Probabilistic forecasting in day-ahead electricity markets: Simulating peak and off-peak prices

In this article we include dependency structures for electricity price forecasting and forecasting evaluation. We work with off-peak and peak time series from the German-Austrian day-ahead price; hence, we analyze bivariate data. We first estimate the mean of the two time series, and then in a second step we estimate the residuals. The mean equation is estimated by ordinary least squares and the elastic net, and the residuals are estimated by maximum likelihood. Our contribution is to include a bivariate jump component in a mean reverting jump diffusion model in the residuals. The models’ forecasts are evaluated with use of four different criteria, including the energy score to measure whether the correlation structure between the time series is properly included. It is observed that the models with bivariate jumps provide better results with the energy score, which means that it is important to consider this structure to properly forecast correlated time series.     

A new bivariate generalized Poisson distribution

In this paper, a new bivariate generalized Poisson distribution (GPD) that allows any type of correlation is defined and studied. The marginal distributions of the bivariate model are the univariate GPDs. The parameters of the bivariate distribution are estimated by using the moment and maximum likelihood methods. Some test statistics are discussed and one numerical data set is used to illustrate the applications of the bivariate model.     

A bivariate count data model for household tourism demand

Households' choice of the number of leisure trips and the total number of overnight stays is empirically studied using Swedish tourism data. A bivariate hurdle approach separating the participation (to travel and stay the night or not) from the quantity (the number of trips and nights) decision is employed. The quantity decision is modelled with a bivariate mixed Poisson lognormal model allowing for both positive as well as negative correlation between count variables. The observed endogenous variables are drawn from a truncated density and estimation is pursued by simulated maximum likelihood. The estimation results indicate a negative correlation between the number of trips and nights. In most cases own price effects are as expected negative, while estimates of cross‐price effects vary between samples. Copyright © 2005 John Wiley & Sons, Ltd.     

Bivariate exponentiated‐exponential geometric regression model

A bivariate exponentiated‐exponential geometric regression model that allows negative, zero, or positive correlation is defined and studied. The model can accommodate under‐ or over‐dispersed count data. The regression model is based on the univariate exponentiated‐exponential geometric distribution, and the marginal means of the bivariate model are functions of the explanatory variables. The parameters of the bivariate regression model are estimated by using the maximum likelihood method. Some test statistics including goodness of fit are discussed. A simulation study is conducted to compare the model with the bivariate generalized Poisson regression model. One numerical data set is used to illustrate the application of the regression model.     

Measures of multivariate asymptotic dependence and their relation to spectral expansions

Asymptotic dependence can be interpreted as the property that realizations of the single components of a random vector occur simultaneously with a high probability. Information about the asymptotic dependence structure can be captured by dependence measures like the tail dependence parameter or the residual dependence index. We introduce these measures in the bivariate framework and extend them to the multivariate case afterwards. Within the extreme value theory one can model asymptotic dependence structures by Pickands dependence functions and spectral expansions. Both in the bivariate and in the multivariate case we also compute the tail dependence parameter and the residual dependence index on the basis of this statistical model. They take a specific shape then and are related to the Pickands dependence function and the exponent of variation of the underlying density expansion.     

Non-normal bivariate densities with normal marginals and linear regression functions *

Summary It is well known that for a bivariate density in order to be bivariate normal, the possession of univariate normal marginal densities alone is not sufficient. In this paper it will be shown by means of some counterexamples that the additional requirement of linearity of the regression functions does not supply a sufficient condition either.     

