Modified inference function for margins for the bivariate clayton copula-based SUN Tobit Model

ABSTRACT

This paper extends the analysis of the bivariate Seemingly Unrelated Regression (SUN) Tobit model by modeling its nonlinear dependence structure through the Clayton copula. The ability to capture/model the lower tail dependence of the SUN Tobit model where some data are censored (generally, left-censored at zero) is an useful feature of the Clayton copula. We propose a modified version of the (classical) Inference Function for Margins (IFS) method by Joe and XP [H. Joe and J.J. XP, The estimation method of inference functions for margins for multivariate models, Tech. Rep. 166, Department of Statistics, University of British Columbia, 1996], which we refer to as Modified Inference Function for Margins (MIFF) method, to obtain the (point) estimates of the marginal and Clayton copula parameters. More specifically, we employ the (frequenting) data augmentation technique at the second stage of the IFS method (the first stage of the MIFF method is equivalent to the first stage of the IFS method) to generate the censored observations and then estimate the Clayton copula parameter. This process (data augmentation and copula parameter estimation) is repeated until convergence. Such modification at the second stage of the usual estimation method is justified in order to obtain continuous marginal distributions, which ensures the uniqueness of the resulting Clayton copula, as stated by Solar's [A. Solar, Fonctions de répartition à n dimensions et leurs marges, Publ. de l'Institut de Statistique de l'Université de Paris 8 (1959), pp. 229–231] theorem; and also to provide an unbiased estimate of the association parameter (the IFS method provides a biased estimate of the Clayton copula parameter in the presence of censored observations in both margins). Since the usual asymptotic approach, that is the computation of the asymptotic covariance matrix of the parameter estimates, is troublesome in this case, we also propose the use of resampling procedures (bootstrap methods, such as standard normal and percentile, by Efron and Tibshirani [B. Efron and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall, New York, 1993] to obtain confidence intervals for the model parameters.

KEYWORDS:

• Bivariate data
• bootstrap confidence intervals
• data augmentation
• Tobit model
• truncation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. Tobit = Tobin(-in) + probit(-prob).

2. Note from Figure 2 that there is a high number of zero observations of the lettuce variable, which corresponds to zero observations of the salad dressings variable (15 pairs of zero observations). This seems to indicate the strongest relationship between the two dependent variables in their lower regions (i.e. for low or no consumption of salad dressings and lettuce), where data are most concentrated. Thus, the use of the Clayton copula is justified in order to accommodate the possible existence of lower tail dependence, as well as positive nonlinear dependence structure.

3. To generate the random numbers from the truncated conditional distribution of the Clayton copula, we can apply the inversion method by Devroye [14, pp. 38–39].

4. The generated/augmented marginal uniform data $u_{ij}^{\mathrm{a}}$ should carry $\left(\omega \right)$ as a superscript $\left(\text{i.e.}{u}_{ij}^{\mathrm{a}\left(\omega \right)}\right)$, but we omit it so as not to clutter the notation.

5. See Joe [25, pp. 301–302] for the form of this matrix.

6. Hinkley [20] suggests that the minimum value of B will depend on the parameter being estimated, but that it will often be at least 100.

7. The Kendall's tau for Clayton copula is equal to $\tau =\theta /(\theta +2)$; see McNeil et al. [30, p. 222].

8. It would be better here if we considered more replications and sample sizes, mainly values of n between 800 and 2000, as indicated by the Bias plots in Figure 3. However, it took about three days, one week and three weeks for the simulation for n = 200, 800, and 2000, respectively, to complete; which represent very long computation times.

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

10. But now with $\mathit{\eta }$ denoting the parameter vector of the bivariate basic SUN Tobit model.

Additional information

Funding

The research is partially funded by the Brazilian organizations, CNPq and FAPESP.