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Using Copulas to Model Time Dependence in StochasticFrontier ModelsChristine Amsler a , Artem Prokhorov b & Peter Schmidt aa Department of Economics , Michigan State University , East Lansing , Michigan , USAb Department of Economics , Concordia University , Montreal , Quebec , CanadaAccepted author version posted online: 09 Aug 2013.Published online: 27 Nov 2013.

To cite this article: Christine Amsler , Artem Prokhorov & Peter Schmidt (2014) Using Copulas to Model Time Dependence inStochastic Frontier Models, Econometric Reviews, 33:5-6, 497-522, DOI: 10.1080/07474938.2013.825126

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Econometric Reviews, 33(5–6):497–522, 2014Copyright © Taylor & Francis Group, LLCISSN: 0747-4938 print/1532-4168 onlineDOI: 10.1080/07474938.2013.825126

USING COPULAS TO MODEL TIME DEPENDENCE IN STOCHASTICFRONTIER MODELS

Christine Amsler1, Artem Prokhorov2, and Peter Schmidt1

1Department of Economics, Michigan State University, East Lansing,Michigan, USA2Department of Economics, Concordia University, Montreal, Quebec, Canada

� We consider stochastic frontier models in a panel data setting where there is dependence overtime. Current methods of modeling time dependence in this setting are either unduly restrictiveor computationally infeasible. Some impose restrictive assumptions on the nature of dependencesuch as the “scaling” property. Others involve T -dimensional integration, where T is the numberof cross-sections, which may be large. Moreover, no known multivariate distribution has theproperty of having commonly used, convenient marginals such as normal/half-normal. We showhow to use copulas to resolve these issues. The range of dependence we allow for is unrestrictedand the computational task involved is easy compared to the alternatives. Also, the resultingestimators are more efficient than those that assume independence over time. We propose twoalternative specifications. One applies a copula function to the distribution of the composederror term. This permits the use of maximum likelyhood estimate (MLE) and generalized methodmoments (GMM). The other applies a copula to the distribution of the one-sided error term.This allows for a simulated MLE and improved estimation of inefficiencies. An applicationdemonstrates the usefulness of our approach.

Keywords Copula; GMM; MLE; MSLE; MSM; Stochastic frontier.

JEL Classification C13.

1. INTRODUCTION

We consider the stochastic frontier model of Aigner et al. (1977) andMeeusen and van den Broeck (1977):

yit = x ′it� + vit − uit = x ′

it� + �it � (1)

Address correspondence to Peter Schmidt, Department of Economics, 486 W. Circle Dr. Room110, Michigan State University, East Lansing, MI 48824, USA; E-mail: [email protected]

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498 C. Amsler et al.

In this model, i denotes individual firms, countries, production units,etc., and t denotes time. (When there is no ambiguity, we will omitthe subscripts.) We assume that the vit ’s are independent and identicallydistributed (i.i.d.) N (0, �2

v). We also assume that the distribution of uit isN (�it , �2

it)+, i.e. it is obtained by truncation to the left at zero of the normal

distribution with mean �it and variance �2it . The parameters (�it , �2

it) maybe constant or they may depend on explanatory variables zit . For simplicity,we will assume in what follows that �it = 0 and �it = �u , for any i and t , sothat uit ∼ N (0, �2

u)+. This is known as the half-normal distribution.

The non-negative error term uit represents technical inefficiency.Typically we assume it is independent over i , but independence over t isnot an attractive assumption. One would expect that inefficiency correlatespositively over time—firms that are relatively inefficient in one time periodwill probably also be inefficient in other time periods.

There are basically four approaches currently available in the literatureto model dependence in inefficiency over t . First, there is the quasimaximum likelihood estimation (QMLE) approach, which basicallyassumes independence of uit over t (see, e.g., Battese and Coelli, 1995).A variation of this approach is to assume that uit ’s are independentconditional on some explanatory variables zit (see, e.g., Wang, 2002).Second, one may assume that uit is time invariant (see, e.g., Pitt and Lee,1981). Then,

uit = ui for all t �

Third, one may assume a multivariate truncated distribution for(ui1, � � � ,uiT ), e.g., multivariate truncated normal (see, e.g., Pitt and Lee,1981). Then, the likelihood becomes a highly complicated function withT -dimensional integrals. Moreover, Horrace (2005) showed that such ajoint distribution function will not have truncated normal marginals. So ifwe wish to have normal/half-normal marginals of the composed error butfollow this approach to allow for dependence over t , it will not provide theright marginals. Finally, one may assume “scaling.” The scaling propertyassumes that uit has the following multiplicative representation:

uit = h(zit ; �)u∗i ,

where u∗i is a random variable, h(zit ; �) is a known function, � is a vector

of unknown parameters, and zit contains variables that affect technicalinefficiency. The variables zit may include functions of inputs xit or variouscharacteristics of the environment in which the firm operates. Because u∗

idoes not vary over t , the variation over time in uit is due only to changesin the observable zit .

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Using Copulas to Model Time Dependence 499

There are many versions of the stochastic frontier model that satisfy thescaling property (see Alvarez et al., 2006 for a survey of such models). Forexample, in Battese and Coelli (1992) and Kumbhakar (1990), zit is just afunction of time (and some parameters). In other cases, e.g., Caudill et al.(1995) and Hadri (1999), zit contains characteristics of the firm. In eithercase, if u∗

i is half-normal, then uit is N (0, �2it)

+, where �2it = �2

uh(zit , �)2. But

the key feature of the assumption is that u∗i does not depend on zit , and

so changes in zit only affect the scale of the distribution of uit (throughthe nonrandom function h(zit ; �)) but not its shape (determined by thedistribution of u∗

i ).The four approaches discussed above have major drawbacks. The

implications of the “scaling” assumptions, that firms with different zit donot differ in the shape of the distribution of technical inefficiency (butdiffer only in the scale) and that all changes in uit over time are dueto changes in zit , are hardly realistic. The assumptions that inefficienciesare not correlated over time or that they are time invariant are even lessrealistic.

In this paper, we develop a tractable method of allowing fordependence without scaling, independence, or time-invariance. We usecopulas to do that. A distinctive feature of copulas is that they allow us tomodel marginal distributions separately from their dependence structure.As a result we have a flexible joint distribution function, whose marginalsare specified by the researcher. The joint distribution can accommodateany degree of dependence. No simplifying assumptions such as scaling ortime-invariance are used in constructing the likelihood. Meanwhile, thetask of maximizing the joint likelihood remains simple because it involvesno T -dimensional integration.

