Multivariate regression analysis of panel data with binary outcomes applied to unemployment data

Summary  In panel studies binary outcome measures together with time stationary and time varying explanatory variables are collected over time on the same individual. Therefore, a regression analysis for this type of data must allow for the correlation among the outcomes of an individual. The multivariate probit model of Ashford and Sowden (1970) was the first regression model for multivariate binary responses. However, a likelihood analysis of the multivariate probit model with general correlation structure for higher dimensions is intractable due to the maximization over high dimensional integrals thus severely restricting ist applicability so far. Czado (1996) developed a Markov Chain Monte Carlo (MCMC) algorithm to overcome this difficulty. In this paper we present an application of this algorithm to unemployment data from the Panel Study of Income Dynamics involving 11 waves of the panel study. In addition we adapt Bayesian model checking techniques based on the posterior predictive distribution (see for example Gelman et al. (1996)) for the multivariate probit model. These help to identify mean and correlation specification which fit the data well. C. Czado was supported by research grant OGP0089858 of the Natural Sciences and Engineering Research Council of Canada.     

A copula-based Markov chain model for the analysis of binary longitudinal data

A fully parametric first-order autoregressive (AR(1)) model is proposed to analyse binary longitudinal data. By using a discretized version of a copula, the modelling approach allows one to construct separate models for the marginal response and for the dependence between adjacent responses. In particular, the transition model that is focused on discretizes the Gaussian copula in such a way that the marginal is a Bernoulli distribution. A probit link is used to take into account concomitant information in the behaviour of the underlying marginal distribution. Fixed and time-varying covariates can be included in the model. The method is simple and is a natural extension of the AR(1) model for Gaussian series. Since the approach put forward is likelihood-based, it allows interpretations and inferences to be made that are not possible with semi-parametric approaches such as those based on generalized estimating equations. Data from a study designed to reduce the exposure of children to the sun are used to illustrate the methods.     

On the estimation of normal copula discrete regression models using the continuous extension and simulated likelihood

The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the asymptotic and small-sample efficiency of two variants of the method for estimating the multivariate normal copula with univariate binary, Poisson, and negative binomial regressions, and show that they lead to biased estimates for the latent correlations, and the univariate marginal parameters that are not regression coefficients. We implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi-Monte Carlo methods. Asymptotic and small-sample efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified multivariate normal copula-based models. An illustrative example is given to show the use of our simulated likelihood method.     

Predictive assessment of copula models

Copulas are powerful explanatory tools for studying dependence patterns in multivariate data. While the primary use of copula models is in multivariate dependence modelling, they also offer predictive value for regression analysis. This article investigates the utility of copula models for model‐based predictions from two angles. We assess whether, where, and by how much various copula models differ in their predictions of a conditional mean and conditional quantiles. From a model selection perspective, we then evaluate the predictive discrepancy between copula models using in‐sample and out‐of‐sample predictions both in bivariate and higher‐dimensional settings. Our findings suggest that some copula models are more difficult to distinguish in terms of their overall predictive power than others, and depending on the quantity of interest, the differences in predictions can be detected only in some targeted regions. The situations where copula‐based regression approaches would be advantageous over traditional ones are discussed using simulated and real data. The Canadian Journal of Statistics 47: 8–26; 2019 © 2018 Statistical Society of Canada     

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).     

Doubly inflated Poisson model using Gaussian copula

Multivariate data are present in many research areas. Its analysis is challenging when assumptions of normality are violated and the data are discrete. The Poisson discrete data can be thought of as very common discrete type, but the inflated and the doubly inflated correspondence are gaining popularity (Sengupta, Chaganty, and Sabo 2015; Lee, Jung, and Jin 2009; Agarwal, Gelfand, and Citron-Pousty 2002).

Our aim is to build a statistical model that can be tractable and used to estimate the model parameters for the multivariate doubly inflated Poisson. To keep the correlation structure, we incorporate ideas from the copula distributions. A multivariate doubly inflated Poisson distribution using Gaussian copula is introduced. Data simulation and parameter estimation algorithms are also provided. Residual checks are carried out to assess any substantial biases. The model dimensionality has been increased to test the performance of the provided estimation method. All results show high-efficiency and promising outcomes in the modeling of discrete data and particularly the doubly inflated Poisson count type data, under a novel modified algorithm.     

Modeling a mixture of ordinal and continuous repeated measures

We study the correlation structure for a mixture of ordinal and continuous repeated measures using a Bayesian approach. We assume a multivariate probit model for the ordinal variables and a normal linear regression for the continuous variables, where latent normal variables underlying the ordinal data are correlated with continuous variables in the model. Due to the probit model assumption, we are required to sample a covariance matrix with some of the diagonal elements equal to one. The key computational idea is to use parameter-extended data augmentation, which involves applying the Metropolis-Hastings algorithm to get a sample from the posterior distribution of the covariance matrix incorporating the relevant restrictions. The methodology is illustrated through a simulated example and through an application to data from the UCLA Brain Injury Research Center.     

