Conditional Copula Function Research Articles

Multi-Objective Interval Prediction of Load Based on the Conditional Copula Function

Compared with the point prediction, the interval prediction of the load could more effectively guarantee the safe operation of the power system. In view of the problem that the correlation between adjacent load data is not fully utilized so that the prediction accuracy is reduced, this paper proposes the conditional copula function interval prediction method, which could make full use of the correlation relationship between adjacent load data so as to obtain the interval prediction result. At the same time, there are the different prediction results of the method under different parameters, and the evaluation results of the two accuracy evaluation indicators containing PICP (prediction interval coverage probability) and the PIAW (prediction interval average width) are inconsistent, the above result that the optimal parameters and prediction results cannot be obtained, therefore, the NSGA-II (Non-dominated Sorting Genetic Algorithm-II) multi-objective optimization algorithm is proposed to seek out the optimal solution set, and by evaluating the solution set, obtain the optimal prediction model parameters and the corresponding prediction results. Finally, the proposed method is applied to the three regions of Shaanxi Province, China to conduct ultra-short-term load prediction, and compare it with the commonly used load interval prediction method such as Gaussian process regression (GPR) algorithm, artificial neural network (ANN), extreme learning machine (ELM) and others, and the results show that the proposed method always has better prediction accuracy when applying it to different regions.

Multi-objective interval prediction of wind power based on conditional copula function

Interval prediction of wind power, which features the upper and lower limits of wind power at a given confidence level, plays a significant role in accurate prediction and stability of the power grid integrated with wind power. However, the conventional methods of interval prediction are commonly based on a hypothetic probability distribution function, which neglects the correlations among various variables, leading to decreased prediction accuracy. Therefore, in this paper, we improve the multi-objective interval prediction based on the conditional copula function, through which we can fully utilize the correlations among variables to improve prediction accuracy without an assumed probability distribution function. We use the multi-objective optimization method of non-dominated sorting genetic algorithm-II (NSGA-II) to obtain the optimal solution set. The particular best solution is weighted by the prediction interval average width (PIAW) and prediction interval coverage probability (PICP) to pick the optimized solution in practical examples. Finally, we apply the proposed method to three wind power plants in different Chinese cities as examples for validation and obtain higher prediction accuracy compared with other methods, i.e., relevance vector machine (RVM), artificial neural network (ANN), and particle swarm optimization kernel extreme learning machine (PSO-KELM). These results demonstrate the superiority and practicability of this method in interval prediction of wind power.

A Novel Nonparametric Estimation for Conditional Copula Functions Based on Bayes Theorem

Conditional copula which measures the conditional dependence among variables, possesses a special position in copula field. In this article, based on Bayes theorem, we derive three kinds of conditional copula functions as the product of the corresponding conditional copula density functions and the corresponding unconditional copula functions (or the cumulative distribution functions). Then, a novel nonparametric method for estimating these conditional copula functions is proposed by the classification of the Monte Carlo Simulation (MCS) samples and by the kernel density estimation. In contrast to other estimation methods for conditional copula functions, the proposed method needs only a set of samples without any parameter or distribution assumption, or other complicated operators (such as estimation of the weights, integral operator, etc.). Therefore, the proposed nonparametric method reduces the computational complexity and possesses more universality for estimating the conditional copula functions. A 2-dimensional normal copula function, a numerical example, a structural system reliability analysis considering the common cause failure and an astrophysics model based on real data are employed to validate the effectiveness of the proposed method. Results show that the proposed nonparametric method is accurate and practical well.

Stochastic extreme downscaling model for an assessment of changes in rainfall intensity-duration-frequency curves over South Korea using multiple regional climate models

