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.

1 Introduction

Let c(w₁,w₂) be a copula density. In this paper, we will propose and use copulas that have the property that the correlation between the copula arguments is zero, but w₁ is correlated with |w₂ − 1/2|. If u and ω are the random variables for which w₁ and w₂ are the corresponding cdf values, this implies that u and ω are dependent but their Spearman’s correlation (rank correlation) is zero.

As a practical motivation for such copulas, suppose that we are interested in estimating a system of equations, where one equation is a production (or cost) function and the other equations are the first-order conditions for cost minimization. In the production frontier literature that dates back to Aigner et al. (1977) and Meeusen and van den Broeck (1977), the production frontier gives the maximal output that can be produced from a vector of inputs. The equation representing the production function contains a one-sided error that represents technical inefficiency, that is, the failure to produce maximal output given the inputs. It is often assumed to be half-normal, though any one-sided distribution is possible. Also, because the first-order conditions for cost minimization will not be satisfied exactly, the corresponding equations contain errors that represent allocative inefficiency, that is, the failure to use the inputs in the correct proportions given input prices. These errors are often assumed to be normal.

As a matter of generic notation, let u ≥ 0 represent technical inefficiency and let ω (taken as a scalar for purposes of this discussion) represent the allocative error in the first-order condition. So u represents the shortfall of output from the frontier, and ω represents the deviation of the actual from the optimal (log) input ratio. If technical inefficiency and allocative inefficiency are independent, there are no particular difficulties involved in deriving a likelihood for the model. See, for example, Schmidt and Lovell (1979). However, if technical and allocative inefficiency are not independent, we need to model this dependence. Schmidt and Lovell (1980) did this in a specific way that will be discussed below, but which was tailored to the normal/half-normal case. More generally, given specific marginal distributions for u and ω, we need to specify a copula so that we can obtain their joint distribution.

We then encounter the issue that common copulas do not capture the type of dependence that the economic model implies. We do not want to model a non-zero covariance between u and ω. For example, a positive correlation between u and ω would imply that firms that are more technically inefficient (larger values of u) have, say, higher capital/labor ratios than more technically efficient firms, which is not what we have in mind. What we want is a positive correlation between u and |ω|, which says that firms that are more technically inefficient have capital/labor ratios that are more in error (either too high or too low) than more technically efficient firms. That is, paraphrasing Schmidt and Lovell (1980, p 96), we need to recognize that, as far as the extent of allocative inefficiency is concerned, what is relevant is not the size of ω, but the size of |ω|.

The same argument can apply in a non-frontier setting. It does not hinge on u ≥ 0. Even if u is a standard zero-mean error (e.g., normal), it may be reasonable to assume that u is correlated with |ω| rather than ω, reflecting the view that firms that are better at using the correct input ratios also on average produce more output from a given set of inputs.

In this paper, we propose a family of copulas that have the desired properties that cov (w₁,w₂) = 0 but cov(w₁,|w₂ − 1/2|) > 0. Here w₁ and w₂ are the uniformly distributed copula arguments, that is, the cdf values of u and ω, respectively. If the distribution of ω is symmetric around zero, then ω = 0 corresponds to w₂ = 1/2. We are not aware of any existing copulas, other than the one implicit in Schmidt and Lovell (1980), that have these properties.

A second contribution of the paper is to give some general results on how to extend a two-dimensional copula to three or more dimensions. This is necessary in our production frontier system when there are more than two inputs, but our results apply more generally to any two-dimensional copula. This is significant because the generalization of two-dimensional copulas to higher dimensions is regarded as one of the most difficult problems in the copula literature.

The plan of the paper is as follows. In Section 2 we give some more specific detail about the economic model we consider and discuss some related literature. In Section 3, we introduce our new family of copulas in the two-dimensional case. Section 4 gives a corresponding family of copulas for the three-dimensional case and discusses the difficulties in extending these results to four or more dimensions. Section 5 provides detail on the evaluation of the simulated likelihood that is used in estimation. Section 6 contain some simulations, and the Section 7 gives an empirical example. Section 8 gives our concluding remarks.

2 A specific production frontier system

Consider the stochastic frontier model

$$y_i = \alpha + x_i^\prime \beta + v_i - u_i = \alpha + x_i^\prime \beta + \varepsilon _i,i = 1, \ldots, N,$$

(1)

where v_i represents statistical noise and u_i ≥ 0 represents technical inefficiency. When x_i is “exogenous” (independent of v_i and u_i), this is the model of Aigner et al. (1977) and Meeusen and van den Broeck (1977), which is commonly called the standard SFM.

We will consider specifically the Cobb--Douglas (log-linear) functional form, in which y_i is the natural log of the output of firm i, and x_i is a k × 1 vector of the natural logs of the inputs. This leads us to the set of equations for the optimal input ratios:

$$x_{i1} - x_{ij} = B_{ij} + \omega _{ij},\quad \,j = 2, \ldots ,K,$$

(2)

where B_ij = p_ij − p_i1 + ln(β₁) − ln(β_j). Here p_ij is the natural log of the price of input j for firm i; β_j is the jth element of β in (1); and ω_ij represents allocative inefficiency.

If we move to the non-statistical world by suppressing ε_i in (1) and the ω_ij in (2), then (2) is the set of first-order conditions for the minimization (with respect to the choice of x_i1,…,x_ik) of the cost of producing output level y_i. Now we return to the statistical world by reintroducing the errors ε_i and ω_ij, and we assume that y_i (not x_i, as in the standard SFM) and the p_ij are exogenous. The inputs x_ij are the solution to Eqs. (1) and (2) and are “endogenous” in the sense that they depend on the errors in the model. As described up to this point, this is the model of Schmidt and Lovell (1979).