One of the advantages of modelling correlation over time is the usualone, namely, that we can obtain estimates of the parameters that are moreefficient than those obtained under a false assumption of independence.However, there is another motivation that is specific to frontiers models,and which applies in one of the two specifications we suggest. We canconstruct an estimate of uit by evaluating the conditional expectation�(uit |�i1, � � � , �iT ) rather than just �(uit | �it). By doing so, we obtain amore precise estimate, and we also smooth the pattern of the estimated uit

over t.The plan of the paper is as follows. Section 2 reviews the traditional

QMLE and introduces a score-based GMM estimator. In Section 3, wedescribe how to use copulas to allow for dependence over time. We useGMM to motivate two new estimators that arise in this setting. Section 4discusses the case that we apply a copula to the composed errors. Wediscuss the computational issues and methods of testing the validity ofthe copula. In Section 5, we discuss the case that assumes that the noisevit is iid and we apply a copula to the technical efficiency term uit . Here

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we discuss computational issues and improved estimation of the technicalinefficiencies. Section 6 provides an empirical illustration of the newcopula-based estimators proposed in the paper. Section 7 concludes.

2. QUASI-MLE AND SCORE BASED GMM

Let fv(v) and fu(u) denote the densities of v and u, respectively. Then,

fv(v) = 1√2��v

e− v2

2�2v ,

fu(u) =√2√

��ue

− u2

2�2u , u ≥ 0, (2)

and the density of the composed error � = v − u can be obtained byconvolution as follows:

f (�) =∫ ∞

0fv(� + u)fu(u)du (3)

= 2�

( ��

)

(−��

�

), (4)

where �2 = �2v + �2

u , � = �u�v, and and are the standard normal density

and distribution functions, respectively (see, e.g., Aigner et al., 1977;Greene, 2003, p. 502).

The QMLE is based on this marginal (single t) density of thecomposed error. Let � = (�, �2, �), and let fit denote the density of thecomposed error term evaluated at �it = vit − uit . Then,

fit(�) = f (vit − uit) = f (yit − x ′it�)� (5)

The QMLE maximizes the sample likelihood obtained under theassumption of independence over i and t . That is,

�QMLE = argmax�

∑i

∑t

ln fit(�)� (6)

It is well known that, under suitable regularity conditions, the QMLEis consistent even if there is dependence over t . The “sandwich” estimatorof the QMLE asymptotic variance matrix is used to obtain the correctstandard errors in this case (see, e.g., Hayashi, 2000, Section 8.7). However,generally the QMLE is inefficient. There is a more efficient estimatorwhich uses the same information as the QMLE.

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To motivate this improved estimator, we rewrite the QMLE as ageneralized method of moments (GMM) estimator. Let sit(�) denote thescore of the density function fit(�), i.e.,

sit(�) = �� ln fit(�), (7)

where �� denotes the gradient with respect to �.Note that sit(�) is a zero mean function when evaluated at the true

parameter value �o , i.e.,

�sit(�o) = 0, for any i and t � (8)

Note that the QMLE of � solves∑i

∑t

sit(�QMLE) = 0,

which is also the first order condition solved by the GMM estimator basedon the moment condition

�∑t

sit(�o) = 0� (9)

Therefore, the QMLE and the GMM estimator based on (9) are identical.The key to improving efficiency of the QMLE is to notice that the

GMM estimator based on (9) implicitly uses a suboptimal weightingmatrix, if sit is correlated over t . The moment condition in (9) sums thezero-mean score functions sit over t whereas the theory of optimal GMMsuggests using a different, data driven weighting mechanism. To be moreprecise, define the vector of T score functions for each individual i :

s∗i (�) =

si1(�)��

siT (�)

�

Note that each element of this vector is zero-mean. Thus,

�s∗i (�o) = 0T , (10)

where 0T is a T × 1 vector of zeros. Under appropriate regularityconditions, applying the optimal GMM machinery to these momentconditions is known to produce an estimator whose asymptotic variancematrix is smaller than that of any other estimator using the same momentconditions, including the QMLE (see, e.g., Hansen, 1982).

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In the setting of a general panel data model, Prokhorov and Schmidt(2009) call the optimal GMM estimator based on (10) the improved QMLE(IQMLE). The IQMLE is the solution to the optimization problem

�IQMLE = argmin�

s∗(�)′W s∗(�), (11)

where s∗(�) = 1N

∑i s

∗i (�) and W is the inverse of the estimated covariance

matrix of s∗i (�o). This estimator is consistent whenever the QMLE is

consistent, but it is in general more efficient. However, neither of the twoestimators allow us to model time dependence in uit .

3. COPULAS AND FULL MLE

Let the bold letters �, u, v denote vectors of dimension T that containthe entire histories of the respective error terms, so, e.g., ui = (ui1, � � � ,uiT ),and let f�, fu, fv denote the respective joint (over all t) densities. Explicitmodelling of time dependence between uit ’s requires a model for themultivariate density f� or fu. We would like this density to accommodatearbitrary dependence and to have the correct marginal densities. That is,if we specify fu we would like it to have half-normal marginals, and if wespecify f� we would like it to accommodate normal/half-normal marginaldensities of the composed error term. Copulas can do this.

Briefly defined, a copula is a multivariate distribution function withuniform marginals (a rigorous treatment of copulas can be found,e.g., in Nelsen, 2006). If we let C(w1, � � � ,wT ) denote the copula cdfand c(w1, � � � ,wT ) denote the copula density, then each marginal wt , t =1, � � � ,T , is assumed to come from a uniform distribution on [0, 1]. Copulafunctions usually have at least one parameter that models dependencebetween the w’s. Let denote a generic vector of such parameters. Eachparametric copula function is known as a copula family. We provide severalexamples of copula families in Appendix A.

An important feature of copula functions is that they differ inthe range of dependence they can cover. Some copulas can cover theentire range of dependence – they are called comprehensive copulas –while others can only accommodate a restricted range of dependence.For example, suppose T = 2 and we measure dependence by Pearson’scorrelation coefficient. Then, the Gaussian, Frank, and Plackett copulasare comprehensive, while the Farlie–Gumbel–Morgenstern copula canmodel correlations only between about −0�3 and +0�3. This will beimportant in empirical applications because the copula-based likelihoodwe construct should be able to capture a fairly high degree of dependencepresent in the data. Clearly, we would like to use comprehensive copulassince we are looking to model an arbitrary degree of dependence.

