Selecting Functional Form in Production Function...

Selecting Functional Form in ProductionFunction Analysis

Ronald C. Griffin, John M. Montgomery, and M. Edward Rister

Functional form selection is a sometimes neglected aspect of applied research inproduction analysis. To provide an improved and uniform basis for form selection, anumber of traditional and popular functional forms are catalogued with respect tointrinsic properties. Guidelines for the conduct of form selection are also discussed.

Key words: flexibility, functional form.

The art and science of applied economics crit-ically depends on model building, and thepracticing economist makes many decisions inthe course of constructing an appropriate mod-el. Within this process, the practitioner mustoften adopt a functional form as the patternfor one or more continuous (possibly piece-wise) physical or economic relationships.Common examples of such relationships areproduction, profit, and cost functions as wellas systems of input and/or output supply anddemand. The continuity of these relationshipsis rarely, if ever, proven but seems sufficientlycompelling to be embraced without questionin nearly all cases. The researcher, however, isnever in a position to know the true functionalform, and, as noted by Hildreth, "the principaldisadvantage of continuous models lies in thebiases which may accrue if an inappropriate[functional] form is used" (p. 64).

The model builder's task is complicated bythe growing number of available functionalforms. A compilation of alternative functionalforms and a comparison of these alternativeson the basis of selected criteria are presentedin this paper. Available selection criteria per-tain to mathematical, statistical, and economicproperties and are useful for formalizing theselection of functional form(s) during model

The authors are an associate professor, a research assistant, andan associate professor, respectively, in the Department of Agri-cultural Economics, Texas A&M University.

This research as conducted by the Texas Agricultural Experi-ment Station (Projects 6447 and 6507) with support from the TexasRice Research Foundation (Econo-Rice Project).

Appreciation is expressed to anonymous reviewers of this Jour-nal and the AJAE for helpful comments during various stages ofthis article's development.

building processes. The use of functional formsin production function applications is empha-sized in order to limit the discussion. Becausethe concept of flexibility is important to formselection, we begin with a brief summary ofthis topic.

Flexibility and Maintained Hypotheses

Recent advances in developing new functionalforms have been dominated by efforts to con-ceive "flexible" forms, and different technicaldefinitions of flexibility have arisen as a resultof these pursuits. Because flexibility is a mul-tidimensioned concept, a given technical def-inition of flexibility may not be adequate inall situations. Local flexibility (sometimesDiewert flexibility or, simply, flexibility) im-plies that an approximating functional formconveys zero error (perfect approximation) foran arbitrary function and its first two deriva-tives at a particular point (Fuss, McFadden,and Mundlak). The locally flexible form placesno restrictions on the value of the function orits first or second derivatives at this point.Therefore, no restrictions are imposed onproperties that can be expressed in terms ofderivatives of second-order or less.

Second-order Taylor series expansions havedominated the field of locally flexible formsbut are not unique in the ability to offer localflexibility (Barnett). Ignoring the complica-tions of statistical estimation by momentarilypresuming that we wish functionally to ap-proximate a known relationship, locally flex-

Western Journal of Agricultural Economics, 12(2): 216-227Copyright 1987 Western Agricultural Economics Association

Selecting Functional Form 217

ible forms can impose large errors in the ap-proximating function and its derivatives awayfrom the point of perfect approximation (Des-potakis). Global flexibility (sometimes Sobo-lev flexibility) is preferred to local flexibilityin that second-order restrictions are every-where absent (Gallant 1981, 1982).

Recent attention to the influence of esti-mation on model development has severelyreduced the attractiveness of locally flexibleforms. Because an estimated Taylor series ex-pansion is a fit to data from the true form,rather than an expansion of the true form, theremay be no actual point where the true functionand its first two gradients are perfectly ap-proximated. Inquiry in this area has producedsome discomforting results. The examples pro-vided by White demonstrate that ordinary leastsquares estimators of Taylor series expansionsare not reliable indicators of the parametervector for the true expansion of a known func-tion. As a consequence of these and other find-ings, predictive properties of locally flexibleforms have been found to be satisfactory (forlarge samples), but inferences involving singleparameter estimates and functional combi-nations of these estimates are not reliable. Be-cause estimation is driven by a global (at leastthroughout the data domain) criterion such asleast sum of squared error, it has been a bitunnatural to tout local flexibility as an advan-tage in economic analyses. The implicationshere are severe for typical applications in pro-duction function analysis.