We assume that the v_i are iid $N\left( {0,\sigma _v^2} \right)$; the u_i are iid $N^ + \left( {0,\sigma _u^2} \right)$, that is, “half-normal”; the ω_i = (ω_i2,…,ω_ik)′ are iid N(0, ∑_ωω); and v is independent of u and ω. The issue of this paper is the relationship of u and ω. In Schmidt and Lovell (1979), it was assumed that u and ω are independent. This implies the joint density of u, v, and ω. The joint density of ε and ω (where ε = v − u) is calculated by an integral that is tractable. We then solve the system for x_i, calculate the Jacobian as equal to $r = \mathop {\sum}\nolimits_{j = 1}^k {\beta _j}$, and obtain the likelihood as given in Eq. (11), p 357 of Schmidt and Lovell (1979).

However, as argued above, independence of u and ω is not an attractive assumption. Schmidt and Lovell (1980) proposed a model with the desired properties that u is uncorrelated with ω, but u is positively correlated with the absolute value of each element of ω. They assumed that u = |u*|, where $\left[ {\begin{array}{*{20}{c}} {u^ \ast } \\ \omega \end{array}} \right]\sim N\left( {0,{\Sigma}} \right)$ and where ${\Sigma} =\left[ {\begin{array}{*{20}{c}} {\sigma _u^2} & {{\Sigma}_{\omega u}^\prime } \\ {{\Sigma}_{\omega u}} & {{\Sigma}_{\omega \omega }} \end{array}} \right]$. This is consistent with the marginal distributions given above, since u is half-normal and ω is multivariate normal. Now u and ω are uncorrelated, and the correlation between u and |ω_j| is $\left( {2/\pi } \right)\left[ {\sqrt {1 - \rho _j^2} + \rho _j\arcsin \left( {\rho _j} \right) - 1} \right] \ge 0$, where ρ_j is the correlation between u* and ω_j. They give the likelihood for this model in Eq. (6), p 88.

Clearly there must be a copula implicit in this construction. It is not hard to see that this copula is the mixture (with weights equal to 1/2) of the Gaussian copula with variance matrix ∑ and the Gaussian copula with variance matrix ${\Sigma}^ \ast = \left[ {\begin{array}{*{20}{c}} {\sigma _u^2} & { - {\Sigma}_{\omega u}^\prime } \\ { - {\Sigma}_{\omega u}} & {{\Sigma}_{\omega \omega }} \end{array}} \right]$. For lack of a better name, we will call this the SL copula. Some details about it are given in Appendix 1.

While this construction depends on the half-normal/normal assumption, the SL copula is a valid copula that can be used regardless of the marginal distributions chosen for u and ω. However, it would be desirable to have alternative copulas to accomplish the same objectives as the SL copula. We can observe the following. Suppose that c(w₁,w₂;θ) is a copula and that c(w₁,w₂;−θ) is also a copula. Then $c_ \ast \left( {w_1,w_2;\theta } \right) = \frac{1}{2}c\left( {w_1,w_2;\theta } \right) + \frac{1}{2}c\left( {w_1,w_2; - \theta } \right)$ is also a copula. We will call it a folded copula. The SL copula is the folded normal copula. More generally, if the value of Spearman’s rho for the copula c(w₁,w₂;θ) is an odd function of θ, then Spearman’s rho equals zero for the folded copula. The normal copula has this property, and that is why the SL copula generates uncorrelated copula arguments. The Student t copula with fixed number of degrees of freedom also has this property. Some copulas that have this property may not yield a useful folded copula. For example, the folded Farlie–Gumbel–Morgenstern (FGM) copula is just the independence copula. However, most common copulas do not have this property. We will therefore construct and propose some alternatives in the next two sections of the paper. These will not be folded copulas but they will have Spearman’s rho equal to zero.

Although our discussion has closely followed the specific models of Schmidt and Lovell (1979, 1980) there are many papers, both theoretical and applied, that consider systems consisting of a production or cost function and a set of first-order conditions for maximization or minimization of a criterion function (e.g., cost minimization or profit maximization). Examples include Christensen and Greene (1976), Greene (1980), Kumbhakar (1987, 1991, 1997), Ferrier and Lovell (1990), and Atkinson and Cornwell (1994). Also, although the Cobb–Douglas assumption is obviously restrictive, we can follow the path of Tsionas and Tran (2016) and just consider the Cobb–Douglas model as the parametric anchorage model for a non-parametric (locally Cobb–Douglas) model which can be estimated by local maximum likelihood. Our new copulas could be useful in any or all of these settings.

3 The APS-2 copulas

In this section we consider the two-dimensional case in which we specify a copula density c(w₁,w₂), where w₁ and w₂ are scalars. In terms of our economic model, this corresponds to the case of two inputs, and correspondingly the relevant random variables are u and scalar ω; then the copula arguments are w₁ = F_u(u) and w₂ = F_ω(ω).

The well-known FGM copula is of the form c(w₁,w₂) = 1 + θ(1 − 2w₁)(1 − 2w₂) with |θ| < 1. This generalizes to the Sarmanov (1966) family of copulas, which are of the form c(w₁,w₂) = 1 + θg(w₁)h(w₂), where ${\int}_0^1 {g\left( s \right)ds} = {\int}_0^1 {h\left( s \right)ds = 0}$, and where the restrictions on θ that are necessary for c to be a density are the restrictions that guarantee that c(w₁,w₂) ≥ 0 for all w₁,w₂ in the unit square. These depend on the specific forms of the functions g and h.

