Ingrid Hobæk Haff

On the simplified pair-copula construction - Simply useful or too simplistic?

Due to their high flexibility, yet simple structure, pair-copula constructions (PCCs) are becoming increasingly popular for constructing continuous multivariate distributions. However, inference requires the simplifying assumption that all the pair-copulae depend on the conditioning variables merely through the two conditional distribution functions that constitute their arguments, and not directly. In terms of standard measures of dependence, we express conditions under which a specific pair-copula decomposition of a multivariate distribution is of this simplified form. Moreover, we show that the simplified PCC in fact is a rather good approximation, even when the simplifying assumption is far from being fulfilled by the actual model.

Parameter estimation for pair-copula constructions

We explore various estimators for the parameters of a pair-copula construction (PCC), among those the stepwise semiparametric (SSP) estimator, designed for this dependence structure. We present its asymptotic properties, as well as the estimation algorithm for the two most common types of PCCs. Compared to the considered alternatives, that is, maximum likelihood, inference functions for margins and semiparametric estimation, SSP is in general asymptotically less efficient. As we show in a few examples, this loss of efficiency may however be rather low. Furthermore, SSP is semiparametrically efficient for the Gaussian copula. More importantly, it is computationally tractable even in high dimensions, as opposed to its competitors. In any case, SSP may provide start values, required by the other estimators. It is also well suited for selecting the pair-copulae of a PCC for a given data set.

Risk estimation using the multivariate normal inverse Gaussian distribution

Appropriate modeling of time-varying dependencies is very important for quantifying financial risk, such as the risk associated with a portfolio of financial assets. Most of the papers analyzing financial returns have focused on the univariate case. The few that are concerned with their multivariate extensions are mainly based on the multivariate normal assumption. The idea of this paper is to use the multivariate normal inverse Gaussian (MNIG) distribution as the conditional distribution for a multivariate GARCH model. The MNIG distribution belongs to a very flexible family of distributions that captures heavy tails and skewness in the distribution of individual stock returns, as well as the asymmetry in the dependence between stocks observed in financial time series data. The usefulness of the MNIG GARCH model is highlighted through a value-at-risk (VAR) application on a portfolio of European, American and Japanese equities. Backtesting shows that for a one-day holding period this model outperforms a Gaussian GARCH model and a Students t GARCH model. Moreover, it is slightly better than a skew Students t GARCH model.

Nonparametric estimation of pair-copula constructions with the empirical pair-copula

A pair-copula construction is a decomposition of a multivariate copula into a structured system, called regular vine, of bivariate copulae or pair-copulae. The standard practice is to model these pair-copulae parametrically, inducing a model risk, with errors potentially propagating throughout the vine structure. The empirical pair-copula provides a nonparametric alternative, which is conjectured to still achieve the parametric convergence rate. Its main advantage for the user is that it does not require the choice of parametric models for each of the pair-copulae constituting the construction. It can be used as a basis for inference on dependence measures, for selecting an appropriate vine structure, and for testing for conditional independence.

Comparison of estimators for pair-copula constructions

We compare two of the most used estimators for the parameters of a pair-copula construction (PCC), namely the semiparametric (SP) and the stepwise semiparametric (SSP) estimators. By construction, the computational speed of the SSP estimator is considerably higher, at the expense of its asymptotic efficiency. Based on an extensive simulation study, we find that the performance of the SSP estimator is overall satisfactory compared to its contender. SSP loses some efficiency with respect to SP with increasing dependence, especially in the top levels of the PCC. On the other hand, the SSP estimator may suffer less under reduced sample sizes and misspecification of the model. Finally, it is the only real alternative for large-dimensional problems. Though it struggles with the top level parameters, the lower order dependences of the resulting estimated PCC mimic the true distribution well. All in all, this study supports the use of SSP in most applications.

How well do regional climate models simulate the spatial dependence of precipitation? An application of pair‐copula constructions

We investigate how well a suite of regional climate models (RCMs) from the ENSEMBLES project represents the residual spatial dependence of daily precipitation. The study area we consider is a 200 km×200 km region in south central Norway, with RCMs driven by ERA-40 boundary conditions at a horizontal resolution of approximately 25 km×25 km. We model the residual spatial dependence with pair-copula constructions, which allows us to assess both the overall and tail dependence in precipitation, including uncertainty estimates. The selected RCMs reproduce the overall dependence rather well, though the discrepancies compared to observations are substantial. All models overestimate the overall dependence in the west-east direction. They also overestimate the upper tail dependence in the north-south direction during winter, and in the west-east direction during summer, whereas they tend to underestimate this dependence in the north-south direction in summer. Moreover, many of the climate models do not simulate the small-scale dependence patterns caused by the pronounced orography well. However, the misrepresented residual spatial dependence does not seem to affect estimates of high quantiles of extreme precipitation aggregated over a few grid boxes. The underestimation of the area-aggregated extreme precipitation is due mainly to the well-known underestimation of the univariate margins for individual grid boxes, suggesting that the correction of RCM biases in precipitation might be feasible

Structure learning in Bayesian Networks using regular vines

Learning the structure of a Bayesian Network from multidimensional data is an important task in many situations, as it allows understanding conditional (in)dependence relations which in turn can be used for prediction. Current methods mostly assume a multivariate normal or a discrete multinomial model. A new greedy learning algorithm for continuous non-Gaussian variables, where marginal distributions can be arbitrary, as well as the dependency structure, is proposed. It exploits the regular vine approximation of the model, which is a tree-based hierarchical construction with pair-copulae as building blocks. It is shown that the networks obtainable with our algorithm belong to a certain subclass of chordal graphs. Chordal graphs representations are often preferred, as they allow very efficient message passing and information propagation in intervention studies. It is illustrated through several examples and real data applications that the possibility of using non-Gaussian margins and a non-linear dependency structure outweighs the restriction to chordal graphs.

Focussed selection of the claim severity distribution

ABSTRACT Risk assessment is a core theme within non-life insurance and estimation of quantiles far out in the upper tail is therefore one of the main applications of the total loss distribution of a non-life insurance portfolio. The choice of claim severity distribution should therefore reflect this. Therefore, we have explored how the focussed information criterion, FIC, aimed at finding the best model for estimating a given parameter of interest, the focus parameter, works as a tool for selecting the claim size distribution. As a quantile cannot be used directly as a focus parameter, we have tried different proxy focus parameters. To see how the FIC performs in this setting, compared to the other commonly used model selection methods AIC and BIC, we have performed a simulation study. In particular, we wanted to investigate the effect of the heaviness of the tail of the claim size distribution and the amount of available data. The performance of the different model selection methods was then evaluated based on the quality of the resulting estimates of the quantiles. Our study shows the best of the focussed criteria is the FIC, based on one single quantile from the claim severity distribution. Further, the performance of the FIC is mostly either comparable to or considerably better than that of the BIC, which is the best performing of the state of the art approaches. In particular, the FIC works well when the data are heavy-tailed, when the sample size is rather low and when the parameter of interest is the quantile far out in the tail of the total loss distribution.