To gauge global flexibility in a manner whichalso acknowledges derivative values, a mea-sure of error is needed which incorporateserrors in derivatives as well as errors in theapproximating function. The Sobolev norm (adistance measure) satisfies this requirement andhas been employed to assess global flexibility(Gallant 1981). In applied work, however, theSobolev norm is intractable for obtaining pa-rameter estimates, so estimation of globallyflexible forms still uses traditional distancemeasures like least squares. Evidence providedby Elbadawi, Gallant, and Souza suggests thatthis produces satisfactory results. This findingsupports the judgment expressed by Rice thatform exceeds norm in importance for approx-imation (pp. 1, 20). It is still true, though, thatthe effects of selecting a particular form arejointly determined by the inseparable influ-ence of form restrictiveness and estimationtechnique.

The concept of maintained hypotheses isfundamental to any concept of flexibility. Aspointed out by Fuss, McFadden, and Mund-lak, maintained hypotheses "are not them-selves tested as part of the analysis, but areassumed true" (p. 222). The choice of func-tional form immediately renders some hy-potheses untestable (maintained) while othersremain testable. "Sometimes the question ofwhether a specific hypothesis should be testedor maintained is critical" (Ladd, p. 9). As anexample, King's 1979 address to the AAEAmembership is most certainly rooted in hisdisappointment with the maintained hypoth-eses of linear models.

Attention to the issue of maintained hy-potheses aids the researcher in developing amore careful and richer analysis. Popular con-cerns pertaining to potential maintained hy-potheses within economic production modelsinclude homogeneity, homotheticity, elasticityof substitution, and concavity. Less restrictivefunctional forms would always be desirable,were it not for the greater information neededto adequately specify such relationships.Greater flexibility can usually be achieved byadding arbitrary and nonredundant terms toany given function, but to do so reduces de-grees of freedom and may increase collinearityand the expense of parameter estimation. Be-cause reductions in maintained hypothesescome at a cost, added flexibility is not alwaysdesirable, and there are likely to be cost-effec-tive opportunities to achieve particular di-mensions of flexibility.

The Choice Set

Based on a review of traditional and popularliterature, twenty functional forms are iden-tified for consideration. The names of each ofthese functions and their algebraic form arelisted in the first two columns of table 1. Note-worthy reference material for these functionsand some of their properties include Headyand Dillon (quadratic, square root, Mitscher-lich, Spillman, resistance, modified resis-tance); Lau (square rootS); Halter, Carter, andHocking (transcendental); Uzawa (CES); Sil-berberg (CES); Diewert (generalized Leontief);

Lau calls this the "generalized version of the 'Generalized Lin-ear Function' " (p. 410). Fuss, McFadden, and Mundlak call it thegeneralized Leontief function.

Griffin, Montgomery, and Rister

218 December 1987

Table 1. Properties of Selected Functional Forms

Western Journal of Agricultural Economics

Function

Leontief

Linear

Quadratica

Cubica b

y=

y=

y=

y=

Generalized Leontiefa

Square roota

Logarithmic

Mitscherlich

Spillman

Cobb-Douglas

Generalized Cobb-Douglas

Transcendental

y=

y=

y=

y=

Functional Form(i, j, k= 1, .. , n)

=min[3,x,, 2x, ... , ./xn] >> 0

=a + ixii

a + ix, + xix,i i j

a + fix, + 2 xixx,

k i+ S 2 S 7yx;^xi j k

=a + ix x + 2 2 6X

= a + 3,iln x,

= a (1 - exp(fi,x))i

(A)

Does x, = 0for all i

Imply y = O?

yes

no

no

no

un

y = aI (1 - ,Ti)i

y= a f xiIi

In y

y:

= + 2 2 6jln((xi + x)/2)

i j(exp( ))= a II x~i(exp(bixi))

(B)Does x, = 0

for anyone i

Imp

(C)

ly y = 0? dy/Ox,

yes 0 or UBC

no UBC

no U

no U

(D)

AsymptoticConvergence

no

no

no

no

yes no UBN no

no no U no

defined undefined UBN no

yes yes UBN if/ << 0

yes yes UBN if 0 << << 1

yes yes UBN

undefined no U

yes yes U

no

no

no

Resistance y-1= a + o /i(6, + x,)-, no no UBN yes

Modified resistancea

y- = a + ~ fix,-' + Z 1 'XF'xij' undefined undefined UBN yes

i joi

CES Y = a + ix-" undefined undefined UBN no

Transloga

In y = a + iBln x, + 6,bi(ln x)(ln x) undefined undefined U no

Generalized quadratic y= fijx>6 -1) yes no UBN no

Generalized power y = a I x{f(x)exp(g(x)) yes yes U no

Generalized Box-Coxa

y(O) = a + /,x;(X) + b1jx(X,)xj(X), no no U no

where y(8) = (y2

- 1)/20and x(X) = (x

x- 1)/X

Augmented Fourier y = fix, + 6 ijx

i"xj no no U no

+ 2 'Yhexp(i hx),Ihl*'H i

where all x, E [0, 21r],_'h = Y + i3', i

2= -- 1

Note: U is unrestricted sign (+, 0, or -); UBC, unrestricted in sign but constant; UBN, unrestricted but nonswitching in sign (e.g., if dy/dxk > 0 for somex, > 0, then dy/Ox, > 0 for all x, > 0); and NG, not in general.a Assuming bj = ,j for all i, j.b Assuming -yk = 7y, = 'Ykii = kji = yjki = 'yj for all i, j, k.c

