A score-driven conditional correlation model for noisy and asynchronous data: an application to high-frequency covariance dynamics

Transcription

1 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 February, 2017 Abstract We propose a new multivariate conditional correlation model able to deal with data featuring both observational noise and asynchronicity. When modelling high-frequency multivariate financial time-series, the presence of both problems and the requirement for positive-definite estimates makes the estimation and forecast of the intraday dynamics of conditional covariance matrices particularly difficult. Our approach tackles all these challenging tasks within a new Gaussian state-space model with score-driven time-varying parameters that can be estimated using standard maximum likelihood methods. Similarly to DCC models, large dimensionality is handled by separating the estimation of correlations from individual volatilities. As an interesting outcome of this approach, intra-day patterns are recovered without the need of any time or cross-sectional averaging, allowing, for instance, to estimate the real-time response of the market covariances to macro-news announcements. JEL codes: C10; C32; C51; D53; D81 Keywords: asynchronous data; microstructure noise; conditional correlation; score-driven models; Kalman filter; dynamic dependence. Buccheri: Scuola Normale Superiore, giuseppe.buccheri@sns.it. Bormetti: Department of Mathematics, University of Bologna, giacomo.bormetti@unibo.it. Corsi: Department of Economics, Ca Foscari University of Venice and City University of London, fulvio.corsi@unive.it. Lillo: Scuola Normale Superiore, fabrizio.lillo@sns.it. We are particularly grateful for suggestions we have received from Maria Elvira Mancino, Davide Delle Monache, Ivan Petrella, Fabrizio Venditti, Giampiero Gallo, Davide Pirino and participants at the XVIII Workshop in Quantitative Finance in Milan. 1

2 1 Introduction Multivariate conditional volatilities and correlation models have been largely applied to low-frequency financial data where prices are synchronized and microstructure effects are negligible. Popular multivariate dynamic time-series models 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). However, these models are misspecified in case data are recorded with observational noise and when observations are not synchronized. Thus, a particularly important field of research where they are difficult to apply is the one related to the study of intraday high-frequency financial data where prices are contaminated by microstructure effects and assets are traded asynchronously. Still, the problem of estimating and forecasting intraday volatilities and correlations of high-frequency asset prices is of crucial interest from both a theoretical and a practical point of view. In this work we propose a new multivariate conditional correlation model able to cope with observations of the underlying process which are asynchronous and affected by noise. This is done by considering a state-space representation with time-varying parameters and treating asynchronicity as a missing value problem. The dynamics of the time-varying parameters is driven by the score of the predictive likelihood, thus leading to a linear and conditionally Gaussian state-space model whose likelihood function can be written down in closed form using the Kalman filter. As described in Delle Monache et al. (2016), the score function is computed by deriving the Kalman filter recursions with respect to the time-varying parameters. This leads to a set of recursions that run in parallel with the Kalman filter. The resulting conditional correlation model uses all the available data, guarantees positive-definite correlation matrices and provides filtered estimates of both the timevarying parameters and the true latent price. Other than being estimated using standard maximum-likelihood techniques, the model allows to deal with large covariance matrices through a two-step maximum likelihood procedure that couples simple univariate models to a parsimonious correlation model. The modelling of asynchronicity as a missing value problem was first developed by Corsi et al. (2015) and Shephard and Xiu (2016) who considered a state-space representation with 2

3 constant parameters for the purpose of estimating the integrated covariance of high-frequency asset prices. Here, we are interested in a different problem, namely the dynamic modelling of the conditional correlation of high-frequency asset prices. In the field of high-frequency financial econometrics, the most tangible effect of asynchronous trading and microstructure noise on covariance estimation is the so-called Epps effect (Epps (1979)), i.e. a downward bias of sample correlations as the sampling interval shrinks. This problem motivated researchers to build estimators of the high-frequency integrated covariance that are robust to noise and asynchronicity. For instance, Hayashi and Yoshida (2005) proposed an unbiased and consistent estimator of the quadratic covariation of a bivariate semimartingale that is observed asynchronously over time. Examples of estimators that also take into account the presence of microstructure noise have been proposed by Aït-Sahalia et al. (2010), Barndorff-Nielsen et al. (2011), Bibinger (2012), Corsi et al. (2015), Shephard and Xiu (2016). Also, there was a parallel interest in the estimation of the intraday instantaneous covariance matrix. This stream of the high-frequency financial econometric literature has received less attention compared to the problem of quadratic covariation estimation and, consequently, is less developed. However, it is similar in spirit to our purpose since we are interested in modelling the intraday dynamics of conditional volatilities and correlations. For instance, Zu and Boswijk (2014) proposed a two-scale realized spot volatility estimator while Bibinger et al. (2014) proposed a spot covariance estimator based on a local method of moments. Generally, allowing for time-varying parameters in a linear and Gaussian state-space representation leads to nonlinear and non Gaussian state-space models. These are parameterdriven models that are estimated using computationally demanding simulation methods (see e.g. Durbin and Koopman (2012)). Observation-driven models are an alternative class of time-varying parameters models. In observation-driven models, parameters change over time based on some nonlinear function of the available observations. This allows having random parameters even though their value is perfectly predictable one-step ahead given the set of past observations. The advantage is that the likelihood function can be tipically written down in closed form and estimation is performed through standard maximum likelihood techniques. See Cox (1981) for a general discussion on observation-driven and parameter- 3

