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This paper shows that a large dimensional vector autoregressive model (VAR) of finite order can generate fractional integration in the marginalized univariate series. We derive high-level assumptions under which the final equation representation of a VAR(1) leads to univariate fractional white noises and verify the validity of these assumptions for two specific models.
This paper proposes a new model with time-varying slope coefficients. Our model, called CHAR, is a Cholesky-GARCH model, based on the Cholesky decomposition of the conditional variance matrix introduced by Pourahmadi (1999) in the context of longitudinal data. We derive stationarity and invertibility conditions and prove consistency and asymptotic normality of the Full and equation-by-equation QML estimators of this model. We then show that this class of models is useful to estimate conditional betas and compare it to the approach proposed by Engle (2016). Finally, we use real data in a portfolio and risk management exercise. We find that the CHAR model outperforms a model with constant betas as well as the dynamic conditional beta model of Engle (2016).
An estimator of the ex-post covariation of log-prices under asynchronicity and microstructure noise is proposed. It uses the Cholesky factorization of the covariance matrix in order to exploit the heterogeneity in trading intensities to estimate the different parameters sequentially with as many observations as possible. The estimator is positive semidefinite by construction. We derive asymptotic results and confirm their good finite sample properties by means of a Monte Carlo simulation. In the application we forecast portfolio Value-at-Risk and sector risk exposures for a portfolio of 52 stocks. We find that the dynamic models utilizing the proposed high-frequency estimator provide statistically and economically superior forecasts.
The properties of dynamic conditional correlation (DCC) models, introduced more than a decade ago, are still not entirely known. This paper fills one of the gaps by deriving weak diffusion limits of a modified version of the classical DCC model. The limiting system of stochastic differential equations is characterized by a diffusion matrix of reduced rank. The degeneracy is due to perfect collinearity between the innovations of the volatility and correlation dynamics. For the special case of constant conditional correlations, a nondegenerate diffusion limit can be obtained. Alternative sets of conditions are considered for the rate of convergence of the parameters, obtaining time-varying but deterministic variances and/or correlations. A Monte Carlo experiment confirms that the often used quasi-approximate maximum likelihood (QAML) method to estimate the diffusion parameters is inconsistent for any fixed frequency, but that it may provide reasonable approximations for sufficiently large frequencies and sample sizes.
We propose a bootstrap-based test of the null hypothesis of equality of two firms’ conditional risk measures (RMs) at a single point in time. The test can be applied to a wide class of conditional risk measures issued from parametric or semiparametric models. Our iterative testing procedure produces a grouped ranking of the RMs, which has direct application for systemic risk analysis. Firms within a group are statistically indistinguishable from each other, but significantly more risky than the firms belonging to lower ranked groups. A Monte Carlo simulation demonstrates that our test has good size and power properties. We apply the procedure to a sample of 94 U.S. financial institutions using ΔCoVaR, MES, and %SRISK. We find that for some periods and RMs, we cannot statistically distinguish the 40 most risky firms due to estimation uncertainty.
Financial asset prices occasionally exhibit large changes. To deal with their occurrence, observed return series are assumed to consist of a conditionally Gaussian ARMA-GARCH type model contaminated by an additive jump component. In this framework, a new test for additive jumps is proposed. The test is based on standardized returns, where the first two conditional moments of the non-contaminated observations are estimated in a robust way. Simulation results indicate that the test has very good finite sample properties, i.e. correct size and high proportion of correct jump detection. The test is applied to daily returns and detects less than 1% of jumps for three exchange rates and between 1% and 3% of jumps for about 50 large capitalization stock returns from the NYSE. Once jumps have been filtered out, all series are found to be conditionally Gaussian. It is also found that simple GARCH-type models estimated using filtered returns deliver more accurate out-of sample forecasts of the conditional variance than GARCH and Generalized Autoregressive Score (GAS) models estimated from raw data.
