise
JEL codes : C58; D53; D81 ∗ Corresponding author: [email protected]. We are particularly grateful for suggestions we have received from Maria ElviraMancino, Davide Delle Monache, Ivan Petrella, Fabrizio Venditti, Giampiero Gallo, Davide Pirino and participants to the IAAE 2017conference in Sapporo, the 10 th SoFiE conference in New York and the VIECO 2017 conference in Wien. Introduction
A large class of conditional covariance models have been proposed in the econometric literature and their use iswidespread in risk and portfolio management at daily or lower frequencies. Popular multivariate dynamic time-seriesmodels include the class of multivariate extensions of the univariate GARCH model of Engle (1982) and Bollerslev(1986) and the Dynamic Conditional Correlation (DCC) model of Engle (2002). A drawback of these models isthat they are misspecified if data are recorded with observational noise and require synchronization in case data areirregularly spaced. As a consequence, they cannot be straightforwardly applied to intraday data, since high-frequencyprices are contaminated by microstructure noise and assets are traded asynchronously. Both effects may lead to ignorea large portion of data and can significantly underestimate correlations. The problem of estimating and forecastingintraday volatilities and correlations is, however, of crucial importance in high-frequency finance. For instance, anhigh-frequency trader is interested in rebalancing the portfolio on an intraday basis and thus needs accurate short-term covariance forecasts. Similarly, the study of the intraday dependencies of financial assets is useful to examinethe reaction of the market to external information and has a theoretical relevance in market microstructure research.We contribute to the literature on intraday covariance estimation by proposing a modelling strategy that canhandle both asynchronous trading and microstructure effects. High-frequency log-prices are modeled through aconditionally norma local-level model where efficient log-prices are affected by measurement errors and the covariancematrices of both the efficient returns and the noise are time-varying. In this state-space representation, asynchronoustrading can be treated as a standard missing value problem. The dynamics of time-varying parameters are driven bythe score of the conditional density (Creal et al. 2013, Harvey 2013). The latter can easily be computed using thestandard Kalman filter, as described by Creal et al. (2008) and Delle Monache et al. (2016).The main advantage of this state-space representation is that it allows to model the covariances of latent efficientreturns using all available observed prices. In standard conditional correlation models, the covariances of observedreturns are instead modelled. While in a low-frequency setting the two approaches are equivalent, at high-frequency,models for observed returns are subject to large data reduction and are affected by microstructure noise. For instance,assume the i -th asset is traded at time t , but is not traded at time t −
1, a circumstance that is very common inpractice. The local-level model can exploit the observation of the price of the i -th asset at time t to reconstructthe efficient price and to update correlations. In contrast, in standard conditional correlation models, the return ofthe i -th asset at time t is treated as a missing value and the information related to the price of the traded asset isneglected. An alternative method would be synchronizing the data, for instance by previous-tick interpolation. Thisleads to a large number of zero returns, which are known to jeopardize the inference. This effect is analogous tothe downward bias of high-frequency sample correlations, the well-known “Epps effect” (Epps 1979), which ariseswhen using previous-tick or other interpolation schemes (see Hayashi and Yoshida 2005 and references therein).Microstructure effects constitute an additional source of bias for standard conditional correlation models. Anotherimmediate consequence of modelling the covariances of latent efficient returns is that correlation estimates are robustto measurement errors. Compared to standard dynamic covariance models, the proposed approach is thus specificallydesigned to deal with high-frequency data and can easily be employed to construct intraday portfolios. To this end,we discuss two alternative parameterizations of the covariance matrix leading to positive-definite estimates.Since Andersen and Bollerslev (1997) and Tsay (2005), it is known that intraday volatilities have the typicalU-shape, being larger at the opening and closing hours of the trading day. In contrast, due to the aforementioneddifficulties, the intraday behavior of correlations has received less attention in the financial econometric literature.2otably exceptions are given by the work of Bibinger et al. (2014), who proposed a nonparametric spot covarianceestimator for intraday data and those of Koopman et al. (2017) and Koopman et al. (2018), which are based ondynamic copula models. The main difference between our approach and the two dynamic copula models is that wemodel the dependencies of latent efficient returns rather than those of observed returns. Our approach therefore doesnot suffer from data reduction when applied to high-frequency data. In addition, we deal explicitly with measurementerrors and thus correlation estimates are not downward-biased due to microstructure effects. Multivariate GARCHgeneralizations have been proposed, among others, by Engle and Kroner (1995), Tse and Tsui (2002), van der Weide(2002), Alexander (2002), Engle (2002), Creal et al. (2011). As underlined above, asynchronicity and market mi-crostructure effects can lead to several unwanted features in the inference of these model. We will examine in detailthe impact of these two effects on the DCC model of Engle (2002) and on the t -GAS model of Creal et al. (2011) inour simulation and empirical study.Score-driven models are a general class of observation-driven models where the dynamics of time-varying pa-rameters are driven by the score of the conditional likelihood. They have been successfully applied in the recenteconometric literature (see e.g. Creal et al. 2011, Creal et al. 2014 and Oh and Patton 2018). One of the mainadvantages of these models is that the conditional likelihood can be written in closed form and thus estimation canbe performed through standard maximum likelihood methods. In a linear-Gaussian state-space representation, thescore (with respect to system matrices) can be computed through an additional filter that runs in parallel with theKalman filter. This method was originally introduced by Creal et al. (2008) and then described in full generalityby Delle Monache et al. (2016). We will use it to model the dynamics of covariances in the local-level model. Theresulting model is condtionally normal and can be estimated through the Kalman filter. As pointed out by Harvey(1991), condtionally normal models are particularly convenient, as they feature nonlinear dynamics while preservingthe possibility of applying the standard Kalman filter.Our Monte-Carlo analysis has three main goals. First, we investigate the finite sample properties of