A conditional copula function based downscaling model in a fully Bayesian framework is developed in this study to evaluate future changes in intensity–duration frequency (IDF) curves in South Korea. The model incorporates a quantile mapping approach for bias correction while integrated Bayesian inference allows accounting for parameter uncertainties. The proposed approach is used to temporally downscale expected changes in daily rainfall, inferred from multiple CORDEX-RCMs based on Representative Concentration Pathways (RCPs) 4.5 and 8.5 scenarios, into sub-daily temporal scales. Among the CORDEX-RCMs, a noticeable increase in rainfall intensity is observed in the HadGem3-RA (9%), RegCM (28%), and SNU_WRF (13%) on average, whereas no noticeable changes are observed in the GRIMs (−2%) for the period 2020–2050. More specifically, a 5–30% increase in rainfall intensity is expected in all of the CORDEX-RCMs for 50-year return values under the RCP 8.5 scenario. Uncertainty in simulated rainfall intensity gradually decreases toward the longer durations, which is largely associated with the enhanced strength of the relationship with the 24-h annual maximum rainfalls (AMRs). A primary advantage of the proposed model is that projected changes in future rainfall intensities are well preserved.

Conditional Correlation Coefficient as a Tool for Analysis of Contagion in Financial Markets and Real Economy Indexes Based on the Synthetic Ratio

We define contagion in financial markets as a significant increase in cross-market linkages after a shock to one or group of countries. Contagion occurs if cross-market co-movement increases significantly after the shock. In this article, the authors attempt to answer the question whether the selected world stock exchanges and economies are infecting each other within the meaning of the definition provided. Conditional copula functions and conditional Spearman's correlation coefficient will be used as a tool. Construction of a synthetic index of world financial markets is introduced. This index is based on the taxonomic distances of chosen stock market from the “best” object, where the best means object which has highest financial parameters.Main goal of this paper is to analyze changes in dependence between US stock market (S&P500) and chosen groups of world stock markets and constructed real economy index.

Analysis of Contagion in Financial Markets Based on the Gradient Measurement of the Growth in Financial Markets and Conditional Copula Functions

We define contagion in financial markets as a significant increase in cross-market linkages after a shock to one country (or group of countries). Contagion occurs if cross-market co-movement increases significantly after the shock.The main goal of this paper is to analyse changes in dependence between a chosen world stock market and the constructed synthetic index. Subsequently the research hypothesis will be verified: dependence between the synthetic stock market index and other stock markets is increasing in periods of a rapid decrease in value of stock market indexes. Positive verification of this hypothesis means that there is a contagion in financial markets.

Time-Varying Dependence in Sovereign and Bank Credit Spreads: Evidence from Europe

This paper shows how a measure of bank-sovereign contagion can be extracted from CDS spreads using conditional Copula functions. I estimate the probability of a European bank to experience extreme upward co-movements in CDS spreads together with its home sovereign. Two main results are obtained. First, bank-sovereign contagion is shown to vary strongly over time and across countries. Contagion is higher when the sovereign suffers from poor public finances. Second, banks with higher exposures to their home sovereign disproportionately benefit from sovereign-bank contagion in that they exhibit significantly lower CDS spreads. The evidence is supportive of a private-to-public risk transfer.

Trading strategies with copulas

A new approach is proposed to identify trading opportunities in the equity market by using the information contained in the bivariate dependence structure of two equities. The relationships between the equity pairs are modelled with bivariate copulas and the fitted copula structures are utilised to identify the trading opportunities. Two trading strategies are considered that take advantage of the relative mispricing between a pair of correlated stocks and involve taking a position on the stocks when they diverge from their historical relationship. The position is then reversed when the two stocks revert to their historical relationship. Only stock-pairs with relatively high correlations are considered. The dependence structures of the chosen stock-pairs very often exhibited both upper- and lower-tail dependence, which implies that copulas with the correct characteristics should be more effective than the more traditional approaches typically applied. To identify trading opportunities, the conditional copula functions are used to derive confidence intervals for the two stocks. It is shown that the number of trading opportunities is highly dependent on the confidence level and it is argued that the chosen confidence level should take the strength of the dependence between the two stocks into account. The backtest results of the pairs-trading strategy are disappointing in that even though the strategy leads to profits in most cases, the profits are largely consumed by the trading costs. The second trading strategy entails using single stock futures and it is shown to have more potential as a statistical arbitrage approach to construct a portfolio.

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.