4 driven models. Score-driven models, also known as generealized autoregressive score models (GAS), are a general class of observation-driven models where the dynamics of time-varying parameters is driven by the score of the conditional likelihood. They were introduced in their full generality by Creal et al. (2013) while Harvey and Chakravarty (2008) proposed a conditional volatility model based on the same principle. Many well-known observation-driven models, such as the GARCH model, the exponential GARCH (EGARCH) model of Nelson (1991), the autoregressive conditional duration (ACD) model of Engle and Russell (1998) can be recovered in this framework. Score-driven models have been successfully applied in the recent econometric literature. In particular, they have been used to model dynamic dependencies among asset returns. Creal et al. (2011) proposed a score-driven heavy-tailed model for timevarying volatilities and correlations and apply it to the study of the dynamics of correlations among daily data. Our model is conditionally Gaussian but allows for an additive noise term. As a consequence, the computation of the score function is different in that it requires the use of the Kalman filter. Moreover, asynchronous data can be efficiently handled in our model. Koopman et al. (2015) considered a score-driven discrete copula distribution to model the intraday dependence structure of high-frequency returns. This allows to investigate the full intraday dependence pattern of the data, beyond correlation structures. However, they ignored the presence of microstructure noise and asynchronous trading. The computation of the score function of time-varying parameters in a linear and Gaussian state-space representation requires the evaluation of some derivatives of the Kalman filter recursions with respect to the time-varying parameters. This approach was described by Creal et al. (2008) for a univariate local-level model with time-varying variances. Instead, we propose a local-level model with time-varying covariance matrices that can be regarded as the multivariate generalization of the latter. Recently, the estimation of state-space models with score-driven parameters has been described in its full generality by Delle Monache et al. (2016) and Delle Monache et al. (2015) in the context of inflation and GDP forecasting. In a similar fashion to DCC models, we estimate covariances in two steps. In the first step individual volatilites are estimated using independent univariate local-level models with time-varying variances and in a further step correlations are estimated conditionally on the 4

5 previous volatility estimates. The problem of estimating a large dimensional model is thus reduced into numerically simpler problems. In addition, restrictions can be imposed on the structure of the parameter space of the correlation model in such a way that the number of static parameters scales well with the cross-sectional dimension. Differently from the DCC model, the covariance matrix is parametrized using spherical coordinates (Rapisarda et al. (2007)). This choice not only guarantees positive-definite estimates, but also avoids the typical problem of over-parametrization present in the DCC and makes the static parameters of the model identifiable. We also discuss two possible extensions of the model in the appendix. The first is the incorporation of autocorrelation in the noise term, which is relevant in certain empirical situations (see e.g. Aït-Sahalia et al. (2011)). The second is a reduced rank version of the model that allows to reduce the number of both dynamic and static parameters when data exhibit a factor structure. By performing extensive Monte-Carlo experiments, we test the performance of the model in estimating several dynamic patterns of volatilities and correlations in a broad range of scenarios that closely mimic real market conditions. We find that our methodology provides superior performances than using standard techniques that are typically employed when dealing with noisy and asynchronous data. Application to transaction data of the 25 most frequently traded assets in the NYSE in 2014 shows that volatilities exhibit the well-known U-shape while correlations are small at the opening and then increase, especially during the first two hours, until the last 15 minutes of the trading day, after which they tend to decrease. We study variations of the intraday patterns over time and how the release of relevant information affects the day-specific dynamics of volatilities and correlations compared to their average patterns. Indeed, since intraday patterns can be recovered separately for each day, our estimator is able to capture in real-time abrupt changes of volatilities and correlations following particular events (e.g. macro-news announcements) occurring on specific days. The remaining part of paper is organized as follows: in Section (2) we introduce the methodology in its full generality. In Section (3) we discuss the estimation method and introduce the correlation model and the univariate variance model. In Section (4) we describe the parametrization of the correlation matrix. Section (5) discusses the results of Monte- Carlo experiments. In Section (6) we show the application of the model to empirical data 5

6 and finally Section (7) concludes. 2 Model 2.1 General framework Let t [0, S] and denote by X t = (X 1 t,..., X n t ) an n-dimensional vector of intraday efficient log-prices. We consider T equally-spaced observation times 0 t 0 < t 1 < < t T 1 S and propose to model X ti, i = 0,..., T 1 as a random walk with heteroskedastic Gaussian innovations: X ti+1 = X ti + η ti, η ti NID(0, Q ti ) (1) Due to microstructure effects, observed prices are contaminated by noise. This implies that we do not directly observe the efficient log-price X t but a blurred version of it. Let Y t i be the n-dimensional vector of observed log-prices. Due to asynchronous trading, the components of Y t i corresponding to assets that are traded at time t i are observed while observations of the remaining components are missing. Let us denote by Y ti the n i, 0 < n i n components 1 of Y t i that are observed at time t i. For i = 0,..., T 1 define an n i n matrix Γ i such that Y ti = Γ i Y t i. We write the observed log-price vector Y ti as: Y ti = Γ i X ti + Γ i ɛ ti, ɛ ti NID(0, H ti ) (2) where ɛ ti is a Gaussian disturbance term which is assumed to be uncorrelated over time and independent on the efficient log-price X ti. The generalization to autocorrelated noise is simple in our setting and is discussed in Appendix (A.1). In line with Corsi et al. (2015) and Shephard and Xiu (2016), we assume that the noise covariance matrix H t is diagonal. 2.2 State-space representation Eq. (2) describes the measurement process of the latent efficient log-price, while eq. (1) describes the dynamics of the latter at discrete times. Thus, the disturbance term ɛ t can 1 The case n i = 0 is discussed in Par. (2.2). 6