Modelling soccer matches using bivariate discrete distributions with general dependence structure

In this paper copulas are used to generate bivariate discrete distributions. These distributions are fitted to soccer data from the English Premier League. An interesting aspect of these data is that the primary variable of interest, the discrete pair shots-for and shots-against, exhibits negative dependence; thus, in particular, we apply bivariate Poisson-related distributions that allow such dependence. The paper focuses on Archimedian copulas, for which the dependence structure is fully determined by a one-dimensional projection that is invariant under marginal transformations. Diagnostic plots for copula fit based on this projection are adapted to deal with discrete variables. Covariates relating to within-match contributions such as numbers of passes and tackles are introduced to explain variability in shot outcomes. The results of this analysis would appear to support the notion that playing the 'beautiful game' is an effective strategy—more passes and crosses contribute to more effective play and more shots on the goal.     

Estimating a generalized correlation coefficient for a generalized bivariate probit model

In this paper we consider semiparametric estimation of a generalized correlation coefficient in a generalized bivariate probit model. The generalized correlation coefficient provides a simple summary statistic measuring the relationship between the two binary decision processes in a general framework. Our semiparametric estimation procedure consists of two steps, combining semiparametric estimators for univariate binary choice models with the method of maximum likelihood for the bivariate probit model with nonparametrically generated regressors. The estimator is shown to be consistent and asymptotically normal. The estimator performs well in our simulation study.     

Compound gamma,beta and F distributions

Summary In this paper a compound gamma distribution has been derived by compounding a gamma distribution with another gamma distribution. The resulting compound gamma distribution has been reduced to the Beta distributions of the first kind and the second kind and to theF distribution by suitable transformations. This includes theLomax distribution as a special case which enjoys a useful property. Moment estimators for two of its parameters are explicitly obtained, which tend to a bivariate normal distribution. The paper contains expressions for a bivariate probability density function, its conditional expectation, conditional variance and the product moment correlation coefficient. Finally, all the parameters of the compound gamma distribution are explicitly expressed in terms of the functions of the moments of the functions of random variables in two different ways. This note is based on a technical report prepared by the author while he was with the Procter and Gamble Company.     

Testing for equality of variance of correlated normal variables

A standard test exists for whether bivariate normal data of arbitrary correlation have equal variances. An extension of this model is useful to test whether two measuring instruments, with which repeated measurements have been made on each of n units, have equal error variance. It is shown that one or two simple F -distributed statistics yield performance comparable with that of the generalized likelihood ratio statistic.     

Distribution of correlation coefficient under mixture normal model

The present paper obtains the nonnull distribution of the product moment correlation coefficient r when sample is drawn from a mixture of two bivariate Gaussian distributions. The moments of 1−r ² have been used to derive the nonnull density of r. Received September 2000     

Nonlinear Granger Causality: Guidelines for Multivariate Analysis

We propose an extension of the bivariate nonparametric Diks–Panchenko Granger non‐causality test to multivariate settings. We first show that the asymptotic theory for the bivariate test fails to apply to the multivariate case, because the kernel density estimator bias and variance cannot both tend to zero at a sufficiently fast rate. To overcome this difficulty we propose to reduce the order of the bias by applying data sharpening prior to calculating the test statistic. We derive the asymptotic properties of the ‘sharpened’ test statistic and investigate its performance numerically. We conclude with an empirical application to the US grain market, using the price of futures on heating degree days as an additional conditioning variable. Copyright © 2015 John Wiley & Sons, Ltd.     

Combining competing forecasts of inflation using a bivariate arch model

This paper sets out the basic structure of the bivariate generalization of Engle's ARCH model. Conditions which guarantee that the conditional covariance matrix is well defined are summarized, as are estimation and hypothesis testing.The process is used to combine forecasts where the weights are allowed to vary over time. Forecast errors from competing models are treated as a bivariate ARCH process so that the conditional covariance matrix adapts over time. At each point in time these conditional estimates of the variances and covariances are used to construct the optimal weights for combining the forecasts. Consequently, when one model is fitting well, its variance will be reduced and its weight will be increased.Two models of US inflation are constructed; one is a stylized monetarist model while the other is a mark-up model. The forecast errors are modeled as a simple bivariate ARCH process. Diagnostic tests reveal that this has overly restricted the parameterization of the covariance matrix. An approximation to the theoretically anticipated factor structure model is then estimated. The results in both cases show the weights varying over the sample period in moderately interpretable fashion.     

