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## A Service of zbw Forecasting Volatility with Copula-Based Time Series Models Based Time Series Models

### Citations

1344 | Comparing predictive accuracy
- Diebold, Mariano
- 1995
(Show Context)
Citation Context ...t|t−1 + b1 + ηt. The accuracy of the volatility forecasts is evaluated using the Mean Squared Prediction Error (MSPE), that is MSPE = 1 P R+P∑ t=R+1 (yt − yt|t−1)2. et al., 2007; Hafner and Manner, 2011, among others). Instead of the copula parameters the copula function itself may also be allowed to vary over time (as in Okimoto, 2008; Chollete et al., 2009; Garcia and Tsafack, 2011). Manner and Reznikova (2011) provide a recent survey. 15 We directly compare the copula-based forecasts with the benchmark HAR specifications by testing the null hypothesis of equal predictive accuracy with the Diebold and Mariano (1995) statistic. Specifically, let yC-RV,t|t−1 and yHAR,t|t−1 denote the two competing one-step ahead forecasts of yt, and define the loss differential dt = e2HAR,t|t−1 − e2C-RV,t|t−1, where e.,t|t−1 = yt − yt|t−1 is the forecast error of the HAR and C-RV models. We then test the null hypothesis of equal predictive accuracy, which corresponds to E[dt] = 0, by means of the t-statistic DM = d√ V (dt)/P , (12) where d is the sample mean of the loss differential dt and V (dt) is an estimate of the variance of dt. Finally, we estimate the forecast encompassing regression yt = b0 + b1yHAR,t|t−1 + b... |

1288 |
An Introduction to Copulas
- Nelsen
- 1999
(Show Context)
Citation Context ...he dependence function, or vice versa, a wide range of conditional distributions can be obtained by combining different marginals F (·) with different copulas C(·, ·). As we aim to focus on the usefulness of copulas to model the nonlinear temporal dependence in realized volatility, we concentrate on possible specifications of the copula function C. As discussed in more detail below, the marginal distribution F is estimated nonparametrically. A wide range of parametric copula specifications is available, with different implications for the dependence structure of volatility, see Joe (1997) and Nelsen (2006) for overviews. Different copula functions may usefully be compared in terms of their so-called ‘quantile dependence’ and the limiting case of tail dependence. A particular measure of quantile dependence is the exceedance probability. This is defined as the conditional probability that IVt exceeds a given quantile q of its marginal distribution given that IVt−1 exceeds that quantile. Specifically, the exceedance probabilities are given by τ(q) = P (ut < q|ut−1 < q) = C(q; q)/q for q ≤ 0.5, P (ut > q|ut−1 > q) = (1− 2q + C(q; q))/(1− q) for q > 0.5. (5) where ut = F (IVt) is the proba... |

787 |
Multivariate Models and Dependence Concepts
- Joe
- 1997
(Show Context)
Citation Context ...the choice of the dependence function, or vice versa, a wide range of conditional distributions can be obtained by combining different marginals F (·) with different copulas C(·, ·). As we aim to focus on the usefulness of copulas to model the nonlinear temporal dependence in realized volatility, we concentrate on possible specifications of the copula function C. As discussed in more detail below, the marginal distribution F is estimated nonparametrically. A wide range of parametric copula specifications is available, with different implications for the dependence structure of volatility, see Joe (1997) and Nelsen (2006) for overviews. Different copula functions may usefully be compared in terms of their so-called ‘quantile dependence’ and the limiting case of tail dependence. A particular measure of quantile dependence is the exceedance probability. This is defined as the conditional probability that IVt exceeds a given quantile q of its marginal distribution given that IVt−1 exceeds that quantile. Specifically, the exceedance probabilities are given by τ(q) = P (ut < q|ut−1 < q) = C(q; q)/q for q ≤ 0.5, P (ut > q|ut−1 > q) = (1− 2q + C(q; q))/(1− q) for q > 0.5. (5) where ut = F ... |

733 | Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institute de Statistique de L’Université de Paris 8 - Sklar - 1959 |

242 | Modelling Asymmetric Exchange Rate Dependence
- Patton
- 2006
(Show Context)
Citation Context ...aneous research by Ning et al. (2010) also suggests copula-based time series models for describing the dynamics of realized volatility measures, but does not consider out-of-sample forecasting. 2 examine which specifications yield the most accurate out-of-sample forecasts of volatility. Second, although the C-RV model allows for possibly nonlinear time series dependence in volatility, the dependence is assumed to be ‘stable’, that is, constant over time. We allow for the possibility of changes in the dependence in the C-RV model by using conditional copulas with time-varying parameters, as in Patton (2006). Our empirical results can be summarized as follows. We find that the C-RV model outperforms the HAR for one-day ahead volatility forecasts in terms of accuracy and in terms of efficiency. Among the copula specifications considered, the Gumbel C-RV model achieves the best forecast performance, which highlights the importance of asymmetry and upper tail dependence for modeling volatility. Although we find substantial variation in the copula parameter estimates over time, conditional copulas do not improve the accuracy of volatility forecasts. The rest of the paper is organized as follows. In S... |