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By a theorem due to Sklar (1959), given the marginals and a jointdistribution H (�1, � � � , �T ), there exists a copula such that

H (�1, � � � , �T ) = C(F1(�1), � � � , FT (�T )) or

h(�1, � � � , �T ) = c(F1(�1), � � � , FT (�T )) · f1(�1) · · · · · fT (�T )�(When the marginals are continuous, the copula is unique.) Conversely, ifwe specify the marginals Ft and the copula C , we have constructed a jointdistribution H that has those marginals and whose dependence structurefollows from the assumed copula. This construction is what we will use inthis paper.

4. APPLICATION OF A COPULA TO (�I 1, � � � , �IT )

We start by using a copula to form f�, the joint (over t) density ofthe composed error. Given the normal/half-normal marginals of �’s and acopula family C(w1, � � � ,wt ; ), the joint density can be written as

f�(�i ; �, ) = c(Fi1(�), � � � , FiT (�); ) · fi1(�) · � � � fiT (�),where, as before, fit(�) ≡ f (�it) = f (yit − x ′

it�) is the pdf of the composederror term evaluated at �it and Fit(�) ≡ ∫ �it

−∞ f (s)ds is the correspondingcdf, and where c is the density corresponding to the copula cdf C . Thisapproach gives rise to two estimators and several computational issues,which we will discuss next.

4.1. Full Maximum Likelihoood Estimator (FMLE) and GMM

Given a sample of size N , the joint density f�(�i ; �, ) permitsconstruction of the sample log-likelihood as follows:

lnL(�, ) =N∑i=1

�ln c(Fi1(�), � � � , FiT (�); ) + ln fi1(�) + · · · + ln fiT (�)� �

(12)

Because the likelihood is based on the joint distribution of the �’s, wewill call the estimator that maximizes the likelihood the full maximumlikelihood estimator (FMLE). That is,

(�FMLE, FMLE) = argmax�,

lnL(�, )� (13)

The first term (copula term) in the summation in Eq. (12) is whatdistinguishes the full log-likelihood from the quasi-log-likelihood—thisterm reflects the dependence between the composed errors at different t .

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From the GMM perspective, the FMLE solves the first order conditionsthat are identical to the first order conditions solved by the GMM estimatorbased on the moment conditions

�[�� ln ci(�o , o) + �� ln fi1(�o) + · · · + �� ln fiT (�o)

� ln ci(�o , o)

]= 0, (14)

where ci(�, ) = c(Fi1(�), � � � , FiT (�); ). Again, the key to developing anew estimator is the observation that the moment conditions in (14)are obtained by applying a specific (not necessarily optimal) weightingscheme. The weighting scheme for the FMLE is optimal if the likelihoodis correctly specified (i.e., the copula and the marginal distributions are allcorrectly specified), but not otherwise, and it is possible for the momentconditions to hold even if the copula is not correctly specified. SeeProkhorov and Schmidt (2009) for further discussion of this point.

Let �i(�, ) ≡

�� ln fi1(�)

� � �� ln fiT (�)�� ln ci(�, )� ln ci(�, )

, and let (�GMM, GMM) be the optimal

GMM estimator based on the moment conditions

��i(�o , o) = 0� (15)

That is, (�GMM, GMM) is the solution to the optimization problem

(�GMM, GMM) = argmin�,

�(�, )′W �(�, ), (16)

where �(�, ) = 1N

∑i �i(�, ) and W is the inverse of the estimated

covariance matrix of �i(�o , o). Prokhorov and Schmidt (2009) considerthis estimator in the setting of a general panel data model and show thatit is consistent so long as the FMLE above is consistent.

Unlike the QMLE and IQMLE, both the FMLE and GMM estimatorsallow for arbitrary time dependence in panel stochastic frontier modelswhile offering important computational advantages over the availablealternatives. We discuss these advantages in the next section.

4.2. Evaluation of Integrals

In this section, we show that by applying a copula to (�i1, � � � , �iT ), wecircumvent the task of a T -dimensional integration imposed by traditionalmethods. In effect, we are replacing that task with the task of T evaluationsof a one-dimensional integral, which is a much simpler task.

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To demonstrate the advantages of our approach, we first consider whathappens if we use a multivariate truncated distribution to form the fulllog-likelihood. Suppose, for example, that the joint distribution of u ismultivariate truncated normal with parameter matrix � (see Pitt and Lee,1981). Then,

fu(u) = (2�)−T /2|�|−1/2 exp{12u′�−1u

}/P0, (17)

where P0 is the probability that u ≥ 0, which can be expressed as thefollowing T -dimensional integral:

P0 =∫ ∞

0· · ·

∫ ∞

0(2�)−T /2|�|−1/2 exp

{12u′�−1u

}du (18)

Given fu(u), the joint density of the composed error vector � can beobtained as yet another T -dimensional integral as follows:

f�(�) =∫ ∞

0· · ·

∫ ∞

0

T∏t=1

1√2��v

e(�t+ut )2

2�2v fu(u)du� (19)

Note that inside this integral there is the joint density fu(u), which isitself calculated using T -dimensional integration. The resulting likelihoodis intractable.

Another difficulty with using the multivariate truncated normaldistribution is that its marginals are not truncated normal distributionsunless the marginals are independent (see Horrace, 2005, Theorems 4and 6). So if we want to maintain the assumption of truncated normalmarginals and allow for dependence by assuming a truncated normal jointdistribution, that approach is not acceptable. Moreover, to our knowledge,no known multivariate distribution, aside from those constructed usingcopulas, has truncated normal marginals. However, these conceptual issuesare obviously distinct from the numerical issues discussed above.

Instead of T -dimensional integrations, the FMLE and GMM estimatorsrequire repeated evaluation of

F (�) ≡∫ �

−∞f (s)ds,

which serves as an argument of the copula function. Evaluation of thisintegral is a standard task in one-dimensional integration that can beaccomplished in standard econometric software by quadrature or by

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simulation.1 Moreover, Tsay et al. (2009) propose an approximation thatcan be used specifically in the setting of a stochatic frontier model to speedup the evaluations.