Semiparametric estimation of conditional copulas

The manner in which two random variables influence one another often depends on covariates. A way to model this dependence is via a conditional copula function. This paper contributes to the study of semiparametric estimation of conditional copulas by starting from a parametric copula function in which the parameter varies with a covariate, and leaving the marginals unspecified. Consequently, the unknown parts in the model are the parameter function and the unknown marginals. The authors use a local pseudo-likelihood with nonparametrically estimated marginals approximating the unknown parameter function locally by a polynomial. Under this general setting, they prove the consistency of the estimators of the parameter function as well as its derivatives; they also establish asymptotic normality. Furthermore, they derive an expression for the theoretical optimal bandwidth and discuss practical bandwidth selection. They illustrate the performance of the estimation procedure with data-driven bandwidth selection via a simulation study and a real-data case.

Estimation of Copula Models With Discrete Margins via Bayesian Data Augmentation

Estimation of copula models with discrete margins can be difficult beyond the bivariate case. We show how this can be achieved by augmenting the likelihood with continuous latent variables, and computing inference using the resulting augmented posterior. To evaluate this, we propose two efficient Markov chain Monte Carlo sampling schemes. One generates the latent variables as a block using a Metropolis–Hastings step with a proposal that is close to its target distribution, the other generates them one at a time. Our method applies to all parametric copulas where the conditional copula functions can be evaluated, not just elliptical copulas as in much previous work. Moreover, the copula parameters can be estimated joint with any marginal parameters, and Bayesian selection ideas can be employed. We establish the effectiveness of the estimation method by modeling consumer behavior in online retail using Archimedean and Gaussian copulas. The example shows that elliptical copulas can be poor at modeling dependence in discrete data, just as they can be in the continuous case. To demonstrate the potential in higher dimensions, we estimate 16-dimensional D-vine copulas for a longitudinal model of usage of a bicycle path in the city of Melbourne, Australia. The estimates reveal an interesting serial dependence structure that can be represented in a parsimonious fashion using Bayesian selection of independence pair-copula components. Finally, we extend our results and method to the case where some margins are discrete and others continuous. Supplemental materials for the article are also available online.

Multivariate and functional covariates and conditional copulas

In this paper the interest is to estimate the dependence between two variables conditionally upon a covariate, through copula modelling. In recent literature nonparametric estimators for conditional copula functions in case of a univariate covariate have been proposed. The aim of this paper is to nonparametrically estimate a conditional copula when the covariate takes on values in more complex spaces. We consider multivariate covariates and functional covariates. We establish weak convergence, and bias and variance properties of the proposed nonparametric estimators. We also briefly discuss nonparametric estimation of conditional association measures such as a conditional Kendall’s tau. The case of functional covariates is of particular interest and challenge, both from theoretical as well as practical point of view. For this setting we provide an illustration with a real data example in which the covariates are spectral curves. A simulation study investigating the finite-sample performances of the discussed estimators is provided.

Estimation of Copula Models with Discrete Margins via Bayesian Data Augmentation

Estimation of copula models with discrete margins can be difficult beyond the bivariate case. We show how this can be achieved by augmenting the likelihood with latent variables, and computing inference using the resulting augmented posterior. To evaluate this we propose two efficient Markov chain Monte Carlo sampling schemes. One generates the latent variables as a block using a Metropolis-Hasting step with a proposal that is close to its target distribution, the other generates them one at a time. Our method applies to all parametric copulas where the conditional copula functions can be evaluated, not just elliptical copulas as in much previous work. Moreover, the copula parameters can be estimated joint with any marginal parameters, and Bayesian selection ideas employed. We establish the effectiveness of the estimation method by modeling consumer behavior in online retail using Archimedean and Gaussian copulas. The example shows that elliptical copulas can be poor at modeling dependence in discrete data, just as they can be in the continuous case. To demonstrate the potential in higher dimensions we estimate 16 dimensional D-vine copulas for a longitudinal model of usage of a bicycle path in the city of Melbourne, Australia. The estimates reveal an interesting serial dependence structure that can be represented in a parsimonious fashion using Bayesian selection of independence pair-copula components. Finally, we extend our results and method to the case where some margins are discrete and others continuous. Supplemental materials for the article are also available online.

