A new family of copulas, with application to estimation of a production frontier system

Abstract

This paper makes two contributions. The first is to propose a new family of copulas for which the copula arguments are uncorrelated but dependent. Specifically, if w₁ and w₂ are the uniform random variables in the copula, they are uncorrelated, but w₁ is correlated with |w₂ − 1/2|. We show how this family of copulas can be applied to the error structure in an econometric production frontier model. The second contribution is to give some general results on how to extend a two-dimensional copula to three or more dimensions. This extension is necessary in our production frontier model when there are multiple inputs, but our results apply more generally to the extension of arbitrary two-dimensional copulas. We also report the results of some simulations and we give an empirical example.

Appendices

Appendix 1

The SL copula

For notational simplicity only, we will consider the case that ω is a scalar.

In the SL model, $\left[ {\begin{array}{*{20}{c}} {u^ \ast } \\ \omega \end{array}} \right]\sim N\left( {0,{\Sigma}} \right)$ where = $\left[ {\begin{array}{*{20}{c}} {\sigma _u^2} & {{\Sigma}_{\omega u}} \\ {{\Sigma}_{\omega u}} & {\sigma _\omega ^2} \end{array}} \right]$. Then u = |u^*|.

The joint density of u* and ω, say g(u*,ω), is the bivariate normal density of N(0,∑) The joint density of u and ω is then

$$\begin{array}{l}h\left( {u,\omega } \right) = g\left( {u,\omega } \right) + g\left( { - u,\omega } \right)\\ \qquad\quad \, = \frac{1}{{2\pi }}\left| {\Sigma} \right|^{ - 1/2}\exp \left[ { - \frac{1}{2}\left( {u,\omega } \right){\Sigma}^{ - 1}\left( {\begin{array}{*{20}{c}} u \\ \omega \end{array}} \right)} \right] \\\qquad\quad + \frac{1}{{2\pi }}\left| {\Sigma} \right|^{ - 1/2}\exp \left[ { - \frac{1}{2}\left( { - u,\omega } \right){\Sigma}^{ - 1}\left( {\begin{array}{*{20}{c}} { - u} \\ \omega \end{array}} \right)} \right]\end{array}$$

as given in Schmidt and Lovell (1980) (Eq. (A.12)).

Now define $\mathop {\sum}\nolimits_ \ast { = \left[ {\begin{array}{*{20}{c}} {\sigma _u^2} & { - {\Sigma}_{\omega u}} \\ { - {\Sigma}_{\omega u}} & {\sigma _\omega ^2} \end{array}} \right]}$. It is easy to verify that |∑_*| = |∑| and that $\left( { - u,\omega } \right){\Sigma}^{ - 1}\left( {\begin{array}{*{20}{c}} { - u} \\ \omega \end{array}} \right) = \left( {u,\omega } \right){\Sigma}_ \ast ^{ - 1}\left( {\begin{array}{*{20}{c}} u \\ \omega \end{array}} \right)$. Therefore$\begin{array}{*{20}{c}} {h\left( {u,\omega } \right) = \frac{1}{{2\pi }}\left| {\Sigma} \right|^{ - 1/2}{\mathrm{exp}}\left[ { - \frac{1}{2}\left( {u,\omega } \right){\Sigma}^{ - 1}\left( {\begin{array}{*{20}{c}} u \\ \omega \end{array}} \right)} \right]} & {^{\prime\prime} {\mathrm{term}}\,1^{\prime\prime} } \end{array}$

$$\begin{array}{*{20}{c}} { + \frac{1}{{2\pi }}\left| {{\Sigma}_ \ast } \right|^{ - 1/2}{\mathrm{exp}}\left[ { - \frac{1}{2}\left( {u,\omega } \right)\mathop {\sum}\nolimits_ \ast ^{ - 1} {\left( {\begin{array}{*{20}{c}} u \\ \omega \end{array}} \right)} } \right]} & {^{\prime\prime} {\mathrm{term}}\,2^{\prime\prime} } \end{array}$$

