Modeling the error terms in stochastic frontier models of production systems requires multivariate distributions with certain characteristics. We argue that canonical vine copulas offer a natural way to model the pairwise dependence between the two main error types that arise in production systems with multiple inputs. We introduce a vine copula construction that permits dependence between the magnitude (but not the sign) of the errors. By using a recently proposed family of copulas, we show how to construct a simulated likelihood based on C-vines. We discuss issues that arise in the estimation of such models and outline why such models better reflect the dependencies that arise in practice. Monte Carlo simulations and a classic empirical application to electricity generation plants illustrate the utility of the proposed approach.
In this article, we consider the following production system:
where the first equation represents a Cobb-Douglas type production function of a generic production unit, with output y and inputs . The set of the equations that follow the production function derive from the first-order conditions of cost minimization and define the optimal input ratios in terms of the input prices . This is the production system introduced in the classic productivity papers by Schmidt and Lovell [21,22] and most recently considered by Amsler et al. . For a textbook exposition of how to derive these equations from the first principles of the economic theory, we refer interested readers to Kumbhakar and Lovell (, Section 126.96.36.199).
The error terms in each of the equations have an economic interpretation. The two errors in first equation are a symmetric component , which accounts for random factors affecting the production potential, e.g., weather, and an asymmetric component u, which represent the technical inefficiency of production, that is, the percentage by which the production unit falls short of the stochastic production frontier. The ’s denote the allocative inefficiencies of the production unit. That is, they are the percentages by which the ratios deviate from their cost-minimizing values. Clearly, production units may be technically inefficient due to allocative inefficiencies and vice versa. Therefore, dependence between u and ’s should be permitted within the production system.
The choice of a parameterization for the dependence between u and ’s is not trivial. A naive approach would simply allow for a nonzero correlation between u and ’s. Such a dependence structure would imply that either too large or too small values of the input ratio , but not both at the same time, are associated with greater inefficiency. However, from an economic standpoint, it is equally likely that technically inefficient firms exhibit both positive and negative deviations from the cost-minimizing input ratios. As such, we require a dependence structure that permits correlation between u and , not between u and . Dependence structures specific to productivity analysis are a new field in dependence modeling that is only starting to attract interest. A new survey of copula-based models of productivity can be found in Prokhorov .
Dependence per se is not new to productivity analysis. Models that permit dependence between u and without copulas have been considered before in the literature . Let where and assume that the errors are jointly normal. Then, Schmidt and Lovell  show that in this case, u is uncorrelated with any but, by the result of Nabeya , u and have a nonzero correlation of the form:
where is the standard deviation of and is the correlation between and .
Much more recently, Amsler et al.  considered a more general dependence framework. A new family of copulas, referred to as APS-T copulas were developed, for which, given any set of marginals , the variables u and , are uncorrelated but u and are correlated. This new copula family is based upon the class of the Sarmanov copulas. The authors also derive a remarkably simple approach to construct copulas for dimensions greater than two. However, this approach to construct high dimensional APS-T copulas restricts the range of dependence that can be covered (see Result 10 in the study by Amsler et al. ).
In this article, we propose a canonical vine decomposition (see Bedford and Cooke ) of the joint distribution of that reflects the desired dependence structure discussed earlier. Specifically, we use the bivariate APS copulas developed by Amsler et al.  and the Gaussian copula as the building blocks of the vine decomposition. Vine copulas simplify the task of constructing multivariate joint densities with a desired dependence structure by representing high-dimensional densities as products of bivariate conditional copulas. The canonical vine is a natural representation in our setting, since we are concerned with modeling bivariate dependence between each pair (u, ). Importantly, our vine copula construction allows for a wider range of dependence than would be possible using the APS-T copula approach of Amsler et al. . We discuss practicalities associated with a maximum simulated likelihood estimation (MSLE) procedure of the production system that uses our vine copula decomposition. Finally, we propose a vine copula-based estimator of the technical inefficiency score that uses information from the estimated allocative inefficiency terms.