165 | Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility
- Andersen, Bollerslev, et al.
- 2007
(Show Context)
Citation Context ...anagement, and derivatives pricing. In recent years, the increasing availability of high-frequency asset price data has led to the development of various different measures of daily volatility based on intraday prices, see McAleer and Medeiros (2008b) and Andersen et al. (2009) for recent surveys. Considerable research effort has also been spent on designing suitable time series models for forecasting these so-called realized volatility measures. In several empirical applications, the heterogeneous autoregressive (HAR) model of Corsi (2009) has been found most successful for this purpose, see Andersen et al. (2007), Corsi (2009), and Busch et al. (2011), among others. The HAR model’s appeal is due to its parsimony and ability to capture the stylized fact of long-memory in (realized) volatility. In addition, the model can be extended in straightforward ways to incorporate other typical features such as leverage effects, jumps, seasonality, as well as impacts of macro-economic announcements, see Martens et al. (2009). A crucial feature of the HAR model is its linearity, in the sense that the dependence between realized volatility on consecutive days is assumed to be constant over time and independent of t... |

107 |
A simple approximate long-memory model of realized volatility
- Corsi
- 2009
(Show Context)
Citation Context ...latility are of crucial importance in portfolio management, risk management, and derivatives pricing. In recent years, the increasing availability of high-frequency asset price data has led to the development of various different measures of daily volatility based on intraday prices, see McAleer and Medeiros (2008b) and Andersen et al. (2009) for recent surveys. Considerable research effort has also been spent on designing suitable time series models for forecasting these so-called realized volatility measures. In several empirical applications, the heterogeneous autoregressive (HAR) model of Corsi (2009) has been found most successful for this purpose, see Andersen et al. (2007), Corsi (2009), and Busch et al. (2011), among others. The HAR model’s appeal is due to its parsimony and ability to capture the stylized fact of long-memory in (realized) volatility. In addition, the model can be extended in straightforward ways to incorporate other typical features such as leverage effects, jumps, seasonality, as well as impacts of macro-economic announcements, see Martens et al. (2009). A crucial feature of the HAR model is its linearity, in the sense that the dependence between realized volatility ... |

85 | Estimation of Copula-based semiparametric time series models, mimeo
- Chen, Fan
- 2002
(Show Context)
Citation Context ...r we examine whether this indeed is the case, in particular from a forecasting perspective.1 We evaluate the forecasting performance of the C-RV model in an empirical application to daily volatility of the S&P500 futures, over the period from January 1995 to December 2006. We employ the realized range developed by Martens and van Dijk (2007) and Christensen and Podolskij (2007) to measure daily volatility. As we focus on the ability of the C-RV model to capture long-memory and nonlinearity in the time series dependence of realized volatility, we adopt the semi-parametric approach advocated by Chen and Fan (2006). This combines nonparametric estimation of the marginal distributions with a parametric copula specification. The adverse effects of misspecified marginals (Fermanian and Scaillet, 2005) are thus avoided, while retaining consistency of estimates of important characteristics of the multivariate distribution, such as moments and quantiles. In the empirical analysis, we address the following two key issues in the specification of CRV models. First, different copula functions imply different types of time series dependence in volatility, in terms of (a)symmetry and tail (in)dependence. This makes... |

81 |
The Economic Value of Volatility Timing using ‘Realized Volatility’,”
- Fleming, Kirby, et al.
- 2003
(Show Context)
Citation Context ...ract to the next, we never compute returns based on prices from two different contracts. The most straightforward estimator of the integrated volatility IVt in (2) is the realized variance, defined as RV Mt = M∑ m=1 r2t,m + r 2 t,o, (7) where M is the number of intra-day intervals used, rt,m is the return during the m-th interval on day t, and rt,o is the overnight return between the closing price on day t and the opening price on day t + 1. The latter is incorporated as Martens (2002) documents that the overnight volatility represents an important fraction of total daily volatility, see also Fleming et al. (2003) and Hansen and Lunde (2005) for discussion. An alternative measure is the realized range, introduced by Martens and van Dijk (2007) and Christensen and Podolskij (2007), defined as RRMt = M∑ m=1 1 4 log 2 (log Ht,m − log Lt,m)2 + r2t,o, (8) where the high price Ht,m and the low price Lt,m are defined as the maximum and minimum of all transaction prices observed during the m-th interval on day t. The realized range exploits the complete price path in the intra-day intervals, while the realized variance only uses the first and last price observations. Consequently, the realized range is a more ... |