the maximumlikelihood estimator. We find that it remains unbiased even in case many observations are missing. We then use themodel as a filter for a misspecified DGP for correlations and compare it to standard dynamic models employed ina low-frequency setting. We find that, in presence of measurement errors and asynchronous observations, standardmethods are subject to a downward bias. The local-level model performs significantly better, as it exploits allavailable data and provides robustness to measurement errors. Finally, we investigate the performance of the modelin presence of fat-tails and asynchronicity and/or noise. To this end, we use the t -GAS model to simulate intradayprices and correlations. After randomly censoring prices, we estimate both the local-level model and the t -GAS. Wefind that, as the number of missing observations increases, the local-level model provides lower in-sample and out-of-sample average losses. The t -GAS, being a conditional correlation model for observed returns, is indeed subject tolarge data reduction in presence of asynchronous observations. An analogous effect is observed when we add noiseto the simulated prices. Therefore, even in presence of extreme fat-tails, in an high-frequency setting the use of thelocal-level model is preferable to that of correctly specified observed return models.We apply the model to transaction data of 10 NYSE stocks. The in-sample analysis based on the AIC revealsthat the local-level model fits data significantly better than standard correlation models for low-frequency data. Wefind the well-known U-shape for volatilities, while correlations reveal an increasing pattern. The rate of increase islarger during the first two hours, then correlations increase at a slower rate and tend to decrease during the last 15minutes of the trading day. Based on the dynamics of the first eigenvalues of the correlation matrix, we interpret this3henomenon as the emergence of a market factor which progressively explains a larger fraction of the total varianceof the market.The local-level model, being robust to asynchronous trading, can be estimated at ultra high frequencies (1-secondin our application) and thus provides a description of the dynamics of covariances at very small time scales. Thisallows as to study the real-time response of correlations to macro-news announcements. Once macro-news arriveon the market, they are instantaneously captured by the score-driven filter, even if very few assets are traded atthat time. The method can thus be employed as a nowcasting tool for high-frequency data. This interesting featureshares some similarities with the macroeconomic literature on nowcasting, where dynamic factor models are used toupdate forecasts of macroeconomic variables based on mixed-frequency observations (see e.g. Giannone et al. 2008and Delle Monache et al. 2016). In the second part of the empirical analysis, we assess the performance of the modelas a nowcasting tool for high-frequency data. We construct intraday out-of-sample portfolios with short investmenthorizons and find that those constructed through the local-level model feature significantly lower risk.The remaining part of the paper is organized as follows: in Section 2 we describe the methodology in its fullgenerality, including the parameterization of correlations and the estimation method; Section 3 discusses the resultsof Monte-Carlo experiments; in Section 4 we provide empirical evidence on the advantages of the model over standardtechniques and study the intraday dynamic behavior of covariances; Section 5 concludes. Let t ∈ [0 , S ] and denote by X t = ( X (1) t , . . . , X ( n ) t ) ′ an n × T equally-spaced observation times 0 ≤ t < t < · · · < t T − ≤ S and propose to model X t i , i = 0 , . . . , T −
1, as arandom walk with heteroskedastic innovations: X t i +1 = X t i + η t i +1 , η t i +1 ∼ (0 , Q t i ) (1)The time-varying matrix Q t i describes the dynamics of volatilities and correlations of the efficient returns and is themain object of interest of this work. The efficient log-price X t i is unobservable because of market microstructureeffects (e.g. bid-ask bounce). Let Y t i be the n × Y t i = X t i + ǫ t i , ǫ t i ∼ (0 , H t i ) (2)where ǫ t i is a measurement error term representing market microstructure effects. The latter is assumed to beindependent from the returns of the efficient log-price. Its variances are allowed to vary over time to capture potentialdynamic effects in microstructure noise. For instance, the bid-ask spread has a well-known intraday pattern, beinglarger at the opening hours and then declining throughout the day (McInish and Wood 1992). Model (1), (2) is atthe basis of traditional market microstructure analysis of trading frictions, asymmetric information and inventorycontrol (Roll 1984, Hasbrouck 1993, Madhavan 2000). 4 .2 State-space representation Eq. (1) describes the martingale dynamics of the efficient log-price process, while eq. (2) is the associated measure-ment equation. We can re-write both equations as: Y t = X t + ǫ t , ǫ t ∼ (0 , H t ) (3) X t +1 = X t + η t , η t ∼ (0 , Q t ) (4)where, without loss of generality, we have set t i +1 − t i = 1. Model (3), (4) is known as local-level model (Harvey1991,Durbin and Koopman 2012). If the two covariance matrices H t and Q t are constant, the local-level modelcan be estimated by quasi-maximum likelihood through the Kalman filter. This is the route of Corsi et al. (2015)and Shephard and Xiu (2017), who proposed a quasi-maximum likelihood estimator of the integrated covariance ofhigh-frequency asset prices. Here we are interested in a different problem, namely the dynamic modelling of thecovariances of both the noise and the efficient returns. We therefore need to specify a dynamic equation for H t and Q t .A convenient way to handle model (3), (4) is assuming that the disturbance terms are condtionally normal (Harvey1991). Let F t − be the σ -field generated by observations of the log-price process up to time t −
1. The condtionallynormal local-level model reads: Y t = X t + ǫ t , ǫ t |F t − ∼ NID(0 , H t ) (5) X t +1 = X t + η t , η t |F t − ∼ NID(0 , Q t ) (6)meaning that, conditionally on the information available at time t −