Simple low order multivariate GARCH models imply marginal processes with a lot of persistence in the form of high order lags. This is not what we find in many situations however, where parsimonious univariate GARCH(1,1) models for instance describe quite well the conditional volatility of some asset returns. In order to explain this paradox, we show that in the presence of common GARCH factors, parsimonious univariate representations can result from large multivariate models generating the conditional variances and conditional covariances/correlations. The diagonal model without any contagion effects in conditional volatilities gives rise to similar conclusions though. Consequently, after having extracted a block of assets representing some form of parsimony, remains the task of determining if we have a set of independent assets or instead a highly dependent system generated with a few factors. To investigate this issue, we first evaluate a reduced rank regressions approach for squared returns that we extend to cross-returns. Second we investigate a likelihood ratio approach, where under the null the matrix parameters have a reduced rank structure. It emerged that the latter approach has quite good properties enabling us to discriminate between a system with seemingly unrelated assets (e.g. a diagonal model) and a model with few common sources of volatility.
In this paper we study various MIDAS models for which the future daily variance is directly related to past observations of intraday predictors. Our goal is to determine if there exists an optimal sampling frequency in terms of variance prediction. Via Monte Carlo simulations we show that in a world without microstructure noise, the best model is the one using the highest available frequency for the predictors. However, in the presence of microstructure noise, the use of very high-frequency predictors may be problematic, leading to poor variance forecasts. The empirical application focuses on two highly liquid assets (i.e., Microsoft and S&P 500). We show that, when using raw intraday squared log-returns for the explanatory variable, there is a “high-frequency wall” – or frequency limit – above which MIDAS-RV forecasts deteriorate or stop improving. An improvement can be obtained when using intraday squared log-returns sampled at a higher frequency, provided they are pre-filtered to account for the presence of jumps, intraday diurnal pattern and/or microstructure noise. Finally, we compare the MIDAS model to other competing variance models including GARCH, GAS, HAR-RV and HAR-RV-J models. We find that the MIDAS model – when it is applied on filtered data –provides equivalent or even better variance forecasts than these models. JEL: C22, C53, G12 / KEY WORDS: Variance Forecasting, MIDAS, High-Frequency Data.
RÉSUMÉ. Nous considérons dans cet article des modèles de régression MIDAS pour examiner l'influence de la fréquence d'échantillonnage des prédicteurs sur la qualité des prévisions de la volatilité quotidienne. L'objectif principal est de vérifier si l'information incorporée par les prédicteurs à haute fréquence améliore la qualité des précisions de volatilité, et si oui, s'il existe une fréquence d'échantillonnage optimale de ces prédicteurs en termes de prédiction de la variance. Nous montrons, via des simulations Monte Carlo, que dans un monde sans bruit de microstructure, le meilleur modèle est celui qui utilise des prédicteurs à la fréquence la plus élevée possible. Cependant, en présence de bruit de microstructure, l'utilisation des měmes prédicteurs à haute fréquence peut ětre problématique, conduisant à des prévisions pauvres de la variance. L'application empirique se concentre sur deux actifs très liquides (Microsoft et S & P 500). Nous montrons que, lors de l'utilisation des rendements intra-journaliers au carré pour la variable explicative, il y a un « mur à haute fréquence » – ou limite de fréquence – au-delà duquel les prévisions des modèles MIDAS-RV se détériorent ou arrětent de s'améliorer. Une amélioration pourrait ětre obtenue lors de l'utilisation des rendements au carré échantillonnés à une fréquence plus élevée, à condition qu'ils soient préfiltrés pour tenir compte de la présence des sauts, de la saisonnalité intra-journalière et/ou du bruit de microstructure. Enfin, nous comparons le modèle MIDAS à d'autres modèles de variance concurrents, y compris les modèles GARCH, GAS, HAR-RV et HAR-RV-J. Nous constatons que le modèle MIDAS – quand il est appliqué sur des données filtrées – fournit des prévisions de variance équivalentes ou měme meilleures que ces modèles.
This paper evaluates the most appropriate ways to model diffusion and jump features of high-frequency exchange rates in the presence of intraday periodicity in volatility. We show that periodic volatility distorts the size and power of conventional tests of Brownian motion, jumps and (in)finite activity. We propose a correction for periodicity that restores the properties of the test statistics. Empirically, the most plausible model for 1-min exchange rate data features Brownian motion and both finite activity and infinite activity jumps. Test rejection rates vary over time, however, indicating time variation in the data generating process. We discuss the implications of results for market microstructure and currency option pricing.