66 |
The Copula-GARCH model of conditional dependencies: An international stock market application‖.
- Jondeau, Rockinger
- 2006
(Show Context)
Citation Context ...abilities documented in Section 3.1 motivates us to consider conditional copulas with time-varying parameters. Following Patton (2006), we specify the dynamics of the copula parameters as a measurable function of past observations, such that estimation and inference for these conditional copulas does not become more complicated than for standard copulas with constant parameters.7 To save space, we only report results 7Alternative approaches to conditional copulas allow the parameters in a given copula function to vary over time in the form of an autoregressive or Markov switching process (see Jondeau and Rockinger, 2006; Bartram 14 for conditional Gumbel copula with time-varying parameter θt given by θt = θ0 + θ1ut−1ut−2, where ut is the normalized EDF of the log realized range for day t. We assess the out-of-sample forecasting results by means of several performance measures to evaluate and compare the various copula specifications and the HAR benchmark model. First, we consider the mean prediction error (ME), that is ME = 1 P R+P∑ t=R+1 (yt − yt|t−1), where yt|t−1 denotes the one-step ahead forecast of the realized measure y for day t. The quality of individual forecasts is further assessed by the Min... |

63 |
A Realized Variance for the Whole Day Based on Intermittent High-Frequency Fata.’
- Hansen, Lunde
- 2005
(Show Context)
Citation Context ... compute returns based on prices from two different contracts. The most straightforward estimator of the integrated volatility IVt in (2) is the realized variance, defined as RV Mt = M∑ m=1 r2t,m + r 2 t,o, (7) where M is the number of intra-day intervals used, rt,m is the return during the m-th interval on day t, and rt,o is the overnight return between the closing price on day t and the opening price on day t + 1. The latter is incorporated as Martens (2002) documents that the overnight volatility represents an important fraction of total daily volatility, see also Fleming et al. (2003) and Hansen and Lunde (2005) for discussion. An alternative measure is the realized range, introduced by Martens and van Dijk (2007) and Christensen and Podolskij (2007), defined as RRMt = M∑ m=1 1 4 log 2 (log Ht,m − log Lt,m)2 + r2t,o, (8) where the high price Ht,m and the low price Lt,m are defined as the maximum and minimum of all transaction prices observed during the m-th interval on day t. The realized range exploits the complete price path in the intra-day intervals, while the realized variance only uses the first and last price observations. Consequently, the realized range is a more efficient estimator than 3Th... |

56 |
Parametric and nonparametric measurement of volatility
- Andersen, Bollerslev, et al.
- 2009
(Show Context)
Citation Context ..., NL-3000 DR Rotterdam, The Netherlands. E-mail: oleg.sokolinskiy@gmail.com ‡Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands. E-mail: djvandijk@ese.eur.nl (corresponding author) 1 Introduction Forecasts of asset price volatility are of crucial importance in portfolio management, risk management, and derivatives pricing. In recent years, the increasing availability of high-frequency asset price data has led to the development of various different measures of daily volatility based on intraday prices, see McAleer and Medeiros (2008b) and Andersen et al. (2009) for recent surveys. Considerable research effort has also been spent on designing suitable time series models for forecasting these so-called realized volatility measures. In several empirical applications, the heterogeneous autoregressive (HAR) model of Corsi (2009) has been found most successful for this purpose, see Andersen et al. (2007), Corsi (2009), and Busch et al. (2011), among others. The HAR model’s appeal is due to its parsimony and ability to capture the stylized fact of long-memory in (realized) volatility. In addition, the model can be extended in straightforward ways to incorp... |

39 | Copula–based models for financial time series’, in
- Patton
- 2009
(Show Context)
Citation Context ...copula-based model for realized volatility (C-RV model) is that we can decompose the joint distribution of current volatility and its first lag into their marginal distributions and a copula function, with the latter characterizing the temporal dependence. As the marginal distributions and the dependence structure can be modeled separately, the copula-based approach allows for a great deal of flexibility in the construction of an appropriate multivariate distribution. Not surprisingly, copulas have quickly gained popularity in economics and particularly fi1 nance, see Cherubini et al. (2004), Patton (2009) and Genest et al. (2009) for recent surveys. Although in finance copulas have been used mostly to describe the contemporaneous dependence between returns on different assets, they can also be used to model nonlinear time series dependence of a single variable. In fact, by combining different marginal distributions with different copula functions, a wide variety of marginal characteristics (including skewness and excess kurtosis) can be modeled, in addition to dependence characteristics such as clustering, asymmetry and tail dependence. Recently, Ibragimov and Lentzas (2008) demonstrate that c... |

