choice of flexible functional forms: review and appraisal · choice of flexible functional forms:...

Choice of Flexible Functional Forms:Review and Appraisal

Gary D. Thompson

Choice between alternative flexible functional forms has received little explicittreatment in many empirical agricultural studies. Theoretical criteria and empiricaltechniques for choosing between flexible functional forms are reviewed. Theoreticaltopics include definitions of flexibility, mathematical expansions, separability, andregular regions. Empirical techniques examined are Monte Carlo analysis, parametricmodeling, bayesian inference, and nonnested hypothesis testing. Comparison of thefull range of theoretical and empirical aspects may provide more credible and reliableempirical estimates when consumer or producer duality assumptions are appropriatein agricultural applications.

Key words: duality, flexible functional forms, mathematical expansions,nonparametric tests.

Choice of functional form has been a pressingissue for empirical production and consumerstudies since the pioneering work of Douglas(Cobb and Douglas, Douglas) and Stone. Testsof the classical theory of the firm based on therestrictive Cobb-Douglas production functionhave been thoroughly criticized (Samuelson,Simon). The development of flexible function-al forms was driven by the search for func-tional forms which imposed fewer maintainedhypotheses. The econometric limitations of theCobb-Douglas functions (Hoch), for example,provided impetus for derivation of the CESand other functional forms (Zellner and Re-vankar). With the subsequent formalization ofthe notion of flexibility (Diewert 1971), a largeset of flexible functional forms (FFF) has be-come available to the empirical researcher (seeappendix).

In some empirical studies, the reasons forchoosing a particular flexible functional formare not explicitly stated. Recent agriculturalproduction duality applications of FFF, for ex-ample, have not addressed in detail the issueof choice among alternative FFF (Antle; Lopez1980; Shumway; Sidhu and Baanante; Weaver

The author is an assistant professor, Department of AgriculturalEconomics, University of Arizona.

The author wishes to thank Ronald C. Griffin, Mark Langwor-thy, and an anonymous referee for helpful comments.

1983). Advocating that the choice of function-al form should be treated explicitly in empir-ical research, Griffin, Montgomery, and Risterhave identified and evaluated criteria forchoosing between competing functional formsin production function analysis.

Following the Griffin, Montgomery, andRister prescription for treating the choice offunctional form explicitly, the focus of this pa-per is on choice of FFF in producer and con-sumer duality settings where cost, profit, orindirect utility functions or systems of equa-tions derived from these functions are to beestimated. The scope of the paper is limitedto FFF because of their recent popularity inapplied studies which use duality theory. The-oretical characteristics of FFF are first dis-cussed, compared, and assessed. Empiricaltechniques for choosing among FFF are thenreviewed and appraised. Conclusions regard-ing the application of an empirical procedurefor choosing among FFF follow.

Pros and Cons of FFF

Duality theory advances have ushered in thewidespread use of FFF for a number of rea-sons. First, with the satisfaction of regularityconditions such as convexity (concavity),monotonicity, and homogeneity, duality re-

Western Journal of Agricultural Economics, 13(2): 169-183Copyright 1988 Western Agricultural Economics Association

Western Journal of Agricultural Economics

suits preclude the need for self-dual functions.Further, the use of derivative properties- Ho-telling's and Shephard's lemmas and Roy'sIdentity-allows for derivation of demand andsupply (or share) functions without solving an-alytically for those functions. Comparativestatics are easily derived from the propertiesof the parent indirect functions. Finally, FFFhave gained popularity because of the en-hanced capacity of nonlinear estimation pro-cedures for nonlinear-in-parameters equationsystems.1

Empirical use of FFF has certain drawbacks:collinearity due to numerous terms involvingtransformations of the same variables and in-teraction among variables; failure to satisfy theregularity conditions over the entire range ofsample observations; and, less important, dif-ficulty in interpreting initial parameter esti-mates. Estimation of nonlinear-in-parameterssystems may involve problems with conver-gence and statistical theory (Lau 1986), whileinterpretation of FFF as approximations to ar-bitrary functions also may cause bias from anestimation standpoint (White, Byron and Bera).

Difficulties in Comparing FFF

Aside from the relative advantages and dis-advantages of FFF, choices between alterna-tive FFF are seldom based on a comparisonof a full range of theoretical and empirical cri-teria. Systematic comparison of FFF is besetby many offsetting and sometimes conflictingtheoretical and empirical criteria. Consider, forexample, the conflict between parameter par-simony (Fuss, McFadden, and Mundlak) andorder of expansion. A third-order expansionof the utility function permits empirical testsof propositions relating to partial strong sep-arability, whereas second-order expansionsyield ambiguous test results (Hayes). Yet thenumber of estimated parameters for a third-order, translog indirect utility function couldbe intractably large for all but the case of a fewgoods. Thus, the theoretically desirable abilityto test more generalized notions of separabilitymay result in empirically undesirable phenom-ena such as collinearity, reduced degrees of

I Nonlinear-in-parameters equation systems are common inconsumer analysis. Nonlinear systems are less common in pro-duction applications. (See Just, Zilberman, and Hochman for aproduction example.)

freedom, and difficulty in interpreting individ-ual parameter estimates.