7 be thought of as summarizing microstructure effects while η t is the idiosyncratic innovation driving X t. We can rewrite both equations as: Y t = Γ t X t + ɛ t, ɛ t NID(0, H t ) (3) X t+1 = X t + η t, η t NID(0, Q t ) (4) where, without loss of generality, we have set t i+1 t i = 1 for i = 0,..., T 1 and H t Γ t H t Γ t. Note that, because of missing values, the dimensionality of the observation vector Y t changes over time. The level of complexity of econometric inference on the discrete-time model (3)-(4) depends on the kind of dynamics of the time-varying parameters H t and Q t. In the language of Cox (1981), we distinguish between parameter-driven and observation-driven models. In the first class of models, time-varying parameters follow a dynamic process with idiosyncratic innovations. In this case it is not possible to write down the likelihood in closed form and parameter estimation requires the use of computationally demanding simulation methods (see e.g. Durbin and Koopman (2012)). In the second class of models, parameters evolve randomly over time but their value is perfectly predictable one-step-ahead since they are deterministic functions of present and past observations. In this case the likelihood can be typically written down in closed form and parameters are estimated through standard maximum likelihood techniques. We model time-varying parameters in an observation-driven framework and assume that H t and Q t are known given the set of observations Y t 1 = {Y 1,..., Y t 1 } of the log-price process up to time t 1 and a vector Θ of static parameters: H t = H t (Y t 1, Θ) (5) Q t = Q t (Y t 1, Θ) (6) Under this assumption, conditional on Y t 1, model (3)-(4) is a linear and Gaussian statespace representation, implying that it is susceptible of treatment with the Kalman filter (see e.g. Harvey (1991)) and its log-likelihood function can be written down in closed form. Denoting by x t t and P t t the conditional mean and variance of X t given Y t and by x t+1 and P t+1 the conditional mean and variance of X t+1 given Y t, the Kalman filter recursions can 7

8 be written as (see e.g. Durbin and Koopman (2012)): In case at time t all the elements of Y t v t = Y t Γ t x t, F t = Γ t (P t + H t )Γ t, K t = P t Γ tf 1 t (7) x t t = x t + K t v t, P t t = P t (I n Γ tk t) (8) x t+1 = x t t, P t+1 = P t t + Q t (9) are missing, the matrix Γ t is no longer defined. In this case, following Durbin and Koopman (2012), we apply the above Kalman filter recursions by setting Γ t equal to an n n matrix of zeros. Note also that, since the latent process X t is non-stationary, the Kalman filter requires diffuse initialization, i.e. we set x 1 = 0 and P 1 = κi n with κ. Given the above Kalman filter recursions, the log-likelihood function log L(Y 1,..., Y T ) = T log p(y t Y t 1 ) (10) i=1 can be computed in the prediction error decomposition form: log L(Y 1,..., Y T ) = nt 2 log 2π 1 2 T t=1 ( log Ft + v tf 1 t v t ) (11) The advantages of having a linear and Gaussian state-space representation are numerous. Beside the fact that the model can be estimated using standard maximum likelihood techniques, another relevant aspect is that Kalman filter recursions still hold in presence of missing observations. The only modification is the introduction of the matrix Γ t which selects the components of Y t that are actually observed at time t. Therefore, the problem of modelling asynchronicity is equivalent to having missing values in the observation vector Y t and these are easily tackled by the Kalman filter. In case H t and Q t are constant over time, model (3)-(4) is a well-known local level (LL) model with constant parameters that is studied, among others, by Durbin and Koopman (2012). In the time invariant case, estimation of the LL model is conveniently performed using the expectation maximization (EM) method as described in Shumway and Stoffer (1982) and Durbin and Koopman (2012). Instead, we are interested in the time-varying case. In order to apply the Kalman filter recursions (7), (8), (9) and write down the loglikelihood function, we need to specify the law of motion of H t and Q t. 8

9 2.3 The LLSD model A general framework for observation-driven models is provided by GAS models. Let us collect all the time-varying parameters of the model in a k-dimensional column vector f t. The one-step-ahead GAS(1,1) update is given by: f t+1 = ω + As t + Bf t (12) where ω is a k 1 vector, A, B are k k matrices and s t is the scaled derivative of the log-density with respect to the time-varying parameter vector f t. We write it as: s t = S t t (13) where [ ] log p(yt f t, Y t 1 ; Θ) t = (14) The vector Θ of static parameters includes the elements of ω, A and B. S t is a scaling matrix that is known at time t. Its role is to correct the size of the score driven step based on the local curvature of the likelihood. Different choices of S t lead to different classes of GAS models. Creal et al. (2013) discussed several choices of S t based on the Fisher information matrix: I t t 1 = E t 1 [ t t] (15) In our application to high-frequency financial data, we will use S t = (I t t 1 ) 1/2, i.e. we take the principal square root matrix of the inverse of I t t 1. Note that E t 1 [s t ] = 0 and, given our choice of the scaling matrix S t, Cov t 1 [s t ] = I k. Therefore, s t is a martingale difference sequence with constant conditional covariance matrix. This implies that model (12) is covariance stationary whenever the eigenvalues of B lie inside the unit circle. Thus, GAS models provide a general rule for updating time-varying parameters based on the full shape of the conditional density of present observations. Compared to other kinds of observation-driven models, GAS models are optimal in the Kullback-Leibler (KL) sense, meaning that the GAS update minimizes the KL divergence between the true conditional density p(y t f t ) of a generic misspecified DGP and the one implied by the observation-driven update p(y t f t ; Θ) (Blasques et al. (2015)). This implies that observation-driven models with f t 9