38 |
Measuring and Forecasting S&P500 Index-Futures Volatility Using HighFrequency Data", The
- Martens
(Show Context)
Citation Context ...aturity, when the trading volume in the second nearby contract becomes larger. We make sure that when changing from one contract to the next, we never compute returns based on prices from two different contracts. The most straightforward estimator of the integrated volatility IVt in (2) is the realized variance, defined as RV Mt = M∑ m=1 r2t,m + r 2 t,o, (7) where M is the number of intra-day intervals used, rt,m is the return during the m-th interval on day t, and rt,o is the overnight return between the closing price on day t and the opening price on day t + 1. The latter is incorporated as Martens (2002) documents that the overnight volatility represents an important fraction of total daily volatility, see also Fleming et al. (2003) and Hansen and Lunde (2005) for discussion. An alternative measure is the realized range, introduced by Martens and van Dijk (2007) and Christensen and Podolskij (2007), defined as RRMt = M∑ m=1 1 4 log 2 (log Ht,m − log Lt,m)2 + r2t,o, (8) where the high price Ht,m and the low price Lt,m are defined as the maximum and minimum of all transaction prices observed during the m-th interval on day t. The realized range exploits the complete price path in the intra-day ... |

37 | Modeling international financial returns with a multivariate regime-switchng copula’,
- Chollete, Heinen, et al.
- 2009
(Show Context)
Citation Context ...ollow the suggestion of Patton and Sheppard (2009) to estimate the MincerZarnowitz regression using Generalized Least Squares (GLS). In this case, this effectively boils down to estimating b0 and b1 with OLS in the regression specification yt yt|t−1 = b0 1 yt|t−1 + b1 + ηt. The accuracy of the volatility forecasts is evaluated using the Mean Squared Prediction Error (MSPE), that is MSPE = 1 P R+P∑ t=R+1 (yt − yt|t−1)2. et al., 2007; Hafner and Manner, 2011, among others). Instead of the copula parameters the copula function itself may also be allowed to vary over time (as in Okimoto, 2008; Chollete et al., 2009; Garcia and Tsafack, 2011). Manner and Reznikova (2011) provide a recent survey. 15 We directly compare the copula-based forecasts with the benchmark HAR specifications by testing the null hypothesis of equal predictive accuracy with the Diebold and Mariano (1995) statistic. Specifically, let yC-RV,t|t−1 and yHAR,t|t−1 denote the two competing one-step ahead forecasts of yt, and define the loss differential dt = e2HAR,t|t−1 − e2C-RV,t|t−1, where e.,t|t−1 = yt − yt|t−1 is the forecast error of the HAR and C-RV models. We then test the null hypothesis of equal predictive accuracy, which corr... |

32 |
Realized volatility: A review. Econometric Reviews,
- McAleer, Medeiros
- 2008
(Show Context)
Citation Context ...iversity Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands. E-mail: oleg.sokolinskiy@gmail.com ‡Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands. E-mail: djvandijk@ese.eur.nl (corresponding author) 1 Introduction Forecasts of asset price volatility are of crucial importance in portfolio management, risk management, and derivatives pricing. In recent years, the increasing availability of high-frequency asset price data has led to the development of various different measures of daily volatility based on intraday prices, see McAleer and Medeiros (2008b) and Andersen et al. (2009) for recent surveys. Considerable research effort has also been spent on designing suitable time series models for forecasting these so-called realized volatility measures. In several empirical applications, the heterogeneous autoregressive (HAR) model of Corsi (2009) has been found most successful for this purpose, see Andersen et al. (2007), Corsi (2009), and Busch et al. (2011), among others. The HAR model’s appeal is due to its parsimony and ability to capture the stylized fact of long-memory in (realized) volatility. In addition, the model can be extended in s... |