Evaluation and comparison of FFF are com-plicated further by use of FFF in consumerand producer applications. Most empirical cri-teria, such as parameter parsimony and easeof interpretation, may not differ across appli-cations; however, theoretical criteria are notalways consistent across consumer and pro-ducer applications. For example, in produc-tion applications an FFF which implies thatvariable inputs are used when no output isproduced may not be acceptable if no produc-tion lags are posited. Yet the same FFF maybe employed in consumer theory to assure con-sistent aggregation across households (Lopez1985, p. 596). Hence, an FFF which may berestrictive in the producer context can be use-ful for consumer applications.

Although no single FFF is unequivocally su-perior with respect to all theoretical and em-pirical criteria, systematic consideration of therelative advantages of each may provide morecompelling grounds for choosing functionalforms. The list of theoretical and empiricalcriteria discussed in the following sections isnot exhaustive, but it is a collection of variouscriteria which appear not to have been con-sidered together. The criteria can serve as achecklist for the applied researcher to considerin light of the particular empirical problem tobe analyzed. Note that FFF have been utilizedalmost exclusively in duality applications.Hence, the theoretical criteria are discussedwithout restricting the implications solely toconsumer or producer duality.

The fourteen FFF selected for comparisonappear in the appendix. The generalizedLeontief and translog have been the most oftenused FFF in empirical studies. The quadraticmean of order rho, the generalized Cobb-Douglas, and the generalized square root qua-dratic may be categorized as embellishmentsin the spirit of second-order approximations.The minflex Laurent forms and the Fourierform were introduced more recently in the lit-erature to deal with difficulties in approxi-mation and estimation. The generalizedMcFadden, generalized Barnett, and general-ized Fuss functions are the most recent addi-tions to the FFF menu which were proposedfor their ease in imposing global curvature con-ditions. Although some of the more recent FFFnest the old FFF as special cases, the newer

170 December 1988

Flexible Functional Forms 171

forms have been proposed to deal with defi-ciencies in the older FFF.

Theoretical Criteria

Theoretical criteria for choosing among FFFare distinguished by their ex ante role in thechoice of functional form (Lau 1986). Ineconometric parlance, theoretical criteria area source of a priori restrictions regarding choiceof functional or algebraic forms for estimation.Although microeconomic theory is nearly al-ways augmented by the researcher's familiaritywith the empirical application in forming apriori restrictions on functional form, theo-retical criteria are discussed per se in the fol-lowing sections for expository purposes.

Definitions of Flexibility

The most fundamental comparison of FFF canbe made according to the two commonly useddefinitions of flexibility. Griffin, Montgomery,and Rister offer a comprehensive treatment offlexibility criteria. The following brief discus-sion of flexibility definitions is intended tohighlight the differences between notions oflocal and global flexibility. Diewert (1971) for-malized the notion of flexibility in functionalforms by defining a second-order approxi-mation to an arbitrary function. In descriptiveterms, Diewert's definition of a flexible func-tional form requires that an FFF have param-eter values such that the FFF and its first- andsecond-order derivatives are equal to the ar-bitrary function and its first- and second-orderderivatives, respectively, for any particularpoint in the domain. Thus, Diewert's flexibil-ity definition refers to a local property. AnyFFF which satisfies this definition may be de-noted Diewert-flexible. Of course, the deriv-ative function of any indirect function (indirectutility, cost, or profit function) will be approx-imated only up to its first derivative evaluatedat any point (Chambers).

Gallant (1981) has proposed the Sobolevnorm as a more attractive measure of flexi-bility than Diewert's definition. The appeal ofthe Sobolev norm is due to its measure of av-erage error of approximation over a chosenorder of derivatives. Sobolev-flexibility is aglobal property, and any functional form dis-playing this property will yield elasticities

closely approximating the true ones (Elbadawi,Gallant, and Souza).

Sobolev-flexibility is attractive because itconfers on the empirical model nonparametricproperties: (a) small average bias approxima-tions (Gallant 1981); (b) consistent estimatorsof substitution elasticities (Elbadawi, Gallant,and Souza); and (c) asymptotically size a test-ing procedures (Gallant 1982). Gallant main-tains that Sobolev-flexibility asymptoticallyremoves the augmenting hypothesis that thetrue model be a member of the family of modelsused in the approximation analysis because theFourier form has desirable nonparametricproperties.

In mathematical and statistical terms, So-bolev-flexibility appears to be a more attrac-tive criterion than Diewert-flexibility. Yet therelative complexity of specifying and estimat-ing a Fourier form has probably contributedto its use in relatively few applications.2 Forestimating the Fourier, theoretical issues re-garding choice of sample size rules for speci-fying the number of parameters to estimate aswell as order of expansion are not clearly set-tled. Difficulties in the calculation of standarderrors for Fourier parameters also may causesome reluctance to use the Fourier form. ThusDiewert-flexibility is the more widely applieddefinition primarily because of the ease in us-ing FFF which satisfy Diewert's definition(column 1 of table 1).

Mathematical Expansions

Comparison of many FFF may be made onthe basis of the class of mathematical expan-sions from which they are derived. None ofthe definitions of flexibility limits FFF to func-tional forms derived from an underlying math-ematical expansion. However, many widelyused FFF may be treated as mathematical ex-pansions about some arbitrary point.