Definition 1 An APS-2 copula is a two-dimensional Sarmanov copula with g(w₁) = 1 − 2w₁ (as in the FGM copula) and $h\left( {w_2} \right) = 1 - k_q^{ - 1}q\left( {w_2} \right),$ where q(s) is integrable on [0,1]; q(s) is symmetric around s = 1/2, that is, q(s) = q(1 − s); q(s) is monotonically decreasing on [0,1/2] and therefore monotonically increasing on [1/2;1]; and $k_q = {\int}_0^1 {q\left( s \right)ds}$ so that ${\int}_0^1 {h\left( s \right)ds = 0}$.

Therefore an APS-2 copula is of the form $c( {w_1,w_2} ) = 1 + \theta ( {1 - 2w_1} )( {1 - k_q^{ - 1}q( {w_2} )} )$ where the function q has the properties given in Definition 1. A restriction on θ will be necessary for this to actually be a copula. This restriction will depend on the form of q.

Result 1 For any APS-2 copula and any value of θ, cov(w₁,w₂) = 0.

The proofs of the results in this section are given in Appendix 2.

Result 1 depends only on the symmetry of q(s) around s = 1/2. It holds not just for g(w₁) = 1 − 2w₁, but for any g(w₁) such that ${\int}_0^1 {g\left( s \right)ds = 0}$, that is, for a larger class of copulas than the APS-2 family.

Result 2 For any APS-2 copula, $cov\left( {w_1,q\left( {w_2} \right)} \right) = \frac{1}{6}\theta k_q^{ - 1}{\mathrm{var}}\left( {q\left( {w_2} \right)} \right).$

Result 2 implies that, in an APS-2 copula, θ is proportional to cov(w₁,q(w₂))/var(q(w₂)), and therefore to $\sqrt {{\mathrm{var}}\left( {q\left( {w_2} \right)} \right)}$· corr(w₁,q(w₂)), where corr(w₁,q(w₂)) is the usual Pearson correlation coefficient.

Results 1 and 2 are for the copula arguments w₁ and w₂. As is true throughout the copula literature, if we consider instead the original variables $u = F_u^{ - 1}\left( {w_1} \right)$ and $\omega = F_\omega ^{ - 1}\left( {w_2} \right)$, there is little that can be said about their Pearson correlation because the transformation from the copula arguments to the original random variables is nonlinear and it depends on the marginal distributions of these variables. However, the Pearson correlation of the copula arguments is equal to the Spearman (rank) correlation of the original random variables, so these results are informative about the rank correlation of the original variables. In particular, if the joint distribution of (u,ω) has an APS-2 copula, then the rank correlation of u and ω is zero.

We can show that the variables u and ω have Pearson correlation equal to zero if their marginal distributions are symmetric and they are linked by an APS-2 copula.

Result 3 Suppose that u and ω have symmetric marginal distributions with finite variance, and that they are linked by an APS-2 copula. Then cov(u, ω) = 0.

In the copula literature there is considerable interest in the tail dependence of various copulas. Upper tail dependence is defined as lim_α→1P(w₁ ≥ α|w₂ ≥ α) and lower tail dependence is defined as lim_α→0P(w₁ ≤ α|w₂ ≤ α). The next result just says that these are both zero for any APS-2 copula.

Result 4 For any APS-2 copula, upper tail dependence and lower tail dependence equal zero.

To proceed beyond Results 1, 2, 3, and 4, we will consider two specific members of the APS-2 family, as follows.

$${\mathrm{APS}}{\mbox{-}}2{\mbox{-}}{\mathrm{A}}\quad c( {w_1,w_2} ) = 1 + \theta ( {1 - 2w_1} )[ {1 - 12( {w_2 - 1/2} )^2} ],\,\,| \theta | \le 1/2$$

(3A)

$${\mathrm{APS}}{\mbox{-}}2 {\mbox{-}}{\mathrm{B}}\quad c( {w_1,w_2} ) = 1 + \theta ( {1 - 2w_1} )( {1 - 4| {w_2 - 1/2} |} ),\,\,| \theta | \le 1$$

(3B)

These are the copula densities. For a Sarmanov copula of the form c(w₁,w₂) = 1 + θg(w₁)h(w₂), the copula cdf is C(w₁,w₂) = w₁,w₂ + θG(w₁)H(w₂), where $G\left( {w_1} \right) = {\int}_0^{w_1} {g\left( s \right)ds}$ and $H\left( {w_2} \right) = {\int}_0^{w_2} {h\left( s \right)ds}$. So, for the APS-2 copula with copula density $c( {w_1,w_2} ) = 1 + \theta ( {1 - 2w_1} )( {1 - k_q^{ - 1}q( {w_2} )} ),$ the copula cdf is $C( {w_1,w_2} ) = w_1w_2 + \theta w_1( {1 - w_1} )( {w_2 - k_q^{ - 1}Q( {w_2} )} ),$ where $Q\left( {w_2} \right) = {\int}_0^{w_2} {q\left( s \right)ds}$. Specifically, for the APS-2-A copula, $k_q^{ - 1} = 12$ and $Q\left( {w_2} \right) = \frac{1}{3}w_2^3 - \frac{1}{2}w_2^2 + \frac{1}{4}w_2$, so that $C(w_1, w_2) = w_1 w_2 + \theta w_1 w_2 (1 - w_1) [1 - (4w_2^2-6w_2+3)]$. Similarly, for the APS-2-B copula, $k_q^{ - 1} = 4$ and

$$Q\left( {w_2} \right) = \left[ {\begin{array}{*{20}{c}} {\frac{1}{2}w_2\left( {1 - w_2} \right),\,\,w_2 \le 1/2} \\ {\frac{1}{4} - \frac{1}{2}w_2\left( {1 - w_2} \right),\,\,w_2 > 1/2} \end{array}} \right.$$

(4)

Result 5 (i) The APS-2-A copula is a copula for |θ|≤1/2. (ii) $cov [ {w_1,( {w_2 - 1/2} )^2} ] = \frac{1}{{90}}\theta$. (iii) ${\mathop{\rm{var}}} [ {( {w_2 - 1/2} )^2] = \frac{1}{{180}}}$. (iv) ${\mathrm{corr}}[{w_1,\left( {w_2 - 1/2} \right)^2}] = \frac{2}{{\sqrt {15} }}\theta \cong 0.516\theta$.

