Extending the inference function for augmented margins method to implement trivariate Clayton copula-based SUR Tobit models

Abstract

In this paper, we perform the analysis of the SUR Tobit model for three left-censored dependent variables by modeling its nonlinear dependence structure through the one-parameter Clayton copula. For unbiased parameter estimation, we propose an extension of the Inference Function for Augmented Margins (IFAM) method to the trivariate case. The interval estimation for the model parameters using resampling procedures is also discussed. We perform simulation and empirical studies, whose satisfactory results indicate the good performance of the proposed model and methods. Our procedure is illustrated using real data on consumption of food items (salad dressings, lettuce, tomato) by Americans.

Keywords:

• Bootstrap confidence intervals
• consumption data
• data augmentation
• trivariate one-parameter Clayton copula
• two-stage maximum likelihood estimation

Acknowledgment

The authors thank the editorial boarding and referees for making interesting comments and suggestions.

Notes

1 The Kendall’s tau for the m-variate Clayton copula with parameter θ is given by ${\tau }_m={\left(2^{m-1}-1\right)}^{-1}\left\{-1+2^m{\prod }_{p=0}^{m-1}\left(1+p\theta \right)/\left(2+p\theta \right)\right\}$ (Genest et al. 2011). After some simple calculations, we find that for m = 3, ${\tau }_3=\theta /\left(\theta +2\right)$.

2 See pages 166, 172, 163, and 181 of Joe (2014) for details on the multivariate Frank, Gumbel, Gaussian and Student’s t copulas, respectively.

3 The augmented residuals are given by the differences between the augmented observed and predicted responses, i.e. $e_{ij}^{\mathrm{a}}=y_{ij}^{\mathrm{a}}-{\mathit{x}}_{ij}^{\prime }{\widehat{\mathit{\beta }}}_{j,\text{IFAM}}$, for $i=1,\dots ,n$ and j = 1, 2, 3, where $y_{ij}^{\mathrm{a}}={\mathit{x}}_{ij}^{\prime }{\widehat{\mathit{\beta }}}_{j,\text{IFAM}}+{\widehat{\sigma }}_{j,\text{IFAM}}{G}^{-1}\left(u_{ij}^{\mathrm{a}}\right)$, with $G^{-1}(.)$ being the inverse function of the $\text{log}\hspace{0.17em}\mathit{\text{istic}}\left(0,1\right)$ c.d.f. given by $G\left(z\right)=1/\left(1+\hspace{0.17em}\text{exp}\hspace{0.17em}\{-z\}\right),z\in \mathrm{I}\mathrm{R}$. Or shortly, $e_{ij}^{\mathrm{a}}={\widehat{\sigma }}_{j,\text{IFAM}}{G}^{-1}\left(u_{ij}^{\mathrm{a}}\right)$.

4 We considered the augmented residuals for computing these test statistics values, since this approach takes into account the presence of covariates in the margins.

5 The trivariate Clayton survival copula can be easily obtained from Equation (2)(2) $$C\left(u_{i1},u_{i2},u_{i3}|\theta \right)={\left(u_{i1}^{-\theta }+u_{i2}^{-\theta }+u_{i3}^{-\theta }-2\right)}^{-\frac{1}{\theta }}$$(2) as shown in Joe (2014, p. 28).

Additional information

Funding

This research is supported by the Brazilian agencies, CNPq and FAPESP.