The Taylor, Laurent, and Fourier expan-sions each have been used to derive FFF. Thegeneralized Leontief, normalized quadratic,and the transcendental logarithmic (translog)functional forms may be interpreted as second-order Taylor-series expansions about different

2Gallant, Chalfant and Gallant, Chalfant, Wohlgenant (1983,1984) and Ewis and Fisher, appear to be the only readily accessiblestudies during the period 1980 to 1985 which utilize the Fourierform.

Thompson


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points with different transformations of thevariables (Fuss, McFadden, and Mundlak;Blackorby, Primont, and Russell).

Recognizing deficiencies in the Taylor-seriesexpansion as a generating function for FFF,Barnett proposed the Laurent expansion forwhich the remainder term varies less over theinterval of convergence (Barnett 1983). Theprincipal mathematical advantage of the Tay-lor-series expansion is that its remainder termconverges to zero within the interval of con-vergence as the number of expansion termsincreases. Although the Laurent remainderterm may not converge to zero at any pointwithin the interval of convergence, the Laurentremainder term varies less within the sameinterval than does the Taylor-series remainderterm (for Laurent and Taylor series of the samefixed order of expansion).

The Fourier approximation, which providesa global rather than a local approximation, isthe basis for a functional form proposed byGallant (1981). Although the logarithmic Fou-rier form is composed of a second-order tran-slog portion plus a trigonometric approxima-tion, the Fourier offers fundamentally differentmathematical properties. Other types of math-ematical expansions, such as the Muntz-Szatz(Barnett and Jonas), have been developed the-oretically but remain to be implemented em-pirically. The type of mathematical expressionfrom which some FFF are derived is sum-marized in column 2 of table 1.

Although each type of expansion possessessome desirable characteristics, the Taylor-se-ries expansion results in FFF which have fewerunconstrained parameters than do the Laurentor Fourier.3 The minflex Laurents have thesame degree of "parametric freedom" as thetranslog and generalized Leontief because theadditional parameters are subject to inequalityrestrictions (Barnett and Lee). The Fourierfunctional form is not readily comparable be-cause the number of estimated parameters mustbe chosen according to a sample size rule toassure consistent estimation (Elbadawi, Gal-lant, and Souza) (column 3 of table 1).

Solely on the basis of mathematical expan-sions, no particular FFF emerges as the mostattractive. Both Taylor- and Laurent-series ex-pansions are Diewert-flexible (Barnett 1983),

3 The comparison of number of parameters is for second-orderexpansions.

whereas the Fourier is Sobolev-flexible.4 Froma strictly mathematical standpoint, Laurent andTaylor series may not provide accurate nu-merical approximations if the data lie outsidethe interval of convergence. The Fourier, incontrast, provides a global approximation inthe sense that it minimizes average bias acrossall data points. Regardless of the type of ex-pansion, the order of the expansion requiredto obtain a "good" approximation is notknown; truncation error is a possible source oferror for Taylor-series, Laurent, and Fourierexpansions alike (Weaver 1983, 1984). Ac-cordingly, third-order functions have been ad-vocated and estimated (Dalal, Hayes). Hence,the theoretical ability of each class of FFF toapproximate satisfactorily-whether locally orglobally-an arbitrary function is not guar-anteed.

One caveat on comparing FFF in terms ofmathematical expansions is that not all FFFare derived directly from a mathematical ex-pansion. The generalized McFadden and gen-eralized Barnett functions are not derived sole-ly from an underlying mathematical expansion(Diewert and Wales). Neither of these two FFFcan be directly compared in terms of expan-sion remainder terms even though each hasbeen proved Diewert-flexible.

Approximations vis-a-vis True Functions

In empirical studies, FFF have been treatedboth as approximations to some unknown truefunction and as true functions despite the factthat Diewert-flexibility and Sobolev-flexibilityare notions based on approximations. How-ever, interpretation of FFF as approximationsrather than as exact functions has generatedcriticism in three areas: estimation, hypothesistesting, and separability properties. The po-tential bias of estimating FFF parameters withordinary least squares (OLS) on the basis ofapproximations made at a particular point hascaused contention (White, Gallant 1981, By-ron and Bera). Difficulties in making statisticalinferences on the basis of approximations alsooccur in consumer demand applications(Hayes, Simmons and Weiserbs). Separabilityrestrictions and tests for separability differ de-

4 Barnett (1983) notes that Gallant's Fourier model has not beenproved to satisfy Diewert's definition of flexibility. Gallant, how-ever, asserts that the Fourier is Diewert-flexible in certain cases(1981, p. 220).

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pending on whether FFF are treated as ap-proximations or true functions.

Separability

The potential restrictions imposed by the sep-arability properties deserve attention in em-pirical studies utilizing FFF. Whether FFFmodels are interpreted as exact functions orapproximations, separability restrictions areessential for consistent aggregation, sequentialoptimization, and specification of marginalsubstitution relationships.