1, the distribution of ǫ t and η t is normal, withknown covariance matrices H t and Q t . The latter are assumed to obey an observation-driven update rule and dependnonlinearly on past observations. The Kalman filter can be applied to compute the likelihood in the usual predictionerror form. As discussed by Harvey (1991), condtionally normal models allow to “inject” nonlinear dynamics in themodel while still preserving the possibility of applying the standard Kalman filter. In our empirical framework, thisis a substantial advantage, given that the Kalman filter can easily handle missing values and can reconstruct theefficient price based on all available observations. In order to estimate the model, we need to specify the law of motion of H t and Q t . Score-driven models (Creal et al.2013, Harvey 2013) are a general class of observation-driven models. In score-driven models, parameters are updatedbased on the score of the conditional density. The GARCH model of Bollerslev (1986), the EGARCH model ofNelson (1991) and the ACD model of Engle and Russell (1998) are examples of models that can be recovered in thisgeneral framework. In addition, score-driven models have information theoretic optimality properties, as shown byBlasques et al. (2015).Motivated by the flexibility of score-driven models, we choose to model the dynamics of H t and Q t based on thescore of the conditional density. The covariance matrix Q t can be decomposed as: Q t = D t R t D t (7)where D t = diag[ Q t ] / is a diagonal matrix of standard deviations and R t is a correlation matrix. This decompositionis common in the econometric literature and is used, for instance, in the DCC model of Engle (2002). For parsimony,5e assume that the covariance matrix H t of the noise is diagonal. This assumption can be relaxed at the expense ofincreasing considerably the number of time-varying parameters. However, as pointed out by Corsi et al. (2015), inthe static case the off-diagonal elements of H t are found to be close to zero. They thus assumed a diagonal covariancematrix for the noise. Shephard and Xiu (2017) made the same assumption.Let f t denote a vector of time-varying parameters. We write: f t = log(diag[ H t ])log(diag[ D t ]) φ t (8)where φ t is a q × R t . The latter will be discussedin Section 2.4. The number of components of f t is thus k = 2 n + q . The update rule in score-driven models is givenby: f t +1 = ω + As t + Bf t (9)where s t is the scaled score vector: s t = ( I t | t − ) − ∇ t , ∇ t = (cid:20) ∂ log p ( Y t | f t , F t − , Θ) ∂f ′ t (cid:21) ′ , I t | t − = E[ ∇ t ∇ ′ t ] (10)The k × ω and the k × k matrices A, B are included in the vector Θ of static parameters to be estimated.The conditional log-likelihood is given by:log p ( Y t | f t , F t − , Θ) = const − (cid:0) log | F t | + v ′ t F − t v t (cid:1) (11)where v t and F t are the Kalman filter prediction error and its covariance matrix, as defined in Appendix A. As shownby Delle Monache et al. (2016), the score ∇ t and the Fisher information matrix I t | t − can be computed as: ∇ t = − h ˙ F ′ t ( I n t ⊗ F − t )vec( I n t − v t v ′ t F − t ) + 2 ˙ v ′ t F − t v t i (12) I t | t − = 12 h ˙ F ′ t ( F − t ⊗ F − t ) ˙ F t + 2 ˙ v ′ t F − t ˙ v t i (13)where n t denotes the number of observations available at time t . Together with v t and F t , the computation of ∇ t and I t | t − requires ˙ v t and ˙ F t , which denote derivatives of v t and F t with respect to the time-varying parameter vector f t .As discussed by Delle Monache et al. (2016), they can be computed through the recursions reported in Appendix A.By running in parallel the Kalman filter and the filter in eq. (9), one can update parameters and computethe conditional log-likelihood in eq. (11). The static parameters Θ are estimated by numerically optimizing thelog-likelihood function: ˆΘ = argmax Θ T X t =1 log p ( Y t | f t , F t − , Θ) (14)Restrictions on the structure of A , B are discussed in the empirical application in Section 4. R t To have a full model specification, we need a parameterization for the correlation matrix R t . We restrict ourattention to parameterizations that guarantee a positive-definite correlation matrix. One possibility is to use hyper-spherical coordinates, as in Creal et al. (2011). Another possibility is given by the equicorrelation parameterizationof Engle and Kelly (2012). 6n the first case, we write the correlation matrix as R t = Z ′ t Z t . The matrix Z t has the form: Z t = c c . . . c n s c s . . . c n s n s s . . . c n s n s n ... ... ... ...0 0 0 . . . Q n − k =1 s kn (15)where c ij = cos θ ij and s ij = sin θ ij . Note that the time index has been omitted for ease of notation. The i -th columnof Z t contains the hyperspherical coordinates of a vector of unit norm in an i -th dimensional subspace of R n , whichis parametrized by i − n ( n − / R t . Weset φ t = [ θ ,t , θ ,t , . . . , θ n − n,t ] ′ in eq. (8), so that q = n ( n − /
2. The number of time-varying parameters is thus k = 2 n + n ( n − / R t is written as: R t = (1 − ρ t ) I n + ρ t J n (16)where J n denotes an n × n matrix of ones. The correlation matrix R t is positive-definite if and only if the parameter ρ t satisfies the constraint − / ( n − ≤ ρ t ≤
1. One possibility to guarantee this constraint is to write: ρ t = 12 (cid:20)(cid:18) − n − (cid:19) + (cid:18) n − (cid:19) tanh( θ t ) (cid:21) (17)as in Koopman et al. (2018). We set φ t = θ t , so that q = 1. The number of time-varying parameters is thus k = 2 n + 1. The equicorrelation parameterization is very parsimonious, as it assumes the same correlation for allcouples of assets. Notwithstanding, it has been proven to be effective in several empirical problems (cf. discussionsin Engle and Kelly 2012). We first study the finite sample properties of the maximum likelihood estimator through Monte-Carlo simulations.We set n = 10, the same number of assets that will be used in our empirical application in Section 4. The number oftime-varying parameters is thus k = 2 n + n ( n − / k = 2 n + 1 = 21 in the equicorrelation parameterization. In the first experiment, we assume that the staticparameters ω , A , B have the following structure: ω = ω h ω d ω r , A = diag A h A d A r , B = diag B h B d B r (18)where ω h , A h , B h are n × ω d , A d , B d are n × ω r , A r , B r are q × ω h = − . ι n , ω d = − . ι n , ω r = 0 . ι q , A h = A d = 0 . ι n , A r = 0 . ι q , B h = B d = 0 . ι n ,7 r = 0 . ι q . The values of ω h , ω d are chosen in such a way that the signal-to-noise ratio (defined as the ratiobetween the unconditional variance of the efficient returns and the unconditional variance of the measurement error)is similar to the one found on empirical data, which is on average equal to 1 (cf. table 5).After simulating the log-prices, we randomly censor observations to mimic asynchronous trading. The probabilityof removing one observation is denoted by λ and is assumed to be the same for the n time-series. To set the initialvalues of time-varying parameters, we estimate a local-level model with constant parameters in the subsamplecomprising the first 100 observations. This can be done through the EM algorithm, as described by Corsi et al.(2015). We then choose f based on eq. (8), with H t , D t , R t equal to those obtained through the EM algorithm. Wesimulate N = 1000 time-series of T = 2000 observations and consider for scenarios characterized by λ = 0, 0 .
3, 0 . .
8. Table 1 reports summary statistics of maximum likelihood estimates in the parameterization with hypersphericalcoordinates. Table 2 reports analogous statistics obtained with the equicorrelation parameterization. λ Mean Std Mean Std Mean Std ω h = − . ι n ω d = − . ι n ω r = 0 . ι q A h = 0 . ι n A d = 0 . ι n A r = 0 . ι q B h = 0 . ι n B d = 0 . ι n B r = 0 . ι q Table 1: Mean and standard deviations of maximum likelihood estimates of the local-level model with score-drivencovariances in the parameterization with hyperspherical coordinates.The results show that, in both parameterizations, maximum likelihood estimates concentrate around the trueparameters. Not surprisingly, missing values lead to an increase of the variance of the maximum likelihood estimator.However, even in the highly asynchronous scenario with λ = 0 .