The proposed vine decomposition is relevant more generally in stochastic frontier models with endogeneity, where we wish to keep the marginal distribution of the inefficiency term prespecified and allow dependence between it and the reduced form errors or endogenous regressors (see, e.g., Amsler et al. , Tran and Tsionas ). Our vine copula decomposition is also of general statistical merit as a way of generating dependent random variables with a specific dependence structure, that is, the dependence structure implied by the bivariate APS2 copulas, from uncorrelated random variables.
The remainder of this article is organized as follows. Section 2 briefly introduces vine copulas. Section 3 reviews the APS copula family proposed by Amsler et al. . Section 4 discusses our proposed vine copula construction. Section 5 discusses the details of MSLE and proposes a nonparametric estimator of the technical inefficiency scores. Sections 6 and 7 present selected simulation results and an empirical application respectively. Section 8 concludes this article.
2 Vine copulas
Copulas are multivariate distributions with uniform marginals (see, e.g., Nelsen  for an early introduction). The well-cited theorem of Sklar  states that any joint distribution of continuous random variables can be written uniquely as a copula function taking as arguments the univariate marginal distributions
In terms of densities, we have
where is the marginal density corresponding to the marginal cdf , c is the T-copula density corresponding to the T-copula function C, and h is the joint density function corresponding to the joint cdf H.
A vine copula decomposition, proposed by Joe , makes use of an equivalent representation of in terms of conditional densities. For example, if , a canonical vine decomposition is
Here, the joint density is decomposed into a product of six 2-copula densities, three of which are acting on unconditional marginal cdf’s, two are acting on conditional cdf’s with one variable in the conditioning set and one is acting on a conditional cdf with two variables in the conditioning set (we provide details in Appendix A). The decomposition is particularly useful in settings with large T. When discussing the bivariate copulas within a vine copula decomposition, we will selectively use the abbreviations and , for distinct indices , where D is the set of indices not including .
An important element of a vine copula construction is the conditional univariate cdf’s used as arguments of the 2-copula densities. For a single variable in the conditioning set, it is easy to see that the conditional cdf can be written in terms of the corresponding copula as follows:
In the case of more than one variable in the conditioning set, Joe  shows that the following formula applies:
Clearly, it is important that the 2-copula densities can be integrated efficiently in each dimension.
Vine copula constructions, including ours, typically assume the so-called simplifying assumption, i.e., that the bivariate copulas that act on conditional cdf’s in the vine copula decomposition do not themselves depend on the values of the conditioning variable(s) (see Haff et al. ). However, even under the simplifying assumption, vine copula constructions are known to be sufficiently general to capture a wide range of dependence even in extremely high dimensions .
3 The APS copula family
Amsler et al.  consider copulas of the form
where and satisfies the restrictions that are necessary for c to be a density (see, e.g., Sarmanov ). The authors characterize the functions and such that , while for some function and call a 2-copula with this property an APS-2 copula.
It turns out that an APS-2 copula is obtained when and , where is integrable on , symmetric around , monotonically decreasing on , and monotonically increasing on , and where . For an APS-2 copula,
so that is proportional to the correlation between and . Amsler et al.  show that if the original random variables have symmetric marginals with finite variance and are linked by an APS-2 copula, then while generally . This property is of particular importance to the production systems we consider in this study.
The two members of the APS-2 copula family studied by Amsler et al.  are expressed as follows:
for which they show that
Extensions to using 2-copulas are usually hard to achieve. Numerous noncompatibility results show that 2-copulas that act on 2-copulas generally do not produce 3- or 4-copulas (see, e.g., Nelsen , pp. 105–107). However, Amsler et al.  show that a valid T-copula can be constructed using 2-copulas as follows:
where is the 2-copula of and . For example, a valid 4-copula has the form
An interesting property of such copulas is that the 2-copulas implied by them are the 2-copulas used to construct them. For example, for the 4-copula, the implied 2-copulas are , and .