When common second-order FFF are in-terpreted as approximations to an unknowntrue function rather than as exact functions,Blackorby, Primont, and Russell (1977) haveproved that such FFF are "separability-inflex-ible." For those second-order FFF, weak sep-arability implies strong separability and, moreimportant, their weakly separable forms (seecolumn 4 of table 1) cannot provide second-order approximations about an arbitrary point.If the FFF is interpreted as an exact function,then testing for the existence of an aggregateinput is also a test of homotheticity for theaggregator function.

For production applications, Lopez indi-cates that those second-order FFF which aredistinguished by a linear transformation of thedependent variable, such as profit, imply aspecial type of additive separability and quasi-homotheticity restrictions (see column 5 of ta-ble 1). Even when weak separability is not im-posed by these linear dependent variable FFF,the restrictions hold.5 Nonlinear FFF, whichare characterized by some nonlinear transfor-mation of the dependent variable, do not im-pose these restrictions.

When FFF are considered as approxima-tions to some arbitrary function characterizedby particular properties, the implied separa-bility restrictions are modified. For the second-order translog approximation to an arbitraryproduction function, Denny and Fuss provethat the tests for weak separability do not im-ply a test of strong separability. Hayes extendsthe precision of the separability tests by deriv-ing them from a third-order translog utilityfunction. A test for distinguishing betweenweak separability and general partial strong

5 In profit function applications, quasi-homotheticity implies thatthe marginal rate of substitution between inputs is invariant tooutput level.

separability is possible with Hayes' third-ordermodel.

The separability properties of more recentlydeveloped FFF, such as the minflex Laurents,generalized Barnett, generalized McFadden(Diewert and Wales), generalized Fuss (Die-wert and Ostensoe), and biquadratic (Diewert),have not been examined. The general qua-dratic function separability results fromBlackorby, Primont, and Russell cannot be di-rectly applied to all these newer FFF becausethey are not solely quadratic functions. Rather,some of these FFF may be considered as non-negative sums of concave functions.

Global Regularity Conditions

Satisfaction of regularity conditions providesanother theoretical criterion for judgingwhether alternative FFF conform to the prop-erties of microeconomic theory. All but one ofthe FFF considered here do not satisfy globalconvexity (concavity) conditions. Only thenormalized quadratic is capable of satisfyingglobal convexity (concavity) restrictions with-out additional constraints in estimation. Thecapability of different FFF to satisfy regularityconditions can be measured by regular regions.Regular regions of different FFF are calculatedby fixing relevant elasticity values at some cho-sen level and then determining the parametervalues for which the regularity conditions hold.In general, the larger is the FFF's regular regionfor a given elasticity of substitution, the moretheoretically compatible is the FFF. If a prioriinformation exists about the magnitude of theelasticities of substitution for a particular ap-plication, size of the regular regions providesa means for discriminating between FFF. Moreusefully, over a wide range of substitution elas-ticity values, a given FFF may have a largerregular region than other FFF.

In what follows, regularity conditions arereferred to solely in terms of the indirect utilityfunction because all of the previous studieshave focused on this consumer case. Regular-ity conditions are (a) monotonicity and (b)quasi-convexity; homogeneity is customarilyimposed. Barnett, Lee, and Wolfe (1985, 1987)have extended Caves and Christensen's orig-inal work to consider the three-good,nonhomothetic case for the generalized Leon-tief, translog, and minflex Laurent generalizedLeontief and minflex Laurent translog indirectutility functions. The minflex Laurent models

174 December 1988


generally possess larger regular regions thaneither the generalized Leontief or translog.6 Thegeneralized Leontief and translog both havelarge regular regions in the neighborhood oftheir globally regular special cases-Leontiefand Cobb-Douglas, respectively-but theirregular regions diminish rapidly for elasticityvalues diverging from these special cases. Theminflex translog tends to possess a larger reg-ular region than the minflex generalized Leon-tief except in cases where there is limited sub-stitutability between pairs of goods (Barnett,Lee, and Wolfe).

No studies have examined regular regionsfor the consumer case of more than three goodsnor have production applications been ex-plored. While the two-dimensional consumercases have an intuitively appealing interpre-tation with indifference curves (see Caves andChristensen), generalization of the regular re-gion notion to three-space requires volumemeasures which are envisioned less easily. Ex-tensions to higher dimensions would requirehigher-order volume measures.

Empirical procedures can be employed toimpose regularity conditions in the estimationof FFF parameters. Lau (1978a) demonstrateda procedure for imposing global concavity(convexity) conditions with econometric esti-mation. Lau also suggested that monotonicitymay be imposed by squaring the appropriateparameters. Jorgenson and Fraumeni usedLau's method of imposing concavity by re-stricting the elements of the Cholesky factor-ization of the matrix of share elasticities de-rived from a price function. The impositionof concavity, however, resulted in setting a largenumber of the elasticities equal to zero. More-over, imposing negative semidefiniteness onthe matrix of own and cross price elasticitiesof the translog cost function can bias the re-sulting estimated elasticities (Diewert andWales).