To calculate the copula, we now need to divide h(u,ω) by the product of the marginal densities of u and ω, that is, by

$$\frac{2}{{\sqrt {2\pi } }}\,\frac{1}{{\sigma _u}}\exp \left( { - \frac{1}{{2\sigma _u^2}}u^2} \right) \cdot \frac{1}{{\sqrt {2\pi } }}\frac{1}{{\sigma _\omega }}\exp \left( { - \frac{1}{{2\sigma _\omega ^2}}\omega ^2} \right).$$

Carrying out this division, the first term above (“term 1”) becomes, by standard algrebra used in the derivation of the normal copula,

$$\frac{1}{2}\left| R \right|^{ - 1/2}\exp \left[ { - \frac{1}{2}\left( {u,\omega } \right)\left( {R^{ - 1} - I} \right)\left( {\begin{array}{*{20}{c}} u \\ \omega \end{array}} \right)} \right] \quad {\mathrm{where}}\,\,R = \left[ {\begin{array}{*{20}{c}} 1 & \rho \\ \rho & 1 \end{array}} \right]$$

$$= \frac{1}{2}\left( {1 - \rho ^2} \right)^{ - \frac{1}{2}}\exp \left[ { - \frac{1}{2}\left( {1 - \rho ^2} \right)^{ - 1}\left( {\rho ^2u^2 + \rho ^2\omega ^2 - 2\rho u\omega } \right)} \right]$$

which is one-half times the normal copula with parameter ρ. Similarly, the second term above (“term 2”) becomes one-half times the normal copula with parameter −ρ.

Appendix 2

Proofs of Results for the APS copulas

Proof of Result 1 We have a copula of the form c(w₁,w₂) = 1 + g(w₁)h(w₂), where ${\int}_0^1 {g\left( s \right)ds} = {\int}_0^1 {h\left( s \right)ds = 0}$, and specifically where $h\left( {w_2} \right) = 1 - k_q^{ - 1}q\left( {w_2} \right)$ with $k_q = {\int}_0^1 {q\left( s \right)ds}$. Define $G\left( {w_1} \right) = {\int}_0^{w_1} {g\left( s \right)ds}$, $H\left( {w_2} \right) = {\int}_0^{w_2} {h\left( s \right)ds}$, $Q\left( {w_2} \right) = {\int}_0^{w_2} {q\left( s \right)ds}$, $G^ \ast = {\int}_0^1 {G\left( {w_1} \right)dw_1}, \,H^ \ast = {\int}_0^1 {H\left( {w_2} \right)dw_2}$, and $Q^ \ast = {\int}_0^1 {Q\left( {w_2} \right)dw_2}$, and note that H(w₂) = w₂−Q(w₂) and k_q = Q(1). A general result for Sarmanov copulas (Rodriguez-Lallena and Ubeda-Flores 2004) is that cov(w₁,w₂) = G^*H^*. The value of G* is θ/6 when g(w₁) = θ(1 − 2w₂), but this does not feature in the proof, which simply establishes that H* = 0.

To show that H* = 0, we use the symmetry of q(s) around s = 1/2, which implies that Q(s) = Q(1) − Q(1 − s) for s > 1/2. Therefore $Q^ \ast = {\int}_0^1 {Q\left( {w_2} \right)dw_2} = {\int}_0^{1/2} {Q\left( {w_2} \right)dw_2} + {\int}_{1/2}^1 {\left[ {Q\left( 1 \right) - Q\left( {1 - w_2} \right)} \right]dw_2} = {\int}_0^{1/2} {Q\left( {w_2} \right)dw_2} + \frac{1}{2}Q\left( 1 \right) - {\int}_{1/2}^1 {Q\left( {1 - w_2} \right)dw_2} = \frac{1}{2}Q\left( 1 \right) = \frac{1}{2}k_q$. Then $H^ \ast = \frac{1}{2} - k_q^{ - 1}Q^ \ast = \frac{1}{2} - k_q^{ - 1}\left( {\frac{1}{2}k_q} \right) = 0$, which implies that cov(w₁,w₂) = 0. ☐

Proof of Result 2

$$\begin{array}{l}E\left[ {w_1q\left( {w_2} \right)} \right] = {\int} {{\int} {w_1q\left( {w_2} \right)dw_1dw_2} + } \left[ {\theta {\int} {w_1\left( {1 - 2w_1} \right)dw_1} } \right]\\ \cdot \left[ {{\int} {q\left( {w_2} \right)\left( {1 - k_q^{ - 1}q\left( {w_2} \right)} \right)} dw_2} \right].\end{array}$$

Here all integrals are from zero to one.

