Maximum likelihood estimation for bivariate SUR Tobit modeling in presence of two right-censored dependent variables

ABSTRACT

This article extends the analysis of the Seemingly Unrelated Regression (SUR) Tobit model for two right-censored dependent variables by modeling its nonlinear dependence structure through the rotated by 180 degrees version of the Clayton copula. An advantage of our approach is to provide unbiased point estimates of the marginal and copula parameters. Moreover, we discuss the construction of confidence intervals using bootstrap resampling procedures. The results of the performed simulation study demonstrate the good performance of the proposed methods. We illustrate our procedures using bivariate customer churn data from a Brazilian commercial bank.

KEYWORDS:

• Bootstrap confidence intervals
• Customer churn
• Frequentist data augmentation
• Latent-dependent variables
• Right-censoring
• Upper tail dependence

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H12
• 62P20

Notes

1 This process is popularly known as merger and acquisition (M&A) or takeover (Hildebrandt 2007).

2 See, for example, Omori and Miyawaki (2010) for examples of Tobit models with unknown and covariate dependent thresholds.

3 That is (wrongly) the same form as in the case of continuous margins.

4 The IFAM estimation method by Louzada and Ferreira (2015, 2016) can also be seen as an alternative to the (classical) EM algorithm, whose development is very challenging for most copula-based models.

5 The truncation dependence invariance property of the rotated Clayton copula can be easily verified, for example, by following the same ideas as in the online short note: http://web.cecs.pdx.edu/∼cgshirl/Documents/Research/Copula_Methods/Clayton%20Copula.pdf.

6 The augmented marginal uniform data u^a_ij should carry (ω) as a superscript (i.e., u^a(ω)_ij), but we omit it so as not to clutter the notation.

7 The Kendall’s tau for the rotated Clayton copula is $\tau =\theta /(\theta +2)$, which is the same for the Clayton copula; see McNeil et al. (2005, p. 222).

8 It would be better here if we considered more replications and sample sizes, especially values of n between 800 and 2,000, as indicated by the Bias plots in Fig. 3. Nevertheless, it took about 3 days, 1 week, and 3 weeks for the simulation for n = 200, 800, and 2,000, respectively, to complete; which represent very long computation times.

9 The augmented residuals are given by the differences between the augmented observed and predicted responses, that is, $e_{ij}^{\mathrm{a}}=y_{ij}^{\mathrm{a}}-{\mathbf{x}}_{ij}^{{}^{\text{'}}}{\widehat{\beta }}_j$, for i = 1, …, n and j = 1, 2, where $y_{ij}^{\mathrm{a}}={\mathbf{x}}_{ij}^{{}^{\text{'}}}{\widehat{\beta }}_j+{\widehat{\sigma }}_j{\Phi }^{-1}\left(u_{ij}^{\mathrm{a}}\right)$, with Φ^{− 1}(.) being the inverse function of the N(0, 1) c.d.f.; or simply, $e_{ij}^{\mathrm{a}}={\widehat{\sigma }}_j{\Phi }^{-1}\left(u_{ij}^{\mathrm{a}}\right)$.

10 The upper tail dependence coefficient for the rotated Clayton copula is equal to the lower tail dependence coefficient for the Clayton copula; see Joe (2014, p. 69).

11 In this case, however, $\eta$ denotes the parameter vector of the basic bivariate SUR Tobit “right-censored” model.