More recently, other procedures for impos-ing global concavity on FFF have been ad-vanced by Gallant and Golub, and Diewertand Wales. Development of the generalizedMcFadden, generalized Barnett, and general-ized Fuss forms by Diewert and Wales wasmotivated by the need to impose globally con-vexity (or concavity). The techniques of as-suring satisfaction of global curvature condi-

6 The minflex Laurent also displays increasing regular regionvolume for trended time-series data.

tions may be classified as system estimation,possibly nonlinear, subject to restrictions.7Gallant and Golub suggest a two-stage opti-mization technique for imposing curvatureconditions at every data point. Diewert andWales propose a restricted estimation tech-nique equivalent to Lau's restrictions on theCholesky factorization of the relevant hessianmatrix. 8

Imposition of curvature restrictions for someFFF may provide more credible elasticity es-timates over the entire range of the sampledata. Clearly, theoretical primal-dual map-pings may be invoked in empirical studies ifthe regularity-curvature, monotonicity, andhomogeneity-conditions are imposed. Froman estimation standpoint, more efficient pa-rameter estimates are obtained from estima-tion subject to restrictions; imposition of er-roneous constraints would result in biased orinconsistent estimates, however.

Review of Theoretical Criteria

The theoretical criteria enumerated could leadto contradictory conclusions in the choice be-tween currently available functional forms. Thedifficulty in using the theoretical criteria iscomplicated insofar as FFF may be interpretedas approximations or true functions. A sum-mary of the theoretical properties in table 1indicates the potential for contradictory the-oretical prescriptions.

From a mathematical standpoint, using well-behaved expansions, such as the Laurent andFourier, provide more desirable approxima-tions. If curvature conditions are also a con-sideration, a potentially fruitful developmentmay be FFF which are the nonnegative sumsof concave functions because these functionsreadily allow the imposition of global curva-ture conditions. However, the separabilityproperties of the newer FFF-minflex Laurenttranslog and generalized Leontief, generalizedMcFadden, and generalized Barnett-have yetto be compared systematically. Thus, as might

7 Hazilla and Kopp imposed regularity conditions by means ofnonlinear restrictions on a long-run cost function. The restrictionswere imposed only at the point of approximation, however, be-cause the cost function was interpreted as an approximation to atrue function.

8 The Cholesky techniques proposed by Lau and by Diewert andWales were used earlier by Wiley, Schmidt, and Bramble. TheCholesky decomposition converts a constrained linear estimationproblem into an unconstrained nonlinear estimation problem.

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be expected, no single FFF appears unambig-uously to dominate all others on theoreticalgrounds. However, the relative theoreticalstrengths and weaknesses of each are apparent.

Empirical Criteria and Techniques

Empirical criteria for judging FFF are char-acterized by their ex post role in the choice offunctional forms. Empirical criteria by defi-nition are less general than theoretical criteriabecause of their application in a specific datasetting; different empirical criteria can be ap-plied in both consumer and production prob-lems, but they are contingent upon the dataanalyzed. Thus, nearly all empirical criteriayield data-specific conclusions.

Empirical criteria for distinguishing be-tween alternative FFF can be catalogued intofour groups: (a) Monte Carlo studies; (b) para-metric models; (c) bayesian analysis; and (d)nonnested hypothesis testing. In Monte Carlostudies, the ability of an FFF to approximatea known underlying technology or preferencemapping is measured. In the latter three cases,however, data-generated measures are used tocompare competing functional forms where theunderlying function is unknown. Knowledgeof the data generating process in Monte Carlostudies permits more hypothetical consider-ation of the approximating abilities of the FFFthan the data-generated criteria. Monte Carloresults may provide some ex ante indicationof an FFF's comparative strengths and weak-nesses, given various data sets.

Monte Carlo Studies

In the first application of Monte Carlo tech-niques for assessing the approximation capa-bilities of FFF, Wales compared the ability oftranslog and generalized Leontief reciprocalindirect utility functions to approximate ahomothetic two-good CES utility function. Thetheoretical results derived from examining reg-ular regions were corroborated: the translogperformed well when substitution elasticitieswere near unity, whereas the generalizedLeontief better approximated substitutionelasticities near zero. Both functional formsviolated regularity conditions-quasi-concav-ity and monotonicity-even though they fit thedata well in terms of R2 and closely estimatedthe true substitution elasticities.

Guilkey and Lovell, and Guilkey and Sicklesextended the Monte Carlo technique tomeasure approximation of true substitutionelasticities, economies of scale, and single vis-a-vis systems estimation techniques. In thesecond study, the single output, three-inputcost function was used to compare the translog,generalized Leontief, and the generalized Cobb-Douglas. Generally, the translog dominated theother two forms, although Guilkey and Sicklesstressed that the better performance of thetranslog did not imply that it was acceptablein all instances. Furthermore, the deviation ofthe estimated substitution elasticities from thetrue elasticities as measured by bias and meanabsolute deviation was not substantially dif-ferent for the three functional forms in somecases. In more complex cases with divergingtrue partial substitution elasticities and com-plementarity, the systems estimator (Zellner'siterative seemingly unrelated regressions) waspreferred.

A recent Monte Carlo study by Chalfant andGallant assessed the ability of the logarithmicFourier functional form to approximate sub-stitution elasticities generated by a three-input,homothetic generalized Box-Cox cost func-tion. The logarithmic Fourier, which nests thetranslog as a special case, approximated elas-ticities of substitution with little bias. Exper-iments with different sample sizes suggestedthat the measured bias was due to errors-in-variables, not to specification bias. Hence, forpurposes of testing economic theory, the Fou-rier appears to be the least ambiguous formfor statistical inference.