Some of the stated conditions are sufficient but not necessary for local concavity (see text).dSame as for quadratic form plus y, = 0 for all i, j, k.If any two of the following conditions are satisfied: (1) all 3, = 0, (2) all 6, = 0, (3) all y,, = 0.

'Yes, but x, may equal zero for only one i.V/2(n

2+ 3n + 3) assuming 0 and X are free.h

See appendix.Linear, quadratic, generalized Leontief, square root, logarithmic, Cobb-Douglas, modified resistance, CES, translog.

Griffin, Montgomery,and Rister Selecting Functional Form 219

Table 1. Extended

(E)

LinearlyHomogenous Ho]

yes

ifa = 0

ifa = 0, ifall bi =0 all

d

(F) (G)ConstantElasticityof Sub-

mothetic stitution

yes a = 0

yes a = oo

, = 0 or NG6, = 0

NG

(H) (I) (J) (K) (L)Linearly Subsumes

No. of Separable WhatDistinct Linearly if any Other

Concavec

Parameters Separable x = 0 Functions

yes n no no

affine n + 1 yes yes

NG 1/(n + l)(n + 2) yes yes Linear

NG (n + 3)(n +2) yes yes Linear,*(n + 1)/6 quadratic

yes NG if all 1/2n(n + 1) yes yes6,> 0

if = 0 NG if / > 0, I/(n + l)(n + 2) yes yes Generalizedall , >- 0 Leontief

yes NG if/ >> 0 n + 1 yes undefined

no NG if a > 0, , < 0, n + 1o Spillmanexp(,ix,) < 1

no NG ifa > 0, n + 1 no no Mitscherlich0 << f < 1,

xi? < 1

yes a = 1 if a > 0, n + 1 yes no0 << << 1,

Y i, < 1

yes no NG /2(n2+ n + 2) yes

fCobb-Douglas

if 6 = 0 if 6 = 0 if a > 0, 2n + 1 yes no Cobb-Douglas

0 << < 1,i, < 1

if = 0 NG if /> 0, 6 > 0 2n + 1 yes yes

if all 6, = 0 NG NG 1/2(n2+ n + 2) yes undefined

yes a = 1/ ifa > 0, i > 0, n + 3 NG undefined Linear,(1 + p) 0 < v < 1 Cobb-Douglas

if all NG if >> 0, all ij > 0 1/2(n + l)(n + 2) yes undefined Cobb-Douglas

NG if all f3 >

0,0 - <- 1,

6 1, v < 1

NG NG NG

if Y f2 i if all ,i = 0 NG=

2aX + 1, or all b = 0,

all Xf = 2 I b, i j and

all X/i = 2,iino no NG

NG

NG

n2

+ 3 NG NG GeneralizedLeontief, CES

indeterminate NG NG Transcendental

NG NG

NG yes yes Linear,quadratic

yes

if a = 0,/=0

no

no

no

if :/i= 1

if 2; , = 1i j

if fli = 1,5=0

if a = 0, 6 =

if a = 0,all 5, = 0

ifa = , v= 1

if fi = 1,all bij = 0

i

if v= 1

y es = 0i

yes

-�


Christensen, Jorgenson, and Lau (translog -transcendental logarithmic); Fuss, McFadden,and Mundlak (translog, generalized Cobb-Douglas, square root); Denny (generalizedquadratic); de Janvry (generalized power); Ap-plebaum (generalized Box-Cox); Berndt andKhaled (generalized Box-Cox); and Gallant(augmented Fourier).

The augmented Fourier form is globally flex-ible, asymptotically, in that no second-orderrestrictions are imposed anywhere in the do-main. It is also nonparametric; the number ofparameters to be estimated varies with samplesize. Because the augmented Fourier form isnot commonly used in production functionanalyses, a few essential details are presentedin an appendix.

The Criteria

As stated above, determination of the truefunctional form of a given relationship is im-possible, so the problem is to choose the bestform for a given task. This problem leads toconsideration of choice criteria, that is, howone functional form may be judged better ormore appropriate than another. These criteriamay be grouped in four categories accordingto whether they relate to maintained hypoth-eses, estimation, data, or application.

