Copula-based regression models with data missing at random

Abstract

The existing literature of copula-based regression assumes that complete data are available, but this assumption is violated in many real applications. The present paper allows the regressand and regressors to be missing at random (MAR). We formulate a generalized regression model which unifies many prominent cases such as the conditional mean and quantile regressions. A semiparametric copula and the target regression curve are estimated via the calibration approach. The consistency and asymptotic normality of the estimated regression curve are proved. We show via Monte Carlo simulations that the proposed approach operates well in finite samples, while a benchmark equal-weight approach fails with substantial bias under MAR. An empirical application on revenues and R&D expenses of German manufacturing firms highlights a practical use of our approach.

Introduction

Regression is the most prevailing method for investigating the relationship between a regressand $Y$ and regressors $\mathit{W}$. Widely used regressions include conditional mean and quantile regressions. Noh et al. [27] proposed a novel approach to estimate a conditional mean regression function by exploiting copulas, where observations are assumed to be independently and identically distributed ($i.i.d$.) and completely observed. Their key insight is that the loss function expressed as a conditional expectation given $\mathit{W}$ can be rewritten as an unconditional expectation involving a parametric copula and nonparametric marginal distributions. The marginal distributions and the copula parameter are estimated via plug-in methods. The flexibility of the semiparametric copula alleviates model specification issues such as how to transform regressors and which cross-products of regressors to include.

Noh et al. [27] spurred extensive research on the copula-based regression. Noh et al. [28] applied the method of Noh et al. [27] to the quantile regression with $i.i.d$. or time series data that are completely observed. De Backer et al. [4] extended the method of Noh et al. [28] to the quantile regression with censored data. Kraus and Czado [19] studied the quantile regression with complete data, using D-vine copulas. Rémillard et al. [30] discussed the asymptotic connection between the estimators of Noh et al. [28] and Kraus and Czado [19]. Chang and Joe [3] proposed an algorithm for computing the conditional distribution function via the vine copula.

Nagler and Vatter [23] unified various copula-based regressions by formulating a general loss function which may not be continuously differentiable. Their generalized regression model includes the conditional mean regression of Noh et al. [27], the conditional quantile regression of Noh et al. [28], and the asymmetric least squares of Newey and Powell [26] as special cases. The unified framework enhances the systematic interpretation of the various existing regressions.

A potential issue left in the existing literature of copula-based regression is that complete data are assumed to be available. There also exists the vast literature where copula itself is a primary target of estimation [11], but the availability of complete data is still assumed in most papers in that literature. The assumption of complete data is violated in many real applications such as applied microeconomics, corporate finance, and survey sampling. In survey sampling, for example, respondents may refuse to report their personal information such as age and salary.

To relax the assumption of complete data, this paper allows both the regressand $Y$ and the regressors $\mathit{W}$ to be missing at random (MAR), a key concept originally explored by Rubin [31]. The MAR condition has been popularly used in econometrics and statistics to identify the parameter of interest [22]. Hamori et al. [13] is one of few works to deal with data missing at random in copula modeling. (Ding and Song [6] proposed an EM algorithm for estimating the Gaussian copula under the MAR condition. Emura et al. [8], Emura and Wang [9], [10] considered the copula inference with truncated survival data. Guo et al. [12] studied the semiparametric estimation of copula models with nonignorable missing data.) Hamori et al. [13] use calibration weights proposed by Chan et al. [2] for both nonparametric marginal distributions and target copula parameters. The calibration estimation is a nonparametric method that balances the empirical moments of covariates between the observed and whole groups. It does not require an explicit specification of the missing mechanism, and delivers consistent inference under MAR.

Inspired by Hamori et al. [13], this paper adopts the calibration estimation to perform the copula-based regression with $\{Y,\mathit{W}\}$ missing at random. As in Nagler and Vatter [23], we formulate the generalized regression model which unifies many prominent regressions. A semiparametric copula and the target regression curve are estimated via the calibration approach. The consistency and asymptotic normality of the estimated regression curve are proved. Our simulation study shows that the proposed approach performs well in finite samples, while a benchmark equal-weight approach fails with substantial bias under MAR.

As an empirical application, we regress the R&D expenses of German manufacturing firms onto their revenues. The revenue is observed for all 500 firms considered, while the R&D expense is observed for only 125 firms. The vast majority of the 375 firms with missing R&D have small revenues. The calibration approach delivers a plausible empirical result by assigning sufficiently large weights on firms with small revenues. The benchmark equal-weight approach, by contrast, delivers a misleading result by discarding the 375 firms with missing R&D and assigning the uniform weight on the 125 firms left. This contrast highlights how the calibration approach achieves valid inference under the MAR mechanism.

The rest of this paper is organized as follows: Section 2 explains our basic framework and notation. The calibration estimation is proposed in Section 3, and its large sample properties are derived in Section 4. Variance estimators and confidence intervals are constructed in Section 5. In Section 6, data-driven selection of a tuning parameter $K$ for sieve basis functions is discussed. The simulation study is performed in Section 7. The empirical application is presented in Section 8. Conclusions are provided in Section 9. Proofs of main theorems are collected in Technical Appendices. In the Online Supplement [14], omitted technical and numerical details are provided.