8, relative errors remain small for all parameters in A , B and for almost all parameters in the intercept ω . Larger relative errors are found for the parameter ω r in theequicorrelation parameterization. We then assume parameters ω , A , B have the following structure: ω = , A = diag A h A d A r , B = diag (19)8 Mean Std Mean Std Mean Std ω h = − . ι n ω d = − . ι n ω r = 0 . ι q A h = 0 . ι n A d = 0 . ι n A r = 0 . ι q B h = 0 . ι n B d = 0 . ι n B r = 0 . ι q Table 2: Mean and standard deviations of maximum likelihood estimates of the local-level model with score-drivencovariances in the parameterization based on the equicorrelation matrix.Under these restrictions, time-varying parameters have non-stationary random-walk dynamics. As will be shownin Section 4.2, this specification provides a better description of intraday high-frequency prices and will be usedto estimate the local-level model on empirical data. We set A h = A d = 0 . ι n , A r = 0 . ι q and report in Table 3summary statistics of maximum likelihood estimates. As in the previous simulation, parameter estimates concentratearound true values and the variance slightly increases with the number of missing values. Overall, maximum likelihoodprovides satisfactory parameter estimates in all the considered scenarios. λ Mean Std Mean Std Mean Std hyperspherical A h = 0 . ι n A d = 0 . ι n A r = 0 . ι q equicorrelation A h = 0 . ι n A d = 0 . ι n A r = 0 . ι q Table 3: Mean and standard deviations of maximum likelihood estimates of the local-level model with score-drivencovariances and random walk-type restrictions for static parameters.9
LLDCCEWMA
LLDCCEWMA
LLDCCEWMA
LLDCCEWMA
Figure 1: Estimates of local-level, DCC and EWMA averaged over all simulations in the scenario δ = 2, λ = 0 . In this Section we aim to assess the effect of noise and asynchronous observations on commonly used dynamiccorrelation models and to show the advantages provided by the proposed modelling strategy. High-frequency pricesare typically synchronized before being analyzed with standard dynamic models. The most popular synchronizationscheme is previous-tick interpolation, consisting in updating missing values with previous available prices. Thisprocedure (and other similar schemes) leads to a downward bias of correlations toward zero. In the literature onrealized covariance estimation, the latter is known as “Epps effect” (Epps 1979). The explanation of the Epps effectis intuitive: synchronization leads to a large number of zero returns, which in turn undermine correlations (seeHayashi and Yoshida 2005 and references therein). As will be shown here, a similar problem arises when estimatingdynamic correlations. The presence of measurement errors is an additional source of bias for correlations. This is notsurprising, as the attenuation bias due to measurement errors occurs in several econometrics and statistics problems,e.g. in error-in-variables models.For simplicity, we consider a bivariate model generated as follows: Y t = X t + ǫ t , ǫ t ∼ N (0 , H ) (20) X t +1 = X t + η t , η t ∼ N (0 , DR t D ) (21)10here H = h I , D = d I and: R t = ρ t ρ t (22)The correlation coefficient ρ t follows these dynamic patterns:1. Sine ρ t = b s sin( c s πt )2. Fast Sine ρ t = b f sin( c f πt )3. Step ρ t = α s − β s ( t 3, 0 . 5, where λ denotes the probability ofremoving an observation.We generate N = 250 simulations of T = 4000 observations for each dynamic pattern and for each scenario. Asa benchmark, we consider the DCC model of Engle (2002) and an EWMA scheme given by:ˆ Q t +1 = γ ˆ Q t + (1 − γ ) r t r ′ t (23)where γ is set equal to 0 . 96. Both DCC and EWMA are applied after synchronizing data through previous-tickinterpolation. Depending on the value of λ , this results in a large number of zero returns. In order to attenuate theeffect of noise and zero returns on these two models, we sample observations at a lower frequency. This procedure iscommonly employed when computing e.g. realized covariance. Indeed, at lower frequencies, returns are less affectedby both microstructure effects and by staleness due to absence of trading. A natural consequence is that a significantpart of data is neglected. However, subsampling greatly improves correlation estimates of DCC and EWMA. Ineach scenario, the new sampling frequency is chosen as the one minimizing the MSE in a pre-simulation study with N = 20 simulations. The selected values range in the interval between 3 and 6 time units.We compute both in-sample and out-of-sample losses of the local-level model and the DCC. To this end, the sampleis divided into two subsamples containing the same number of observations. The latter is equal to T sub = 2000 inthe local-level model, while it depends on the sampling frequency in the case of the DCC. The local-level modeland the DCC are estimated in the first subsample, where in-sample losses are computed. The second subsampleis then used to compute out-of-sample losses. The parameterization used in the local-level model is the one basedon hyperspherical coordinates. Figure 1 shows, in the scenario δ = 2 , λ = 0 . 5, correlation estimates of the three11 LLDCCEWMA LLDCCEWMA LLDCCEWMA LLDCCEWMA Figure 2: Estimates of local-level, DCC and EWMA averaged over all simulations in the scenario δ = 0 . λ = 0.models averaged over all simulations. Even after sampling at lower frequencies, DCC and EWMA estimates arebiased toward zero. This is mainly a consequence of asynchronicity, which is high in this scenario. Figure 2 shows,in the scenario δ = 0 . λ = 0, average estimates of the three models. DCC and EWMA have a similar downwardbias. Compared to the previous scenario, the bias is now entirely due to measurement errors, as asynchronicity isabsent. It is instead evident that the local-level model is much less affected by the two effects. The use of standarddynamic covariance models thus leads to data reduction, as a consequence of sampling at lower frequencies, andunderestimates (absolute value of) correlations.Table 4 reports in-sample and out-of-sample MSE of the three models in all the simulated scenarios. The local-level significantly outperforms the DCC in all the scenarios. Compared to the DCC, the performance of the EWMAis closer to that of the local-level in patterns “sine”, “fast sine”, “step”. However, with only one exception (out-of-sample MSE of “fast sine” in scenario λ = 0 . δ = 2) the MSE of the local-level is lower than that of the EWMA.For patterns “ramp” and “model”, characterized by frequent changes and stochasticity, the local-level performssubstantially better than the EWMA. 12 odel Sine Fast Step Ramp Model Sine Fast Step Ramp Model Sine Fast Step Ramp Model δ = 0 . δ = 1 δ = 2 λ = 0LL 0.0367 0.0586 0.0302 0.0217 0.0361 0.0263 0.0413 0.0234 0.0200 0.0279 0.0197 0.0302 0.0185 0.0178 0.0224DCC 0.0628 0.0711 0.0533 0.0422 0.1222 0.0455 0.0575 0.0409 0.0295 0.0760 0.0344 0.0457 0.0327 0.0225 0.0474EWMA 0.0474 0.0643 0.0388 0.0402 0.1066 0.0376 0.0586 0.0317 0.0290 0.0641 0.0321 0.0555 0.0277 0.0230 0.0417LL 0.0438 0.0620 0.0435 0.0237 0.0458 0.0302 0.0453 0.0370 0.0216 0.0309 0.0219 0.0325 0.0290 0.0187 0.0226DCC 0.0655 0.0718 0.0570 0.0421 0.1283 0.0471 0.0582 0.0455 0.0296 0.0811 0.0352 0.0459 0.0376 0.0228 0.0516EWMA 0.0489 0.0677 0.0462 0.0399 0.1068 0.0405 0.0631 0.0402 0.0291 0.0630 0.0362 0.0609 0.0370 0.0235 0.0400 λ = 0 . λ = 0 . Table 4: MSE of local-level (LL), DCC and EWMA estimates in the 9 scenarios obtained by combining the three missing value scenarios ( λ = 0 , . , . 