Concerns regarding maintained hypothesesform one set of objectives whereby appropri-ateness can be assessed. If the maintained hy-potheses implied by a certain function are ac-ceptable, or even useful, then the function maybe deemed appropriate. In the absence of astrong theoretical or empirical basis for adopt-ing a given maintained hypothesis, however,a functional form which is unrestrictive withrespect to this hypothesis may be consideredappropriate.

Second, functional form has implications forstatistical processes of parameter estimation.Data availability, data properties, and theavailability of computing resources can affectthe choice of functional form for statistical es-timation. Moreover, some forms do not per-mit parameter estimation by linear leastsquares procedures, and alternative proce-dures typically offer less information concern-ing estimator properties. Criteria relating tomaintained hypotheses and statistical esti-mation are discussed below, and importantfindings have been condensed in table 1.

A third category of selection criteria in-volves data-specific considerations (goodness-of-fit and general conformity to data). Suchconcerns are necessarily omitted from this dis-cussion because comparisons among functionforms require the use of a specific dataset andfindings would not be general. In addition, thereis a significant body of literature concerningtesting of nested and non-nested models (seeJudge et al.).

The fourth grouping of selection criteria per-tains to application-specific characteristics. Forexample, if the resulting equation is to be usedin simulation or optimization procedures, theremay be other desirable properties for func-tional form. These considerations are largelyexcluded from the present discussion becausethe number of potential criteria within thiscategory is quite large and highly customized.

For ease of presentation, the discussion em-phasizes selection of a production functionrather than a demand, profit, or cost function.The rationale for this choice stems partiallyfrom the fact that most recent advances in con-ceiving flexible forms are published as cost ordemand applications. Much of the discussionpresented here can be readily applied to costor demand studies. System-related properties,such as symmetry, are not considered becauseof the production function emphasis.

Some functional forms allow as special casesthe assumption of all properties of other func-tional forms. The more flexible forms may beappropriate when information regarding thenature of the relationship does not permit cer-tain hypotheses to be maintained. Those func-tional forms subsumed by others are indicatedin the last column of table 1. Employing anonlinear transformation of coordinates, thegeneralized Box-Cox form subsumes nine oth-er forms as nested cases. They are: translog(0 = X = O); CES (vO = X, all Xf = 26,, bi = 0for all i # j); modified resistance (6 = -1/2, X =-1, all b6 = 0); Cobb-Douglas (0 = X = 0,Z /i = 1, all bi = 0); logarithmic (0 = /2, X =0); square root and generalized Leontief (0 =X = 1/2); and quadratic and linear (0 = 1/2, X =1).

Maintained Hypotheses

Column A of table 1 concerns the assumedeffect of inputs specified in the function butnot applied (i.e., those inputs with a zero level).Assuming that one or more inputs are hy-

220 December 1987


pothesized to be related to output and includedas exogenous variables, column A indicates forwhich functional forms output would be zeroif all input levels were zero. Such a maintainedhypothesis may not conform to observed phe-nomena. In agronomic fertility studies, for ex-ample, one often encounters experimental datawhere nutrient applications are observed, butsoil nutrient levels are not. Zero yields are im-posed by particular functional forms when ob-servations pertain to applied nutrients and nonutrients are applied. The hypothesis repre-sented by column A thereby underscores therelationship between variable specification andfunctional form.

A similar but more relevant property is de-scribed in column B, which indicates for whichfunctions inputs are essential; that is, no out-put occurs if at least one of the inputs of amultiple-input model is not applied. Again,the importance of this property primarily de-pends on how the inputs of a specific modelare defined.

First derivatives of production functions areimportant in nearly all applications. Becausesome are cumbersome to compute, derivativesfor all twenty forms are given in table 2. Col-umn C of table 1 indicates maintained hy-potheses concerning the sign of marginal prod-uct. Functions in table 1 are characterized bymarginal products which are unrestricted insign (U), unrestricted but nonswitching in sign(UBN), or unrestricted in sign but constant invalue (UBC). Functional forms with marginalproducts that are unrestricted but nonswitch-ing in sign allow successive units of appliedinput to increase, decrease, or not change totaloutput, but any change in total output mustbe in the same direction as the change pro-duced by the previous unit of input. Thus, thenine UBN functional forms (as well as the twoUBC ones) do not allow model estimation todetermine at what input level output begins todecrease (assuming the data supports this test-able hypothesis) but rather maintains thehypothesis of everywhere positive (or every-where negative) marginal productivity.

Column D indicates which functional formsmaintain the hypothesis of asymptotic con-vergence of output towards a maximum as in-put levels are increased. In such cases, outputincreases as the level of input increases, re-gardless of the data employed to estimate thechange in output due to a change in the levelof input.