The first term on the r.h.s. of this equation is ${\int} {w_1dw_1} \cdot {\int} {q\left( {w_2} \right)dw_2} = \frac{1}{2}Q\left( 1 \right) = \frac{1}{2}k_q$.

The first term in brackets following the “+” sign equals $\theta \left( {\frac{1}{2} - \frac{2}{3}} \right) = - \frac{1}{6}\theta$.

The second term in brackets equals

$$\begin{array}{l}{\int} {q\left( {w_2} \right)dw_2 - } k_q^{ - 1}{\int} {\,} q\left( {w_2} \right)^2dw_2 = k_q - k_q^{ - 1}Eq\left( {w_2} \right)^2\\ = k_q - k_q^{ - 1}\left[ {{\mathrm{var}}\left( {q\left( {w_2} \right)} \right) + k_q^2} \right] = - k_q^{ - 1}{\mathrm{var}}\left( {q\left( {w_2} \right)} \right)\end{array}$$

Combining terms, $E \left[ {w_1q\left( {w_2} \right)} \right] =\frac{1}{2}k_q + \frac{1}{6}\theta k_q^{ - 1}{\mathrm{var}}\left( {q\left( {w_2} \right)} \right)$ and therefore

$${\mathop{\rm{cov}}} \left[ {w_1q\left( {w_2} \right)} \right] = \frac{1}{6}\theta k_q^{ - 1}{\mathrm{var}}\left( {q\left( {w_2} \right)} \right).\quad \square$$

Proof of Result 3 We have the APS-2 copula $c( {w_1,w_2} ) = 1 + \theta ( {1 - 2w_1} )( {1 - k_q^{ - 1}q( {w_2} )} )$ where $w_1 = F_u\left( u \right)$ and $w_2 = F_\omega \left( \omega \right)$. For notational simplicity only, suppose that $E\left( u \right) = E\left( \omega \right) = 0$. (Otherwise we just have to do the analysis below in terms of deviations from means.) Then

$${\mathrm{cov}}\left( {u,\omega } \right) = E\left( {u\omega } \right) = {\int}_{\! - \infty }^\infty {{\int}_{\! - \infty }^\infty u\omega {f_u\left( u \right)f_\omega \left( \omega \right)c\left( {F_u\left( u \right),\,F_\omega \left( \omega \right)} \right)du\,d\omega } }$$

$$\begin{array}{l}\begin{array}{*{20}{c}} { = \displaystyle{\int}_{ - \infty }^\infty {{\int}_{ - \infty }^\infty {u\omega f_u\left( u \right)f_\omega \left( \omega \right)du\,d\omega } } } & {\left[ {{\mathrm{term}}\,1} \right]} \end{array}\\ \end{array}$$

$$\begin{array}{*{20}{c}} { + \theta \displaystyle{\int}_0^\infty {{\int}_0^\infty {u\omega f_u\left( u \right)f_\omega \left( \omega \right)\left( {1 - 2w_1} \right)\left( {1 - k_q^{ - 1}q\left( {w_2} \right)} \right)\,du\,d\omega } } } & {\left[ {{\mathrm{term}}\,{\mathrm{2}}} \right]} \end{array}$$

$$\begin{array}{*{20}{c}} { + \theta \displaystyle{\int}_{ - \infty }^0 {{\int}_{ - \infty }^0 {u\omega f_u\left( u \right)f_\omega \left( \omega \right)\left( {1 - 2w_1} \right)\left( {1 - k_q^{ - 1}q\left( {w_2} \right)} \right)\,du\,d\omega } } } & {\left[ {{\mathrm{term}}\,{\mathrm{3}}} \right]} \end{array}$$