Dixon, Garcia, and Anderson conductedMonte Carlo simulations to evaluate the use-fulness of pretests in testing the behavioral as-sumptions and regularity conditions for thetranslog and generalized Leontief profit func-tions. While concluding that such pretests aregenerally not useful validation tools, they not-ed difficulty in choosing between the translogand generalized Leontief functional forms. Thegeneralized Leontief consistently underesti-mated substitution elasticities, while the trans-log elasticity estimates displayed extreme vari-ations about the true means.

The Monte Carlo results discussed are spe-cific to the data generated and the microeco-nomic context analyzed. Both consumer (in-direct utility function) and production (costand profit function) applications have beenconsidered. Thus, the results of the studies are

176 December 1988


not directly comparable on economic grounds.Experimental design differs considerably acrossthese Monte Carlo studies: data generation,sample size, number of replications, and sto-chastic disturbance specifications varythroughout.

Fruitful ground for future studies lies in thecomparison of a wider range of FFF, such asthe generalized Box-Cox, minflex Laurent,Fourier, and nonnegatively summed concavefunctions while varying assumptions regardingdata generation and sample size.

Parametric Modeling

Parametric methods have been proposed toassess the plausibility of different functionalforms in fitting actual data. The generalizedBox-Cox function has been used for paramet-ric testing; with suitable parameter restric-tions, the generalized Box-Cox nests the trans-log, generalized Leontief, and the generalizedsquare-root quadratic functions as special cases(see Griffin, Montgomery, and Rister). Hence,rather than estimating any one of the specialcases with the functional form as a maintainedhypothesis, the functional form can be testedthrough hypothesis tests of the appropriate pa-rameter restrictions.

Comparison of the parametric test resultsacross studies using different data sets indi-cates that many commonly used functionalforms are rejected. Using updated productiondata from Berndt and Christensen, Appel-baum rejected the translog and generalizedLeontief in primal and dual share equationmodels but failed to reject the generalizedsquare root quadratic in the dual setting. Usingsimilar production data from Berndt andWood, Bemdt and Khaled could not reject thegeneralized Leontief but rejected the general-ized square root quadratic and probably re-jected the translog for approximating the costfunction. 9 Using aggregate agricultural inputdata, Chalfant rejected all three functionalforms in a cost function context.

The primary limitation of the generalizedBox-Cox as previously formulated is that para-

9 The Bemdt and Khaled formulation of the generalized Box-Cox does not allow for direct testing of the translog because por-tions of the likelihood function of the generalized Box-Cox becomedegenerate when parameter values yield the translog special case(Berndt and Khaled, p. 1227). Tests for parameter values ap-proaching those of the translog lead to rejection of those functionalforms.

metric tests can only discriminate among asubset of FFF. Laurent, Fourier, generalizedMcFadden, and generalized Barnett functionscannot be tested as special cases of the gen-eralized Box-Cox. 10 Further, the usual caveatsregarding power of hypothesis tests are appli-cable with the use of the generalized Box-Coxmodel.

Bayesian Analysis

An alternative to parametric testing is bayes-ian analysis by which a posteriori comparisonsof FFF can be made. The attraction of thismethod is that it allows comparison of fun-damentally different models on the basis ofactual data; nonnested models can be com-pared on the basis of diffuse priors.

In a consumer application, Berndt, Dar-rough, and Diewert compared the translog,generalized Leontief, and generalized Cobb-Douglas in estimating market demand sharesfor Canadian consumption data. With andwithout symmetry restrictions imposed, thetranslog was preferred a posteriori while thegeneralized Leontief and generalized Cobb-Douglas had nearly identical log likelihoodvalues in both cases. For U.S. time-series foodconsumption data, Wohlgenant estimated two-good demand functions with the generalizedLeontief, translog, and Fourier forms. TheFourier form dominated the translog and gen-eralized Leontief forms with posterior oddsratio of 9.83:1 and 41.95:1, respectively. TheFourier also performed favorably when own-price and income elasticities of demand werecompared at each sample point. In a produc-tion study using the Berndt and Wood data onaggregate U.S. manufacturing, Rossi com-pared the translog and logarithmic Fourier fora three-input cost share system exclusive ofthe cost function. Two methods to calculatethe posterior odds ratio were used, and thelogarithmic Fourier was preferred by a ratioof approximately 3:2.

Whether applied in a consumer or produc-tion setting, bayesian analysis affords a con-venient means for discriminating among com-peting FFF on the basis of actual data. Yet,there exists potential for conflict between mi-

10 The logarithmic Fourier nests the translog, the minflex trans-log nests the translog, and the minflex generalized Leontief neststhe generalized Leontief. Hypothesis tests of subsets of the param-eters in each of these models could be used to test the special cases.However, none nests a wide range of alternative models.

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croeconomic theoretical results and the esti-mated parameters of the FFF chosen a pos-teriori. The functional form with the highestposterior odds ratio could possess, for exam-ple, own-price elasticities which are positiveor violate other regularity conditions. The ap-plicability of microlevel theoretical results inaggregate or market studies would have to beassessed in light of the bayesian results.