Wang et al. (2011) tabulate several quantiles of F (�) using analternative parameterization, in which � is replaced with �v

√1 + �2. They

obtain their quantiles by simulation. We chose the parameterization(�, �) rather than (�v , �) because it bounds the variance of each errorcomponent by the value of �2. This is convenient because given � thequantiles do not become large negative numbers as � increases (the cdfdoes not flatten out). Our cdf is effectively on the domain [−4, 4] for all�. Also when applying copulas to �, we prefer using numerical integrationrather than simulation.2

4.3. Maximum Simulated Likelihood (MSLE) and GenerationMethod of Simulated Moments (GMSM)

Integral evaluations by simulation generate an approximation errorwhich needs to be accounted for. This section does that.

The basic idea of simulation-based methods is that under suitableregularity conditions a simulator will converge to the object it simulates.Let f (�) and F (�) denote the simulation-based estimators of f (�) andF (�), respectively. We are interested in the properties of the MLE andGMM estimators that use the estimators instead of the true values of thetwo functions. Such estimation methods fall into the class of MaximumSimulated Likelihood (MSLE) and Generalized Method of SimulatedMoments (GMSM) estimators.

Statistical properties of MSLE and GMSM are well-studied (see, e.g.,McFadden, 1989; Gouriéroux and Monfort, 1991; McFadden and Ruud,1994). For example, it is well known that the so-called direct simulator

f (�) = 1S

S∑s=1

(� + us),

1The normal/half-normal error structure permits an additional shortcut when evaluating F (�),the composed error cdf. By the algebraic properties of the cdf, for a fixed �, F�(�) = F1(�/�),where the subscript denotes the value of � used in evaluting F . Therefore, we can express quantilesof F (�) for any � as � times the corresponding quantile for � = 1. This means that we need toevaluate the integral only at � = 1. Then, we can obtain its value at any other � by calculating theproduct.

2Our experience suggests that numeral integration by quadrature is somewhat faster thansimulation for the same level of precision. A GAUSS code, which can be used to obtain any numberof quantiles at desired precision using both methods, is available from the authors.

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where is the density of v and where us are draws from N (0, �2u)

+,s = 1, � � � , S , converges to f (�) as S → ∞, and

F (�) = 1N

∑e≤�

f (e),

converges to F (�) as N → ∞. Moreover, by a well-known result fromGouriéroux and Monfort (1991), the estimator obtained by maximizingthe simulated log-likelihood∑

i

(ln c(Fi1(�), � � � , FiT (�); ) + ln fi1(�) + · · · + ln fiT (�)) (20)

has the same asymptotic distribution as the FMLE provided that

S → ∞N → ∞√

N /S → ∞�

So the simulation error is ignorable asymptotically if enough simulationdraws are used.

A similar result is available for the GMSM estimator. McFadden (1989)shows that, under usual regularity conditions, the GMM estimator basedon the moment conditions (15), in which fit is replaced with fit and Fitis replaced with Fit , is consistent for � even for fixed S . However, theasymptotic variance matrix of the GMSM estimator has to be adjusted toreflect the simulation error unless S → ∞. Compared to the MSLE, theGMSM estimator is more complicated and is known to be numericallyunstable (see, e.g., McFadden and Ruud, 1994). Moreover, in settings withlarge T and relatively small N as in our application, the GMSM maygenerate too many moment conditions to handle given the sample size.Therefore, the MSLE appears more practical.

4.4. Choosing a Copula

There are many tests that can help choose a copula (see, e.g., Genestet al., 2009, for a review of copula goodness of fit tests). This sectionproposes a variation of the test of Prokhorov and Schmidt (2009), which isparticularly well suited for the case when we apply a copula to (�i1, � � � , �iT ).The test is focused on the copula based moment conditions which we aredoubtful about, while maintaining the validity of the marginals.3 The test

3We modify the original test of Prokhorov and Schmidt (2009) to make it suitable for usewith the standard QMLE in the first step rather than IQMLE.

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proceeds as follows. In the first step, we obtain �QMLE. In the second step,we apply GMM with a specific form of the weighting matrix to the copula-based moment conditions

�

[�� ln ci(�QMLE, )� ln ci(�QMLE, )

]= 0, (21)

where the required form of the weighting matrix is given in Appendix B.This produces an estimate of —denote it by —and the optimized valueof the GMM criterion function—denote it by Q (�QMLE, ). To constuct thetest statistic, we look at the product of the sample size and Q (�QMLE, ).The claim is that, under the null of a correct copula, N · Q (�QMLE, ) isasymptotically distributed according to the �2 distribution with the degreesof freedom equal to the dimension of �. We will reject the null hypothesisthat the chosen copula is valid for large values of the test statistic. Theproof of this claim is given in Appendix B.

4.5. Meta-Elliptical Copulas

For the moment, assume we observe the composed errors �it . Thatis not true, of course, but we can estimate them by the QMLE residuals.Suppose for the moment that we assume a Gaussian copula. Then forany pair of time indices s, t we can derive an estimate of the correlationparameter st in the copula via the estimated value of Kendall’s tau.Specifically, for bivariate normal data it is well known that � = 2

�arcsin( ),

or = sin(�2�), where � is Kendall’s tau and is the usual correlationcoefficient. So if we estimate � we can use this relationship to estimate .However, since Kendall’s tau is invariant to monotonic transformation ofthe data, this result requires only that the copula be normal, not that themarginals be normal. In fact, Fang et al. (2002) point out that it requiresonly that the copula belong to the meta-elliptical family, which includesthe multivariate normal (see Fang et al., 2002, Definition 1.2, p. 3 for adefinition of this family).

The FMLE requires a joint maximization of the likelihood with respectto the parameters of the marginal distributions (�) and the copulaparameters ( ). However, if we assume a Gaussian copula, or any othermeta-elliptical copula, we can instead use a three-step procedure thatshould be computationally less demanding. First, estimate � by QMLE toobtain � and �it . Second, estimate the copula parameters from the �it

via the estimate of Kendall’s tau. Explicitly, for each pair of time indicess, t , estimate �st in the usual way (involving counting concordant anddiscordant pairs of cross-sectional observations, that is, of indices i , j), andthen define st = sin(�2 �st). Third, form the likelihood as a function of �,

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but with the copula parameters fixed at , and maximize this to obtaina new estimate �. Of course, this procedure could be iterated further. Thedifficulty with such a multi-step approach is that it leads to complicatedasymptotic distribution theory. The estimation error in each step will ingeneral affect the asymptotic distribution of the estimates in subsequentsteps. However, this may be a very reasonable way to obtain starting valuesfor the computation of the FMLE.