30 | Measuring volatility with the realized range. - Martens, Dijk - 2007 |

28 | Realized range-based estimation of integrated variance.
- Christensen, Podolskij
- 2007
(Show Context)
Citation Context ...gether with the ability to capture nonlinear dependence in a flexible and parsimonious way, this makes copula-based time series models a possible contender to conventional approaches for modeling realized volatility, such as the HAR model. In this paper we examine whether this indeed is the case, in particular from a forecasting perspective.1 We evaluate the forecasting performance of the C-RV model in an empirical application to daily volatility of the S&P500 futures, over the period from January 1995 to December 2006. We employ the realized range developed by Martens and van Dijk (2007) and Christensen and Podolskij (2007) to measure daily volatility. As we focus on the ability of the C-RV model to capture long-memory and nonlinearity in the time series dependence of realized volatility, we adopt the semi-parametric approach advocated by Chen and Fan (2006). This combines nonparametric estimation of the marginal distributions with a parametric copula specification. The adverse effects of misspecified marginals (Fermanian and Scaillet, 2005) are thus avoided, while retaining consistency of estimates of important characteristics of the multivariate distribution, such as moments and quantiles. In the empirical ana... |

28 | Some statistical pitfalls in copula modeling for financial applications.
- Fermanian, Scaillet
- 2005
(Show Context)
Citation Context ...n to daily volatility of the S&P500 futures, over the period from January 1995 to December 2006. We employ the realized range developed by Martens and van Dijk (2007) and Christensen and Podolskij (2007) to measure daily volatility. As we focus on the ability of the C-RV model to capture long-memory and nonlinearity in the time series dependence of realized volatility, we adopt the semi-parametric approach advocated by Chen and Fan (2006). This combines nonparametric estimation of the marginal distributions with a parametric copula specification. The adverse effects of misspecified marginals (Fermanian and Scaillet, 2005) are thus avoided, while retaining consistency of estimates of important characteristics of the multivariate distribution, such as moments and quantiles. In the empirical analysis, we address the following two key issues in the specification of CRV models. First, different copula functions imply different types of time series dependence in volatility, in terms of (a)symmetry and tail (in)dependence. This makes the choice of a copula specification an important issue in practice. We consider a variety of copula functions and 1Independent, contemporaneous research by Ning et al. (2010) also sugge... |

25 | 2009): “Evaluating Volatility and Correlation Forecasts
- Patton, Sheppard
(Show Context)
Citation Context ...f-sample forecasting results by means of several performance measures to evaluate and compare the various copula specifications and the HAR benchmark model. First, we consider the mean prediction error (ME), that is ME = 1 P R+P∑ t=R+1 (yt − yt|t−1), where yt|t−1 denotes the one-step ahead forecast of the realized measure y for day t. The quality of individual forecasts is further assessed by the Mincer-Zarnowitz type regression yt = b0 + b1yt|t−1 + ηt. (11) In (11), b0 and b1 should be equal to 0 and 1, respectively, for the forecast to be considered efficient. We follow the suggestion of Patton and Sheppard (2009) to estimate the MincerZarnowitz regression using Generalized Least Squares (GLS). In this case, this effectively boils down to estimating b0 and b1 with OLS in the regression specification yt yt|t−1 = b0 1 yt|t−1 + b1 + ηt. The accuracy of the volatility forecasts is evaluated using the Mean Squared Prediction Error (MSPE), that is MSPE = 1 P R+P∑ t=R+1 (yt − yt|t−1)2. et al., 2007; Hafner and Manner, 2011, among others). Instead of the copula parameters the copula function itself may also be allowed to vary over time (as in Okimoto, 2008; Chollete et al., 2009; Garcia and Tsafack, 2011). ... |

23 | A Multiple Regime Smooth Transition Heterogeneous Autoregressive Model for Long Memory and Asymmetries
- McAleer, Medeiros
- 2008
(Show Context)
Citation Context ...iversity Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands. E-mail: oleg.sokolinskiy@gmail.com ‡Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands. E-mail: djvandijk@ese.eur.nl (corresponding author) 1 Introduction Forecasts of asset price volatility are of crucial importance in portfolio management, risk management, and derivatives pricing. In recent years, the increasing availability of high-frequency asset price data has led to the development of various different measures of daily volatility based on intraday prices, see McAleer and Medeiros (2008b) and Andersen et al. (2009) for recent surveys. Considerable research effort has also been spent on designing suitable time series models for forecasting these so-called realized volatility measures. In several empirical applications, the heterogeneous autoregressive (HAR) model of Corsi (2009) has been found most successful for this purpose, see Andersen et al. (2007), Corsi (2009), and Busch et al. (2011), among others. The HAR model’s appeal is due to its parsimony and ability to capture the stylized fact of long-memory in (realized) volatility. In addition, the model can be extended in s... |