On the basis of this result, Amsler et al.  define the APS-T copula as follows:
Two members of this family would arise if we use the bivariate APS-2A and APS-2B copulas for any with associated dependence parameter . Again, the key feature of this family is that , while in general, , . As such, Amsler et al.  apply the APS-T copula to model the dependence between technical and allocative inefficiency in production systems.
The proposed four-dimensional copula is expressed as follows:
where u is the technical inefficiency and, is allocative inefficiency.
Two important limitations of the APS-T copula restrict the range of dependence between T random variables it can accommodate. First, the Sarmanov specification is a perturbation of independence, which is generally not a comprehensive copula. For example, the Eyraud-Farlie-Gumbel-Morgenstern copula, which is a member of the Sarmanov class, cannot accommodate dependence outside the Kendall range of .
Second, by construction, the APS-T family ( ) can accommodate no dependence beyond that implied by pairwise copulas. This follows from Eq. (4) since all low-dimensional marginals of APS-T are also expressed in terms of . For example, contains terms of the form but no terms of the form , for some function m. This places additional restrictions on the type and strength of dependence that can be estimated using APS-T.
4 Vine copulas for a production system
We propose to use vine copulas to construct the joint density of the error terms in the system (1) and (2). To begin, we follow standard conventions from the prior literature and assume that the marginal distributions of v and are normal and that u follows a half-normal distribution. We use F and f to denote a cdf and pdf, respectively, and the subscripts will denote which error’s distribution is in question, e.g., denote the cdf of u and , respectively, while is the cdf of .
For example, if , the vine decomposition (Eq. (3)) of the joint density of can be written as follows:
We assume that all the random variables are continuous, and hence, the 2-copulas in this vine decomposition are unique. Since we wish to preserve the key property of the APS copula family for the pairs , a natural choice of is an APS-2 copula. For the symmetric errors ( ’s), we assume joint normality, and hence, a conventional choice for and is the Gaussian copula. Since the bivariate Gaussian copula is exchangeable, the various potential orderings of in Eq. (6) will all result in equivalent specifications.
A conceptual difference from the APS-4 copula is that the vine copula approach places virtually no restriction on the type of dependence (aside from that between u and ) that can be accommodated. For example, an APS-T copula must be of the structure given in Eq. (5), while our vine decomposition is subject to no such restriction. Moreover, since the vine decomposition of the joint density uses bivariate APS-2 copulas, it can cover a theoretically wider range of dependence than the equivalent APS-T copula construction, because the equivalent APS-T copula is subject to restrictions on (see Result 10 in the study by Amsler et al. ). Our vine approach is also more general than the approach based on joint normality used by Schmidt and Lovell .
The use of APS-2 copulas in the vine decomposition provides a number of important computational advantages because the conditional distributions have simple closed-form expressions. To see this, differentiate the APS-2 copula function (or integrate the APS-2 copula density) with respect to the second argument. For example, for u and , the two members of the APS-2 copula family imply the following distributions:
Then, the required conditional distributions can be written as follows:
The joint (conditional) normality assumption on ’s also leads to a simple formula for conditional cdf’s. For example, the conditional copula of and given u can be written as follows:
where denotes a bivariate normal cdf, denotes the inverse of a standard normal cdf, and is the correlation between and . Thus, the conditional distribution function for given u and can be written as follows:
Other conditional distributions that serve as arguments of the bivariate copula densities in vine decompositions of the form Eq. (7) can be derived similarly.
5 Estimation of parameters and technical inefficiencies
We now seek to construct a likelihood for the production system based upon the joint density of and maximize this likelihood to obtain the parameter vector. It is customary to assume that , being the random noise, is independent of u and ’s. Then, the joint density of , u, ’s can be written as follows:
where the copula term assumes a vine decomposition similar to Eq. (7).