Two related properties exhibited by certainforms are homogeneity and homotheticity.Functional forms homogenous of degree 1, saidto be linearly homogenous, are indicated incolumn E. If production is found to be prof-itable for some input combination in a linearlyhomogenous production function, profit canbe increased without bound by increasing in-puts proportionately.

Homothetic forms are indicated in columnF of table 1. The interesting characteristic ofhomothetic production functions is that themarginal rate of technical substitution (MRTS)remains constant as all inputs are increasedproportionately. When the marginal rate oftechnical substitution is constant, as for homo-thetic functions, any change in the value ofoutput will affect the optimum input levelsproportionately as long as input prices are un-changed.

Related to the concept of the marginal rateof technical substitution is the concept of theelasticity of substitution. In general, MRTSjwill increase in magnitude as xi is increased(and xj is decreased) along any given isoquant.This rate of change can be measured by theelasticity of substitution.2 Those functionalforms assuming constant elasticity of substi-tution are indicated in column G.

The generalized constant elasticity of sub-stitution (CES) production function, which isdefined not only to have constant elasticity ofsubstitution but also to be homogenous of de-gree v when a = 0, represents a much-studiedclass of production functions (Arrow et al.;McFadden; Uzawa). The concept of elasticityof substitution and the CES class of productionfunctions have been applied primarily to theproblem of the distribution of income amonggenerally defined or highly aggregated factorsof production (Arrow et al.; Behrman; Ner-love). With regard to production functionanalysis, however, Silberberg states that"knowledge of a at any point would undoubt-edly be a useful technological datum for em-

2 For any change along an isoquant, d(xi/xj) is the change in theuse of xi as compared with that of Xj, and d(dx/dxj) is the corre-sponding change in the marginal rate of technical substitution."The ratio of these differentials, expressed in proportional termsto make them independent of units of measurement, is defined asthe elasticity of substitution between the factors at the combinationof factors considered" (Allen, p. 341). The elasticity of substitutionbetween xi and xj is given by

(f=/I) d(x,/xj)"j (x,/ j) d(f/l)

where fi = dy/9xi and f = ay/dx,.



Table 2. First Derivatives of Selected Functional Forms

Function Ox,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Leontief

Linear

Quadratic

Cubic

Generalized Leontief

Square root

Logarithmic

Mitscherlich

Spillman

Cobb-Douglas

Generalized Cobb-Douglas

Transcendental

0 or f,

fi

fi + 2,ijx

j joi k

fixi

(1i + Z aijX2)x

? '

e i Xi

-a1,exp(81x,) II (1 - exp(jx))joi

-apxiln(i) 1I (1 - jpx)joi

aiXi-1 nxif

y Z 2&,/(x, + xj)

a(bi + ilx,) II xaiexp(bjxj)

Resistance

Modified resistance

CES

Translog

Generalized quadratic

,iY2

(6i + xi)-2

y2(xi,-2 + 2ijXi2Xi-1)

j=i

v+p

ivxI'--P[a + ' oiXiP] p

y[1i + 2 ijln(xi)]/xi

v-6

vx;-'y v [ f F.jx.xl-, + (1-) j f i '-,xI]i J

Generalized power

Generalized Box-Cox

Augmented Fourier

x- 'y'-20[Xi + 2 bij(Xx - 1)]/X

Pi + S 2ijx + i ; hihexp(i z hix,)j Ihl*'H i

pirical work" in that the statistic describes therelative ease with which one input may be sub-stituted for another (p. 317).

A final property, concavity, is important inat least two respects. It is in one sense pertinentto description of the production process, re-flecting a situation in which output increasesat a decreasing rate (or decreases at an increas-ing rate) as the level of input is increased. Theproperty may be of primary importance, how-ever, in the context of economic optimization;only if the function is concave can input levelsthat maximize profit be computed from first-order equations.

As indicated in column H of table 1, esti-mated parameters of the model may need tobe examined before the researcher can ascer-tain whether concavity is present. The impor-tance of concavity for profit maximization im-plies that such an examination should alwaysbe conducted. While the conditions listed forconcavity in table 1 are necessary and sufficientfor global concavity, that is, for concavityeverywhere in the nonnegative orthant, manyof these conditions are not necessary for localconcavity. In particular, the indicated condi-tions are merely sufficient for local concavityin the case of the square root, resistance, mod-

222 December 1987

ayaxiFunction


ified resistance, CES, generalized Leontief,translog, and generalized quadratic forms.

Statistical Estimation

Certain properties embodied in functionalforms have important implications for themathematical procedures employed in the sta-tistical estimation of parameters. Pertinentproperties included in table 1 are the numberof distinct parameters (column I), and linearseparability of the parameters (columns J andK).