Result 6 (i) The APS-2-B copula is a copula for |θ| ≤ 1. (ii) ${\mathop{\rm{cov}}} \left[ {w_1,|w_2 - 1/2|} \right] = \frac{1}{{72}}\theta$. (iii) ${\mathop{\rm{var}}} \left( {\left| {w_2 - 1/2} \right|} \right) = \frac{1}{{48}}$. (iv) ${\mathrm{corr}}\left[ {w_1,|w_2 - 1/2|} \right] = \frac{1}{3}\theta$.

These copulas are clearly not comprehensive in the range of correlation (between w₁ and $\left( {w_2 - \frac{1}{2}} \right)^2$ or between w₁ and $\left| {w_2 - \frac{1}{2}} \right|$) that they can cover. For APS-2-A, the maximal (absolute) correlation is (0.516)(0.5) = 0.258. For APS-2-B, it is 1/3. Many popular copulas (e.g., FGM) also have limited ranges of correlation that they can accommodate. The practical implications of this would depend on the specific empirical application.

4 The three-dimensional case

4.1 Some general comments

We now consider the three-dimensional case. In terms of our economic model, this would correspond to the case of three inputs, and therefore two equations for the optimal input ratios, as in Eq. (2) above. We have three random variables u,ω₂ and ω₃, and correspondingly three copula arguments, ${w_1}={F_u}(u),\;{w_2}=F_{\omega_2}(\omega_2)$, and ${w_3}=F_{\omega_3}(\omega_3)$. We want w₂ and w₃ to follow any standard bivariate copula, such as bivariate normal, and we want w₁ to be linked to w₂ and w₃ as in the APS-2 copulas. That is, as before, we want w₁ to be uncorrelated with w₂ (and w₃) but correlated with |w₂ − 1/2|.

Most of the copula literature covers the two-dimensional case. Moving from two-dimensional copulas to copulas of three or more dimensions is non-trivial. As Nelsen (2006, p 105) notes, “Constructing n-copulas is difficult. Few of the procedures discussed earlier … have n-dimensional analogs.” The problem is that there is inevitably an infinity of possibilities.

To illustrate this issue, start with the two-dimensional FGM copula c₁₂ ≡ c(w₁,w₂) = 1 + θ(1 − 2w₁)(1 − 2w₂). Now consider the following three-dimensional copulas:

$$c_{123}^A = 1 + \theta _{123}\left( {1 - 2w_1} \right)\left( {1 - 2w_2} \right)\left( {1 - 2w_3} \right)$$

(5A)

$$\begin{array}{l}c_{123}^B = 1 + \theta _{12}\left( {1 - 2w_1} \right)\left( {1 - 2w_2} \right) + \theta _{13}\left( {1 - 2w_1} \right)\left( {1 - 2w_3} \right)\\ \qquad+ \,\theta _{23}\left( {1 - 2w_2} \right)\left( {1 - 2w_3} \right)\end{array}$$

(5B)

$$\begin{array}{l}c_{123}^C = 1 + \theta _{12}\left( {1 - 2w_1} \right)\left( {1 - 2w_2} \right) + \theta _{13}\left( {1 - 2w_1} \right)\left( {1 - 2w_3} \right)\\\qquad + \,\theta _{23}\left( {1 - 2w_2} \right)\left( {1 - 2w_3} \right) + \theta _{123}\left( {1 - 2w_1} \right)\left( {1 - 2w_2} \right)\left( {1 - 2w_3} \right)\end{array}$$

(5C)

The last of these, $c_{123}^C$, is given in Nelsen (2006, p 108). So far as we are aware, the other two are new. In any case, for suitable values of the θ’s, these are all copulas; they are densities, and their two-dimensional marginals are two-dimensional copulas (“2-copulas”). For $c_{123}^A$, the implied 2-copulas are uniform, for example, ${\int} {c_{123}^A\,dw_1 = 1}$. So $c_{123}^A$ is a distribution in which the three w’s are pairwise independent but not jointly independent. For $c_{123}^B$ and $c_{123}^C$, the implied 2-copulas are FGM, for example, ${\int} {c_{123}^B\,dw_1} = {\int} {c_{123}^C\,dw_1} = 1 + \theta _{23}\left( {1 - 2w_2} \right)\left( {1 - 2w_3} \right)$. So $c_{123}^B$ and $c_{123}^C$ are different joint distributions that have the same marginals of order two and one. The problem is that it is not clear which of these is in some sense more natural.

4.2 Some general results

Suppose very generally that we wish to extend a 2-copula to a 3-copula. An intuitively reasonable possibility is to use a copula as an argument in a copula. More specifically, suppose that c^o and c are 2-copulas, and we define

$$c^ \ast \left( {w_1,w_2,w_3} \right) = c^o\left( {c\left( {w_1,w_2} \right),w_3} \right).$$

(6)

That is, we use the copula c^o to link the copula c to a third random variable w₃ (which could be another copula). This may be intuitively reasonable, but unfortunately it does not generally yield a 3-copula. This is the so-called compatibility problem, discussed by Nelsen (2006, pp 105–107), for which there are quite a few results, most of them negative. That discussion is in terms of copula cdf’s, not densities, but the same negative conclusion holds for densities as in (6). For example, suppose that c is an arbitrary copula and c^o is FGM. So

$$c^ \ast \left( {w_1,w_2,w_3} \right) = 1 + \theta \left[ {1 - 2c\left( {w_1,w_2} \right)} \right]\left( {1 - 2w_3} \right).$$

Then ∫c*(w₁,w₂,w₃)dw₃ = 1 but

$${\int} {c^ \ast \left( {w_1,w_2,w_3} \right)dw_1 = 1 + \theta \left( {1 - 2w_3} \right)\left[ {1 - 2{\int} {c\left( {w_1,w_2} \right)dw_1} } \right] = 1 - \theta \left( {1 - 2w_3} \right)}$$

which is not a 2-copula. (And a similar argument applies to the integral with respect to w₂.)