As noted by Amsler et al. , u and are not observed separately. To apply MLE to estimate the parameters of our production system, we need the joint density of and ’s. Let . Then, as long as is available, Amsler et al.  obtain the required joint density using expectation over the distribution of u as follows:
where is the joint density of and ’s and is the expectation with respect to . The joint density can be approximated to a desired precision by simulation after taking the average over a large number of random draws of u.
The vine decomposition permits an additional representation of given by Eq. (7) and the assumption that the copulas are APS-2 copulas and the conditional copulas linking ’s are Gaussian. For example, for ,
where denotes the Gaussian copula density evaluated at . The Gaussian density can be written as follows:
where . As mentioned earlier, we can use and for APS-2A and for APS-2B, .
By using the Jacobian of the transformation from to given by Schmidt and Lovell (, p. 88), we can define a maximum simulated likelihood estimation (MSLE) based on maximizing the following log-likelihood function:
where contains all the model parameters to be estimated, , , and , . The parameter vector contains the parameters of the production function as well as the distributional parameters of the error terms, including the copula parameters. For example, if , this includes four parameters in the production function , six parameters from the marginal distributions of the error terms (means and variances of each , variance of and variance of u), plus three dependence parameters ( and from the APS-2 copulas and , respectively, and from the Gaussian copula ). Note, that if , then even if .
Once is estimated, we can obtain the technical inefficiency scores using the well-known formula of Jondrow et al. ,
where , , and , and is the standard normal density function. We use the residuals in place of , to evaluate the conditional expectation. Amsler et al.  note that the availability of allocative inefficiency terms allows for an improvement in the precision . Unfortunately, their approach is restricted to models that allow for dependence between and ’s not between u and ’s
However, in our setting, it is possible to use the approach proposed by Amsler et al. (, Section 5.2) for panel stochastic frontier models. This approach makes use of the fact that, once we estimate the model, we know the joint distribution of , as given by the copula, and can simulate from it. Then, we can use any nonparametric smoother to estimate the conditional expectation.
Let and let index that draws from the joint distribution (copula). The Nadaraya-Watson estimator of technical inefficiency scores can be written as follows:
where is a univariate kernel function and denotes the error-specific bandwidth parameter, different for each element of . In practice, one would often use the Gaussian kernel with the rule-of-thumb value for , where is the standard deviation of the simulated draws for the relevant variable.
Multivariate (nonproduct) kernels using certain features of the joint distribution of can in principle be more effective in the nonparametric estimation of the conditional expectation. They may mitigate the curse of dimensionality inherent in high-dimensional tasks. However, because the number of factors of production does not typically exceed three, often due to an aggregation into land, labor and capital, and because we can obtain any number of draws from the joint distribution of , we are not concerned with the curse of dimensionality of the estimator in (8). In cases that require a larger J, practitioners may wish to use nonparametric regression techniques that are more suitable for higher dimensions, such as the local linear forest of Friedberg et al.  (see, e.g., Amsler et al. ).
As mentioned earlier, we would evaluate this estimator at the values of the residuals but now we have both the residuals , and , both as defined earlier. It is a standard result in nonparametrics that, under , and , the new estimator converges to . Theoretically optimal bandwidth choices for a twice differentiable conditional expectation function are known to be of order ; in practice, univariate rules of thumb such as the one mentioned earlier would be used. A classic survey discussing consistency of simulated likelihood-based estimators is Gourieroux and Monfort .
6 Monte Carlo simulations
We evaluate the performance of the proposed vine construction in terms of the parameter specific mean squared error (MSE) and the aggregate MSE over all parameters. The production function and endogenous regressor equations used to generate the data in our simulations are as follows:
The bivariate copula used to generate dependence between u and is APS-2B with parameter for and for . The marginal distributions of the error terms are , , and , are bivariate normal with correlation parameter . First, we generate three dependent uniforms from the canonical vine APS-2B copula using the algorithm described in, e.g., Aas et al.  and Czado . Then, we invert the respective marginal distribution functions to generate the vector . Second, we generate and independently as and use these together with the simulated allocative inefficiency terms to generate and . Finally, y is generated using , the simulated technical inefficiency u, and the simulated random noise v, according to Eq. (10).
