Moments of standardized Fernandez–Steel skewed distributions: Applications to the estimation of GARCH-type models

Trottier, Denis-Alexandre; Ardia, David

doi:10.1016/j.frl.2016.05.006

Cited by 34 publications

(9 citation statements)

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“…The quantities E[η t,k I{η t,k < 0}] and E[η 2 t,k I{η t,k < 0}] required for the covariance-stationarity conditions in the GJR and TGARCH models, and the quantity E [|η k,t−1 |] required in the conditional variance equation of the EGARCH model, are implemented following Trottier and Ardia (2016).…”

Section: Tgarch Modelmentioning

confidence: 99%

“…Fernández and Steel (1998) provide a simple way to introduce skewness into any unimodal standardized distribution, via the additional parameter ξ > 0; if ξ = 1 the distribution turns out to be symmetric. Trottier and Ardia (2016) derive the moments of the standardized Fernandez-Steel skewed distributions which are needed in the estimation of the EGARCH, GJR, and TGARCH models. We refer the reader to that publication for details.…”

Section: Ged Distributionmentioning

confidence: 99%

“…where CSC k denotes the covariance-stationarity condition and PC k the positivity condition in regime k; see Trottier and Ardia (2016). ξ k,1 is the asymmetry parameter and ξ k,2 the tail parameter of the skewed Student-t distribution in regime k. f N (•; µ, Σ) denotes the multivariate normal density with mean vector µ and covariance matrix Σ.…”

Section: Model Estimationmentioning

confidence: 99%

See 2 more Smart Citations

Markov-Switching GARCH Models in R: The MSGARCH Package

Ardia¹,

Bluteau²,

Boudt³

et al. 2019

J. Stat. Soft.

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Section: Tgarch Modelmentioning

confidence: 99%

Section: Ged Distributionmentioning

confidence: 99%

Section: Model Estimationmentioning

confidence: 99%

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Markov-Switching GARCH Models in R: The MSGARCH Package

Ardia¹,

Bluteau²,

Boudt³

et al. 2019

J. Stat. Soft.

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“…A standardized skew distribution is given by

\begin{matrix} p^{*} false(ϵ_{t} false| ξ, ϑ false) & = \frac{2 σ_{ξ}}{ξ + \frac{1}{ξ}} p ([, \frac{σ_{ξ} ϵ_{t} + μ_{ξ}}{ξ} I false{ϵ_{t} ⩾ - μ_{ξ} false/ σ_{ξ} false}) \\ (], + ξ false(σ_{ξ} ϵ_{t} + μ_{ξ} false) I false{ϵ_{t} < - μ_{ξ} false/ σ_{ξ} false} false| ϑ), \end{matrix}

where ξ >0 is the skewness parameter, ϑ denotes the parameters of the initial distribution and

\begin{matrix} μ_{ξ} \equiv m_{1} ((), ξ - \frac{1}{ξ}), & σ_{ξ}^{2} \equiv false(1 - m_{1}^{2} false) ((), ξ^{2} + \frac{1}{ξ^{2}}) + 2 m_{1}^{2} - 1, \\ m_{1} \equiv 2 \int_{0}^{\infty} u p false(u false| ϑ false) normald u . \end{matrix}

The previous expressions can be used to obtain a skew version of any symmetric unimodal density function which has zero mean and unit variance. We use formulas in Trottier and Ardia () for computing moments of skew distributions.…”

Section: Garch‐type Modelsmentioning

confidence: 99%

The impact of parameter and model uncertainty on market risk predictions from GARCH‐type models

Ardia

Kolly

Trottier

2017

Journal of Forecasting

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We study the effect of parameter and model uncertainty on the left‐tail of predictive densities and in particular on VaR forecasts. To this end, we evaluate the predictive performance of several GARCH‐type models estimated via Bayesian and maximum likelihood techniques. In addition to individual models, several combination methods are considered, such as Bayesian model averaging and (censored) optimal pooling for linear, log or beta linear pools. Daily returns for a set of stock market indexes are predicted over about 13 years from the early 2000s. We find that Bayesian predictive densities improve the VaR backtest at the 1% risk level for single models and for linear and log pools. We also find that the robust VaR backtest exhibited by linear and log pools is better than the backtest of single models at the 5% risk level. Finally, the equally weighted linear pool of Bayesian predictives tends to be the best VaR forecaster in a set of 42 forecasting techniques.