10 a GAS-like update (e.g. the GARCH model) are optimal in the KL sense, while observationdriven models whose update rule is not in the GAS class (e.g. the DCC model) are not. Moreover, Koopman et al. (2016) showed that score-driven time series models have a similar forecasting performance as correctly specified non-linear non-gaussian state space models over a wide range of model specifications. Motivated by the generality and flexibility of score driven approaches, we propose to model dynamically volatilities and correlations in the multivariate LL model using the score of the predictive likelihood. The resulting model is a multivariate local level with score-driven time-varying parameters (LLSD). The time-varying parameters in the LLSD are the elements of the diagonal noise covariance matrix H t and those in the latent state covariance matrix Q t. As in the DCC model and in the multivariate t-gas model of Creal et al. (2011), we decompose Q t as: Q t = D t R t D t (16) where R t is the correlation matrix and D t is the diagonal matrix of standard deviations. Differently from the DCC model, the correlation matrix R t is decomposed as: R t = Z tz t (17) where Z t is an n n matrix containing k Z = n(n 1)/2 time-varying parameters. The parametrization of Z t will be discussed in detail in Section (4). For the time being, note that the present decomposition of Q t and R t guarantees that both matrices are positivedefinite by construction. The k-dimensional time-varying parameter vector f t includes the 2n elements of H t, D t and the k Z elements of Z t and therefore k = 2n + n(n 1)/2 is the number of time-varying parameters in the LLSD model. 3 Estimation A major challenge in modelling time-varying correlations is handling large dimensionality. The DCC model has a clear computational advantage over multivariate GARCH models in that the number of parameters to be estimated in the correlation process is independent on 10

11 the cross-sectional dimension n. This is possible if one uses a two-step maximum likelihood procedure where one first separately estimates n GARCH models and then inputs the estimated variances into the DCC. If one assumes that correlations have the same persistence, the number of static parameters to be estimated in the last step will be independent on the dimension n. Clearly, this estimation procedure implies a loss of efficiency since variances are estimated without any regard to the multivariate information contained in the dataset. However, for practical implementation, it turns out to be very convenient since it allows to considerably reduce the number of parameters that have to be estimated at the same time. Thanks to the particular structure of the LL model, this relevant feature of the DCC can be incorporated into the LLSD model. In the language of Harvey (1991), the multivariate LL model is a particular case of a Seemingly Unrelated Time Series Equation (SUTSE) model in that each series is modelled as in the univariate case but disturbances are correlated across series. Therefore, one can employ n univariate LLSD models to separately estimate the time-varying parameters in H t and D t, i.e. the time-varying variances of the noise and the standard deviations of the latent state. In a further step, the time-varying correlations in R t can be estimated as if H t and D t were known and given by the previous estimates. In the following paragraphs we describe in detail how the model is estimated using the two-step maximum likelihood procedure. First, we discuss the estimation of the time-varying parameters in Z t. The k Z -dimensional time-varying parameter vector will be denoted as ft Z. The results that will be recovered in this paragraph are independent on the parametrization of Z t. We refer to the dynamics of ft Z as the correlation model. Second, we discuss the estimation of the noise covariance matrix H t and the matrix of standard deviations D t of the latent process. We will consider n independent univariate LLSD models where the timevarying parameter vector is a 2-dimensional vector that we will denote as f t,i for i = 1,..., n. This univariate specification coincides with the one proposed by Creal et al. (2008). In practice, one first estimates the n univariate models and then uses the estimated matrices H t and D t as an input to the correlation model. However, since some results related to the univariate model can be recovered as a particular case of those related to the correlation model, we first discuss the latter and then introduce the univariate model. Before showing the main results, we introduce some notation. We use to denote the 11

12 Kronecker product between two matrices. The operator vec( ), applied to an m n matrix A, stacks the columns of A into an mn column vector while the operator vech( ), applied to a symmetric n n matrix B, stacks all the n(n 1)/2 upper (or lower) diagonal elements into a column vector. We also introduce the commutation matrix K mn, i.e. the mn mn matrix such that K mn veca = veca 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) The correlation model Let us consider the multivariate LL model (3), (4) and assume that H t and D t are known. As underlined above, f Z t is a k Z -dimensional column vector that includes all the time-varying parameters in Z t. The k Z 1 score vector t and the k Z k Z Fisher information matrix I t t 1 appearing in eq. (12) can be computed in the following way. Denoting by v t and F t the derivatives of v t and F t with respect to f t, namely F t = vec(f t )/ f t, v t = v t / f t, it can be shown that: t = 1 2 I t t 1 = 1 2 [ F ] t(i nt Ft 1 )vec(i nt v t v tf t 1 ) + 2 v tf t 1 v t Ft 1 ) F ] t + 2 v tf t 1 v t [ F t(f t 1 A componentwise version of the above formulas is present in Creal et al. (2008) while the multivariate formulation presented here was derived in Delle Monache et al. (2016). as: where: The two terms F t and v t needed to evaluate t and I t t 1 can be computed recursively (18) (19) v t = Γ t ẋ t (20) F t = (Γ t Γ t ) P t (21) ẋ t+1 = ẋ t + (v t I n ) K t + K t v t (22) P t+1 = P t (K t Γ t I n ) P t (I n P t Γ t)k nn K t + Q t (23) K t = (Ft 1 Γ t I n ) P t (Ft 1 K t ) F t (24) 12