An apparent solution is to remove the factor of 2 in the term 2c(w₁,w₂).

Result 7 Suppose that c(w₁,w₂) is a 2-copula and define

$$c^ \ast \left( {w_1,w_2,w_3} \right) = 1 + \theta \left[ {1 - c\left( {w_1,w_2} \right)} \right]\left( {1 - 2w_3} \right).$$

(7)

Then, for values of θ such that c*(w₁,w₂,w₃) ≥ 0 for all w₁,w₂,w₃, c* is a 3-copula.

The proof of Result 7 is simply to calculate that ∫c*(w₁,w₂,w₃)dw₁ = ∫c*(w₁,w₂,w₃)dw₂ = ∫c*(w₁,w₂,w₃)dw₃ = 1, so that all three implied 2-copulas are the uniform (independence) copula. So we have joint dependence but pairwise independence, which is not what we want. The copula $c_{123}^A$ in Eq. (5A) is of the form of (7) and suffers from this same problem, as noted above.

Another possible extension of a 2-copula to a 3-copula is given by the following result.

Result 8 Suppose that c(w₁,w₂) is a 2-copula and define

$$c^ \ast \left( {w_1,w_2,w_3} \right) = c\left( {w_1,w_2} \right) + \theta \left[ {1 - c\left( {w_1,w_2} \right)} \right]\left( {1 - 2w_3} \right)$$

(8)

Then, for values of θ such that c*(w₁,w₂,w₃) ≥ 0 for all (w₁,w₂,w₃) c* is a copula.

It is easy to calculate that ∫c*(w₁,w₂,w₃)dw₁ = ∫c*(w₁,w₂,w₃)dw₂ = 1 and that ∫c*(w₁,w₂,w₃)dw₃ = c(w₁,w₂), all of which are 2-copulas. So the 2-copula for w₁,w₂ that we started with is preserved, but the other two 2-copulas are the independence copula, which is restrictive, and in our case not what we want.

The purpose of the last two examples is to stress that it is not hard to extend a 2-copula to a 3-copula, but the resulting 3-copula may not have the properties that we want. However, we are now ready to give a positive and (we hope) useful result.

Result 9 Let c₁₂(w₁,w₂), c₁₃(w₁,w₃), and c₂₃(w₂,w₃) be 2-copulas. Define

$$c^ \ast \left( {w_1,w_2,w_3} \right) = 1 + \left( {c_{12} - 1} \right) + \left( {c_{13} - 1} \right) + \left( {c_{23} - 1} \right)$$

(9)

Then if c* is a density, it is a 3-copula, and the implied 2-copulas are c₁₂,c₁₃, and c₂₃.

The proof is trivial. Simply calculate, for example, ∫c*(w₁,w₂,w₃)dw₁ = 1 + 0 + 0 + (c₂₃ − 1) = c₂₃.

This is a very simple construction, but so far as we are aware it is original. Because the integral of c* equals one, the requirement that c* be a density is just the requirement that c*(w₁,w₂,w₃) ≥ 0 for all w₁,w₂,w₃ in the unit cube.

The result is important because it shows how, if we start with 2-copulas that capture the bivariate dependence between any two of w₁,w₂,w₃, we can construct a 3-copula that gives their joint distribution, and does so in such a way that the form of the bivariate dependence is preserved.

The FGM 3-copula $c_{123}^B$ in Eq. (5B) above is of this form.

The construction in Result 9 generalizes to higher dimensions. For example, in the four-dimensional case, we could construct c*(w₁,w₂,w₃,w₄) = 1 + (c₁₂ − 1) + (c₁₃ − 1) + (c₁₄ − 1) + (c₂₃ − 1) + (c₂₄ − 1) + (c₃₄ − 1). If this is a density, it is a copula, its 3-copulas are of the form given in Result 9, and its 2-copulas are the c_ij with which we started. However, this is not the only option for extending Result 9 to four dimensions. We discuss this issue in Appendix 3.

4.3 The APS-3 copulas

We now return to the special case of our economic model with three inputs, and therefore two equations that give the optimal input ratios. We have three random variables u,ω₂ and ω₃, and correspondingly three copula arguments, w₁ = F_u(u), w₂ = $F_{\omega_2}$(ω₂), and w₃ = $F_{\omega_3}$(ω₃). We want w₂ and w₃ to follow any standard bivariate copula, such as bivariate normal, and we want w₁ to be linked to w₂ and w₃ as in the APS-2 copulas. We can use Result 9 to accomplish this.

