Abstract
An approach to the modelling of volatile time series using a class of uniformity-preserving transforms for uniform random variables is proposed. V-transforms describe the relationship between quantiles of the stationary distribution of the time series and quantiles of the distribution of a predictable volatility proxy variable. They can be represented as copulas and permit the formulation and estimation of models that combine arbitrary marginal distributions with copula processes for the dynamics of the volatility proxy. The idea is illustrated using a Gaussian ARMA copula process and the resulting model is shown to replicate many of the stylized facts of financial return series and to facilitate the calculation of marginal and conditional characteristics of the model including quantile measures of risk. Estimation is carried out by adapting the exact maximum likelihood approach to the estimation of ARMA processes, and the model is shown to be competitive with standard GARCH in an empirical application to Bitcoin return data.
Highlights
We show that a class of uniformity-preserving transformations for uniform random variables can facilitate the application of copula modelling to time series exhibiting the serial dependence characteristics that are typical of volatile financial return data
This paper has proposed a new approach to volatile financial time series in which v-transforms are used to describe the relationship between quantiles of the return distribution and quantiles of the distribution of a predictable volatility proxy variable
We have characterized v-transforms mathematically and shown that the stochastic inverse of a v-transform may be used to construct stationary models for return series where arbitrary marginal distributions may be coupled with dynamic copula models for the serial dependence in the volatility proxy
Summary
We show that a class of uniformity-preserving transformations for uniform random variables can facilitate the application of copula modelling to time series exhibiting the serial dependence characteristics that are typical of volatile financial return data. The standard GARCH mechanism serves to create power-law behaviour in the marginal distribution, even when the innovations come from a lighter-tailed distribution such as Gaussian (Mikosch and Starica 2000) While such models work well for many return series, they may not be sufficiently flexible to describe all possible combinations of marginal and serial dependence behaviour encountered in applications. The right-hand plot of Figure 1 shows the sample autocorrelation function (acf) of the data given by zt = Φ−1(vt) where Φ is the standard normal cdf This reveals a persistent pattern of positive serial correlation which can be modelled by the implied ARMA copula.
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