Nonnested Hypothesis Testing

Nonnested hypothesis testing allows the re-searcher to make pair-wise comparisons be-tween competing models. A conceptual differ-ence between nonnested testing and bayesiananalysis is that nonnested testing allows for allproposed models to be rejected on the basis ofthe data. For FFF interpreted as approxima-tions, nonnested tests provide the possibilityof rejecting all approximations to the unknownunderlying function. With bayesian analysis,in contrast, the models would be ranked onthe basis of posterior odds ratios so that somealternative model would probably be deemedthe most plausible.

Although myriad nonnested tests have beenproposed recently, the Cox test for nonnested,nonlinear equations systems (Pesaran andDeaton) is the most general for testing alter-native FFF in budget or cost (profit) shareequation systems. The only apparent limita-tion for the Cox test is that all competing func-tional forms must be specified with the sametransformations of the dependent variables.The Cox statistic has been developed for test-ing linear and log-linear single equation regres-sion models (Aneuryn-Evans and Deaton).However, empirical applications of the Coxtest for linear and logarithmic dependent vari-ables have not been made to either single-equation nonlinear regressions or multipleequation regressions. Share equation and de-mand/supply equation systems which have dif-ferent transformations of the dependent vari-ables (e.g., shares vs. single variables) couldnot be used directly to perform a nonnestedtest with the Cox statistic.

Few empirical studies have used the Coxstatistic to compare functional forms (Pesaranand Deaton; Deaton). Alternative functionalforms, not flexible functional forms, have beentested in the single-equation agricultural pro-duction function applications (Ackello-Ogutu,Paris, and Williams). The extent to which non-

nested testing results coincide with bayesianresults is a potential methodological and em-pirical question.

Review of Empirical Criteria

Overall assessment of the compatibility of dif-ferent FFF with empirical data does not leadto the acceptance of any clearly superior func-tional form. Monte Carlo studies offer the mostgeneral empirical means for choosing amongFFF by comparing their abilities to approxi-mate underlying partial substitution elastici-ties. However, no studies of a wide range ofprospective FFF have been published. The rel-ative attractiveness of FFF in both productionand consumer Monte Carlo applications mayalso yield useful results for applied researchers.

Dixon, Garcia, and Anderson shed doubt onthe usefulness of pretests in assessing main-tained behavioral hypotheses. Nonparametricalternatives to the parametric tests analyzedby Dixon, Garcia, and Anderson might be auseful pretest for maintained behavioral hy-potheses. Varian (1982, 1984), for example,has developed nonparametric procedureswhereby data can be tested for consistency withutility-maximizing, cost-minimizing, or prof-it-maximizing behavior. If the nonparametrictests did not reject the behavioral hypotheses,the parametric analysis with FFF could be pur-sued with more confidence (see Barnhart andWhitney).

Unless a more general composite model thanthe generalized Box-Cox is found, the mostpromising techniques for testing alternativeFFF are bayesian inference and nonnested hy-pothesis testing. In practice, calculation of pos-terior odds may require fewer, less complicat-ed operations than are necessary for derivinga Cox statistic. When appropriate, nonnestedtests based on instrumental variable esti-mators could also be used (see Godfrey andPesaran). The fundamental methodologicaladvantage of nonnested testing is that no par-ticular functional form is necessarily acceptedon the basis of the data; all prospective FFFmay be rejected in pair-wise tests. Whetherbayesian or nonnested techniques are used, theability to test models with different transfor-mations of the dependent variable-logarith-mic versus linear, for example-is necessaryfor discriminating between a wide range of cur-rently available FFF.

178 December 1988


Conclusions

An extensive array of theoretical criteria andempirical techniques for choosing among FFFhas been reviewed. Ample opportunity existsin many empirical studies to compare com-peting FFF more systematically on the basisof these theoretical and empirical measures.Although any comparison of all available func-tional forms by all these measures may notyield an unambiguous choice, the relative ad-vantages of competing FFF in the particularempirical problem should provide more com-pelling grounds for choice of flexible functionalforms.

As Griffin, Montgomery, and Rister haveremarked, the choice of functional form shouldbe included explicitly in empirical studies be-cause nearly all econometric model specifica-tions may be considered as approximations ofsome unknown underlying data generationprocess. One avenue for formalizing the selec-tion process of FFF in duality modeling mightbe the following empirical testing procedure:

(a) Test the behavioral assumptions of theduality model (utility maximization, cost min-imization, or profit maximization) using thenonparametric testing procedures from Vari-an. Choice among alternative flexible func-tional forms is valid conditional upon the be-havioral assumptions being appropriate. If thedata are not consistent with the behavioral as-sumptions, one would have to judge whetherthe inconsistencies are caused by measurementerror or to cross-sectional and time-series het-erogeneity (Hanoch and Rothschild). In thecase of measurement error inconsistencies, theviolating data points might be adjusted or thesample might be censored to purge the incon-sistent data points (see Barnhart and Whitney).Cross-sectional and time-series heterogeneity,such as differing firm endowments, regionaldifferences, and technical progress, might callfor alternative models which account for theinconsistencies.

