Kurtosis transformation and kurtosis ordering

Leptokurtic distributions can be generated by applying certain non-linear transformations to a standard normal random variable. Within this work we derive general conditions for these transformations which guarantee that the generated distributions are ordered with respect to the partial ordering of van Zwet for symmetric distributions and the partial ordering of MacGillivray for arbitrary distributions. In addition, we propose a general power transformation which nests the H-, J- and K-transformations which have already been proposed in the literature. Within this class of power transformations the above mentioned condition can be easily verified and the power can be interpreted as parameter of leptokurtosis. --Power kurtosis transformation,leptokurtosis,kurtosis ordering

Augoregressive Conditional Kurtosis

This paper proposes a new model for autoregressive conditional heteroscedasticity and kurtosis. Via a time-varying degrees of freedom parameter, the conditional variance and conditional kurtosis are permitted to evolve separately. The model uses only the standard Student’s t density and consequently can be estimated simply using maximum likelihood. The method is applied to a set of four daily financial asset return series comprising US and UK stocks and bonds, and significant evidence in favour of the presence of autoregressive conditional kurtosis is observed. Various extensions to the basic model are examined, and show that conditional kurtosis appears to be positively but not significantly related to returns, and that the response of kurtosis to good and bad news is not significantly asymmetric. A multivariate model for conditional heteroscedasticity and conditional kurtosis, which can provide useful information on the co-movements between the higher moments of series, is also proposed.conditional kurtosis, GARCH, fourth moment, fat trails, student's t distribution

On the accuracy of simulations of turbulence

The widely recognized issue of adequate spatial resolution in numerical simulations of turbulence is studied in the context of two-dimensional magnetohydrodynamics. The familiar criterion that the dissipation scale should be resolved enables accurate computation of the spectrum, but fails for precise determination of higher-order statistical quantities. Examination of two straightforward diagnostics, the maximum of the kurtosis and the scale-dependent kurtosis, enables the development of simple tests for assessing adequacy of spatial resolution. The efficacy of the tests is confirmed by examining a sample problem, the distribution of magnetic reconnection rates in turbulence. Oversampling the Kolmogorov dissipation scale by a factor of 3 allows accurate computation of the kurtosis, the scale-dependent kurtosis, and the reconnection rates. These tests may provide useful guidance for resolution requirements in many plasma computations involving turbulence and reconnection

Investigation of intermittency in magnetohydrodynamics and solar wind turbulence: scale-dependent kurtosis

The behavior of scale-dependent (or filtered) kurtosis is studied in the solar wind using magnetic field measurements from the ACE and Cluster spacecraft at 1 AU. It is also analyzed numerically with high-resolution magnetohydrodynamic spectral simulations. In each case the filtered kurtosis increases with wavenumber, implying the presence of coherent structures at the smallest scales. This phase coupling is related to intermittency in solar wind turbulence and the emergence of non-Gaussian statistics. However, it is inhibited by the presence of upstream waves and other phase-randomizing structures, which act to reduce the growth of kurtosis

Modelling the Density of Inflation Using Autoregressive Conditional Heteroscedasticity, Skewness, and Kurtosis Models

The paper aimed at modelling the density of inflation based on time-varying conditional variance, skewness and kurtosis model developed by Leon, Rubio, and Serna (2005) who model higher-order moments as GARCH-type processes by applying a Gram-Charlier series expansion of the normal density function. Additionally, it extended their work by allowing both conditional skewness and kurtosis to have an asymmetry term. The results revealed the significant persistence in conditional variance, skewness and kurtosis which indicate high asymmetry of inflation. Additionally, diagnostic tests reveal that models with nonconstant volatility, skewness and kurtosis are superior to models that keep them invariant.inflation targeting, conditional volatility, skewness and kurtosis, modelling uncertainty of inflation

Autorregresive conditional volatility, skewness and kurtosis

This paper proposes a GARCH-type model allowing for time-varying volatility, skewness and kurtosis. The model is estimated assuming a Gram-Charlier series expansion of the normal density function for the error term, which is easier to estimate than the non-central t distribution proposed by Harvey and Siddique (1999). Moreover, this approach accounts for time-varying skewness and kurtosis while the approach by Harvey and Siddique (1999) only accounts for nonnormal skewness. We apply this method to daily returns of a variety of stock indices and exchange rates. Our results indicate a significant presence of conditional skewness and kurtosis. It is also found that specifications allowing for time-varying skewness and kurtosis outperform specifications with constant third and fourth moments.skewness and kurtosis, conditional volatility, Gram-Charlier series expansion, stock indices