5) to each ofthe three signal scenarios ( δ = 0 . , , .3 Comparison with fat-tail return models High-frequency data can exhibit extreme movements during flash crashes or in correspondence of macro-news an-nouncements. It is therefore important to investigate the quality of correlation estimates provided by the proposedapproach in presence of outliers. Creal et al. (2011) introduced a score-driven dynamic correlation model, named t -GAS, based on multivariate Student- t distribution. The conditional density of log-returns in the t -GAS is: p ( r t | Σ t , ν ) = Γ( ν + n )Γ( ν )[( ν − π ] n/ | Σ t | / (cid:18) r ′ t Σ t r t ν − (cid:19) − ( ν + n ) / (24)The covariances in Σ t obey the usual score-driven update rule. Different parameterizations for Σ t are possible, suchas the one discussed in Section 2.4 based on hyperspherical coordinates.In this simulation study, we use eq. (24) as a data generating process for log-returns. After computing log-prices,we randomly censor them to mimic asynchronous trading. We then estimate both the local-level model with score-driven covariances and the t -GAS. There are two possible methods to estimate the t -GAS in presence of missingvalues. The first consists in synchronizing data through an interpolation scheme, e.g. the previous-tick. We have seenin the previous application that this generally leads to a downward bias. The second method consists in computingthe score with respect to the marginal density of observed data (Lucas et al. 2016). In the case of the multivariateStudent- t distribution, this is particularly simple, since it is known that the marginals are Student- t densities withthe same number of degrees of freedom. Specifically, assume X ∼ t ν (0 , Σ) ∈ R n and partition the elements in X , Σas: X = X X , Σ = Σ , Σ Σ , Σ (25)where X ∈ R p , p < n and Σ ∈ R p × p . The marginal density of X is then a t ν (0 , Σ ). See e.g. Golam Kibria and Joarder(2017) for details. Missing values can therefore be handled as in the Kalman filter, i.e. by computing the score ofthe marginal density t ν (0 , Γ t Σ t Γ ′ t ), where Γ t is a selection matrix with ones in the columns corresponding to tradedassets.This method avoids the introduction of artificial zero returns. However, it induces significant data reduction, asonly consecutive trades can be used to compute returns. For instance, if the i -th assets is traded at time t , but isnot traded at time t − 1, the return r i,t is treated as a missing value and the information related to the price at time t is lost. The local-level instead exploits the observation of the price at time t to reconstruct the vector of efficientprices and to update covariances.To illustrate this concept, we simulate log-returns through a t -GAS model with n = 10 and, in order to emphasizethe effect of fat-tails, we set ν = 3. Ten different scenarios are considered, characterized by an increasing level ofasynchronicity ( λ = 0 , . , . . . , . n = 250 simulations of time-series of T = 4000 observations. Asin the previous study, in-sample estimates are computed in the first subsample of T sub = 2000 observations, whileout-of-sample estimates are computed in the second subsample. As a loss measure, we use the Frobenius distance,defined as: || ˆΣ t − Σ t || F = q Tr[( ˆΣ t − Σ t ) ] (26)The parameterization of R t used in both models is the one based on hyperspherical coordinates, and the staticparameters driving the dynamic elements in D t and R t have the restrictions in eq. (18).Figure 3 shows on the left the results of the simulation study. There is an evident trade-off between fat-tails andasynchronicity. Not surprisingly, if data are perfectly synchronized, the t -GAS model performs better. However,14 Fraction of missing values A v g . F r oben i u s l o ss In-sample t-GASIn-sample LLOut-of-sample t-GASOut-of-sample LL 10 Signal-to-noise ratio A v g . F r oben i u s l o ss In-sample t-GASIn-sample LLOut-of-sample t-GASOut-of-sample LL Figure 3: Left: in-sample and out-of-sample Frobenius losses of t -GAS and local-level model as a function of theprobability of missing values λ . The DGP is a t -GAS with ν = 3. Right: in-sample and out-of-sample Frobeniuslosses of t -GAS and local-level model as a function of the signal-to-noise ratio. The DGP is a t -GAS with ν = 3contaminated by a Student- t distributed measurement error with ν err = 3.once missing values come into play, the local-level model tends to improve its relative performance. In particular,for λ > . 5, it provides better in-sample and out-of-sample estimates. Note that the average losses of t -GAS rapidlyincrease with λ . For λ = 0 . 9, the in-sample loss of t -GAS is ∼ 24 times larger than the one at λ = 0, while for thelocal-level it is just ∼ . t -GAS in presenceof missing values is due to the aforementioned effect of data reduction. A similar tradeoff is observed when we add anoise to the simulated prices. In a second experiment, we ignore asynchronicity, but contaminate the prices simulatedthrough the t -GAS with a zero-mean Student- t distributed measurement error with ν err = 3. We consider differentlevels of signal-to-noise ratio, from δ = 0 . δ = 100. As shown on the right in figure 3, for δ < 20, the average lossof the local-level model is significantly lower that that of the t -GAS, both in-sample and out-of-sample. This is dueto the fact that, in the local-level, we model the covariances of latent returns, rather than those of observed returns.Of course, adding noise and asynchronicity together would lead to further increase the relative difference betweenthe two models.The results of this analysis suggest that the choice of the model depends on the tradeoff between outliers andasynchronicity/noise. As will be evident in our empirical application, high-frequency data are characterized by highlevels of asynchronicity, with λ > . 8, and are significantly noisy, having δ < 1. The use of the local-level model istherefore more suited in an high-frequency setting, where asynchronicity and microstructure effects are extremelyimportant. Our dataset contains 1-second transaction data of 10 most frequently traded NYSE assets in 2014. Data are relatedto transactions from 02-01-2014 to 31-12-2014, including a total of 251 business days. The exchange opens at 9.30and closes at 16.00 local time, so that the number of seconds per day is T = 23400. We perform the standard15rocedures described by Barndorff-Nielsen et al. (2009) to clean the data. Table 5 shows summary statistics of the10 assets, including the average duration (in seconds) between trades, the average probability of missing values λ avg (computed as the average fraction of trades per day) and the average number of trades per day. We also report theaverage signal-to-noise ratio δ avg , computed from the estimated matrices D t , H t . Specifically, δ avg is defined as theratio D t,ii /H t,ii , i = 1 , . . . , 10, averaged over all timestamps.The numbers in the table provide a first evidence about the relevance of asynchronous trading and microstructureeffects in high-frequency data. The average probability of missing values is greater than 0.8, indicating large levels ofsparsity even for the liquid assets included in this dataset. The average signal-to-noise is lower than 1 for most of theassets. This implies that also microstructure effects play a relevant role. As done in the simulation study in Section3, we thus aim to show the advantages of the proposed method over standard techniques ignoring these effects. Symbol Asset Avg. duration (sec) λ avg Avg. n. of trades δ avg XOM Exxon Mobil Corporation 5.434 0.816 4304 1.178C Citigroup Inc 6.135 0.836 3832 1.246JPM JPMorgan Chase 6.250 0.840 3743 0.999HAL Halliburton Company 6.369 0.843 3690 0.872CVX Chevron 6.579 0.848 3553 0.850DIS Walt Disney 6.622 0.849 3543 0.846JNJ Johnson & Johnson 6.666 0.850 3529 0.809SLB Schlumberger Limited 6.802 0.853 3454 0.613DAL Delta Air Lines 6.993 0.857 3348 0.766WMT Walmart 7.042 0.858 3325 0.698 