Specifically, we define the APS-3-A and APS-3-B copulas as follows:

$${\mathrm{APS}}{\mbox{-}}3{\mbox{-}}{\mathrm{A}}\quad c^ \ast \left( {w_1,w_2,w_3} \right) = 1 + \left( {c_{12} - 1} \right) + \left( {c_{13} - 1} \right) + \left( {c_{23} - 1} \right)$$

(10A)

where

$$c_{12}\left( {w_1,w_2} \right) = 1 + \theta _{12}\left( {1 - 2w_1} \right)\left[ {1 - 12\left( {w_2 - \frac{1}{2}} \right)^2} \right]$$

$$c_{13}\left( {w_1,w_3} \right) = 1 + \theta _{13}\left( {1 - 2w_1} \right)\left[ {1 - 12\left( {w_3 - \frac{1}{2}} \right)^2} \right]$$

$$c_{23}(w_{2},w_{3})={\mathrm{bivariate}}\;{\mathrm{normal}}\;{\mathrm{copula}}$$

$${\mathrm{APS}}{\mbox{-}}3{\mbox{-}}{\mathrm{B}}\quad c^ \ast \left( {w_1,w_2,w_3} \right) = 1 + \left( {c_{12} - 1} \right) + \left( {c_{13} - 1} \right) + \left( {c_{23} - 1} \right)$$

(10B)

where

$$c_{12}\left( {w_1,w_2} \right) = 1 + \theta _{12}\left( {1 - 2w_1} \right)\left( {1 - 4\left| {w_2 - \frac{1}{2}} \right|} \right)$$

$$c_{13}\left( {w_1,w_3} \right) = 1 + \theta _{13}\left( {1 - 2w_1} \right)\left( {1 - 4\left| {w_3 - \frac{1}{2}} \right|} \right)$$

$$c_{23}(w_{2},w_{3})={\mathrm{bivariate}}\;{\mathrm{normal}}\;{\mathrm{copula}}$$

For these to be copulas, they must be densities, that is, we must have c*(w₁,w₂,w₃) ≥ 0 for all w₁,w₂,w₃. This will require the following restrictions on θ₁₂, θ₁₃ and the correlation parameter ρ in the bivariate normal copula.

Result 10 Let δ = (1 − ρ²)^−1/2 exp$\left( { - \frac{1}{2}\frac{{\rho ^2}}{{\left( {1 - \rho ^2} \right)}}} \right)$ where ρ is the correlation parameter in the bivariate normal copula. Then: (1) APS-3-A is a copula if |θ₁₂| + |θ₁₃| ≤ δ/2. (2) APS-3-B is a copula if |θ₁₂| + |θ₁₃| ≤ δ.

The proof is given in Appendix 2.

Note that δ = 1 when ρ = 0; δ = 0.97743 when ρ = ±0.5; and δ = 0.68514 when ρ = ±0.8. Stronger correlation between w₂ and w₃ reduces the allowable range of θ₁₂ and θ₁₃.

5 Some remarks on simulation of the likelihood

We now return to the problem of the estimation of the model of Section 2. To form a likelihood we need the joint density of ε, ω₂, and ω₃, which we will denote as $f_{\varepsilon, \omega_2,\omega_3}(\varepsilon,\omega_2,\omega_3)$.

We can obtain the joint density of u, ω₂, and ω₃ by specifying their marginal densities and a copula. That is,

$$f_{u,\omega _2,\omega _3}\left( {u,\omega _2,\omega _3} \right) = c^ \ast \left( {w_1,w_2,w_3} \right) \cdot f_u\left( u \right) \cdot f_{\omega _2}\left( {\omega _2} \right) \cdot f_{\omega _3}\left( {\omega _3} \right)$$

(11)

where as before w₁ = F_u(u), $w_{2}=F_{\omega_{2}}(\omega_{2})$, and $w_{3}=F_{\omega_{3}}(\omega_{3})$. Here c*(w₁,w₂,w₃) could be any copula, for example, the APS-3-A or APS-3-B copula as given in Eqs. (10A) and (10B) above. The marginal densities f_u(u),$f_{\omega_{2}}(\omega_{2})$, and $f_{\omega_{3}}(\omega_{3})$ could be anything, though what we have in mind for our model is half-normal, normal and normal.

Since v is independent of u, ω₂, and ω₃, the joint density of v, u, ω₂, and ω₃ is

$$f_{v,u,\omega _2,\omega _3}\left( {v,u,\omega _2,\omega _3} \right) = f_v\left( v \right) \cdot c^ \ast \left( {w_1,w_2,w_3} \right) \cdot f_u\left( u \right) \cdot f_{\omega _2}\left( {\omega _2} \right) \cdot f_{\omega _3}\left( {\omega _3} \right).$$

(12)

Then

$$\begin{array}{l}f_{\varepsilon ,\omega _2,\omega _3}\left( {\varepsilon ,\omega _2,\omega _3} \right)\, = {\int} {f_{v,u,\omega _2,\omega _3}\left( {u + \varepsilon ,u,\omega _2,\omega _3} \right)du} \\\qquad\qquad\qquad = {\int} {c^ \ast } \left( {w_1,w_2,w_3} \right) \cdot f_u\left( u \right) \cdot f_{\omega _2}\left( {\omega _2} \right) \cdot f_{\omega _3}\left( {\omega _3} \right) \cdot f_v\left( {u + \varepsilon } \right)du\\\qquad\qquad\qquad = f_{\omega _3}\left( {\omega _3} \right) \cdot f_{\omega _2}\left( {\omega _2} \right) \cdot \int c^ \ast \left( {w_1,w_2,w_3} \right) \cdot f_v\left( {u + \varepsilon } \right) \cdot f_u\left( u \right)du\end{array}$$

(13)

(Note that c* remains inside the integral sign because w₁ is a function of u.)

The integral in (13) is generally intractable. However, we can write this as

$$f_{\varepsilon ,\omega _2,\omega _3}\left( {\varepsilon ,\omega _2,\omega _3} \right) = f_{\omega _2}\left( {\omega _2} \right) \cdot f_{\omega _3}\left( {\omega _3} \right) \cdot E_u\left[ {c^ \ast \left( {w_1,w_2,w_3} \right) \cdot f_v\left( {u + \varepsilon } \right)} \right]$$

(14)

where E_u represents the expectation with respect to the distribution of u. This expectation can be evaluated (approximated) by taking the average over a large number of draws from the distribution of u. The log likelihood for the model can then be obtained by summing (over observations) the log of the simulated densities in (14). This leads to the method of simulated likelihood, for which a standard reference is Greene (2003).