Eqs. (10) and (11) can be viewed as a simplified version of Eqs. (1) and (2), where ’s represent log input ratios, ’s represent log price ratios, and y denotes the log ratio of output to the numeraire input. The assumption of the Cobb-Douglass production function and cost minimization implied by Eqs. (1) and (2) translate into restrictions on ’s and , but we do not impose those restrictions in simulations except for setting the true values of the parameters close to realistic values.
The true parameter values are , , , and . We consider three combinations of copula parameter values , corresponding to a low, medium, and high degree of dependence between u and . We study two sample sizes and conduct 1,000 replications. A sample size of 500 is used to evaluate the expectation in Eq. (8).
We compare our vine copula-based model with three other alternative models. Table 1 reports the simulation results. Vine2A and Vine2B are the models that use the proposed vine copula constructions with APS-2A and APS-2B copulas, respectively. The APS3A and APS3B estimators use high-dimensional APS-T copulas rather than vines. The QMLE estimator is based on the assumption of independence between u and ’s (see Schmidt and Lovell ).
Gaussian is the estimator that assumes a Gaussian copula for , rather than APS-3A, APS-3B, or vine constructions with APS-2A and APS-2B. This is different from Schmidt and Lovell  who assume joint normality of as discussed in Section 1. However, the estimates based on Schmidt and Lovell  are similar and are not reported. For more details, on the copula implied by the Schmidt and Lovell  approach, see the study by Amsler et al. .
We find that in the case of near independence (upper panel), all six estimators perform similarly for both sample sizes, with QMLE being not much worse and sometimes better than the other estimators in terms of aggregate MSE. As the strength of the dependence increases (middle panel), the two vine copula-based estimators Vine2A and Vine2B behave similarly to the APS3A and APS3B models for and marginally better for . All APS copula-based models perform better than QMLE. In the case of strongest dependence (lower panel), Vine2A and Vine2B show superior performance when compared with APS3A and APS3B, respectively (particularly for the larger sample size). The parameter estimates from the vine copula-based models are considerably more accurate than those from the QMLE and Gaussian copula models. Parameter-specific MSEs show that this behavior is not limited to just a few parameters but is prevalent uniformly across all parameters. It is perhaps remarkable that estimates of are always associated with the largest MSE, regardless of the estimator, sample size, or dependence strength. The Gaussian estimator is dominated by all the other estimators, since this dependence structure is always miss-specified.
The last two lines of each panel contain the MSE for the two variants of technical inefficiency predictions. We have the set of ’s used in the data generating process and a corresponding set of predictions and computed using the two different conditioning sets and associated estimators in each iteration. For each iteration, we calculate MSEs:
The line showing is obtained using the average of the latter values over 1,000 replications. The row showing is obtained using the former. Perhaps, surprisingly, for all dependence strengths, we obtain very similar values of the MSE, even though one estimator is nonparametric and the other uses an analytic expression. There is some evidence that the estimator that uses ’s in the conditioning set provides a better estimate of the technical inefficiency scores as the strength of dependence increases, although the differences are generally not large.