Table 5: Summary statistics and average signal-to-noise ratio of high-frequency transaction data employed in theempirical analysis. Restriction type 23 Jan 11 Apr 27 May 6 Aug 4 Sep 16 Oct I − . − . − . − . − . − . II − . − . − . − . − . − . Table 6: AIC ( × ) statistics computed on five randomly selected trading days with the parameter restriction I based on eq. (18) and the restriction II based on eq. (19).The vector ω and the matrices A , B in the dynamic equation (9) have dimensions k × k × k , respectively.For n = 10, k is equal to 21 in the equicorrelation parameterization, while it is equal to 65 in the parameterizationbased on hyperspherical coordinates. It is therefore necessary to impose restrictions on the parameter space in orderto estimate the model. We perform an AIC test to choose among different restrictions. Specifically, we compare theparameter structure I in eq. (18) to the parameter structure II in eq (19) Table 6 reports AIC statistics of the local-level model estimated on five randomly selected trading days with the two restrictions. The parameterization used inthis test is the one based on hyperspherical coordinates, however similar results are obtained with the equicorrelation16 an Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan-7-6-5-4-3-2-101 10 AIC LL hyper AIC t-GASAIC DCC Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan-5000500100015002000 AIC equi - AIC hyper Figure 4: Left: AIC of local-level with hyperspherical coordinates, t -GAS and DCC. Right: differences between AICof local-level with equicorrelation parameterization and local-level with hyperspherical coordinates.parameterization. The AIC provided by the random walk-type restriction II is always lower. As will be shown innext sections, the intuitive reason behind this result is that intraday covariances have non-stationary dynamics anddo not mean revert to an unconditional level. We therefore implement this restriction when estimating the local-levelmodel in all subsequent analyses. We estimate the local-level model on each business day of 2014 (the average estimation time on Matlab 2018 is 28.3min on an i7-2600 CPU at 3.40GHz). In order to initialize time-varying parameters, we proceed as in Section 3.1,i.e. we estimate the static version of the model in a pre-sample including the first 15 minutes of the trading day.The starting values of time-varying parameters are then set equal to the static estimates. As a parameterizationfor correlations, we use both hyperspherical coordinates and equicorrelations. We compare covariance estimatesobtained through the local-level model to the covariances of the t -GAS model. As in Creal et al. (2011), correlationsin the t -GAS are parameterized using hyperspherical coordinates. In presence of asynchronous data, the t -GAScan be estimated by previous-tick synchronization or using the missing value approach of Lucas et al. (2016). Asdiscussed in Sections 3.2, 3.3, the first method leads to a downward bias, while the second implies data reduction,as a consequence of modelling returns rather than prices. We use the missing value approach, as zero returnsdramatically jeopardize the estimation of correlations. The same random walk-type restrictions on static parametersare implemented for the t -GAS. As an additional benchmark, we use the DCC model estimated through previous-tickinterpolation. When data are sampled at the largest available frequency, the correlations provided by the DCC turnout to be flat and close to zero. This is due to both microstructure effects and zero returns. In order to attenuatethese two effects, we aggregate data at a lower frequency. We use a sampling frequency of 20 seconds. Indeed, athigher frequencies correlations are significantly biased, while at lower frequencies the inference becomes extremelyinefficient due to data reduction.Figure 4 shows on the left the daily AIC provided by the local-level with hyperspherical coordinates, the t -GAS and the DCC. The AIC of the local-level model is significantly lower than that of t -GAS and DCC. Thesedifferences are mainly due to data reduction and provide a quantitative assessment of the relevance of this effect.17 -4 LL hyper LL equi t-GASDCC 10:00 11:00 12:00 13:00 14:00 15:00 16:000.150.20.250.30.350.40.450.50.55 LL hyper LL equi t-GASDCC Figure 5: Intraday average volatilities (left) and correlations (right) of local-level with hyperspherical coordinates,local-level with equicorrelation parameterization, t -GAS and DCC. Volatilities are average over all the 10 assets whilecorrelations are averaged over all couples of assets.On the right, we plot the daily difference between the AIC of the equicorrelation parameterization and the AIC ofhyperspherical coordinates. The difference is positive in most of the days of the sample, indicating better in-samplefit for hyperspherical coordinates. However, note that the relative difference is small, meaning that the loss due tothe equicorrelation assumption does not affect significantly the quality of the fit.Figure 5 shows the estimated intraday volatilities and correlations averaged over the whole sample of 251 businessdays. Volatilities are averaged over the 10 assets, while correlations are averaged over all couples of assets. Theintraday pattern of volatilities provided by the t -GAS is higher than that of the two local-level models. The reasonis that the estimated volatilities in the t -GAS are affected by microstructure noise, while the local-level sets theefficient return volatilities in D t apart from the noise variances in H t . Microstructure effects thus lead to largervariances in the t -GAS. The average volatility estimated by the DCC is closer to that of the two-local level models.Indeed, microstructure effects in the DCC are attenuated by sampling at a lower frequency. However, previous-tickinterpolation leads to correlations which are significantly biased toward zero. The average correlations of the t -GASare significantly less biased compared to DCC correlations, confirming the ability of the missing value approach toavoid the distortions due to the introduction of artificial zero returns. Nevertheless, they suffer from the additionalbias due to microstructure effects. Overall, the in-sample analysis provides clear evidence that the proposed approachis more suited when dealing with intraday covariances.The intraday patterns of the local-level model with hyperspherical coordinates are close to those provided by theequicorrelation parameterization, in accordance with the result in figure 4. Larger deviations are observed in thecorrelations, in particular in the first part of the trading day, where the average equicorrelation increases at a slightlyhigher rate, and during the last few minutes, where the decrease is more pronounced. We now study in more detail the estimates provided by the local-level model with the aim to extract meaningfulinformation on the behavior of intraday covariances. To gain insights on the level of heterogeneity of correlations, weperform the analysis using the parameterization based on hyperspherical coordinates. We first examine the variation18f intraday patterns over time. To this purpose, we take averages of the estimated time-varying parameters overassets or couples of assets. For t = 1 , . . . , j = 1 , . . . , d jt = 1 n n X i =1 D jt ( i, i ) , ˜ h jt = 1 n n X i =1 q H jt ( i, i )˜ δ t = 1 n n X i =1 D jt ( i, i ) H jt ( i, i ) , ˜ ρ jt = 2 n ( n − X p>q R jt ( p, q )where n = 10. Similarly, to examine the variation of intraday patterns among different assets, we take averages oftime-varying parameters over time. For t = 1 , . . . , i, p = 1 , . . . , q < p we compute:¯ d it = 1 N N X j =1 D jt ( i, i ) , ¯ h it = 1 N N X j =1 q H jt ( i, i ) , ¯ δ it = 1 N N X j =1 D jt ( i, i ) H jt ( i, i ) , ¯ ρ p,qt = 1 N N X j =1 R jt ( p, q )where N = 251. The average intraday pattern can be computed by averaging the variables labeled with a “tilde”over j = 1 , . . . , 251 or, equivalently, by averaging the variables labeled with a bar over i, p = 1 , . . . , n and q < p . Foreach timestamps t = 1 , . . . , D t exhibits the well known U-shape. Volatilities are large at thebeginning of the trading day. They gradually decline until the last few minutes, when they steeply increase. Theaverage intraday pattern of microstructure noise volatility has two regimes: a steep decline from 9:30 to 10:00 anda slow decline from 10:00 until the end of the trading day. This dynamic behavior is close to the typical intradaypattern of bid-ask spreads (see e.g. McInish and Wood 1992), confirming that the local-level is consistently settingapart the efficient log-price process from microstructure effects. The average signal-to-noise ratio exhibits an U-shapeas well: it is larger at the beginning of the day ( δ ∼ 10) and at the end ( δ ∼ δ ∼ . R t . We notethat the explanatory power of the first eigenvalue, associated to the market factor, progressively increases duringthe day. At 15:45, it accounts for ∼ 50% of the total variance. This implies that, while at the beginning of the dayasset dynamics are dominated by idiosyncratic risk, a systematic component emerges in the second part of the day.This systematic component is associated to market risk, as all the remaining eigenvalues decrease with time. Theseresults are in agreement with the empirical findings of Allez and Bouchaud (2011), who employed standard samplecorrelations to study the intraday evolution of dependencies among asset prices. An increase of correlations duringthe trading day was also found by Bibinger et al. (2014) and Koopman et al. (2018).19 -4 Average D t -4 Average H t1/2 t t Figure 6: Average intraday patterns of the estimated time-varying parameters D t , H / t , δ t and R t . For each t = 1 , . . . , D t , H / t , δ t and R t (over assets or couples of assets).Comparing figures 6 and 7, we note that both the U-shape of efficient returns volatility and the decreasing patternof noise volatility is common to all assets and is stable in time. Similarly, the increase of correlations throughout theday and their sudden drop in the last 15 minutes is common to all the stocks and to all days. There is, however,a remarkable variation on the level of correlations over different days, especially at the end of the trading day. Forinstance, at 15:30, the 10% decile of ˜ ρ t in figure 6 is ∼ . 3, while the 90% decile is ∼ . 7. From figure 7, we notethat even among couples of assets, correlations present a certain degree of heterogeneity. This explains the betterin-sample fit found for the hyperspherical coordinates over the equicorrelation parameterization.While the U-shape and the increasing pattern of correlations are stable in time, there are other patterns which areobserved in correspondence of specific events. We plot in figures 9, 10 the quantities ˜ d jt , ˜ h jt , ˜ δ jt and ˜ ρ jt computed incorrespondence of two meetings of the Federal Open Market Committee (FOMC) in 2014, the first on 30-04-2014 andthe second on 18-06-2014. Compared to the average intraday patterns, we observe significant deviations in the intervalbetween 14:00 and 15:00. This time window coincides with the press conference in which economic news are releasedby the central bank. The local-level model, exploiting all available 1-second data, instantaneously updates covariancesand allows to reconstruct in high resolution the dynamic interaction of covariances with external information. This20 -4 Average D t -4 Average H t1/2 t t Figure 7: Average intraday patterns of the estimated time-varying parameters D t , H / t , δ t and R t . For each t = 1 , . . . , D t , H / t , δ t and R t (over days).mechanism shares some similarities with the macroeconomic literature on nowcasting, where dynamic factor modelsbased on Kalman filter with mixed-frequency observations are used to update forecasts of macroeconomic variables(see e.g. Giannone et al. 2008 and Delle Monache et al. 2016). Similarly, the local-level model can be employed asa nowcasting tool for high-frequency data. FOMC events are characterized by an increase of volatilities at 14:00,followed by a rapid decline in the subsequent minutes. The increase of microstructure noise volatility at 14:00 is inagreement with the fact that bid-ask spreads are observed to increase during public announcements. After 15:00,volatilities get back to their average pattern. Correlations significantly increase at 14:00 and turn back to theiraverage pattern at the end of the trading day. Note that, due to data reduction, standard return models would notprovide such high resolution description of covariance dynamics. In this Section we examine the performance of the model as a nowcasting tool for high-frequency data. To thispurpose, we construct intraday global minimum variance portfolios and compare the ex-post realized variance of theportfolio constructed through the local-level model with that of portfolios constructed through alternative methods.21 Figure 8: Average intraday pattern of the first five eigenvalues of the correlation matrix R t .Intraday risk and portfolio management has became increasingly popular due to high-frequency trading, which todayaccounts for about 50% of total trades in the US equity market (see e.g. Karmakar and Paul 2018). Local-level (hyper) Local-level (equi) t -GAS DCC EWMAAvg. portfolio variance 3 . . . . . M (out of 250) 209 173 74 39 35Avg. p -value 0 . . . . . Table 7: Average variance ( × ) of 1-minute global minimum variance portfolios constructed through out-of-samplecovariance forecasts of local-level model (with both hyperspherical coordinates and equicorrelations), t -GAS andDCC. We also report the number of times each model is included in the MCS at 10% confidence level and the dailyaverage of p -values of MCS tests.We adopt the following strategy: on each day j = 2 , . . . , j − 1. We rebalance the portfolios every minute, ending up with 390 portfoliosper day. Following Engle and Colacito (2006) and Patton and Sheppard (2009), we choose as “best covarianceestimator” the one minimizing the ex-post portfolio variance, computed as: σ j = X k =1 ( ˆ w ′ k,j r k,j ) (27)where ˆ w k,j denotes the solution of the global minimum variance problem at minute k of day j and r k,j is the vector of1-minute returns at minute k of day j . We employ the same dataset of n = 10 assets used in the previous analysis. Asbenchmarks, we consider the t -GAS model, the DCC and the EWMA. The latter two are estimated by synchronizing22 -4 Average D t patternD t (30-04-2014) 10:00 11:00 12:00 13:00 14:00 15:00 16:0023456789101112 10 -5 Average H t1/2 patternH t1/2 (30-04-2014)10:00 11:00 12:00 13:00 14:00 15:00 16:00024681012 Average t pattern t (30-04-2014) 10:00 11:00 12:00 13:00 14:00 15:00 16:000.20.30.40.50.60.70.8 Average R t patternR t (30-04-2014) Figure 9: We plot ˜ d jt , ˜ h jt , ˜ δ jt , ˜ ρ jt computed for j = 82, corresponding to 30-04-2014prices at the frequency of one minute, while the t -GAS is estimated through the missing value approach. The ex-post realized portfolio variances provided by the different methods are compared through the Model Confidence Set(MCS) of Hansen et al. (2011). Specifically, we perform a MCS analysis on each day of the dataset.Table 7 reports the results of the analysis. The first line shows the average of σ j , over j = 2 , . . . , t -GAS. The average variance of DCC and EWMA is significantly larger. The secondline shows the number of days each model is included in the MCS at 10% confidence level. The local-level modelbased on hyperspherical coordinates is included 209 times, while the one based on equicorrelations is included 173times. This result confirms what we found in the in-sample analysis, namely that hyperspherical coordinates providelower average losses, but the deterioration due to the equicorrelation assumption is not excessive. As a consequenceof large data reduction and microstructure effects, the t -GAS, the DCC and the EWMA are only included 74, 39and 35 times, respectively. Finally, we report on the third line the average p -values of MCS tests, which furthercorroborates the previous findings. 