23 |
New evidence of asymmetric dependence structures in international equity markets.
- Okimoto
- 2008
(Show Context)
Citation Context ...efficient. We follow the suggestion of Patton and Sheppard (2009) to estimate the MincerZarnowitz regression using Generalized Least Squares (GLS). In this case, this effectively boils down to estimating b0 and b1 with OLS in the regression specification yt yt|t−1 = b0 1 yt|t−1 + b1 + ηt. The accuracy of the volatility forecasts is evaluated using the Mean Squared Prediction Error (MSPE), that is MSPE = 1 P R+P∑ t=R+1 (yt − yt|t−1)2. et al., 2007; Hafner and Manner, 2011, among others). Instead of the copula parameters the copula function itself may also be allowed to vary over time (as in Okimoto, 2008; Chollete et al., 2009; Garcia and Tsafack, 2011). Manner and Reznikova (2011) provide a recent survey. 15 We directly compare the copula-based forecasts with the benchmark HAR specifications by testing the null hypothesis of equal predictive accuracy with the Diebold and Mariano (1995) statistic. Specifically, let yC-RV,t|t−1 and yHAR,t|t−1 denote the two competing one-step ahead forecasts of yt, and define the loss differential dt = e2HAR,t|t−1 − e2C-RV,t|t−1, where e.,t|t−1 = yt − yt|t−1 is the forecast error of the HAR and C-RV models. We then test the null hypothesis of equal predicti... |

22 |
Dynamic stochastic copula models: Estimation, inference and applications.
- Hafner, Manner
- 2011
(Show Context)
Citation Context ...tz type regression yt = b0 + b1yt|t−1 + ηt. (11) In (11), b0 and b1 should be equal to 0 and 1, respectively, for the forecast to be considered efficient. We follow the suggestion of Patton and Sheppard (2009) to estimate the MincerZarnowitz regression using Generalized Least Squares (GLS). In this case, this effectively boils down to estimating b0 and b1 with OLS in the regression specification yt yt|t−1 = b0 1 yt|t−1 + b1 + ηt. The accuracy of the volatility forecasts is evaluated using the Mean Squared Prediction Error (MSPE), that is MSPE = 1 P R+P∑ t=R+1 (yt − yt|t−1)2. et al., 2007; Hafner and Manner, 2011, among others). Instead of the copula parameters the copula function itself may also be allowed to vary over time (as in Okimoto, 2008; Chollete et al., 2009; Garcia and Tsafack, 2011). Manner and Reznikova (2011) provide a recent survey. 15 We directly compare the copula-based forecasts with the benchmark HAR specifications by testing the null hypothesis of equal predictive accuracy with the Diebold and Mariano (1995) statistic. Specifically, let yC-RV,t|t−1 and yHAR,t|t−1 denote the two competing one-step ahead forecasts of yt, and define the loss differential dt = e2HAR,t|t−1 − e2C-RV,t|... |

20 | The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets.
- Busch, Christensen, et al.
- 2011
(Show Context)
Citation Context ...ent years, the increasing availability of high-frequency asset price data has led to the development of various different measures of daily volatility based on intraday prices, see McAleer and Medeiros (2008b) and Andersen et al. (2009) for recent surveys. Considerable research effort has also been spent on designing suitable time series models for forecasting these so-called realized volatility measures. In several empirical applications, the heterogeneous autoregressive (HAR) model of Corsi (2009) has been found most successful for this purpose, see Andersen et al. (2007), Corsi (2009), and Busch et al. (2011), among others. The HAR model’s appeal is due to its parsimony and ability to capture the stylized fact of long-memory in (realized) volatility. In addition, the model can be extended in straightforward ways to incorporate other typical features such as leverage effects, jumps, seasonality, as well as impacts of macro-economic announcements, see Martens et al. (2009). A crucial feature of the HAR model is its linearity, in the sense that the dependence between realized volatility on consecutive days is assumed to be constant over time and independent of the level of volatility. In principle, t... |

19 | The euro and European financial market dependence. - Bartram, Taylor, et al. - 2007 |