13 Q t = (D t D t )Ṙt (25) Ṙ t = (Żt I n )K nn Ż t + (I n Z t)żt (26) The above formulas are recovered by deriving the Kalman filter recursions (7), (8) and (9) with respect to f t and are a particular case of the general recursions in presence of missings appearing in Delle Monache et al. (2016). Note that D t and H t are not derived since they are freezed and equal to the output of the univariate estimation. The computation of the Fisher information matrix I t t 1 by direct differentiation of the Kalman filter equations is known to become numerically unstable as I t t 1 becomes close to singular. As suggested by Creal et al. (2013), in order to obtain a numerically stable scheme, we introduce an EWMA smoothing method. That is, instead of I t t 1, we use its smoothed version I s t t 1 : I s t t 1 = λi s t 1 t 2 + (1 λ)i t t 1, λ [0, 1] (27) If λ is equal to zero, we recover the standard unstable scheme. If λ > 0, some weight is given to past values and the smoothed matrix I s t t 1 turns out to be less erratic. The parameter λ can be considered as part of the static parameter vector Θ and is estimated by maximum likelihood. At time t, the one-step-ahead prediction of the time-varying vector f Z t GAS(1,1) updating scheme: is given by the f Z t+1 = ω Z + A Z s t + B Z f Z t (28) where ω Z, A Z and B Z are static parameters of appropriate dimensions that are estimated through maximum-likelihood. Similarly to the DCC model, the dimension of the parameter space can be reduced by choosing A Z and B Z to be diagonal, eventually with the same element in the main diagonal. The last choice makes the dimension of the parameter space of the correlation model independent on the cross-section dimension n. We will discuss in detail the constraints on the vector ω Z and matrices A Z and B Z in our empirical application to NYSE tick data. The scaled score s t is computed using eq. (18), (19) and the derivative recursions (20)- (23) which run in parallel with the Kalman filter recursions (7), (8) and (9). The conditional 13

14 log-likelihood at time t is given in the prediction error decomposition form: l Z t = n 2 log 2π 1 2 ( log Ft + v tf 1 t v t ) Therefore, given a time-series of observed log-price vectors Y 1,..., Y T and a point Θ Z parameter space, the log-likelihood function is written as: (29) in the log L Z (Y 1,..., Y T Θ Z) = T lt Z (30) t=1 where l Z t are evaluated as in (29) using the Kalman filter recursions and the GAS update equation with parameters given by Θ Z. The maximum likelihood estimate of the static parameter vector Θ Z is simply obtained by numerically optimizing the log-likelihood function: ˆΘ Z = argmax log L Z (Y 1,..., Y T Θ) (31) Θ Given a linear and conditionally Gaussian state-space representation, regularity conditions under which the MLE of the static parameters is consistent and asymptotically normal are discussed in detail by Harvey (1991). The most relevant condition to be satisfied is that Θ is identifiable. In the LLSD this is the case since, as we will discuss in detail in Section (4), the use of the TAP parametrization avoids the introduction of redundant parameters. This assumption, together with additional regularity conditions on the derivatives of the log-likelihood function, guarantees that ˆΘ Z is consistent and, as T where Θ Z denotes the true parameter vector and I ΘZ computed on Θ Z. s( ˆΘZ Θ Z ) d N(0, I 1 Θ Z ) (32) is the asymptotic information matrix However, since the LLSD is estimated using a two-step maximum likelihood procedure, consistency of the second step depends on consistency of the first step. Following the same argument in Engle (2002), under the assumption that the log-likelihood function log L Z is continuous in a neighborhood of the true parameters, consistency of the first step is sufficient to guarantee consistency of the second step. Also, the asymptotic error covariance matrix must be corrected using the two-step Generalized Method of Moments in Newey and McFadden (1986). Let us denote by Θ V a vector collecting all the static parameters of 14

15 the n univariate models and by l V t (Y t Θ V ) = n i=1 l t,i the sum at time t of the conditional log-likelihood functions of the n univariate models in eq. (42). The moment conditions corresponding to the first and second steps are: T ΘV lt V (Y t Θ V ) = 0, t=1 T ΘZ lt Z (Y t Θ Z, ˆΘ V ) = 0 (33) where ˆΘ V is the value of Θ V satisfying the moment conditions at the first step. Define: where we have denoted by Θ Z and Θ V t=1 m(y, Θ V ) = ΘV l V t (Y Θ V ) g(y, Θ Z, Θ V ) = ΘZ l Z t (Y Θ Z, Θ V ) G ΘZ = E[ ΘZ g(y, Θ Z, Θ V )] G ΘV = E[ ΘV g(y, Θ Z, Θ V )] g(y ) = g(y, Θ Z, Θ V ) M = E[ ΘV m(y, Θ V )] ψ(y ) = M 1 m(y, Θ V ) the true values of the static parameters. Under standard regularity assumptions that are summarized in Theorem 3.4 of Newey and McFadden (1986), the asymptotic error covariance matrix V is given by Theorem 6.1 of Newey and McFadden (1986): V = G 1 Θ Z E{[g(Y ) + G ΘV ψ(y )][g(y ) + G ΘV ψ(y )] }G 1 Θ Z (34) The univariate LLSD model We show here how the time-varying parameters in H t and D t can be estimated separately using the univariate version of the LLSD model. If n = 1, model (3)-(4) becomes a univariate LL model with time-varying variances: Y t,i = X t,i + ɛ t,i, ɛ t N(0, σt,i) 2 (35) X t+1,i = X t,i + η t,i, η t N(0, ξt,i) 2 (36) for i = 1,..., n. This model is the same as the one introduced by Creal et al. (2008). The time-varying parameters are σt,i 2 and ξt,i. 2 In order to recover positive estimates of the 15