(b) If the data are consistent with the be-havioral assumptions, test other appropriatetheoretical properties of the data such as re-turns to scale, homotheticity, and separabilityusing similar nonparametric tests. The appro-priate theoretical properties to be tested will,of course, vary according to the particularproblem.

(c) Choose flexible functional forms which

can embody the behavioral assumptions andproperties not rejected by the nonparametrictests. Parametric tests of the relevant theoret-ical properties for each flexible functional formmay then be performed. If the parametric testsdo not reject the theoretical properties, theproperties can then be imposed in subsequentestimation.

(d) Choose among flexible functional formsusing one or both of the following techniques:

(i) Bayesian analysis: compare the alter-native FFF on the basis of posterior odds.The principal advantage of bayesian analy-sis is that FFF with different transformationsof the dependent variables may be com-pared.

(ii) Nonnested hypothesis tests: performpair-wise comparisons of the alternative FFFusing classical statistical techniques. Themain disadvantage is that systems of equa-tions with different transformations of thedependent variables may not be comparedwith current formulations of the nonnestedtests. With nonnested tests, no particularmodel might emerge as the best. Note thatbayesian and nonnested test results mightconflict.(e) Check the robustness of the economic

measures, such as price and partial substitu-tion elasticities calculated from the flexiblefunctional form(s), to determine the sensitivityof the measures to the choice of flexible func-tional form.

Presumably, more credible parameter andelasticity estimates may be obtained by theseexplicit comparisons of flexible functionalforms when the maintained hypotheses ofduality analysis are appropriate. The empiricalselection procedure also appears to be be con-sistent with this journal's new policy of makingeconometric specification searches more ex-plicit. Whether the benefits of such a formalselection procedure would outweigh the costsin the context of a data set of questionablequality is a judgment that the researcher mustmake.

[Received September 1987; final revisionreceived July 1988.]

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Thompson


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Appendix

Flexible Functional Forms

(1) Quadratic Form

0(y) = ao + a'x + 1/2x'Bx(a) Normalized Quadratic: (Lau 1978a, p. 194)

(y) = y a'= [a, . .. a,,] x' = [xl ..... x,B= {bi} b, = bji -ij

(b) Generalized Leontief: (Diewert 1971)q(y) = y a'= x'= [X/1 2 . .. Xnl]

B= {bj} b, = bji -vij(c) Translog: (Christensen, Jorgenson, and Lau)

0(y) = In y a' = [a,, .. ., an] x' = [ln x, . . ., n xnB = {bj} b, = bji, ij

(2) Quadratic Mean of Order Rho: (Denny, Kadiyala)\ I/P

Y=

z z bjj~p/2 jp/2i i i '

·=( a ix?2 + C C b, xpx ~

(3) Generalized Cobb-Douglas: (Diewert 1973b)

Y = I n (1/2x + l/2X)bj bij = bji -'iji j

(4) Generalized Square Root Quadratic: (Diewert 1971)

y = (x'Bx)' 2 x'= [xi .. .. , Xn] B = {bi} b, = bji, ij(5) Generalized Box-Cox: (nonhomothetic) (Bemdt and Khaled, Applebaum)

-y(b) = 1 + XG(X)} Iwhere

G(X) = a + S a,P,(X) + 1/2 S S bP,(X)P,(X)i i j

and

P,(X) = P(/2)-IX/2(6) Symmetric Generalized McFadden: (Diewert and Wales)

O(y) = g(x) + bx, + cx,

g(x) = /2x'Bx/O'x 0 > 0 x' = [x, . . ., Xn B = {bi} bi = bi -ij(7) Generalized Barnett: (Diewert and Wales)

() = g(x) + b,x, + ( cj)

N N N N

g(x) = x bx'1/,x'. -- X dox2xr/'x-1'2i=l j=l i=2 j=2

i j i j

N N bej = bji - 0l- , : 2 rejx 1 1 dj J= dj, > O

i=2

e > 00(8) Generalized Fuss: (Diewert and Ostensoe)

y= 1/2a'zX'AXX

I+ /2'xzC'Bzz-I + x'Cz

+ 3'xb'zz I + V/2f'xboz' + x'c

a' > 0 ' > 0 A = {a} a- = aj ijalj = al, = 0

182 December 1988


b'= [0, b2.., bM] B = {b,} b= bji ijbl= b,i= 0

c'= [c, .... CN] C = {ci}(9) Biquadratic: (Diewert 1986)

N N-I N--

y = anX + 1/2 biXnXiX-nn=l n=l i=l

bni = bi 1 < n < i < N(10) Minflex Laurents: (Bamett 1983, 1985)

0(y) = ao + 2 2 a,x, + 2 ajjx 2 + 2 (a2xixj - bxjFx,- l)

(a) Minflex Laurent Generalized Leontief:

0(Y) = Y xi = x;,:(b) Minflex Laurent Translog:

0(y) = In y x, = In x,(11) Fourier: (Gallant 1981)

A J

y Vo + b'x + l2x'Bx + ~ Voa + 2 2 [vjacos(jXk'ax)- vjasin(jXk'x)]a=i j=i

andA

B=

- VXa a.

a=1

Thompson

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