$$\begin{array}{*{20}{c}} { + \theta \displaystyle{\int}_{ - \infty }^0 {{\int}_0^\infty {u\omega f_u\left( u \right)f_\omega \left( \omega \right)\left( {1 - 2w_1} \right)\left( {1 - k_q^{ - 1}q\left( {w_2} \right)} \right)\,du\,d\omega } } } & {\left[ {{\mathrm{term}}\,{\mathrm{4}}} \right]} \end{array}$$

$$\begin{array}{*{20}{c}} { + \theta \displaystyle{\int}_0^\infty {{\int}_{ - \infty }^0 {u\omega f_u\left( u \right)f_\omega \left( \omega \right)\left( {1 - 2w_1} \right)\left( {1 - k_q^{ - 1}q\left( {w_2} \right)} \right)\,du\,d\omega } } } & {\left[ {{\mathrm{term}}\,{\mathrm{5}}} \right]} \end{array}$$

where again, for visual simplicity, w₁ = F_u(u) and w₂ = F_ω(ω).

Term 1 equals zero because E(u) = E(ω) = 0.

Term 2 equals the negative of term 3 (i.e., they sum to zero). Because u and ω are symmetric, f_u(u) = f_u(−u) and f_ω(ω) = f_ω(−ω). Also F_u(u) = 1 − F_u(−u) so that 1 − 2F_u(−u) = −[1 − 2F_u(u)]. Finally, q(s) is symmetric around $s = \frac{1}{2}$, so that q(s) = q(1−s) and therefore q(F_ω(−ω)) = q(1 − F_ω(−ω)) = q(F_ω(ω)). This implies that the value of the integrand in term 2 for any u,ω pair (e.g., (0.3, 0.4)) is the negative of the value of the integrand in term 3 of the corresponding pair (e.g., (−0.3, −0.4)), and thus the two terms sum to zero.

Similarly term 4 and term 5 sum to zero, and then cov(u,ω) = 0. ☐

Proof of Result 4 Lower tail dependence equals P(w₁ ≤ α|w₂ ≤ α) = P(w₁ ≤ α, w₂ ≤ α)/P(w₂ ≤ α) = $\{ {\alpha ^2 + \theta \alpha ( {1 - \alpha } )[ {\alpha - k_q^{ - 1}Q( \alpha )} ]} \}/\alpha = \alpha + \theta ( {1 - \alpha } )[ {\alpha - k_q^{ - 1}Q( \alpha )} ]$ and this goes to zero as α → 0. Upper tail dependence equals P(w₁ ≥ α|w₂ ≥ α) = P(w₁ ≥ α, w₂ ≥ α)/P(w₂ ≥ α) = $\{ {( {1 - \alpha } )^2 + \theta \alpha ( {1 - \alpha } )[ {\alpha - k_q^{ - 1}Q( \alpha )} ]} \}/( {1 - \alpha } ) = ( {1 - \alpha } ) + \theta \alpha [ {\alpha - k_q^{ - 1}Q( \alpha )} ]$ and this goes to zero as α → 1. ☐

Proof of Result 5

(i) It is easy to verify that the marginals of c(w₁,w₂) are uniform, so we just need to verify that c(w₁,w₂) ≥ 0 for all w₁,w₂ in the unit cube. We have −1 ≤ 1 − 2w₁ ≤ 1 and −2 ≤ 1 − 12(w₂ − 1/2)² ≤ 1, so −2 ≤ (1 − 2w₁)[1 − 12(w₂ − 1/2)²] ≤ 2. Therefore c(w₁,w₂) ≥ 0 if −1/2 ≤ θ ≤ 1/2.

(ii), (iii) Some useful integrals (integrals are from zero to one):

• (a) ${\int} {\left( {w_2 - 1/2} \right)^2dw_2 = \frac{1}{{12}}}.$

• (b) ${\int} {\left( {w_2 - 1/2} \right)^4dw_2 = \frac{1}{{80}}}$

Therefore ${\mathop{\rm{var}}} \left( {w_2 - 1/2} \right)^2 = \frac{1}{{12}} - \left( {\frac{1}{{80}}} \right)^2 = \frac{1}{{180}}$ which is (iii). To establish (ii), use Result 2 to obtain ${\mathop{\rm{cov}}} \left( {w_1,q\left( {w_2} \right)} \right) = \frac{1}{6}\theta \cdot 12 \cdot \frac{1}{{180}} = \frac{1}{{90}}\theta$.