In the special case of the APS-3 copulas, the expression in (14) can be rewritten as follows. Let g(w) = (1 − 2w) and $h\left( w \right) = 1 - 12\left( {w - \frac{1}{2}} \right)^2$ [for the APS-3-A copula] or $h\left( w \right) = \left( {1 - 4\left| {w - \frac{1}{2}} \right|} \right)$ [for the APS-3-B copula]. Note that c₁₂(w₁,w₂) = 1 + θ₁₂g(w₁)h(w₂); c₁₃(w₁,w₃) = 1 + θ₁₃g(w₁)h(w₃), and c*(w₁,w₂,w₃) = c₁₂ + c₁₂ + c₂₃ − 2 = θ₁₂g(w₁)h(w₂) + θ₁₃g(w₁)h(w₃) + c₂₃ = g(w₁)[θ₁₂h(w₂) + θ₁₃h(w₃)] + c₂₃. Inserting this expression for c* into (14), we obtain $f_{\varepsilon,\omega_2,\omega_3}$(ε,ω₂,ω₃) = $f_{\omega_2}$(ω₂)·$f_{\omega_3}$(ω₃)·[θ₁₂h(w₂) + θ₁₃h(w₃)]·E_u[g(w₁)·f_v(u + ε)] + c₂₃·$f_{\omega_3}$(ω₃)·$f_{\omega_2}$(ω₂)·E_uf_v(u + ε). But c₂₃·$f_{\omega_3}$(ω₃)·$f_{\omega_2}$(ω₂) = the bivariate normal density φ(ω₂,ω₃), and E_uf_v(u + ε) = f_ε(ε). So

$$\begin{array}{l}f_{\varepsilon ,\omega _2,\omega _3}\left( {\varepsilon ,\omega _2,\omega _3} \right) = f_\varepsilon \left( \varepsilon \right) \cdot \varphi \left( {\omega _2,\omega _3} \right)\\ \qquad\qquad\qquad\quad\,\,+ \,f_{\omega _2}\left( {\omega _2} \right) \cdot f_{\omega _3}\left( {\omega _3} \right) \cdot \left[ {\theta _{12}h\left( {w_2} \right) + \theta _{13}h\left( {w_3} \right)} \right] \cdot E_u\left[ {g\left( {w_1} \right) \cdot f_v\left( {u + \varepsilon } \right)} \right]\end{array}$$

(15)

This expectation is simpler and may be easier to simulate than the expectation in (14) above.

6 Monte Carlo simulations

We conduct a set of Monte Carlo simulations that mimic the production frontier system with two inputs. We also conduct Monte Carlo simulations that follow the model of Tran and Tsionas (2015). The results for these latter simulations are in the Supplemental Material for this paper, or at https://sites.google.com/site/artembprokhorovv0/papers/APSsupplement.pdf. This supplement also explains how we draw observations from the APS-2-A copula.

Dropping the observation subscript, the structural equations for our production frontier system are

$$y = \ln\alpha + \beta _1x_1 + \beta _2x_2 + v - u,$$

$$x_1 - x_2 = p_2 - p_1 + \ln\beta _1 - \ln\beta _2 + \omega.$$

Here the log output (y) is exogenous, the log prices (p’s) are exogenous, and the log inputs (x’s) are endogenous. The reduced form equations (input demands) are (e.g., Schmidt and Lovell (1979), Eqs. (3A) and (8)):

$$x_1 = \ln k_1 + \frac{1}{r}y + \ln\left[ {\mathop {\prod}\nolimits_{ {i = 1} }^2 {P_i^{\beta _i/r}/P_1} } \right] - \frac{1}{r}\left( {v - u} \right) + \frac{{\beta _2}}{r}\omega ,$$

$$x_2 = \ln k_2 + \frac{1}{r}y + \ln\left[ {\mathop {\prod}\nolimits_{ {i = 1} }^2 {P_i^{\beta _i/r}/P_2} } \right] - \frac{1}{r}\left( {v - u} \right) - \omega + \frac{{\beta _2}}{r}\omega ,$$

where $k_j = \beta _j\left[ {\alpha \mathop {\prod}\nolimits_{i = 1}^2 {\beta _i^{\beta _i}} } \right]^{ - 1/r}$; r = β₁ + β₂ = returns to scale; and P_i = exp(p_i).

We have normal v, normal ω and half-normal u; v is independent of u and ω, and u and ω are linked by the APS-2-A copula.

The parameters are $\alpha ,\beta _1,\beta _2,\sigma _u^2,\sigma _v^2,\sigma _\omega ^2$ and the copula parameter θ. We pick α = 1, β₁ = β₂ = 0.45. This implies that k_j = 1. This also implies that returns to scale are slightly decreasing. To illustrate the effect of changing the error variance we consider two cases: $\sigma _v^2 = \sigma _u^2 = \sigma _\omega ^2 = 0.1$ (low error variance) and $\sigma _v^2 = \sigma _u^2 = \sigma _\omega ^2 = 1$ (high error variance).

We generate y as standard normal and (p₁,p₂) as standard bivariate normal with ρ = 0.5. These series are generated once, not once per replication. Then for each replication x₁ and x₂ are generated according to the input demand equations.

We consider three sample sizes n = {100, 200, 1000} and two values of θ = {0, 0.4} The number of replications is 1000.