5. APPLICATION OF A COPULA TO (UI1, � � � ,UIT)

We now consider the alternative copula specification. We apply acopula to form the joint distribution of (ui1, � � � ,uiT ) rather than that of(�i1, � � � , �iT ). As we mentioned before, a T -dimensional integration willbe needed to form the likelihood in this case. This will be similar tothe example of the multivariate truncated normal in the previous section,but the use of a copula will permit us to avoid some of the difficultiesassociated with that approach.

As before, let fu(u; �, ) denote the copula-based joint density of theone-sided error vector, and let fu(u; �) denote the marginal density of anindividual one-sided error term. Then,

fu(u; �, ) = c(Fu(u1; �), � � � , Fu(uT ; �); ) · fu(u1; �) · · · fu(uT ; �), (22)

where Fu(u; �) ≡ ∫ u0 fu(s; �)ds is the cdf of the half-normal error term.

The joint density of the composed error vector � can be expressed asfollows:

f�(�; �, ) =∫ ∞

0· · ·

∫ ∞

0fv(� + u; �)fu(u; �, )du1 · · · duT = �u(�, )fv(� + u; �),

(23)

where �u(�, ) denotes the expectation with respect to fu(u; �, ) and fv(v; �)is the multivariate normal pdf of v, and where (as before) it is assumedthat all of the v’s are independent and have equal variance �2

v . Asin the multivariate truncated normal example, this is a T -dimensionalintegral which has no analytical form and is not easy to evaluate usingnumerical methods. Unlike that example, however, it involves only oneT -dimensional integral and it has the form of an expectation over adistribution we can easily sample from. We discuss the estimators that areavailable in this setting, and the issues that arise implementing them, inthe next section.

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5.1. Full MSLE

The key benefit of using a copula in this case is that we can simulatefrom the distribution of u and estimate f�(�) by averaging fv(� + u; �) overdraws from that distribution.

Let S denote the number of simulations. The direct simulator of f�(�)is

f�(�; �, ) = 1S

S∑s=1

fv(� + us(�, ); �),

where us(�, ) is a draw from fu(u; �, ) constructed in (22). Then, the fullsimulated log-likelihood is

lnLs(�, ) =∑i

ln f�(�i ; �, ),

where, as before, �i = (�i1, � � � , �iT ) and �it = yit − x ′it�. The full maximum

simulated likelihood estimator (FMSLE) is then given by

(�FMSL, FMSL) = argmax�,

lnLs(�, )� (24)

This method is analogous to the simulation-based estimation ofunivariate densities discussed in a previous section. This is a multivariateextension of that method. Similar asymptotic arguments suggest that theasymptotic distribution of (�FMSL, FMSL) is identical to the FMLE based on(23), provided that S → ∞, N → ∞,

√N /S → ∞ and suitable regularity

conditions hold (see, e.g., Gouriéroux and Monfort, 1991). Note that wedo not have a GMSM estimator here because we simulate the joint densityf� directly.

Section 4.5 discussed some computational simplifications when a meta-elliptical copula was assumed for �. These simplifications do not apply herebecause we do not “observe” u. That is, it is not expressible as a residual,since it is intrinsically confounded with v. We can estimate u, as discussedin the next section, but not consistently.

Implementation of the FMSL estimator depends on our ability todraw observations from the copula c that underlies the joint densityfu(u; �, ) in Eq. (22) above. In our empirical example given below, weuse the Gaussian copula and drawing from it is equivalent to drawingfrom a multivariate normal distribution. (That is, one draws observationsfrom a multivariate standard normal with desired correlation structure,and then applies the standard normal cdf to these normals to convertthem to uniform random variables.) This is a simple task. However, inthe non-normal case we have the sometimes difficult task of drawing

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observations from a multivariate distribution whose density is available inclosed form. One possibility would be generic Markov Chain Monte Carlomethods such as the Metropolis–Hastings algorithm (see, e.g., Robert andCasella, 2004; Gill, 2008). However, the standard method in the copulaliterature—sometimes called conditional sampling—is to iteratively invertthe conditional distributions implied by a copula (see, e.g., Li et al.,1996; Cherubini et al., 2004, Section 6.3). Besides deriving the conditionaldistributions and their inverses, the method usually employs numericalalgorithms to find roots of equations involving the conditional cdf’s. Theerror in these numerical approximations accumulates as the number ofdimensions grows, and therefore the performance of these techniques ispoor for large dimensional copulas (see, e.g., Devroye, 1986, Chapter 11).However, when the number of dimensions is moderate (no more than 10),the performance seems acceptable and some software packages containsuch random number generators (see, e.g., Kojadinovic and Yan, 2010).For example, the copula R package has this routine implemented for upto 6–10 dimensions depending on the copula family. Moreover, simplerand more reliable methods exist for two subclasses of copula families,the Archimedian and elliptical copulas (see, e.g., Cherubini et al., 2004,Sections 6.2 and 6.4).

Another approach that could be applied is importance sampling. Herethe generic problem is to evaluate �g (u), where in our case g (u) = fv(� +u; �) as in equation (23) above. Suppose that the density of u is f (u). Ifwe can draw from f , we can simulate this as 1

S

∑s g (u

s), as above. However,now suppose that it is very difficult or impossible to draw from f , butthere is another density h(u) that is easy to draw from. (This could be, forexample, the joint density implied by the Gaussian copula.) Then we canwrite

�g (u) =∫

g (u)f (u)du =∫

g (u)f (u)h(u)

h(u)du,

and we can simulate this as

1S

∑s

g (us)f (us)

h(us),

where now the us are drawn from h rather than from f . We shouldpick h on the basis that it is easy to draw from but that it shouldresemble f as closely as possible. See, e.g., Bee (2010) for a discussion andimplementation of this procedure.

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5.2. Estimation of Technical Inefficiencies

The use of a copula to form fu permits the construction of a new andimproved estimator of technical inefficiencies ui , i = 1, � � � ,N .

In a single cross section (single t) setting, Jondrow et al. (1982)propose estimating uit by the conditional expectation �uit |�it , where

�uit |�it = ��

1 + �2

[

(�it �

�

)1 −

(�it �

�

) − �it�

�

]�

We have defined above four consistent estimators of �. Using eachof them, we can easily construct an analoguous estimator of �uit | �it

(see, e.g., Greene, 2005). However, this estimator does not explicitlyincorporate information about the dependence of the uit ’s over time.We propose estimating technical inefficiencies by �(u |�)–the expectedtechnical inefficiency conditional on the composed error vector. The pointis that, when we have correlation over time, all of the values �i1, �i2, � � � , �iT

are informative about uit . By conditioning on the larger set of relevantvariables, we improve the precision of the estimate of uit (for all t). We alsowill smooth the estimated uit over t .

