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Marketing researchers are increasingly taking advantage of the instrumental variable (IV)-free Gaussian copula approach. They use this method to identify and correct endogeneity when estimating regression models with non-experimental data. The Gaussian copula approach’s original presentation and performance demonstration via a series of simulation studies focused primarily on regression models without intercept. However, marketing and other disciplines’ researchers mainly use regression models with intercept. This research expands our knowledge of the Gaussian copula approach to regression models with intercept and to multilevel models. The results of our simulation studies reveal a fundamental bias and concerns about statistical power at smaller sample sizes and when the approach’s primary assumptions are not fully met. This key finding opposes the method’s potential advantages and raises concerns about its appropriate use in prior studies. As a remedy, we derive boundary conditions and guidelines that contribute to the Gaussian copula approach’s proper use. Thereby, this research contributes to ensuring the validity of results and conclusions of empirical research applying the Gaussian copula approach.

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.

Endogeneity in regression models is a key marketing research concern. The Gaussian copula approach offers an instrumental variable (IV)-free technique to mitigate endogeneity bias in regression models. Previous research revealed substantial finite sample bias when applying this method to regression models with an intercept. This is particularly problematic as models in marketing studies almost always require an intercept. To resolve this limitation, our research determines the bias's sources, making several methodological advances in the process. First , we show that the cumulative distribution function estimation's quality strongly affects the Gaussian copula approach's performance. Second , we use this insight to develop an adjusted estimator that improves the Gaussian copula approach's finite sample performance in regression models with (and without) an intercept. Third , as a broader contribution, we extend the framework for copula estimation to models with multiple endogenous variables on continuous scales and exogenous variables on discrete and continuous scales, and non-linearities such as interaction terms. Fourth , simulation studies confirm that the new adjusted estimator outperforms the established ones. Further simulations also underscore that our extended framework allows researchers to validly deal with multiple endogenous and exogenous regressors, and the interactions between them. Fifth , we demonstrate the adjusted estimator and the general framework's systematic application, using an empirical marketing example with real-world data. These contributions enable researchers in marketing and other disciplines to effectively address endogeneity problems in their models by using the improved Gaussian copula approach.

This article examines whether investors receive compensation for holding crash-sensitive stocks. We capture the crash sensitivity of stocks by their lower-tail dependence (LTD) with the market based on copulas. We find that stocks with strong LTD have higher average future returns than stocks with weak LTD. This effect cannot be explained by traditional risk factors and is different from the impact of beta, downside beta, coskewness, cokurtosis, and Kelly and Jiang’s (2014) tail risk beta. Hence, our findings are consistent with the notion that investors are crash-averse.

This article proposes a new class of copula-based dynamic models for high-dimensional conditional distributions, facilitating the estimation of a wide variety of measures of systemic risk. Our proposed models draw on successful ideas from the literature on modeling high-dimensional covariance matrices and on recent work on models for general time-varying distributions. Our use of copula-based models enables the estimation of the joint model in stages, greatly reducing the computational burden. We use the proposed new models to study a collection of daily credit default swap (CDS) spreads on 100 U.S. firms over the period 2006 to 2012. We find that while the probability of distress for individual firms has greatly reduced since the financial crisis of 2008-2009, the joint probability of distress (a measure of systemic risk) is substantially higher now than in the precrisis period. Supplementary materials for this article are available online.

In this paper, we introduce quantile coherency to measure general dependence structures emerging in the joint distribution in the frequency domain and argue that this type of dependence is natural for economic time series but remains invisible when only the traditional analysis is employed. We define estimators that capture the general dependence structure, provide a detailed analysis of their asymptotic properties, and discuss how to conduct inference for a general class of possibly nonlinear processes. In an empirical illustration we examine the dependence of bivariate stock market returns and shed new light on measurement of tail risk in financial markets. We also provide a modelling exercise to illustrate how applied researchers can benefit from using quantile coherency when assessing time series models.

Although email marketing is highly profitable and widely used by marketers, it has received limited attention in the marketing literature. Extant research has focused on either customers' email responses or the \"average\" effect of emails on purchases. In this article, the authors use data from a U.S. home improvement retailer to study customers' email open and purchase behaviors by using a unified hidden Markov and copula framework. Contrary to conventional wisdom, the authors find that email-active customers are not necessarily active in purchases, and vice versa. Furthermore, the number of emails sent by the retailer has a nonlinear effect on both the retailer's short- and long-term profitability. Through a counterfactual study, the authors provide a decision support system to guide retailers in making optimal email contact decisions. This study shows that sending the right number of emails is vital for long-term profitability. For example, sending four (ten) emails instead of the optimal number of seven emails can cause the retailer to lose 32% (16%) of its lifetime profit per customer.

We investigate the performance of VaR (Value at Risk) forecasts, considering different multivariate models: HS (Historical Simulation), DCC-GARCH (Dynamic Conditional Correlation-Generalized Autoregressive Conditional Heteroskedasticity) with normal and Student's distribution, GO-GARCH (Generalized Orthogonal-Generalized Autoregressive Conditional Heteroskedasticity), and copulas Vine (C-Vine, D-Vine, and R-Vine). For copula models, we consider that marginal distribution follow normal, Student's and skewed Student's distribution. We assessed the performance of the models using stocks belonging to the Ibovespa index during the period from January 2012 to April 2022. We build portfolios with 6 and 12 stocks considering two strategies to form the portfolio weights. We use a rolling estimation window of 500 and 1000 observations and 1%, 2.5%, and 5% as significance levels for the risk estimation. To evaluate the quality of the risk forecasts, we compute the realized loss and cost. Our results show that the performance of the models is sensitive to the use of different significance levels, rolling windows, and strategies to determine portfolio weights. Furthermore, we find that the model that presents the best trade-off between the costs from risk overestimation and underestimation does not coincide with the model suggested by the realized loss.

We propose a method to correct for sample selection in quantile regression models. Selection is modeled via the cumulative distribution function, or copula, of the percentile error in the outcome equation and the error in the participation decision. Copula parameters are estimated by minimizing a method-of-moments criterion. Given these parameter estimates, the percentile levels of the outcome are readjusted to correct for selection, and quantile parameters are estimated by minimizing a rotated \"check\" function. We apply the method to correct wage percentiles for selection into employment, using data for the UK for the period 1978-2000. We also extend the method to account for the presence of equilibrium effects when performing counterfactual exercises.

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.