We report the averages and standard deviations (over replications) for four estimators. QMLE is the estimator obtained under the assumption of independence between u and ω. It is the estimator of Schmidt and Lovell (1979). SL80 is the estimator of Schmidt and Lovell (1980). APS-2-A and APS-2-B are the estimators that use the APS-2-A and APS-2-B copula, respectively, to link u and ω.

The main point of these simulations was not really to compare these different estimators, but rather just to see whether the APS-2-A model can be estimated reliably. It appears that the answer is yes. Having said that, we will now discuss and compare the estimators in more detail.

Table 1 contains the results for the case when $\sigma _v^2 = \sigma _u^2 = \sigma _\omega ^2 = 0.1$. Consider first the results for θ = 0. In this case u and ω are independent, and all four models are correctly specified, but SL80, APS-2-A and APS-2-B each estimate an extra parameter that QMLE sets equal to zero. The results show that the intercept, slope and variance parameters are estimated quite well in general. The standard deviations of $\sigma _u^2$ and θ are large but appear to decrease at a root-n rate. APS-2-A and APS-2-B underestimate $\sigma _v^2$ and overestimate $\sigma _u^2$ for small samples. The choice of APS-2-A vs APS-2-B does not matter much. While we might expect QMLE to be more efficient than the other estimators, because it imposes a restriction that is true, this is not evident in the results.

Table 1 Simulation results for the low error variance case

Next consider the case that θ = 0.4. Now APS-2-A is correctly specified and the other three models are misspecified. We might expect bias due to this misspecification but it is not evident in the results. The estimate of θ for APS-2-A is biased downward in small samples, but the bias disappears as n grows.

Table 2 contains the results for the case when $\sigma _v^2 = \sigma _u^2 = \sigma _\omega ^2 = 1$. These results are not qualitatively different from those in Table 1. Higher error variances translate into higher standard deviations of the slope and intercept estimates, but they have little effect on the comparisons of the various estimators.

Table 2 Simulation results for the high error variance case

7 Empirical example

We now present the results of an empirical example, which is intended to illustrate the feasibility and applicability of the APS methods that we have suggested.

We use a dataset previously used by Rungsuriyawiboon and Stefanou (2007). The data came from the Energy Information Administration, the Federal Energy Regulatory Commission and the Bureau of Labor Statistics, and covers 72 electric utilities between 1986 and 1999. We are grateful to Spiro Stefanou for providing us with the data and an accompanying data description. A similar application but using older data (the data used by Schmidt and Lovell 1979, 1980) is available in the online Supplement.

The output variable is net steam electric power generated using fossil fuel (coal, oil, gas), in megawatt-hours. There are three inputs: fuel, labor and maintenance (L&M), and capital. The price of fuel aggregate is an average price of fuels in dollars per BTU. The price of labor and maintenance is a cost-share weighted price of labor and maintenance, where the price of labor is company-wide average wage rate and the price of maintenance is a price index of electrical supplies from the Bureau of Labor Statistics. The price of capital is the yield of the latest debt issued by the firm. Quantities of fuel, L&M and capital can be computed by dividing the reported fuel expenses, aggregate L&M costs and capital costs by the relevant prices.

The dataset contained 1008 observations. However, we dropped all of the 1999 observations, because to construct a 1999 price index for input prices and the dollar-valued inputs as in Eq. (22) of Rungsuriyawiboon and Stefanou (2007) we would need the prices for 2000, which we do not have. We also dropped two observations (firm 10, years 1997 and 1998) because of missing values. Therefore we ended up with 934 observations.

There is some doubt about how Rungsuriyawiboon and Stefanou transformed the original price data prior to estimation. We followed their description as closely as we could, but our transformation is not identical to theirs. Our summary statistics are close to theirs but not identical. Since our intent is to construct a good dataset, not to reproduce their results, we did not aggressively pursue these differences.

We ignore the panel nature of the data in estimation. That is, the estimates below do not include time or firm dummies, nor do they allow for correlation due to random time or firm effects. However, we do report robust standard errors for our estimates that would be asymptotically valid in the presence of random time or firm effects.

Our results are presented in Table 3. The first two sets of results are for the model of Schmidt and Lovell (1979), which we could also call the QMLE, and the model of Schmidt and Lovell (1980). The next two sets of results are for the APS-3-A and APS-3-B models.

Table 3 Estimates of the system of Eqs. (1) and (2)

The results clearly indicate dependence between u and (ω₂,ω₃). The restriction that would reduce the SL80, APS-3-A and APS-3-B models to the SL79 model is strongly rejected by the likelihood ratio test in all three cases. The SL80, APS-3-A and APS-3-B models all have the same number of parameters, so it is fair to compare likelihood values, and this comparison marginally favors the APS-3-B model.

From an economic perspective, the results look reasonable. The marginal products of all three inputs are positive and returns to scale are close to unity. The only possible anomaly is the large and significant value of μ₂. This, coupled with a value of μ₃ that is close to zero, indicates a large systematic underutilization of input two (L&M) relative to its cost-minimizing value. We do not have an explanation for this result.

The results for the four models are quite similar to each other. So the choice of the copula does not make much difference in the results in this case, and the main interest in the application is that it demonstrates the feasibility of estimating the models based on the APS-3 copulas by simulated MLE.

8 Concluding remarks

In this paper we propose a new family of copulas for which the copula arguments are uncorrelated but dependent. We want to use this copula to construct random variables that are uncorrelated, but where the first random variable is correlated with the absolute value of the second. We show how this family of copulas can be applied to the error structure in an econometric production frontier model, and we give an empirical application.

Our family of copulas can be two- or three-dimensional. We provide a simple method of extending our two-dimensional copulas to three (or more) dimensions in such a way that the two-dimensional copulas are preserved. This method can be used in a general setting, and so it represents a solution to what is regarded in the copula literature as a difficult problem.

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.