Class Of Multivariate Models Research Articles

Higher-order asymptotic refinements in a multivariate regression model with general parameterization

This paper derives a general Bartlett correction formula to improve the inference based on the likelihood ratio test in a multivariate model under a quite general parameterization, where the mean vector and the variance-covariance matrix can share the same vector of parameters. This approach includes a number of models as special cases such as non-linear regression models, errors-in-variables models, mixed-effects models with non-linear fixed effects, and mixtures of the previous models. We also employ the Skovgaard adjustment to the likelihood ratio statistic in this class of multivariate models and derive a general expression of the correction factor based on Skovgaard approach. Monte Carlo simulation experiments are carried out to verify the performance of the improved tests, and the numerical results confirm that the modified tests are more reliable than the usual likelihood ratio test. Applications to real data are also presented for illustrative purposes.

Some benefits of standardisation for conditional extremes

AbstractA key aspect where extreme values methods differ from standard statistical models is through having asymptotic theory to provide a theoretical justification for the nature of the models used for extrapolation. In multivariate extremes, many different asymptotic theories have been proposed, partly as a consequence of the lack of ordering property with vector random variables. One class of multivariate models, based on conditional limit theory as one variable becomes extreme has received wide practical usage. The underpinning value of this approach has been supported by further theoretical characterisations of the limiting relationships. However, the paper “Conditional extreme value models: fallacies and pitfalls” by Holger Drees and Anja Janßen provides a number of counterexamples to these results. This paper studies these counterexamples in a conditional extremes framework which involves marginal standardisation to a common exponentially decaying tailed marginal distribution. Our calculations show that some of the issues identified can be addressed in this way.

Multifactor Stochastic Variance Models in Risk Management: Maximum Entropy Approach and Levy Processes

There is extensive empirical evidence that historical distributions of daily changes for stock prices, interest rates, foreign exchange rates, commodity prices and other underlyings have high peaks, heavy tails and non-zero skewness contrary to the normal distribution. These risk factors exhibit jumps, their volatility varies stochastically with clustering. Above distributional properties have significant impact on Risk Management, specifically on Value-at-Risk (VaR) calculations. This paper presents a class of multivariate models with the stochastic variance driven by Levy processes. The models with correlation structure in the stochastic variance allow for different shape and tail behavior of the marginal risk factor distributions, exact fit into the risk factor correlation structure, and proper non-linear scaling of VaR for different holding periods. In one-dimensional case, a pure jump Gamma process and other Levy processes for the stochastic variance are derived from the Maximum Entropy principle. Corresponding stochastic processes for the risk factors possess marginal distributions with wide range of heavy tails, from exponential to polynomial. Ornstein-Uhlenbeck type processes for the stochastic variance and corresponding term structure of the risk factor kurtosis and quantiles are investigated. In multi-dimensional case, the effective calibration and Monte Carlo simulation procedures are considered. Presented empirical evidence for different markets confirms a good agreement between the model and actual historical risk factor distributions.

BOUNDED UNCERTAINTY MODELS IN FINANCE: PARAMETER ESTIMATION AND FORECASTING

In this paper we consider the problem of modelling observed data using a class of multivariate models with unknown-but-bounded (ubb) noise and uncertainty. Standard ARX models with additive and multiplicative bounded noise belong to the considered class, as well as the deterministic counterpart of ARCH models extensively used in econometrics. We outline a method to fit these models based on historical data, and discuss the issues of set-valued forecasting.

Multivariate Dispersion Models Generated From Gaussian Copula

In this paper a class of multivariate dispersion models generated from the multivariate Gaussian copula is presented. Being a multivariate extension of Jørgensen's (1987a) dispersion models, this class of multivariate models is parametrized by marginal position, dispersion and dependence parameters, producing a large variety of multivariate discrete and continuous models including the multivariate normal as a special case. Properties of the multivariate distributions are investigated, some of which are similar to those of the multivariate normal distribution, which makes these models potentially useful for the analysis of correlated non‐normal data in a way analogous to that of multivariate normal data. As an example, we illustrate an application of the models to the regression analysis of longitudinal data, and establish an asymptotic relationship between the likelihood equation and the generalized estimating equation of Liang & Zeger (1986).

Generalized multivariate models for longitudinal data

A general class of multivariate models is proposed for unbalanced and incomplete longitudinal data. The proposed model is an extension of the seemingly unrelated regression model (Zellner, 1962). The advantage of this model is discussed regarding its applicability to a larger class of problems and the ease of estimation. The application of the model includes the model for the time varying covariates proposed by Patel (1988) and growth curve models. Two estimation methods are considered; one method is the generalized least squares method based on Zellner's noniterative two-stage estimation and the other is the iterative maximum likelihood estimation method using the EM algorithm (Dempster,Laird,and Rubin,1977). Simulation studies are conducted to compare the small sample properties of the two estimators.