On estimating extremal dependence structures by parametric spectral measures

Estimation of extreme value copulas is often required in situations where available data are sparse. Parametric methods may then be the preferred approach. A possible way of defining parametric families that are simple and, at the same time, cover a large variety of multivariate extremal dependence structures is to build models based on spectral measures. This approach is considered here. Parametric families of spectral measures are defined as convex hulls of suitable basis elements, and parameters are estimated by projecting an initial nonparametric estimator on these finite-dimensional spaces. Asymptotic distributions are derived for the estimated parameters and the resulting estimates of the spectral measure and the extreme value copula. Finite sample properties are illustrated by a simulation study.     

Vine copula models with GLM and sparsity

Vine copula provides a flexible tool to capture asymmetry in modeling multivariate distributions. Nevertheless, its flexibility is achieved at the expense of exponentially increasing complexity of the model. To alleviate this issue, the simplifying assumption (SA) is commonly adapted in specific applications of vine copula models. In this paper, generalized linear models (GLMs) are proposed for the parameters in conditional bivariate copulas to relax the SA. In the spirit of the principle of parsimony, a regularization methodology is developed to control the number of parameters, leading to sparse vine copula models. The conventional vine copula with the SA, the proposed GLM-based vine copula, and the sparse vine copula are applied to several financial datasets, and the results show that our proposed models outperform the one with SA significantly in terms of the Bayesian information criterion.     

Likelihood ratio procedures and tests of fit in parametric and semiparametric copula models with censored data

Copula models for multivariate lifetimes have become widely used in areas such as biomedicine, finance and insurance. This paper fills some gaps in existing methodology for copula parameters and model assessment. We consider procedures based on likelihood and pseudolikelihood ratio statistics and introduce semiparametric maximum likelihood estimation leading to semiparametric versions. For cases where standard asymptotic approximations do not hold, we propose an efficient simulation technique for obtaining p-values. We apply these methods to tests for a copula model, based on embedding it in a larger copula family. It is shown that the likelihood and pseudolikelihood ratio tests are consistent even when the expanded copula model is misspecified. Power comparisons with two other tests of fit indicate that model expansion provides a convenient, powerful and robust approach. The methods are illustrated on an application concerning the time to loss of vision in the two eyes of an individual.     

Asymmetric Forecast Densities for U.S. Macroeconomic Variables from a Gaussian Copula Model of Cross-Sectional and Serial Dependence

Most existing reduced-form macroeconomic multivariate time series models employ elliptical disturbances, so that the forecast densities produced are symmetric. In this article, we use a copula model with asymmetric margins to produce forecast densities with the scope for severe departures from symmetry. Empirical and skew t distributions are employed for the margins, and a high-dimensional Gaussian copula is used to jointly capture cross-sectional and (multivariate) serial dependence. The copula parameter matrix is given by the correlation matrix of a latent stationary and Markov vector autoregression (VAR). We show that the likelihood can be evaluated efficiently using the unique partial correlations, and estimate the copula using Bayesian methods. We examine the forecasting performance of the model for four U.S. macroeconomic variables between 1975:Q1 and 2011:Q2 using quarterly real-time data. We find that the point and density forecasts from the copula model are competitive with those from a Bayesian VAR. During the recent recession the forecast densities exhibit substantial asymmetry, avoiding some of the pitfalls of the symmetric forecast densities from the Bayesian VAR. We show that the asymmetries in the predictive distributions of GDP growth and inflation are similar to those found in the probabilistic forecasts from the Survey of Professional Forecasters. Last, we find that unlike the linear VAR model, our fitted Gaussian copula models exhibit nonlinear dependencies between some macroeconomic variables. This article has online supplementary material.     

Ordered response models

Summary We discuss regression models for ordered responses, such as ratings of bonds, schooling attainment, or measures of subjective well-being. Commonly used models in this context are the ordered logit and ordered probit regression models. They are based on an underlying latent model with single index function and constant thresholds. We argue that these approaches are overly restrictive and preclude a flexible estimation of the effect of regressors on the discrete outcome probabilities. For example, the signs of the marginal probability effects can only change once when moving from the smallest category to the largest one. We then discuss several alternative models that overcome these limitations. An application illustrates the benefit of these alternatives. We are grateful to an anonymous referee for valuable comments.     

Nonparametric estimation of copula-based measures of multivariate association from contingency tables

Nonparametric estimation of copula-based measures of multivariate association in a continuous random vector X=(X₁, …, X_d) is usually based on complete continuous data. In many practical applications, however, these types of data are not readily available; instead aggregated ordinal observations are given, for example, ordinal ratings based on a latent continuous scale. This article introduces a purely nonparametric and data-driven estimator of the unknown copula density and the corresponding copula based on multivariate contingency tables. Estimators for multivariate Spearman's rho and Kendall's tau are based thereon. The properties of these estimators in samples of medium and large size are evaluated in a simulation study. An increasing bias can be observed along with an increasing degree of association between the components. As it is to be expected, the bias is severely influenced by the amount of information available. Additionally, the influence of sample size is only marginal. We further give an empirical illustration based on daily returns of five German stocks.     