The conditional expectation �(u|�) is with respect to the conditionaldensity f (u|�), which we do not know. However, we now know how to formthe joint distribution of (u1, � � � ,uT ) using a copula and we can draw fromthat distribution.

As before, we omit the subscript i for simplicity and write

u =u1

��uT

=�1

��T

−v1

��vT

= � − v,

where ut ∼ N (0, �2u)

+, v ∼ N (0, �2v�T ), and u and v are independent.

Assume we apply a copula to u as discussed above, and we now have fu(u).Then,

fu(u; �2u , ) = c

[Fu(u1; �2

u), � � � , Fu(uT ; �2u);

] ·T∏t=1

fu(ut ; �2u)

Note that fu and Fu denote the pdf and cdf, respectively, of the half-normaldistribution N (0, �2

u)+. Assume for now that (�2

u , �2v , ) are known. The key

observation is that we can draw from the joint distribution of u and v.First, we draw from the copula-based distribution of u. This is done as

follows:

1) Draw �w1, � � � ,wT � from copula C ;

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2) Generate ut = F −1(wt), t = 1, � � � ,T , where F −1(w) is the inverse cdfof Fu(u).

We obtain S such draws us .Second, we draw vs ∼ N (0, �2

v�T ) and obtain �s = vs − us , s = 1, � � � , S .We now have draws from the joint distribution of (u,�) and can estimate�(u|�) by any nonparametric method, e.g., by the multivariate version ofthe Nadaraya–Watson estimator

�(u|�) =∑S

s=1 usK(�s−�h

)∑Ss=1 K

(�s−�h

) ,

where K(�s−�h

) = ∏Tt=1 K

(�ts−�t

h

), K (·) is a kernel function and h is the

bandwidth parameter. It is a standard result that

�(u|�) → �(u|�) as S → ∞, h → 0 and ShT → ∞Obviously, this procedure is not feasible unless we know the values of

(�2u , �

2v , ). But consistent estimates of these parameters are available from

the FMSLE. Similarly, to obtain the estimated inefficiencies, we evaluatethe estimated conditional expectation at the residuals �it = yit − x ′

it �, where� also comes from FMSLE.

As alternatives, one may use the QMLE, IQMLE, FMLE, or GMMestimates of (�′, �2

u , �2v). However, these estimators do not produce an

estimate of the copula parameter for u.4 A way around this difficulty is todraw from the joint distribution of (u,�) for different values of until acertain property of the data, e.g., sample correlation (or Kendall’s �) of�it over time, is matched by the simulated sample. Because of the variouscriteria that can be used in matching (especially if T is large and if is nota scalar), this estimator is less attractive than the one using the FMSLE.

6. EMPIRICAL APPLICATION: INDONESIAN RICE FARMPRODUCTION

We demonstrate the use of copulas with a well-known data set fromIndonesian rice farms (see, e.g., Erwidodo, 1990; Lee and Schmidt, 1993;Horrace and Schmidt, 1996, 2000). The data set is a panel of 171Indonesian rice farmers for whom we have six annual observations of riceoutput (in kg), amount of seeds (in kg), amount of urea (in kg), labor(in hours), land (in hectares), whether pesticides were used, which of

4The FMLE and GMM produce estimates of the dependence parameter in the copula for �

but this is not the parameter we need.

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TABLE 1 Indonesian rice production: QMLE and Gaussian copula-based FMLE and FMSLE

HS(2000) QMLE FMLE FMSLE

Est. Est. St.Err Rob.St.Err. Est. St.Err. Est. St.Err.

CONST 5�9540 5�5763 0�2000 0�2773 5�4499 0�1810 5�5291 0�1915SEED 0�0583 0�1449 0�0275 0�0358 0�1226 0�0229 0�1468 0�0244UREA 0�1028 0�1131 0�0167 0�0243 0�1212 0�0149 0�1122 0�0150TSP 0�0034 0�0763 0�0111 0�0119 0�0636 0�0122 0�0775 0�0131LABOR 0�1970 0�1999 0�0284 0�0313 0�2224 0�0310 0�2029 0�0321LAND 0�6374 0�4911 0�0299 0�0418 0�4815 0�0240 0�4885 0�0279DP 0�0138 0�0064 0�0276 0�0320 0�0266 0�0269 0�0068 0�0296DV1 −0�0861 0�1670 0�0380 0�0357 0�1334 0�0415 0�1711 0�0476DV2 0�0853 0�1212 0�0517 0�0505 0�1381 0�0579 0�1217 0�0628DR1 0�2173 −0�0596 0�0548 0�0612 −0�1049 0�0720 −0�0597 0�0617DR2 −0�0330 −0�1160 0�0522 0�0510 −0�1609 0�0667 −0�1158 0�0672DR3 0�1385 −0�1342 0�0337 0�0432 −0�1118 0�0453 −0�1311 0�0380DR4 0�0817 −0�1457 0�0340 0�0428 −0�1632 0�0444 −0�1443 0�0370DR5 0�1829 −0�0481 0�0430 0�0587 −0�0847 0�0458 −0�0480 0�0382� 0�2744 0�4439 0�0253 0�0306 0�4360 0�0313 0�4427 0�0328� 0�5482 1�3723 0�2655 0�3825 1�2193 0�2702 1�3604 0�2731

lnL – −342.32 −283.95 −320.86

three rice varieties was planted and in which of six regions the farmer’svillage is located. Therefore, there are 13 explanatory variables besides theconstant.

We start with the results from the QMLE defined in (6). They aregiven in Table 1. For reference, we also give the ML estimates obtainedby Horrace and Schmidt (2000) using just the first cross section of thedata. Our results are based on all six cross sections. It is important toaccount for potential correlation between the cross sections. The robuststandard errors we report for the QMLE do this. They are obtained usingthe “sandwich” formula, while the first set of standard errors are obtainedusing the estimated Hessian.

The second set of results are the FMLE defined in (13). We use theGaussian copula to construct a multivariate distribution of �. The Gaussiancopula is obtained from the multivariate normal distribution with thecorrelation matrix R by plugging in the inverse of the univariate standardnormal distribution function (x) (see Appendix A). It is easy to show(see, e.g., Cherubini et al., 2004, p. 148) that the resulting copula densitycan be written as follows:

c(w1, � � � ,wT ;R) = 1√|R| e− 1

2 �′(R−1−I )�,

where � = (−1(w1), � � � ,−1(wT ))′. In the Gaussian copula, the

dependence structure between the marginals is represented by the

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correlation matrix R. So our FMLE results contain estimates of a 6 × 6correlation matrix (which we do not report).