16 variance terms, we consider the two time-varying parameter vectors: ( ) ( ) log σ 2 f t,i = t,i σ 2, log ξt,i 2 ft,i = t,i ξt,i 2 which are related by the following link-function: ( ) exp f 1 f t,i = L(f t,i ) = t,i exp ft,i 2 (37) (38) The Kalman filter recursions are simply given as a special case of (7), (8) and (9): v t,i = Y t,i x t,i, F t,i = P t,i + f 1 t,i, K t,i = P t,i F t,i (39) x t t,i = x t,i + K t,i v t,i, P t t,i = P t,i (1 K t,i ) (40) x t+1,i = x t t,i, P t+1,i = P t,i (1 K t,i ) + f 2 t,i (41) The conditional log-likelihood at time t is computed as: l t,i = 1 2 log 2π 1 ( ) log F t,i + v2 t,i 2 F t,i In case at time t observation Y t,i is missing, the above Kalman filter recursions and the expression (42) of the conditional log-likelihood are still valid, provided that one takes K t,i = 0, v t,i = 0 and F t,i = 1. The vector f t,i is driven by the scaled score of the predictive likelihood: (42) s t,i = S t,i t,i (43) where t,i = ( log l t,i )/( f t,i) and S t,i = (I i t t 1 ) 1/2, I i t t 1 = E t 1[ t,i t,i]. Denoting by J L the Jacobian matrix of the link function J L = exp f 1 t,i 0 0 exp f 2 t,i (44) we have: t,i = J L t,i, I i t t 1 = J L Ĩ i t t 1J L (45) where t,i = ( log l t,i )/( f t,i) and Ĩi t t 1 = E t 1[ t,i t,i ]. The latter are computed as special cases of (18) and (19): t,i = 1 2 [ ( ( ) 2 F vt,i t,i 1 F t,i 16 F t,i ) 2 v t,i v t,i F t,i ] (46)

17 Ĩ t t 1,i = 1 2 [ F t,i F ] t,i + 2 v t,i v t,i Ft 2 F t (47) Note that now dotted variables denote derivatives with respect to f t. Similarly to the multivariate case, the latter are computed by deriving the Kalman filter recursions (39), (40) and (41). One obtains: v t,i = ẋ t,i (48) F t,i = P t,i + (1, 0) (49) P K t,i = t,i K t,i F t,i (50) F t,i ẋ t+1,i = ẋ t,i (1 K t,i ) + K t,i v t,i (51) P t+1,i = P t,i (1 K t,i ) P t,i K t,i + (0, 1) (52) Therefore, the GAS(1,1) update equation in the univariate LLSD is given by: f t+1,i = ω i + A i s t,i + B i f t,i (53) where ω i is a two-dimensional column vector, while A i and B i are two-dimensional square matrices. They are part of the vector Θ i of static parameters that is estimated by maximum likelihood. As in the multivariate case, we introduce an EWMA smoothing scheme for Ĩ t t 1,i and the corresponding smoothing parameter λ i is estimated together with the GAS parameters. Constraints on ω i, A i and B i depend on the specific data at hand and will be discussed in detail in our empirical application. As in the correlation model, the likelihood is maximized numerically. Once all of the n univariate models have been estimated, one can construct the two matrices H t and D t as: exp f t,1 1 0 (exp f t,1) 2 1/2 0 H t =....., D t =..... (54) 0 exp ft,n 1 0 (exp ft,n) 2 1/2 4 The TAP parametrization for R t In the DCC model, the correlation matrix R t is decomposed as 1 t P t 1 t where P t is a symmetric and positive definite matrix and t is a diagonal matrix whose nonzero entries 17

18 are the square roots of the diagonal elements of P t. This guarantees that R t is a correlation matrix, meaning that it is positive definite with all the entries in the interval [ 1, 1] and all the diagonal elements being equal to 1. However, this decomposition leads to overparametrization since the total number of parameters of P t is n(n + 1)/2 while the number of independent parameters of R t is n(n 1)/2. As a consequence, the Fisher information matrix (19) turns out to be singular and the static parameters are not identifiable 2. In our correlation model, the scaling matrix S t appearing in the GAS dynamics is related to the inverse of the information matrix. Also, when applied to the LLSD, the DCC-like parametrization of the correlation matrix does not guarantee positive definite estimates of P t. A parametrization that turns out to be very convenient in this case is the triangular angles parametrization (TAP) introduced in Jaeckel and Rebonato (1999) and used by Creal et al. (2011) in modelling time-varying correlations. In the TAP parametrization, the generic element z ij,t of the matrix Z t is given by: 1 i = j = 1 z ij = cos θ 1j i = 1, j > 1 i 1 k=1 sin θ kj cos θ ij 1 j>i + i 1 k=1 sin θ kj1 i=j i > 1 where we have suppressed the subscript t for ease of notation. Thus, denoting by c ij = cos θ ij and s ij = sin θ ij, the matrix Z t has the following form: 1 c 12 c c 1n 0 s 12 c 23 s c 2n s 1n Z t = 0 0 s 23 s c 3n s 2n s 1n n 1 k=1 s kn The i-th column of Z t contains the spherical coordinates of a vector of unit norm in an i-dimensional subspace of R n, which is parametrized by i 1 angles. Therefore, we have a total of n(n 1)/2 angles [θ 12,t, θ 13,t,..., θ n 1n,t ]. Multiplication of Z t by its transpose results in taking scalar products of unit vectors, which is always in the interval [ 1, 1]. 2 Usually, the n redundant parameters are fixed by targeting the unconditional correlation implied by the DCC model to the sample correlation matrix. 18 (55) (56)