(iv) ${\mathrm{corr}}\,\left( {w_1,q\left( {w_2} \right)} \right) = \frac{{\frac{1}{{90}}\theta }}{{\sqrt {1/12} \sqrt {1/180} }} = \frac{2}{{\sqrt {15} }}\theta$. ☐

Proof of Result 6

(i) Once again it is easy to verify that the marginals of c(w₁,w₂) are uniform, so we just need to verify that c(w₁,w₂) ≥ 0 for all w₁,w₂ in the unit cube. We have −1 ≤ 1 − 2w₁ ≤ 1 and −1 ≤ 1 − 4|w₂ − 1/2| ≤ 1, so −1 ≤ (1 − 2w₁)[1 − 4]|w₂ − 1/2| ≤ 1. Therefore c(w₁,w₂) ≥ 0 if −1 ≤ θ ≤ 1.

(ii), (iii) Some useful integrals (integrals are from zero to one):

• (a) ${\int} {\left| {w_2 - 1/2} \right|dw_2 = \frac{1}{4}.}$

• (b) ${\int} {\left| {w_2 - 1/2} \right|^2dw_2 = \frac{1}{{12}}}$

Therefore ${\mathop{\rm{var}}} \left( {\left| {w_2 - 1/2} \right|} \right) = \frac{1}{{12}} - \left( {\frac{1}{4}} \right)^2 = \frac{1}{{48}}$ which is (iii). Then use Result 2 to obtain ${\mathop{\rm{cov}}} \left( {w_1,q\left( {w_2} \right)} \right) = \frac{1}{6}\theta \cdot 4 \cdot \frac{1}{{48}} = \frac{1}{{72}}\theta$.

(iv) ${\mathrm{corr}}\left( {w_1,q\left( {w_2} \right)} \right) = \frac{{\frac{1}{{72}}\theta }}{{\sqrt {1/12} \sqrt {1/48} }} = \frac{1}{3}\theta$. ☐

Proof of Result 10 First consider APS-3-A. We have c*(w₁,w₂,w₃) = (c₁₂ − 1) + (c₁₃ − 1) + c₂₃. Here $c_{12} - 1 = \theta _{12}( {1 - 2w_1} )[ {1 - 12( {w_2 - \frac{1}{2}} )^2} ]$. We have −1 ≤ (1 − 2w₁) ≤ 1 and $- 2 \le 1 - 12\left( {w_2 - \frac{1}{2}} \right)^2 \le 1$. So the smallest that c₁₂ − 1 can be is −2|θ₁₂|. Similarly the smallest that c₁₃−1 can be is −2|θ₁₃|. Finally, $\delta = \left( {1 - \rho ^2} \right)^{ - 1/2}\exp \left( { - \frac{1}{2}\frac{{\rho ^2}}{{\left( {1 - \rho ^2} \right)}}} \right)$ ≤ bivariate normal copula = $\left( {1 - \rho ^2} \right)^{ - 1/2}\exp \left[ { - \frac{1}{2}\frac{1}{{\left( {1 - \rho ^2} \right)}}\left( {\rho w_2 - \rho w_3} \right)^2} \right]$. So a sufficient condition for c* to be a copula is −2|θ₁₂| − 2|θ₁₃| + δ ≥ 0, or |θ₁₂|+|θ₁₃| ≤ δ/2. To see why this is necessary, note that the lower bound is attainable, for example at w₁ = 0,w₂ = 1, and w₃ = 0. ☐

The proof of the result for APS-3-B follows exactly the same lines so it is omitted.