The last set of estimates are the FMSLE defined in (24), where weuse the Gaussian copula to construct and sample from a multivariatedistribution of u. We use S = 1000 draws in evaluating f�(�) by simulation.As in FMLE, the results include 15 estimated correlations. Now these arecorrelations over t of −1(Fu(ut)), not of −1(F�(�t)), though the twoestimated correlation matrices (not reported here) turn out to be verysimilar.

For both the FMLE and the FMSLE, we considered some restrictedversions of the correlation matrix, such as those implied by equicorrelationand stationarity. These restrictions were rejected by the relevant likelihoodratio tests.

The coefficient estimates in Table 1 are relatively similar across thethree specifications. The FMSLE gives coefficients that are more similar tothe QMLE than the FMLE coefficients are. The standard errors are moredifferent. Interestingly, the two MLE specifications do not give uniformlysmaller standard errors than the QMLE (robust) standard errors. So theefficiency gain from modelling dependence, which was one of our twostated advantages to the copula approach, is not very important in thisapplication.

Next we consider the two moment based estimators, namely theIQMLE defined in (11) and the GMM estimator defined in (16). Theseestimators have some theoretical advantages such as higher asymptoticefficiency that the QMLE but, in practice, they may be infeasible. In thisapplication, for example, we have 16 parameters, 6 time periods, andonly 171 cross-sectional observations. The vector of moment conditions onwhich the IQMLE is based is 96 × 1. Moreover, the additional vector ofcopula-based moment conditions employed by the GMM estimator is 15 ×1 for the parameters of R and 16 × 1 for the parameters of the marginals.So we have as many as 127 moment conditions. Obtaining stable GMMestimates for this problem with only 171 observations proved impossible.

Finally, we use the procedure outlined in Section 5.2 to estimatetechnical inefficiencies. As a benchmark we report the inefficiencyestimates obtained using the analytical expression for �uit |�it derivedby Jondrow et al. (1982). These estimates are reported in Table 2. Tosave space we report results only for ten of 171 farms, namely thosethat are at the 0�05, 0�15, � � � , 0�95 fractiles of the sample distribution ofinefficiencies for t = 1. The first two columns report the rank and thequantile of the selected farms. The third and the fourth columns containthe sample standard deviation and the sample mean (over t), respectively,of the inefficiency estimates, which are reported in columns 5 through 10.The last row of the table reports the correponding column averages overthe ten farms.

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TABLE 2 Indonesian rice production: Inefficiency estimates based on JLMS formula

Rank Fractile � ¯u t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

9 0.05 0.0523 0.2674 0.1207 0.1419 0.7718 0.1287 0.1952 0.245826 0.15 0.0018 0.1899 0.1421 0.1852 0.2514 0.2423 0.1531 0.165443 0.25 0.0082 0.2299 0.1713 0.1219 0.1812 0.3819 0.3199 0.202960 0.35 0.0123 0.3270 0.1902 0.2438 0.4935 0.4518 0.3251 0.257777 0.45 0.0142 0.3252 0.2079 0.4997 0.2089 0.2148 0.4339 0.386294 0.55 0.0240 0.3184 0.2254 0.5454 0.1435 0.5180 0.2596 0.2183

111 0.65 0.0100 0.3746 0.2549 0.292 0.3316 0.3898 0.4165 0.5629128 0.75 0.0064 0.2964 0.2873 0.1908 0.4496 0.3165 0.2387 0.2955145 0.85 0.0458 0.4403 0.3373 0.2782 0.8637 0.5235 0.2132 0.4260162 0.95 0.0456 0.4113 0.4050 0.3997 0.1788 0.8499 0.2499 0.3847

Average 0.0221 0.3180 0.2342 0.2899 0.3874 0.4017 0.2805 0.3145

Table 3 reports the same results when we use the procedure ofSection 5.2, based on the estimates that assumed a copula for u.

A few important observations can be made from the tables. First, thereis a substantial reduction in the variability of inefficiencies over t when weuse the copula based estimator. The sample variation � of the inefficiencyestimates over time obtained using the traditional method is several timeshigher for all farms. The improved estimates of inefficiencies uit varyless over t reflecting the positive correlation between them. For example,for the firm with rank 94, inefficiencies range from 0.1435 to 0.5180 inTable 2, but only from 0.2013 to 0.3238 in Table 3. The timing of the highand low efficiencies tend to agree, but the spread is less in Table 3. In ourview the temporal changes in inefficiencies in Table 3 are more believablethan those in Table 2, and in this empirical application that is the mainadvantage of allowing for temporal dependence.

TABLE 3 Indonesian rice production: inefficiency estimates based on a copula for u

Rank Fractile � ¯u t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

9 0.05 0.0060 0.2543 0.1480 0.2155 0.2228 0.3785 0.2279 0.333026 0.15 0.0008 0.1932 0.1734 0.1856 0.2050 0.2516 0.1759 0.167543 0.25 0.0046 0.2154 0.1876 0.1720 0.2616 0.3467 0.1519 0.172760 0.35 0.0002 0.2165 0.2051 0.2123 0.2297 0.2371 0.2075 0.207177 0.45 0.0073 0.2796 0.2226 0.2559 0.1434 0.2951 0.3999 0.360494 0.55 0.0021 0.2627 0.2403 0.2174 0.2013 0.3238 0.3115 0.2821

111 0.65 0.0025 0.3087 0.2541 0.3080 0.3818 0.3643 0.2503 0.2939128 0.75 0.0037 0.3016 0.2826 0.2690 0.2013 0.3078 0.3629 0.3862145 0.85 0.0048 0.3079 0.3115 0.3038 0.3023 0.4468 0.2618 0.2213162 0.95 0.0034 0.3491 0.4043 0.3677 0.2740 0.4374 0.2996 0.3118

Average 0.0036 0.2689 0.2430 0.2507 0.2423 0.3389 0.2649 0.2736

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Neither method produces consistently higher or consistently lowerinefficiencies for any of the selected farms. However, overall the copulabased estimator seems to produces a slightly lower average estimatedinefficiency. So far as we known this is just a result for this application andnot a general result.