23 -4 Average D t patternD t (18-06-2014) 10:00 11:00 12:00 13:00 14:00 15:00 16:00456789101112 10 -5 Average H t1/2 patternH t1/2 (18-06-2014)10:00 11:00 12:00 13:00 14:00 15:00 16:00024681012 Average t pattern t (18-06-2014) 10:00 11:00 12:00 13:00 14:00 15:00 16:000.20.250.30.350.40.450.50.550.60.650.7 Average R t patternR t (18-06-2014) Figure 10: We plot ˜ d jt , ˜ h jt , ˜ δ jt , ˜ ρ jt computed for j = 116, corresponding to 18-06-2014 In this paper we analyzed the problem of intraday covariance modelling with noisy and asynchronous prices andproposed a modelling strategy that allows to handle these two effects. Specifically, we proposed to model intradaydata through a local-level model with time-varying covariance matrices. The dynamics of covariances are driven bythe score of the conditional density, allowing to estimate the model in closed form. Asynchronous trading is treatedas a standard missing value problem in state-space models.The main advantage of this approach is that we model the covariances of latent efficient returns rather thanthose of observed returns. In an high-frequency setting, where observations are asynchronous, models for observedreturns are subject to large data reduction if a missing value approach is adopted. On the other hand, if data aresynchronized through previous-tick or other interpolation schemes, a large number of “artificial” zero returns areintroduced, which lead to a downward bias on correlations. Another advantage is that the estimated covariances arerobust to microstructure effects, as the model includes an additive measurement error term.In our simulation analysis we studied the finite sample properties of the maximum-likelihood estimator and foundthat it remains unbiased in scenarios characterized by high levels of asynchronicity. We then showed the advantagesof the proposed methodology over standard correlation models for observed returns. In particular, in presence ofmissing values, the use of the methodology is preferable even under a misspecified data generating process with a24at-tail conditional density. The reason lies in the ability of the model to use all available prices in order to reconstructthe efficient price dynamics and the covariances. In contrast, correctly specified return models are subject to largedata reduction and the quality of their estimates rapidly deteriorates with increasing asynchronicity.The advantages of the methodology are then assessed on empirical data with 1-second transaction data of 10NYSE assets. The in-sample analyses based on the AIC reveals that benchmark return models are heavily affectedby data reduction and that correlations are biased due to market microstructure effects. The analysis of the intradaycovariances shows that, while the first trading hours are dominated by idiosyncratic risk, a market risk factorprogressively emerges in the second part of the trading day, with the correlations among all couples of assets increasinguntil the last few minutes of trading. The fact that all available prices are used to build covariances allows to captureinstantaneously fast changes on covariance dynamics due to macro-news announcements, such as those releasedduring FOMC announcements.Finally, we examined whether the above mentioned advantages are able to produce out-of-sample forecast gains.To this purpose, we designed an intraday out-of-sample portfolio test and compared the ex-post realized varianceof competing estimators. We found that, in most of the days in our dataset, the variance of portfolios constructedthrough the local-level is significantly lower than other portfolio variances. 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The operator vec[ · ], applied to an m × n matrix A , stacks the columns of A into an mn × · ] applied to an n × n matrix stacks its diagonal elements into an n × n × n × n matrix with the elements of the vector in the main diagonal. We also introduce the commutationmatrix C mn , i.e. the mn × mn matrix such that C mn vec A = vec A ′ for every m × n matrix A . The derivative of an m × n matrix function F ( X ) with respect to the p × q matrix X is defined as in Abadir and Magnus (2005), i.e. as the mn × pq matrix computed as ∂ vec( F ( X )) /∂ vec( X ) ′ . A Computation of ˙ v t and ˙ F t Let us define a t = E t − [ X t ] and P t = Cov t − [ X t ]. Due to asynchronous trading, Y t is a vector with n t ≤ n components. Wedefine the n t × n selection matrix Γ t with ones in the columns corresponding to observed prices. The Kalman filter recursionsfor the local-level model (3), (4) are given by: v t = Y t − Γ t a t a t +1 = a t + K t v t F t = Γ t ( P t + H t )Γ ′ t P t +1 = P t ( I n − K t Γ t ) ′ + Q t (A.1)where K t = P t Γ ′ t F − t . If at time t all observations are missing, we set a t +1 = a t and P t +1 = P t + Q , as discussed inDurbin and Koopman (2012). It is convenient to introduce the auxiliary vector of time-varying parameters:˜ f t = diag[ H t ]diag[ D t ] φ t (A.2)The latter is related to f t by the following link-function:˜ f t = L ( f t ) = exp f (1) t ...exp f (2 n ) t f (2 n +1) t ... f ( k ) t (A.3)The Jacobian of the transformation is: J L = (cid:18) ∂ ˜ f t ∂f t ′ (cid:19) = H t n × n n × q n × n D t n × q q × n q × n I q (A.4)Note that, using the chain rule, ∇ t and I t | t − can be expressed as: ∇ t = J L ˜ ∇ t , I t | t − = J L ˜ I t | t − J L (A.5)where: ˜ ∇ t = (cid:20) ∂ log p ( Y t | ˜ f t , F t − , Θ) ∂ ˜ f ′ t (cid:21) ′ , ˜ I t | t − = E[ ˜ ∇ t ˜ ∇ ′ t ] (A.6)which can be computed as in (12), (13), but deriving with respect to ˜ f t rather than f t . We thus focus on ˙ v t = ∂v t /∂ ˜ f ′ t and˙ F t = ∂ vec( F t ) /∂ ˜ f ′ t . As a particular case of the general recursions appearing in Delle Monache et al. (2016), we obtain:˙ v t = − Γ t ˙ a t (A.7)˙ F t = (Γ t ⊗ Γ t )( ˙ P t + ˙ H t ) (A.8) here: ˙ a t +1 = ˙ a t + ( v ′ t ⊗ I n ) ˙ K t + K t ˙ v t (A.9)˙ P t +1 = ˙ P t − ( K t Γ t ⊗ I n ) ˙ P t − ( I n ⊗ P t Γ ′ t ) C nn t ˙ K t + ˙ Q t (A.10)˙ K t = ( F − t Γ t ⊗ I n ) ˙ P t − ( F − t ⊗ K t ) ˙ F t (A.11)˙ Q t = [( D t R t ⊗ I n ) + ( I n ⊗ D t R t )] ˙ D t + ( D t ⊗ D t ) ˙ R t (A.12)Here ˙ H t = ∂ vec( H t ) ∂ ˜ f ′ t , ˙ D t = ∂ vec( D t ) ∂ ˜ f ′ t are n × k matrices given by:˙ H t = . . . . . . . . . . . . . . . | {z } n . . . | {z } n + q (A.13)˙ D t = 12 . . . D t, . . . . . . . . . D t, . . . . . . . . . | {z } n . . . . . . D t,nn | {z } n . . . | {z } q (A.14)The computation of ˙ R t depends on the parameterization. We distinguish the case where the hyperspherical coordinatesare used and the case where the equicorrelation parameterization is used. In the first case, we have:˙ R t = [( Z ′ t ⊗ I n ) C nn + ( I n ⊗ Z ′ t )] ˙ Z t (A.15)The derivative of the element Z ij,t with respect to the hyperspherical angle θ lm,t is given by: ∂Z ij ∂θ lm = i > j, j = m, l ≥ m, l > i − Z ij tan θ ij i < j, l = i Z ij tan θ ij i ≤ j, l < i (A.16)Note that the time index was suppressed for ease of notation. In the second case we have:˙ R t = [0 n × n , ˙ ρ t vec( −I n + J n )] (A.17)where: ˙ ρ t = 12 (cid:18) n − (cid:19) θ t (A.18)(A.18)