7 Empirical illustration
We illustrate the use of our vine copula-based estimator using classic electricity generation data from 111 privately-owned steam-electric power plants constructed in the United States between 1947 and 1965 (see Cowing [8,9] for details). The output is measured in kWh of electricity generated in the first year of operation, inputs are capital as measured by actual cost of construction, fuel as measured in British Thermal Units (BTU) of actual consumption of coal, oil, and gas in the first year, and labor as measured by the total number of employees times 2,000 h. We also have input prices: firm’s bond rate prior to plant construction, actual price of a BTU of fuel, and regional industry salary rate averaged over two years prior to plant opening. Summary statistics for the data are given in Table 2, where output and inputs have been logged.
|Price of capital||−3.329||−3.387||0.192||−3.594||−2.947|
|Price of fuel||−1.337||−1.241||0.313||−2.797||−0.877|
|Price of labor||0.800||0.829||0.247||0.300||1.278|
We use a Cobb-Douglas type production function to mimic equations (1) and (2), with , , , and . We report parameter estimates together with standard errors. As benchmarks, we reproduce the estimates of Schmidt and Lovell , (SL80) which assumes joint normality, and Amsler et al.  (APS16), which assume that v is correlated with , but u is independent of v and . We also report the estimates based on the study by Amsler et al. , which use the APS-3 copulas (APS3A and APS3B). Table 3 presents the results. All of the parameters are as defined earlier; in addition, ’s are the means of the allocative inefficiencies ’s, which are permitted to be nonzero. The parameter estimates and standard errors associated with our vine copula-based estimators (Vine2A and Vine2B) are reported in the last four columns.
|Est||Std Err||Est||Std Err||Est||Std Err||Est||Std Err||Est||Std Err||Est||Std Err||Est||Std Err|
We start by noticing that the estimated input–output elasticities are similar for all the estimators that allow for dependence between u and ’s (SL80, APS3A&B, Vine2A&B) and different from APS16. Moreover, there are substantial differences in the dependence parameter estimates. APS3A&B and Vine2A&B show positive dependence between u and and negative between u and , while SL80 shows positive dependence for both. The correlations for all copula-based estimators are, for the most part, large in magnitude but statistically insignificant, while SL80 shows a very weak but statistically significant correlation.
Finally, we note the similarity in the three versions of the technical inefficiency score estimates. The statistics and are the conditional mean and variance, computed as averages (over observations) of the closed-form expressions in the studies by Jondrow et al.  and Bera and Sharma , respectively. The estimates and are the nonparametric (Nadaraya-Watson) versions of the same quantities. The estimates and use Nadaraya-Watson for the enlarged conditioning set. The values associated with the APS16 model are visibly different from the estimators based on the APS copula models and SL80. Again, perhaps surprisingly, it makes little difference whether the estimator is parametric or nonparametric and whether we condition on or .
Technically inefficient firms are likely to exhibit both positive and negative deviations from their cost minimizing input ratios. As such, stochastic frontier models of production systems should incorporate a dependence structure that permits correlation between technical inefficiency and the absolute value of allocative inefficiency. We propose a vine copula construction that allows for this unique dependence structure and argue that our approach admits a more comprehensive coverage of dependence than the multivariate APS-T copula proposed by Amsler et al. . Moreover, we discuss the MSLE parameter estimation procedure and implement an improved estimator of the technical inefficiency score, permitted by the fact that we can condition on a larger set of error terms.
Helpful comments from Christine Amsler, Brendan Beare, Peter Schmidt, and the participants of SETA2019 and INFORMS2019 are gratefully acknowledged. The use of the University of Sydney’s high performance computing cluster, Artemis, is acknowledged.
Funding information: Research for this article was supported by a grant from the Russian Science Foundation (Project No. 20-18-00365).
Conflict of interest: Prof. Artem Prokhorov is a member of the Editorial Advisory Board for “Dependence Modeling,” although had no involvement in the review of this manuscript or the final editorial decision.
Appendix A C-vine decomposition
where the conditional densities are represented in terms of copulas and marginals as follows:
Substituting into Eq. (A.1) produces the result in the main text.
There is clearly more than one way to applying the conditioning argument (see, e.g., Aas et al. ). The one we present corresponds to what is known as a C-vine (canonical vine) decomposition. A D-vine (drawable vine) decomposition, for example, has the form . However, a D-vine decomposition do not align with our goal of representing the dependence between u and each via .
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