Most functions, in their complete forms, re-quire a geometrically increasing number of pa-rameters to be estimated as the number of vari-ables of main effect is increased. This isprimarily because of the large number of in-teractions specified among variables of maineffect. For example, if two inputs are hypoth-esized to affect output, a linear function re-quires only three parameters to estimate thateffect, whereas a complete cubic function re-quires estimation of ten parameters. If ten in-puts are hypothesized to affect output, a linearfunction requires that eleven parameters beestimated, but a cubic function requires esti-mation of 286 parameters. 3

Fuss, McFadden, and Mundlak have ex-pressed concern that "excess" parameterswould "exacerbate" multicollinearity presentin survey data (p. 224). If multicollinearity ishigh, the variance of the parameter estimatesis increased such that it may be impossible todetermine how much variation in the endog-enous variable is explained by different exog-enous variables. Higher parameter variancesalso enlarge confidence intervals for applica-tions (such as the computation of an optimalinput program) involving subsets of the esti-mated parameters. If, however, multicollin-earity is irrelevant because the primary pur-pose of the model is for prediction (Maddala,p. 186), then the issue of having many param-eters can be quite unimportant. Of-possibleconcern to the researcher may be the expenseof estimating a large number of parameters orthe loss of degrees of freedom. Either of theseproblems could actually preclude the use ofcertain forms, depending on the number ofvariables of main effect, computing resourcesavailable to the researcher, and/or size of the

3 In some cases, additional maintained hypotheses can be intro-duced to eliminate certain interactions. For example, any real in-teraction between the quantities of irrigation water and harvestlabor in crop production may be thought, a priori, to be absent.

dataset. The total number of parameters to beestimated by a given form may be calculatedwith the formulas presented in column I.

Ease of analysis and availability of resourcesmay also limit the modeling process to func-tional forms containing parameters that are alllinearly separable (indicated in column J) andcan therefore be estimated by common andwell-developed linear least squares regressiontechniques. Whereas direct estimation of non-linearly separable forms may be possible bysuch techniques as maximum likelihood esti-mation, Fuss, McFadden, and Mundlak notethat "linear-in-parameter systems have a com-putational cost advantage, and have, in addi-tion, the advantage of a more fully developedstatistical theory" (p. 225). Linear least squaresregression provides information regarding thesmall-sample accuracy and precision of esti-mates, which may not be available from othertechniques. The value of this information mustbe weighed against any advantages in the useof nonlinearly separable forms. On the otherhand, the gain in information for linear leastsquares estimators may be artificial, since thisinformation results from several important as-sumptions.

In addition, the nature of the dataset mustbe considered if a technique such as ordinaryleast squares is to be employed. Column K oftable 1 shows which functional forms are lin-early separable in parameters when any inputlevels take on zero values. For example, eventhough column J identifies the generalizedCobb-Douglas form as being linearly separa-ble, if the dataset used for estimating the func-tion contains an observation in which none ofone input is applied, ordinary least squarescannot be conducted.

It is notable that the generalized Box-Coxform is not linear in parameters, and this facttends to limit applications. If coefficients 0 andX are chosen a priori, the form becomes linearin parameters. When this is done, the Box-Coxform becomes one of the subsumed functionalforms, and its generality is sacrificed.

Formalizing the Selection ofFunctional Form

This paper describes why the researcher shouldbe concerned with the functional form of acontinuous input-output model and providessome guidance for assessing whether somefunctional forms may be considered more ap-



propriate than others. As a result of the math-ematical properties inherent in each functionalform, hypotheses are maintained about theproduction relationship whenever a specificfunction is selected for estimation. 4 Any po-tential hypothesis which is not maintained andcan be expressed as one or more linear rela-tions among function parameters will be test-able.5 Such relations are indicated in table 1.Inherent mathematical properties of a givenfunctional form can have important implica-tions for statistical estimation and economicapplication.

A functional form may be appropriate be-cause of the correspondence of maintained hy-potheses with generally held theories of thetrue input-output relationship, possibility andease of statistical estimation, possibility andease of application, general conformity to data,or a combination of these criteria. None of themany criteria guarantee that the true relation-ship will be discovered, nor do any allow atotally objective choice to be made. Subjectivejudgment is a necessary aspect of choice re-garding functional form. Thus, formalizationof the selection process requires deliberativechoice and frank presentation.