19 The advantage of this parametrization is that the number of free parameters is equal to the number of independent parameters of the correlation matrix. Thus, the Fisher information matrix is full-rank and the static parameters in the GAS dynamics are identifiable. Note that, because of the uniqueness of the Cholesky decomposition, Z t turns out to be equal to the Cholesky factor. Moreover, as pointed out by Rapisarda et al. (2007) every correlation matrix can be written in the TAP form. The only drawback in dealing with the TAP parametrization is that the variables that are modelled are angles and therefore their dynamics is not easy to interpret directly. However, through eq. (17), the dynamics of the angles is immediately converted into the corresponding dynamics of correlations. At time t, the vector f Z t of time-varying parameters is a k Z = n(n 1)/2 dimensional vector containing all the angles θ 12,t, θ 13,t,..., θ n 1n,t. In order to perform recursion (26), we need to know the derivative of Z t with respect to f Z t. This is easily computed by observing that: z ij θ lm = 0 i > j, j m, l m, l > i z ij tan θ ij i < j, l = i 5 Simulation study z ij tan θ ij i j, l < i In order to test the forecasting performance of the model, we design two different simulation settings, one for the correlation model and a different one for the univariate variance model. In the former case, the Monte-Carlo experiments that will be performed are similar to those conducted by Engle (2002) and Creal et al. (2011). While keeping variances constant, we estimate a bivariate LLSD over a set of deterministic and stochastic correlation patterns. Different levels of noise and asynchronicity are simulated in order to closely mimic real market conditions. A relevant advantage of the LLSD is that the conditional correlation matrix is estimated using all the available data. This implies more efficient estimates compared to standard synchronization schemes where a significant fraction of the observations is tipically neglected. (57) We examine the impact of asynchronicity in a higher-dimensional experiment 19

20 where the LLSD estimates are compared to those provided by the DCC model estimated on a synchronized sample. The chosen synchronization scheme is the refresh-time described in Barndorff-Nielsen et al. (2011). The DCC model and other existing multivariate conditional correlation models are misspecified in presence of additive noise. In order to assess the impact of the noise term, we simulate a bivariate LL as in (3), (4) with T = 5000 timestamps. The two matrices H t and D t are kept constant while the correlation in R t follows the deterministic path in figures (1a) - (1d). We denote by δ = ξ 2 i /σ 2 i, i = 1, 2 the signal-to-noise ratio that is assumed to be the same for the two simulated series. If δ is very large (e.g. δ = 10) the estimate provided by the DCC is close to that of the LLSD. However, as δ decreases, the DCC exhibits a downward bias because it is not able to separate the diagonal noise term ɛ t from the innovation term η t in the latent process. Note that, except for the very large value δ = 10, the other values are commonly observed in real markets, as we will see in the empirical application in Section (6). In order to attenuate the effect of the noise on the estimates provided by the DCC, we proceed in two different ways. The first is to compute returns by subsampling at larger timescales, thus neglecting part of the available observations. This has the effect of increasing the signal-to-noise ratio. We name as DCC sub the correlation estimates obtained by the DCC on the new sample with returns computed on a time-scale K > 1. The second method is to apply a pre-averaging procedure (Jacod et al. (2009)). Indeed, since ɛ t is uncorrelated, the variance of the noise term of 1/K K 1 j=0 Y t i j, is 1/K times the variance of the noise of Y ti. Therefore, the effect of the pre-averaging is, again, to increase the signal-to-noise ratio. We name the correlation estimates obtained by the DCC on the pre-averaged sample as DCC pre. In both cases the tuning parameter K is chosen as the one minimizing the mean square error in formula (58) over a sample of N = 100 independent realizations. In figures (1e), (1f) we show the estimates provided by the LLSD, DCC sub and DCC pre for δ = 1 and δ = 0.5, respectively. Differently from the previous case, DCC sub and DCC pre are now able to capture the dynamics of the simulated correlation pattern. 20

21 (a) δ = 10 (b) δ = 2 (c) δ = 1 (d) δ = 0.5 (e) δ = 1 (f) δ = 0.5 Figure 1: (a)-(d) Examples of correlation estimates on a bivariate time-series of T = 5000 timestamps provided by the LLSD and the DCC using different levels of noise. (e), (f) Estimates provided by the LLSD, DCC sub and DCC pre for large noise scenarios. 21

22 In the bivariate and univariate tests, our measures of accuracy are the mean squared error (MSE) and the mean absolute error (MAE) which are computed as: MSE = 1 T T (θ t ˆθ t ) 2, MAE = 1 T t=1 T θ t ˆθ t (58) t=1 where θ t generically denote the simulated variable, i.e. the correlation in the first test and the variance in the second test, while ˆθ t denotes the estimated variable. In the higher dimensional experiment, we use the Frobenius norm F and the 1 norm 1 defined as: R t ˆR n n t F = (R t,ij ˆR t,ij ) 2, R t ˆR n t 1 = max R t,ij ˆR t,ij (59) 1 j n i=1 j=1 where R t is the simulated correlation matrix and ˆR t is the estimated one. We then average the two over all the timestamps. i=1 5.1 Monte-Carlo design I: time-varying correlations We assume a trading day of 6.5 hours and simulate N = 250 bivariate time series of T = observations from the time-varying local level model (3), (4). Following Engle (2002) and Creal et al. (2011), we keep the two variances σt,i, 2 ξt,i, 2 i = 1, 2 constant while the offdiagonal element of the correlation matrix R t, that we will denote as ρ t, evolves dynamically according to the following patterns: 1. Sine ρ t = a s + b s sin(c s πt) 2. Fast Sine ρ t = a f + b f sin(c f πt) 3. Step ρ t = α s β s (t > γ s ) 4. Ramp ρ t = 1 b r mod (t + a r, b r ) 5. Model ρ t = exp (h t )/(1 + exp (h t )) where h t follows an AR(1) process: h t+1 = c m + b m h t + a m φ t, φ t N(0, 1) 22