Measuring the Marginal Systemic Risk Contribution Using Copula

In this paper we present a closed analytical formula for the calculation of the CoVaR (respectively Delta CoVaR), which is a macro risk measure introduced by Adrian and Brunnermeier to analyze and quantify the marginal contribution of a given financial institutions to the systemic risk. We derive a closed analytical expression of the CoVaR, by using a link between the conditional copula function and the CoVaR through their probabilistic interpretations. We conclude this article by applying our formula to compute CoVaR and Delta CoVaR in the Gaussian setting and presenting some interesting related results. We show that in the Gaussian framework the CoVaR and the Delta CoVaR may be expressed as a linear function of the value at risk of the corresponding single financial institution.

Intellectual Capital and Profitability in the Leather Set Up, Leather Artifacts, Traveling Products and Footwear Sector in Brazil

Empirical studies show that asset returns are non-normal and also time varying. These findings explain why the normality VaR and unconditional VaR perform poorly in market risk measurement and management. In the literature, the bivariate conditional copula function can be used to represent a joint distribution function of correlated asset returns whose marginal distributions can be of any form, and to accommodate time varying behavior of asset returns. This study extends the technique further to be able to measure the risk of a portfolio of any number of assets. The extension is successful because the study can endorse a positive definite correlation matrix of the conditional copula function by applying Semidefinite Programming (SDP). The study applies the method in risk measurement of Thai Government bond portfolios. In principle, the method should show improved performance over previous empirical studies, but the results from the Kupiec test show that the conditional copula method fails to estimate the VaR of these portfolios. This failure may have been caused by the misspecification of marginal distribution. ABSTRACT Once several studies have proved that the market value of the companies has dis- tanced themselves from the book value, researchers have been searching the causes that explain the high discrepancy. One research stream focuses the intangible re- sources, and among these, the impact that the intellectual capital shows on the value creation. This study has tested the model proposed by Professor Ante Pulic in order to verify if the intellectual capital exerts a positive impact on the profitability of the compa- nies. The sample reached the Leather Set Up, Leather Artifacts, Travelling Products, and Footwear Sector in Brazil, of which companies are picked up from the data basis of the Brazilian Institute of Geography and Statistics (IBGE) that carries out the Annual Industrial Research (PIA), with the national code of economic industry (CNAE) 19. The hypotheses tested were: (I) a positive relationship between profitability and intellectual capital exists; (II) a positive relationship between profitability and stock of intellectual capital exists; (III) a positive relationship between profitability and efficiency of the capi- tal employed exists; (IV) a positive relationship between profitability and efficiency of the human capital exists; (V) a positive relationship between profitability and efficiency of the structural capital exists. Panel data analysis was used in order to corroborate the hypotheses. The obtained results put forward a statistically significant and positive rela- tionship between (the stock and the flow) the intellectual capital and the profitability, measured by Return on Assets (ROA), in static models based on minimum square er- rors. The dynamic models have not been corroborated. ABSTRACT This paper examines empirically the value of early exercise by testing the ability of two American put valuation models to predict the early exercise premium for the S&P 100 American put options. An accuracy test and a quality test are performed on (1) the MacMillan (1986) & Barone-Adesi and Whaley (1987) model, and (2) the Carr, Jarrow and Myneni (1992) model. The test results show that early exercise premium is significant regardless of moneyness. Moreover, consistent with the theory, the value of early exercise is significantly negatively related to moneyness and interest rates and significantly positively related to time to maturity and to the volatility of the underlying index. Both American put valuation models examined misprice the early exercise premium embedded in American put prices.

Modeling electricity spot and futures price dependence: A multifrequency approach

Electricity prices are known to exhibit multifractal properties. We accommodate this finding by investigating multifractal models for electricity prices. In this paper we propose a flexible Copula-MSM (Markov Switching Multifractal) approach for modeling spot and weekly futures price dynamics. By using a conditional copula function, the framework allows us to separately model the dependence structure, while enabling use of multifractal stochastic volatility models to characterize fluctuations in marginal returns. An empirical experiment is carried out using data from Nord Pool. A study of volatility forecasting performance for electricity spot prices reveals that multifractal techniques are a competitive alternative to GARCH models. We also demonstrate how the Copula-MSM model can be employed for finding optimal portfolios, which minimizes the Conditional Value-at-Risk.