Having selected two or more estimable func-tional forms with acceptable theoretical andapplication properties, the researcher may wishto base final selection upon a statistical crite-rion. Such criteria entail data-specific consid-erations and may comprise a decision rule assimple as choosing the model with the highestcoefficient of multiple determination. Strictlystatistical information can be used for the pro-cess of model selection since several formal/informal opportunities are available for bothnested and nonnested testing. Judge et al. notethat tests of nested models measure "how wellthe models fit the data after some adjustmentfor parsimony" (p. 862). Tests of nonnestedmodels ask the question: Is the performanceof model 1 "consistent with the truth" of mod-el 2 (Pesaran and Deaton, p. 678)? Thus, thetesting of each model against the evidence pro-vided by the other will not necessarily allow

4 Also, properties which are not inherent to a given form canoften be incorporated as maintained hypotheses by the appropriatechoice of linear restrictions. Table 1 indicates the appropriate re-strictions for the considered criteria. Moreover, different functionalforms can be simultaneously developed in order to test differenthypotheses (as in Shumway's paper).

5 However, such tests require the additional (augmenting) hy-pothesis that the true form is of the class under consideration(Gallant 1984).

the investigator to choose one model over theother. While it is clear that such data-relatedobjectives can form an important set of cri-teria, allowing statistical evidence to dictatemodel selection is regarded as a questionableand often unnecessary practice.

The following observation by Hildreth is ofinterest: "It is particularly disconcerting that,in many instances in which several alternativeassumptions [as to functional form] have beeninvestigated, alternative fitted equations haveresulted which differ little in terms of conven-tional statistical criteria such as multiple cor-relation coefficients or F tests of the deviation,but differ much in their economic implication"(p. 64). Given the possible differences in eco-nomic implications, it is often advisable toexplore the sensitivity of calculated economicoptima to the choice of functional form. As aninteresting example, in an analysis of a singleapplication of nitrogen and potassium on corn,Bay and Schoney have assessed the costs of"incorrectly" choosing among four possiblefunctional forms. Similarly, in an analysis ofmultiple nitrogen applications on rice, Griffinet al. have examined the sensitivity of optimalfertilizer programs to functional form.

Conclusions

It is often difficult to ascertain why particularfunctional forms are chosen for the modelspresented in published economic research.Possibly, researchers confine their attentionsto particular functional forms because they aremost comfortable and experienced with theseforms or because these forms are in vogue. Infairness, there are probably many exampleswhere the research process is extensive andformal in the selection of functional form, buta description of these procedures is omittedfrom published results. These omissions maybe due to a propensity of authors, reviewers,or editors to consider such material extra-neous. Applied economists can enrich theiranalysis by selecting functional forms from asbroad a choice set as possible and by consid-ering a number of selection criteria (other thanmerely data-related criteria). Formal state-ment of these procedures in published outputis desirable so that readers might better gaugeresearch results (particularly sensitivity toform).

The need "to make research processes overt"

224 December 1987


(Tweeten, p. 549) pertains as much to the choiceof functional form as to other aspects of com-monly reported research methodology. Thatthe true form cannot be known and that it isimpossible to measure how well any chosenform approximates the true form lends im-portance to the selection process. Formaliza-tion of this process requires the developmentof an explicit and, hopefully, exhaustive choiceset and some acceptable criteria for conductingthe selection. The objective of this paper hasbeen to develop such a choice set and offer aninitial listing of plausible criteria. Extension ofthis compilation and discussion to settingsother than production function applicationsremains as a needed endeavor.

[Received December 1985; final revisionreceived August 1987.]

References

Allen, R. G. D. Mathematical Analysis for Economists.New York: St. Martin's Press, 1938.

Applebaum, E. "On the Choice of Functional Forms."Int. Econ. Rev. 20(1979):449-58.

Arrow, K. J., H. B. Chenery, B. S. Minkas, and R. M.Solow. "Capital Labor Substitution and EconomicEfficiency." Rev. Econ. and Statist. 63(1961):225-50.

Barnett, W. A. "New Indices of Money Supply and theFlexible Laurent Demand System." J. Bus. and Econ.Statist. 1(1983):7-23.

Bay, T. F., and R. A. Schoney. "Data Analysis with Com-puter Graphics: Production Functions." Amer. J. Agr.Econ. 64(1982):289-97.

Behrman, J. R. "Selectoral Elasticities of SubstitutionBetween Capital and Labor in a Developing Econo-my: Time Series Analysis in the Case of PostwarChile." Econometrics 40(1972):311-26.

Berndt, E. R., and M. S. Khaled. "Parametric Productiv-ity Measurement and Choice Among Flexible Func-tional Forms." J. Polit. Econ. 87(1979):1220-45.

Christensen, L. R., D. W. Jorgenson, and L. J. Lau."Transcendental Logarithmic Production Frontiers."Rev. Econ. and Statist. 55(1973):28-45.

de Janvry, A. "The Generalized Power Function." Amer.J. Agr. Econ. 54(1972):234-37.

Denny, M. "The Relationship Between Forms of the Pro-duction Function." Can. J. Econ. 7(1974):21-31.