17 | Dependence Structure and Extreme Comovements in International Equity and Bond Markets.
- Garcia, Tsafack
- 2011
(Show Context)
Citation Context ... Patton and Sheppard (2009) to estimate the MincerZarnowitz regression using Generalized Least Squares (GLS). In this case, this effectively boils down to estimating b0 and b1 with OLS in the regression specification yt yt|t−1 = b0 1 yt|t−1 + b1 + ηt. The accuracy of the volatility forecasts is evaluated using the Mean Squared Prediction Error (MSPE), that is MSPE = 1 P R+P∑ t=R+1 (yt − yt|t−1)2. et al., 2007; Hafner and Manner, 2011, among others). Instead of the copula parameters the copula function itself may also be allowed to vary over time (as in Okimoto, 2008; Chollete et al., 2009; Garcia and Tsafack, 2011). Manner and Reznikova (2011) provide a recent survey. 15 We directly compare the copula-based forecasts with the benchmark HAR specifications by testing the null hypothesis of equal predictive accuracy with the Diebold and Mariano (1995) statistic. Specifically, let yC-RV,t|t−1 and yHAR,t|t−1 denote the two competing one-step ahead forecasts of yt, and define the loss differential dt = e2HAR,t|t−1 − e2C-RV,t|t−1, where e.,t|t−1 = yt − yt|t−1 is the forecast error of the HAR and C-RV models. We then test the null hypothesis of equal predictive accuracy, which corresponds to E[dt] = 0, by me... |

16 | Forecasting S&P 500 volatility: long memory, level shifts, leverage effects, day-of-the week seasonality, and macroeconomic announcements - Martens, Dijk, et al. - 2009 |

15 | Copulas and temporal dependence.
- Beare
- 2010
(Show Context)
Citation Context ...ibe the contemporaneous dependence between returns on different assets, they can also be used to model nonlinear time series dependence of a single variable. In fact, by combining different marginal distributions with different copula functions, a wide variety of marginal characteristics (including skewness and excess kurtosis) can be modeled, in addition to dependence characteristics such as clustering, asymmetry and tail dependence. Recently, Ibragimov and Lentzas (2008) demonstrate that copula-based time series models can also display long memory properties, see also Chen et al. (2009) and Beare (2010). Together with the ability to capture nonlinear dependence in a flexible and parsimonious way, this makes copula-based time series models a possible contender to conventional approaches for modeling realized volatility, such as the HAR model. In this paper we examine whether this indeed is the case, in particular from a forecasting perspective.1 We evaluate the forecasting performance of the C-RV model in an empirical application to daily volatility of the S&P500 futures, over the period from January 1995 to December 2006. We employ the realized range developed by Martens and van Dijk (2007) ... |

15 |
The volatility of realized volatility. Econometric Reviews,
- Corsi, Mittnik, et al.
- 2008
(Show Context)
Citation Context ...a variety of copula specifications, namely Gaussian, Student’s t, Frank, Gumbel, Clayton, and Clayton-survival, as well as mixtures of the Clayton or Gumbel copula with its survival counterpart. Second, we examine the forecasting performance of C-RV models relative to conventional modeling approaches for realized measures. As a benchmark, we construct forecasts from the logarithmic HAR model of Corsi (2009), given by logRRt = β0 + β1logRRt−1,1 + β2logRRt−1,5 + β3logRRt−1,22 + εt, (10) where logRRt−1,L ≡ 1L ∑L l=1 logRRt−l is the average logarithmic realized range between days t − L and t − 1 (Corsi et al., 2008). The HAR model has been found to be most successful for forecasting empirical measures of integrated volatility in several empirical applications (see Andersen et al. (2007), Corsi et al. (2008), Corsi (2009), and Busch et al. (2011), among others). Third, the instability in the exceedance probabilities documented in Section 3.1 motivates us to consider conditional copulas with time-varying parameters. Following Patton (2006), we specify the dynamics of the copula parameters as a measurable function of past observations, such that estimation and inference for these conditional copulas does no... |

15 |
The advent of copulas in finance,”
- Genest, Gendron, et al.
- 2009
(Show Context)
Citation Context ... for realized volatility (C-RV model) is that we can decompose the joint distribution of current volatility and its first lag into their marginal distributions and a copula function, with the latter characterizing the temporal dependence. As the marginal distributions and the dependence structure can be modeled separately, the copula-based approach allows for a great deal of flexibility in the construction of an appropriate multivariate distribution. Not surprisingly, copulas have quickly gained popularity in economics and particularly fi1 nance, see Cherubini et al. (2004), Patton (2009) and Genest et al. (2009) for recent surveys. Although in finance copulas have been used mostly to describe the contemporaneous dependence between returns on different assets, they can also be used to model nonlinear time series dependence of a single variable. In fact, by combining different marginal distributions with different copula functions, a wide variety of marginal characteristics (including skewness and excess kurtosis) can be modeled, in addition to dependence characteristics such as clustering, asymmetry and tail dependence. Recently, Ibragimov and Lentzas (2008) demonstrate that copula-based time series m... |