Appendix 3

Generalization of Result 9 to higher dimensions

Consider the four-dimensional case. We start with two-dimensional copulas c₁₂, c₁₃,c₁₄,c₂₃,c₂₄, and c₃₄. We can construct three-dimensional copulas as in Result 9. We can construct a four-dimensional copula as

$$\begin{array}{l}c^ \ast \left( {w_1,w_2,w_3,w_4} \right) = 1 + \left( {c_{12} - 1} \right) + \left( {c_{13} - 1} \right) + \left( {c_{14} - 1} \right) \\\qquad\qquad\qquad\qquad+ \left( {c_{23} - 1} \right) + \left( {c_{24} - 1} \right) + \left( {c_{34} - 1} \right).\end{array}$$

The implied 3-copulas are as given in Result 9, for example,

$$\begin{array}{l}\displaystyle \int {c^ \ast \left( {w_1,w_2,w_3,w_4} \right)dw_1} = 1 + 0 + 0 + 0 + \left( {c_{23} - 1} \right)\\ \qquad\qquad\qquad\qquad\qquad\,\,+ \left( {c_{24} - 1} \right) + \left( {c_{34} - 1} \right)\\ \qquad\qquad\qquad\qquad\qquad\,\,= c^ \ast \left( {w_2,w_3,w_4} \right).\end{array}$$

The implied 2-copulas are therefore the two-copulas with which we started, for example, c₁₂,c₁₃, etc.

This extends to arbitrary dimensionality d. We can define a d-copula c*(w₁,…,w_d) using $\left( {\begin{array}{*{20}{c}} d \\ 2 \end{array}} \right)$ bivariate copulas c_ij as follows:

$$c^ \ast \left( {w_1, \ldots ,w_d} \right) = 1 + \mathop {\sum}\limits_{1 \le i < j \le d} {\left( {c_{ij} - 1} \right)}.$$

Returning for purposes of discussion to the four-dimensional case, the copula c*(w₁,w₂,w₃,w₄) is a copula if it is a density (i.e., it is non-negative) and there is no issue if we are satisfied with the lower dimensional copulas that it implies. But this may not always be the case. For example, suppose that we have a production frontier system as in Eqs. (1) and (2) but now we have four inputs, so that we have four random errors (u, ω₂, ω₃, and ω₄) instead of three. It might be natural to want (ω₂,ω₃,ω₄) to be trivariate normal, that is, to be marginally normal and to have the trivariate normal copula. But c*(w₁,w₂,w₃,w₄) as defined above does not imply a trivariate normal 3-copula, even if c₂₃, c₂₄, and c₃₄ are all bivariate normal copulas.

An alternative construction is as follows. Let c^o(w₂,w₃,w₄) be a trivariate normal copula, and c₁₂, c₁₃, and c₁₄ be the desired 2-copulas linking w₁ to w₂, w₃, and w₄. Then define the 4-copula

$$\begin{array}{l}c^o\left( {w_1,w_2,w_3,w_4} \right) = 1 + \left( {c_{12} - 1} \right) + \left( {c_{13} - 1} \right) + \left( {c_{14} - 1} \right) \\\qquad\qquad\qquad\qquad+ \left( {c^o\left( {w_2,w_3,w_4} \right) - 1} \right).\end{array}$$

If c₂₃,c₂₄, and c₃₄ are all bivariate normal copulas, then the 2-copulas implied by c^o are the same as those implied by c*, and so are the 3-copulas that involve w₁. But the 3-copula for w₂, w₃, and w₄ is different, because c^o(w₁,w₂,w₃,w₄) implies the 3-copula c^o(w₂,w₃,w₄), whereas c*(w₁,w₂,w₃,w₄) implies the 3-copula 1 + (c₂₃ − 1) + (c₂₄ − 1) + (c₃₄ − 1), which is not a trivariate normal copula even if its constituent 2-copulas are all bivariate normal.

Another way to construct a four-copula from lower dimensional copulas is to use a vine copula, as in Joe (1996) and Aas et al. (2009). These require specification of two-dimensional marginal and conditional copulas. In general there are many such vine copulas because they depend on the vine structure and the numbering of the variables. However, in our problem there is arguably a natural structure where the two-dimensional marginals are APS-2 and the conditional copulas are Gaussian. Thus the first variable is the half-normal error, and the remaining three variables are multivariate normal. Because of the multivariate normal assumption, the ordering of the last three variables does not matter. The benefit is that this representation results in a somewhat simpler functional form of the density than other vine representations.