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“…I{η t,k < 0}] required for the covariance-stationarity conditions in the GJR and TGARCH models, and the quantity E [|η k,t−1 |] required in the conditional variance equation of the EGARCH model, are implemented followingTrottier and Ardia (2016).…”

mentioning

confidence: 99%

Markov-Switching GARCH Models in R: The MSGARCH Package

et al. 2016

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We describe the package MSGARCH, which implements Markov-switching GARCH (generalized autoregressive conditional heteroscedasticity) models in R with efficient C++ object-oriented programming. Markov-switching GARCH models have become popular methods to account for regime changes in the conditional variance dynamics of time series. The package MSGARCH allows the user to perform simulations as well as maximum likelihood and Bayesian Markov chain Monte Carlo estimations of a very large class of Markov-switching GARCH-type models. The package also provides methods to make single-step and multi-step ahead forecasts of the complete conditional density of the variable of interest. Risk management tools to estimate conditional volatility, value-at-risk, and expected-shortfall are also available. We illustrate the broad functionality of the MSGARCH package using exchange rate and stock market return data. ∼ D(0, 1, ξ k ).2 Furthermore, assuming a non zero state dependent mean would generally imply E[ytyt−1] = 0, which is against the model we specified.3 As explained below, the parametric formulation of the conditional distribution D(0, h k,t , ξ k ) can be different across regimes. In this case, the notation D k (0, h k,t , ξ k ) would be more appropriate. The same applies for the h(·) function in (2). We keep the simpler notation to improve readability. Also, for t = 1, we initialize the regime probabilities and the conditional variances at their unconditional levels. To simplify exposition, we use henceforth for t = 1 the same notation as for general t, since there is no confusion possible. Conditional variance dynamicsAs in Haas et al. (2004a), the conditional variance of y t is assumed to follow a GARCHtype model. Hence, conditionally on regime s t = k, h k,t is available as a function of the past observation, y t−1 , past variance h k,t−1 , and the additional regime-dependent vector of parameters θ k :where h(·) is a I t−1 -measurable function that defines the filter for the conditional variance and also ensures its positiveness. In the MSGARCH package, the initial value of the variance recursions, that is h k,1 (k = 1, . . . , K), are set equal to the unconditional variance in regime k.Depending on the form of h(·), we obtain different scedastic specifications. In the R package MSGARCH, we follow this specification in order to reduce model complexity. Finally, when K = 1, we recover single-regime GARCH-type models identified by the form of h(·). 4 The mixture of GARCH model presented in Haas et al. (2004b) allows for interactions between the mixture component variances. Here, we report the case referred to as "Diagonal" by Haas et al. (2004b). Journal of Statistical Software 5Below we briefly present the scedastic specifications available in the R package MSGARCH. Each of them is identified with a label used in the code for defining a model specification. Similarly, model coefficients are also identified with labels. ARCH modelThe ARCH model of Engle (1982) is given by:for k = 1, . . . , K. In this case, we have θ...

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Moments of standardized Fernandez–Steel skewed distributions: Applications to the estimation of GARCH-type models

Cited by 34 publications

References 8 publications

Markov-Switching GARCH Models in R: The MSGARCH Package

Markov-Switching GARCH Models in R: The MSGARCH Package

The impact of parameter and model uncertainty on market risk predictions from GARCH‐type models

Markov-Switching GARCH Models in R: The MSGARCH Package

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