7. CONCLUDING REMARKS

In this paper we have proposed a new copula-based method of allowingfor dependence between cross sections in panel stochastic frontier models.Compared to available alternatives, this method is simple and flexible. Itallows arbitrary dependence and does not involve much computationalcost. We propose several estimators based on this method, illustrate howone can test for copula validity, and suggest a simulation-based method ofestimating technical inefficiencies.

The advantages of the proposed approach are not only computational.By assuming a copula we assume a joint distribution. This should lead toestimates that are more efficient than QMLE. It also leads to a smoothingof the pattern of the inefficiencies over time.

An interesting extension of this paper would be to compare therobustness of the copula-based estimators we propose with that of thealternative estimators based on the scaling property. We leave this questionfor future research.

APPENDIX A: SOME COPULA FAMILIES

Independence or product copula:

C(w1, � � � ,wT ) = w1 × · · · × wT �

Gaussian or Normal copula:

C(w1, � � � ,wT ;R) = T (−1(w1), � � � ,−1(wT );R),

where denotes the standard normal cdf and R denotes the correlationmatrix.

Gumbel copula:

C(w1, � � � ,wT ; ) = exp[−((− lnw1)

+ · · · + (− lnwT ) )1/

], ∈ [1,∞)�

Clayton copula:

C(w1, � � � ,wT ; ) = max[(w−

1 + · · · + w− T )−1/ , 0

], ∈ [−1,∞)�0��

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Farlie–Gumbel–Morgenstern (FGM) copula:

C(w1, � � � ,wT ; ) =T∏t=1

wt

(1 +

T∑t=2

∑1≤j1<···<jk≤T

j1��jk (1 − wj1) � � � (1 − wjk )

),

where j ∈ [−1, 1].General copula by inversion:

i) Start with cdf’s K (x1, � � � , xT ), w1 = F1(x1), � � � ,wT = FT (xT );ii) Obtain x1 = F −1

1 (w1), � � � , xT = F −1T (wT ) and

C(w1, � � � ,wT ) = K (F −11 (w1), � � � , F −1

T (wT ))�

Archimedean copulas:

i) Start with a generator function � : (0, 1) → [0,∞],�′ < 0 and �′′ > 0;ii) Obtain

C(w1, � � � ,wT ) = �−1(�(w1) + · · · + �(wT ))

iii) Multivariate copulas can often be obtained recursively from a bivariatecopula (not generally true for other families): e.g., Gumbel copulais Archimedean with �(t) = (− log t) , Frank copula – with �(t) =ln 1−e−

1−e− t

APPENDIX B: COPULA VALIDITY TEST

The test is similar to the test in Theorem 9 of Prokhorov and Schmidt(2009). We adapt the test to the first step QMLE.

For simplicity, we consider T = 2. For t = 1, 2 and i = 1, � � � ,N , letfti(�) = f (�ti ; �), ci(�, ) = c(F (�1i ; �), F (�2i ; �); ),

gi(�) = �� ln f1i(�) + �� ln f2i(�)� ,

ri(�, ) =[�� ln ci(�, )� ln ci(�, )

]�

Also, let

g (�) ≡ 1N

N∑i=1

gi(�), r (�, ) ≡ 1N

N∑i=1

ri(�, )�

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Let (�o , o) denote the true parameter values, and define the followingmatrices:

Vo11 ≡ �g (�o)g (�o)′,

Vo22 ≡ �r (�o , o)r (�o , o)

′,

Vo12 = Vo

21′ ≡ �g (�o)r (�o , o)

′,

Do11 ≡ ��g (�o),

Do21 ≡ ��r (�o , o),

Do22 ≡ �� r (�o , o),

where expectations are with respect to the true joint density h(�1, �2).

Proposition 1. Let �QMLE be the QMLE of � and let be obtained by minimizingr (�QMLE, )′B−1

o r (�QMLE, ), where

Bo = Vo22 − Do

21Do′11

–1Vo12 − Vo

21Do11

–1Do21

′

+ Do21(D

o′11V

o11

–1Do11)

–1Do21

′�

Then,

N r (�QMLE, )′B-1o r (�QMLE, )

a∼ �2p , (25)

where p is the dimension of �.

First note that, by standard QMLE results, �QMLE satisfies

√N (�QMLE − �o) = −(Do

11)−1

√N g (�o) + op(1)� (26)

The first order condition for can be written as

[� ′ r (�QMLE, )]′B−1

o r (�QMLE, ) = 0,

Do′22B−1

o

√N r (�QMLE, ) = op(1)� (27)

Now, by the mean-value theorem, we have

√N r (�QMLE, ) = √

N r (�o , o) + Do21

√N (�QMLE − �o)

+ Do22

√N ( − o) + op(1)� (28)

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Substituting (26) into (28), pre-multiplying by Do22

′B−1o , and solving for√

N ( − o) using (27) yields

√N ( − o) = −(Do′

22B−1Do22)

−1Do22

′B−1o

√N r (�o , o)

+ (Do22

′B−1o Do

22)−1Do

22′B−1

o Do21D

o11

−1√N g (�o) + op(1)� (29)

Substituting (29) and (26) into (28) and simplifying results in

√N r (�QMLE, ) = Ro

√N (�o , o) + op(1), (30)

where

Ro = � − Do22(D

o22

′B−1o Do

22)−1Do

22′B−1

o

(�o , o) = r (�o , o) − Do21D

o11

−1g (�o)�

Note that√N (�o , o) ∼ �(0,Bo), and thus B−1/2

o

√N (�o , o) ∼

�(0, �). Also, note that Ro′B−1

o Ro = B− 1

2o [I − B

− 12

o Do22(D

o22

′B−1o Do

22)−1Do

22′

B− 1

2o ]B− 1

2o .

Thus, the test statistic in (25) can be written as

N (�, )′�(�, ), (31)

i.e., as a quadratic form in standard normals with the coefficient matrix

� = Ip+q − B−1/2o Do

22(Do22

′B−1o Do

22)−1Do

22′B−1/2

o � (32)

This matrix is idempotent: it is the projection matrix orthogonal to

B− 1

2o Do

22. The �2-test in (25) follows immediately because tr (�) = p + q −rank(Do

22) = p.

ACKNOWLEDGMENTS

Helpful comments from participants of the Canadian EconometricsStudy Group Meetings, the International Conference on Panel Data, andthe European Workshop on Efficiency and Productivity Analysis as well asseminar participants at University of New South Wales and Ewha WomansUniversity are gratefully acknowledged. This research has been supportedby the Social Sciences and Humanities Research Council of Canada.

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