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We investigate the dynamics of correlations present between pairs of industry indices of US stocks traded in US markets by studying correlation based networks and spectral properties of the correlation matrix. The study is performed by using 49 industry index time series computed by K. French and E. Fama during the time period from July 1969 to December 2011 that is spanning more than 40 years. We show that the correlation between industry indices presents both a fast and a slow dynamics. The slow dynamics has a time scale longer than five years showing that a different degree of diversification of the investment is possible in different periods of time. On top to this slow dynamics, we also detect a fast dynamics associated with exogenous or endogenous events. The fast time scale we use is a monthly time scale and the evaluation time period is a 3 month time period. By investigating the correlation dynamics monthly, we are able to detect two examples of fast variations in the first and second eigenvalue of the correlation matrix. The first occurs during the dot-com bubble (from March 1999 to April 2001) and the second occurs during the period of highest impact of the subprime crisis

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The analysis of the intraday dynamics of correlations among high-frequency returns is challenging due to the presence of asynchronous trading and market microstructure noise. Both effects may lead to significant data reduction 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 covariance matrices 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 on 1-second NYSE data, shows that opening hours are dominated by idiosyncratic risk and that a market factor progressively emerges in the second part of the day. The method can be used as a nowcasting tool for highfrequency data, allowing to study the real-time response of covariances to macro-news announcements and to build intraday portfolios with very short optimization horizons. . 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 to the IAAE 2017 conference in Sapporo, the 10 th SoFiE conference in New York and the VIECO 2017 conference in Wien. 1A large class of conditional covariance models have been proposed in the econometric literature and their use is widespread in risk and portfolio management at daily or lower frequencies. Popular multivariate dynamic time-series models include the class of multivariate extensions of the univariate GARCH model of Engle (1982) andBollerslev (1986) and the Dynamic Conditional Correlation (DCC) model of Engle (2002). A drawback of these models is that they are misspecified if data are recorded with observational noise and require synchronization in case data are irregularly spaced. As a consequence, they cannot be straightforwardly applied to intraday data, since high-frequency prices are contaminated by microstructure noise and assets are traded asynchronously. Both effects may lead to ignore a large portion of data and can significantly underestimate correlations. The problem of estimating and forecasting intraday volatilities and correlations is, however, of crucial importance in high-frequency finance. For instance, an high-frequency trader is interested in rebalancing the portfolio on an intraday basis and thus needs accurate shortterm covariance forecasts. Similarly, the study of the intraday dependencies of financial assets is useful to examine the 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 can handle both asynchronous trading and microstructure effects. High-frequency log-prices are modeled t...

In this note we consider a generalisation to the metric setting of the recent work [20].In particular, we show that under relatively weak conditions on a metric measure space (X, d, ν), it holds true that u(x) − u(y) d(x, y)where s is a generalised dimension associated to X and [•] L p w is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.Formula (1.3) has been later generalised for anisotropic norms on R N in [29].

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Despite their effectiveness, linear models for realized variance neglect measurement errors on integrated variance and exhibit several forms of misspecification due to the inherent nonlinear dynamics of volatility. We propose new extensions of the popular approximate long-memory heterogeneous autoregressive (HAR) model apt to disentangle these effects and quantify their separate impact on volatility forecasts. By combining the asymptotic theory of the realized variance estimator with the Kalman filter and by introducing time-varying HAR parameters, we build new models that account for: (i) measurement errors (HARK), (ii) nonlinear dependencies (SHAR) and (iii) both measurement errors and nonlinearities (SHARK). The proposed models are simply estimated through standard maximum likelihood methods and are shown, both on simulated and real data, to provide better out-of-sample forecasts compared to standard HAR specifications and other competing approaches.

We investigate the dynamics of correlations present between pairs of industry indices of US stocks traded in US markets by studying correlation based networks and spectral properties of the correlation matrix. The study is performed by using 49 industry index time series computed by K. French and E. Fama during the time period from July 1969 to December 2011 that is spanning more than 40 years. We show that the correlation between industry indices presents both a fast and a slow dynamics. The slow dynamics has a time scale longer than five years showing that a different degree of diversification of the investment is possible in different periods of time. On top to this slow dynamics, we also detect a fast dynamics associated with exogenous or endogenous events. The fast time scale we use is a monthly time scale and the evaluation time period is a 3 month time period. By investigating the correlation dynamics monthly, we are able to detect two examples of fast variations in the first and second eigenvalue of the correlation matrix. The first occurs during the dot-com bubble (from March 1999 to April 2001) and the second occurs during the period of highest impact of the subprime crisis (