A generalisation of Gumbel's bivariate logistic distribution

Summary The bivariate distributionF(x, y)=1/[1+exp(–x)+exp(–y)] was examined byGumbel. We have generalised this expression by raising it to an arbitarary power. Such a distribution may occur as a mixture of bivariate extreme-value distribution. As well as giving its basic properties, we have paid special attention to measures of correlation alternative to the product-moment, namely, Kendall's and Spearman's rank correlations and the product-moment correlation calculated after transforming the marginal distributions into Normal ones. An application to the multifactorial model of disease transmission is outlined.     

Bivariate and multivariate scaled association models. An application to homogamy of social origin and education in Hungary between 1930 and 1979

In this article we compare bivariate and multivariate models for homogamy of social origin and education to test whether bivariate models of homogamy lead to biased results. We use data on Hungarian couples married between 1930 and 1979 and loglinear models of scaled association. The results indicate some differences between bivariate and multivariate analyses. At each point of time bivariate models overestimate homogamy, both with respect to education and social origin. However, results on trends in time do not differ much between the two analyses. The exception is the period 1940–1959, in which bivariate analysis showed decreasing educational homogamy, and multivariate analysis showed an increasing trend. The latter finding can be explained by declining homogamy of social origin, as well as the weaker reproduction and cross-effects in this period.     

Large sample properties for a class of copulas in bivariate survival analysis

This work is concerned with asymptotic properties of the bivariate survival function estimator using the functional relationship between marginal survival functions and a class of copulas for the dependence structure. Specifically, we study consistency and weak convergence of the bivariate survival function estimator obtained considering a two-step procedure of estimation. The obtained results are found from a key decomposition of the bivariate survival function in quantities that can be studied separately. In particular, we use relating results to almost sure and weak convergence of estimators, almost sure convergence of uniformly equicontinuous functions, and the delta method for functionals.     

Modelling and Estimation for Bivariate Financial Returns

Maximum likelihood estimates are obtained for long data sets of bivariate financial returns using mixing representation of the bivariate (skew) Variance Gamma (VG) and two (skew) t distributions. By analysing simulated and real data, issues such as asymptotic lower tail dependence and competitiveness of the three models are illustrated. A brief review of the properties of the models is included. The present paper is a companion to papers in this journal by Demarta & McNeil and Finlay & Seneta.     

Fisher information in order statistics and their concomitants in bivariate censored samples

We evaluate the Fisher information (FI) contained in a collection of order statistics and their concomitants from a bivariate random sample. Special attention is given to Type II censored samples. We present a general decomposition result and recurrence relations that are useful in finding the FI in all types of censored samples. We also obtain some asymptotic results for the FI. For the bivariate normal parent, we obtain explicit and asymptotic expressions for the elements of the FI matrix for Type II censored samples. We discuss implications of our findings on inference on the bivariate normal parameters, especially on the correlation. The first author’s research was supported in part by National Institutes of Health, USA, Grant # M01 RR00034 and the second author’s research was supported by a training grant from the Egyptian government     

Beyond spatial representation

The correspondence between theory and observation is often evaluated by a comparison between a hypothesized constraint matrix and the spatial representation of a pxp similarity matrix. This comparison of constraint and proximity matrices assumes the accurate translation of similarities to proximities. If the translation is not exact (i.e., a stress or alienation coefficient greater than zero), the hypothesized structure is evaluated using a false representation of the observed data. The proposed model eliminates the need for spatial representation by making a direct comparison between the hypothesized constraint matrix and the multivariate structure of the bivariate similarities. Goodness of fit indices are used for three model comparisons; (1) single data set, one hypothesized structure; (2) single data set, two hypothesized structures; and (3) two data sets, one hypothesized structure.