Measures of biomarker dependence using a copula-based multivariate epsilon–skew–normal family of distributions

In this note we develop a new multivariate copula model based on epsilon–skew–normal marginal densities for the purpose of examining biomarker dependency structures. We illustrate the flexibility and utility of this model via a variety of graphical tools and a data analysis example pertaining to salivary biomarker. The multivariate normal model is a sub-model of the multivariate epsilon–skew–normal distribution.     

Comparison of Goodness-of-Fit Measures in Probit Regression Model

This article examines several goodness-of-fit measures in the binary probit regression model. Existing pseudo-R ² measures are reviewed, two modified and one new pseudo-R ² measure are proposed. For the probit regression model, empirical comparisons are made for different goodness-of-fit measures with the squared sample correlation coefficient of the observed response and the predicted probabilities. As an illustration, the goodness-of-fit measures are applied to a “paid labor force” data set.     

ESTIMATORS BASED ON KENDALL'S TAU IN MULTIVARIATE COPULA MODELS

The estimation of a real‐valued dependence parameter in a multivariate copula model is considered. Rank‐based procedures are often used in this context to guard against possible misspecification of the marginal distributions. A standard approach consists of maximizing the pseudo‐likelihood. Here, we investigate alternative estimators based on the inversion of two multivariate extensions of Kendall's tau developed by Kendall and Babington Smith, and by Joe. The former, which amounts to the average value of tau over all pairs of variables, is often referred to as the coefficient of agreement. Existing results concerning the finite‐ and large‐sample properties of this coefficient are summarized, and new, parallel findings are provided for the multivariate version of tau due to Joe, along with illustrations. The performance of the estimators resulting from the inversion of these two versions of Kendall's tau is compared in the context of copula models through simulations.     

Modeling Dependence in High Dimensions With Factor Copulas

This article presents flexible new models for the dependence structure, or copula, of economic variables based on a latent factor structure. The proposed models are particularly attractive for relatively high-dimensional applications, involving 50 or more variables, and can be combined with semiparametric marginal distributions to obtain flexible multivariate distributions. Factor copulas generally lack a closed-form density, but we obtain analytical results for the implied tail dependence using extreme value theory, and we verify that simulation-based estimation using rank statistics is reliable even in high dimensions. We consider “scree” plots to aid the choice of the number of factors in the model. The model is applied to daily returns on all 100 constituents of the S&P 100 index, and we find significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms. We also show that factor copula models provide superior estimates of some measures of systemic risk. Supplementary materials for this article are available online.     

Tail-weighted measures of dependence

Multivariate copula models are commonly used in place of Gaussian dependence models when plots of the data suggest tail dependence and tail asymmetry. In these cases, it is useful to have simple statistics to summarize the strength of dependence in different joint tails. Measures of monotone association such as Kendall's tau and Spearman's rho are insufficient to distinguish commonly used parametric bivariate families with different tail properties. We propose lower and upper tail-weighted bivariate measures of dependence as additional scalar measures to distinguish bivariate copulas with roughly the same overall monotone dependence. These measures allow the efficient estimation of strength of dependence in the joint tails and can be used as a guide for selection of bivariate linking copulas in vine and factor models as well as for assessing the adequacy of fit of multivariate copula models. We apply the tail-weighted measures of dependence to a financial data set and show that the measures better discriminate models with different tail properties compared to commonly used risk measures – the portfolio value-at-risk and conditional tail expectation.     

Focused information criteria for copulas

In this paper, we extend the focused information criterion (FIC) to copula models. Copulas are often used for applications where the joint tail behavior of the variables is of particular interest, and selecting a copula that captures this well is then essential. Traditional model selection methods such as the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) aim at finding the overall best‐fitting model, which is not necessarily the one best suited for the application at hand. The FIC, on the other hand, evaluates and ranks candidate models based on the precision of their point estimates of a context‐given focus parameter. This could be any quantity of particular interest, for example, the mean, a correlation, conditional probabilities, or measures of tail dependence. We derive FIC formulae for the maximum likelihood estimator, the two‐stage maximum likelihood estimator, and the so‐called pseudo‐maximum‐likelihood (PML) estimator combined with parametric margins. Furthermore, we confirm the validity of the AIC formula for the PML estimator combined with parametric margins. To study the numerical behavior of FIC, we have carried out a simulation study, and we have also analyzed a multivariate data set pertaining to abalones. The results from the study show that the FIC successfully ranks candidate models in terms of their performance, defined as how well they estimate the focus parameter. In terms of estimation precision, FIC clearly outperforms AIC, especially when the focus parameter relates to only a specific part of the model, such as the conditional upper‐tail probability.