23 The parameters appearing in the dynamics of the correlation coefficient ρ t are chosen in the following way: a s = a f = 0, b s = b f = 0.4, c s = 7/(2T ), c f = 101/(2T ), α s = 0.25, β s = 0.5, γ s = T/2, b r = T/20, a r = b r /2, b m = 0.997, c m = 0.4(1 b m ), a m = The variance ξt,i 2 of the latent price process is constant and equal to 0.1 for all the simulated patterns. Instead, the variance σt,i 2 of the noise is computed based on the chosen signalto-noise ratio δ which is varied in the simulations in order to reproduce different scenarios: δ = 2 (high signal), δ = 1 (moderate signal) and δ = 0.5 (low signal). As we will see in Section (6), these three scenarios are close to those observed in real data. Asynchronicity is reproduced by censoring the simulated observations using Poisson sampling. The probability of missing values is set equal to Λ = 0 (no missing values), Λ = 0.5 (moderate missing) and Λ = 0.8 (high missing). The last two scenarios are commonly observed in real transaction data as will be shown in the next section. In the simulations each signal-to-noise ratio scenario is combined to each missing values scenario in order to study the behaviour of the estimator under a multiplicity of realistic market conditions. Figures (2a) - (2e) show the simulated correlation patterns and the corresponding LLSD estimates in the scenario δ = 1, Λ = 0.5. In order to estimate the LLSD, we first estimate the parameter constant version of the local level model using the EM algorithm (as described in Corsi et al. (2015)). Then, the two-step maximum likelihood procedure is applied. We first estimate the univariate variance model separately for each of the two series. In doing this, we fix ω i R 2, i = 1, 2 by targeting the unconditional expectation in the GAS process to the constant parameters σ i and ξ i found by the EM algorithm. Thus, we have: ω i = (I 2 B i ) f i (60) where B i is diagonal and f i = (log σ i 2, log ξ i 2 ). The estimated time-varying variances are then input into the correlation model. The latter has a covariance stationary GAS specification for all the five patterns. In particular, we impose the constraint B Z < 1 in the numerical maximization and fix ω Z by targeting the unconditional correlation to the constant level ρ estimated by the EM algorithm: ω Z = (1 B Z ) f (61) 23

24 (a) Sine (b) Step (c) Fast sine (d) Ramp (e) Model (f) CIR Figure 2: (a)-(e) Examples of correlation estimates provided by the LLSD for the five simulated patterns in Section (5.1) in the scenario Λ = 0.5, δ = 1 for a bivariate timeseries of T = timestamps. (f) LLSD estimate of the time-varying variance of the latent log-price process. The horizontal dashed line denotes the value of the noise variance. 24

25 where f = arccos ρ. Therefore, there are three elements in the vector Θ Z of static parameters: the two scalar GAS parameters A Z and B Z and the smoothing parameter λ. The average MSE (MAE) of the LLSD is compared to that of DCC sub and DCC pre in Table (1). In the two scenarios Λ = 0.5 and Λ = 0.8 the DCC sub and DCC pre models are estimated on a synchronized sample obtained using the refresh-time scheme. The LLSD outperforms the DCC sub and DCC pre on all of the simulated scenarios while the DCC pre outperforms the DCC sub in most of the cases. In the no missing scenario, the relative difference between the LLSD and the other two estimators is large for all the simulated patterns. Indeed, the Kalman filter is able to optimally estimate the latent state, thus allowing to capture the dynamics of correlations even in presence of very noisy observations. In the two scenarios (Λ = 0.5, 0.8), the effect of the noise diminishes because the average duration between observations increases. However, the use of the refresh-time implies neglecting a significant part of the available data and at the same time alters the instantaneous correlation structure of the observations. Indeed, we note that the relative difference between the LLSD and the other two estimators is larger in case of patterns with abrupt changes (e.g. step and ramp) and in stochastic patterns (model) while it is more moderate in case of smooth patterns (e.g. sine and fast sine). In order to provide a better understanding of the advantages of our methodology in dealing with asynchronous data, we design a new Monte-Carlo experiment where 5 dimensional time-series of T = timestamps are simulated using the DCC as a DGP. The simulated observations are then randomly censored as in the previous experiment. We consider the following scenarios characterized by increasing probabilities of missing values: Λ 1 = [0.1, 0.1, 0.1, 0.1, 0.1] Λ 2 = [0.1, 0.1, 0.1, 0.1, 0.8] Λ 3 = [0.3, 0.3, 0.3, 0.3, 0.3] Λ 4 = [0.5, 0.5, 0.5, 0.5, 0.5] Λ 5 = [0.8, 0.8, 0.8, 0.8, 0.8] 25

26 Estimator Sine Fast Step Ramp Model Sine Fast Step Ramp Model Sine Fast Step Ramp Model δ = 0.5 δ = 1 δ = 2 Λ = 0 LLSD DCCsub DCCpre LLSD DCCsub DCCpre Λ = 0.5 LLSD DCCsub DCCpre LLSD DCCsub DCCpre Λ = 0.8 LLSD DCCsub DCCpre LLSD DCCsub DCCpre Table 1: Results of the Monte-Carlo experiment I for all the 9 scenarios described in Section (5.1). For each scenario and for each of the five simulated correlation patterns, we show the average MSE (MAE) of the LLSD, DCCsub and DCCpre estimated on bivariate time-series of T = timestamps and based on N = 250 independent repetitions. 26