13 | Efficient estimation of copula-based semiparametric markov models.
- Chen, Wu, et al.
- 2009
(Show Context)
Citation Context ...en used mostly to describe the contemporaneous dependence between returns on different assets, they can also be used to model nonlinear time series dependence of a single variable. In fact, by combining different marginal distributions with different copula functions, a wide variety of marginal characteristics (including skewness and excess kurtosis) can be modeled, in addition to dependence characteristics such as clustering, asymmetry and tail dependence. Recently, Ibragimov and Lentzas (2008) demonstrate that copula-based time series models can also display long memory properties, see also Chen et al. (2009) and Beare (2010). Together with the ability to capture nonlinear dependence in a flexible and parsimonious way, this makes copula-based time series models a possible contender to conventional approaches for modeling realized volatility, such as the HAR model. In this paper we examine whether this indeed is the case, in particular from a forecasting perspective.1 We evaluate the forecasting performance of the C-RV model in an empirical application to daily volatility of the S&P500 futures, over the period from January 1995 to December 2006. We employ the realized range developed by Martens and... |

9 |
A survey on time-varying copulas: Specification, simulations, and estimation. Econometric Reviews, page forthcoming.
- Manner, Reznikova
- 2011
(Show Context)
Citation Context ... to estimate the MincerZarnowitz regression using Generalized Least Squares (GLS). In this case, this effectively boils down to estimating b0 and b1 with OLS in the regression specification yt yt|t−1 = b0 1 yt|t−1 + b1 + ηt. The accuracy of the volatility forecasts is evaluated using the Mean Squared Prediction Error (MSPE), that is MSPE = 1 P R+P∑ t=R+1 (yt − yt|t−1)2. et al., 2007; Hafner and Manner, 2011, among others). Instead of the copula parameters the copula function itself may also be allowed to vary over time (as in Okimoto, 2008; Chollete et al., 2009; Garcia and Tsafack, 2011). Manner and Reznikova (2011) provide a recent survey. 15 We directly compare the copula-based forecasts with the benchmark HAR specifications by testing the null hypothesis of equal predictive accuracy with the Diebold and Mariano (1995) statistic. Specifically, let yC-RV,t|t−1 and yHAR,t|t−1 denote the two competing one-step ahead forecasts of yt, and define the loss differential dt = e2HAR,t|t−1 − e2C-RV,t|t−1, where e.,t|t−1 = yt − yt|t−1 is the forecast error of the HAR and C-RV models. We then test the null hypothesis of equal predictive accuracy, which corresponds to E[dt] = 0, by means of the t-statistic DM = d... |

3 |
Copulas and long memory.
- Ibragimov, Lentzas
- 2008
(Show Context)
Citation Context ...nce, see Cherubini et al. (2004), Patton (2009) and Genest et al. (2009) for recent surveys. Although in finance copulas have been used mostly to describe the contemporaneous dependence between returns on different assets, they can also be used to model nonlinear time series dependence of a single variable. In fact, by combining different marginal distributions with different copula functions, a wide variety of marginal characteristics (including skewness and excess kurtosis) can be modeled, in addition to dependence characteristics such as clustering, asymmetry and tail dependence. Recently, Ibragimov and Lentzas (2008) demonstrate that copula-based time series models can also display long memory properties, see also Chen et al. (2009) and Beare (2010). Together with the ability to capture nonlinear dependence in a flexible and parsimonious way, this makes copula-based time series models a possible contender to conventional approaches for modeling realized volatility, such as the HAR model. In this paper we examine whether this indeed is the case, in particular from a forecasting perspective.1 We evaluate the forecasting performance of the C-RV model in an empirical application to daily volatility of the S&P... |

1 |
Modeling asymmetric volatility clusters using copulas and high frequency data. Ryerson Working Paper No.
- Ning, Xu, et al.
- 2010
(Show Context)
Citation Context ...Fermanian and Scaillet, 2005) are thus avoided, while retaining consistency of estimates of important characteristics of the multivariate distribution, such as moments and quantiles. In the empirical analysis, we address the following two key issues in the specification of CRV models. First, different copula functions imply different types of time series dependence in volatility, in terms of (a)symmetry and tail (in)dependence. This makes the choice of a copula specification an important issue in practice. We consider a variety of copula functions and 1Independent, contemporaneous research by Ning et al. (2010) also suggests copula-based time series models for describing the dynamics of realized volatility measures, but does not consider out-of-sample forecasting. 2 examine which specifications yield the most accurate out-of-sample forecasts of volatility. Second, although the C-RV model allows for possibly nonlinear time series dependence in volatility, the dependence is assumed to be ‘stable’, that is, constant over time. We allow for the possibility of changes in the dependence in the C-RV model by using conditional copulas with time-varying parameters, as in Patton (2006). Our empirical results ... |