Despotakis, K. A. "Economic Performance of FlexibleFunctional Forms." Eur. Econ. Rev. 30(1986): 1107-43.

Diewert, W. E. "An Application of the Shephard DualityTheorem: A Generalized Leontief Production Func-tion." J. Polit. Econ. 79(1971):481-507.

Elbadawi, I., A. R. Gallant, and G. Souza. "An Elasticity

Can Be Estimated Consistently without A PrioriKnowledge of Functional Form." Econometrica51(1983):1731-51.

Fuss, M., D. McFadden, and Y. Mundlak. "A Survey ofFunctional Forms in the Economic Analysis of Pro-duction." Production Economics: A Dual Approach toTheory and Applications, ed. M. Fuss and D. Mc-Fadden, pp. 219-68. Amsterdam: North-HollandPublishing Co., 1978.

Gallant, A. R. "On the Bias in Flexible Functional Formsand an Essentially Unbiased Form." J. Econometrics15(1981):211-45.

. "The Fourier Flexible Form." Amer. J. Agr. Econ.66(1984):204-15.

. "Unbiased Determination of Production Tech-nologies." J. Econometrics 20(1982):285-323.

Griffin, R. C., M. E. Rister, J. M. Montgomery, and F. T.Turner. "Scheduling Inputs with Production Func-tions: Optimal Nitrogen Programs for Rice." S. J. Agr.Econ. 17(1985):159-68.

Halter, A. N., H. O. Carter, and J. G. Hocking. "A Noteon the Transcendental Production Function." J. FarmEcon. 39(1957):966-74.

Heady, E. 0., and J. L. Dillon, eds. Agricultural Produc-tion Functions. Ames: Iowa State University Press,1961.

Hildreth, C. G. "Discrete Models with Qualitative De-scriptions." Methodological Procedures in the Eco-nomic Analysis ofFertilizer Use Data, ed. E. L. Baum,E. O. Heady, and J. Blackmore, pp. 62-75. Ames:Iowa State College Press, 1955.

Judge, G. G., W. E. Griffiths, R. C. Hill, H. Liitkepohl,and T. Lee. The Theory andPractice ofEconometrics,2nd ed. New York: John Wiley & Sons, 1985.

King, R. A. "Choices and Consequences." Amer. J. Agr.Econ. 61(1979):839-48.

Ladd, G. W. "Artistic Tools for Scientific Minds." Amer.J. Agr. Econ. 61(1979): 1-11.

Lau, L. J. "Testing and Imposing Monotonicity, Con-vexity and Quasi-Convexity Constraints." ProductionEconomics: A Dual Approach to Theory and Appli-cations, ed. M. Fuss and D. McFadden, pp. 409-53.Amsterdam: North-Holland Publishing Co., 1978.

McFadden, D. "Constant Elasticity of Substitution Pro-duction Functions." Rev. Econ. Stud. 30(1963):73-83.

Maddala, G. S. Econometrics. New York: McGraw-HillBook Co., 1977.

Nerlove, M. "Recent Empirical Studies of the CES andRelated Production Functions." The Theory and Em-pirical Analysis of Production, ed. M. Brown, pp. 55-122. New York: Columbia University Press, 1967.

Pesaran, M. H., and A. S. Deaton. "Testing Non-NestedNonlinear Regression Models." Econometrica46(1978):677-94.

Rees, C. S., S. M. Shah, and C. V. Stanojevic. Theoryand Applications of Fourier Analysis. New York: M.Dekker, 1981.

Rice, J. R. The Approximation ofFunctions, vol. 1. Read-ing MA: Addison-Wesley Publishing Co., 1964.



Rudin, W. Principles of Mathematical Analysis, 2nd ed.New York: McGraw-Hill Book Co., 1964.

Shumway, C. R. "Supply, Demand, and Technology ina Multiproduct Industry: Texas Field Crops." Amer.J. Agr. Econ. 65(1983):748-60.

Silberberg, E. The Structure of Economics: A Mathemat-ical Analysis. New York: McGraw-Hill Book Co.,1978.

Tolstov, G. P. Fourier Series. Englewood Cliffs NJ: Pren-tice-Hall, 1962.

Tweeten, L. "Hypothesis Testing in Economic Science."Amer. J. Agr. Econ. 65(1983):548-52.

Uzawa, H. "Production Functions with Constant Elas-ticity of Substitution." Rev. Econ. Stud. 29(1962):291-99.

White, H. "Using Least Squares to Approximate Un-known Regression Functions." Int. Econ. Rev.21(1980):149-70.

Appendix

The Augmented Fourier Form

Because complex-valued trigonometric polyno-mials of order H,

(A. 1) y(x)= 'Yh exp(i ihix,) ,Ihl\-H i

can be used to approximate any known periodicfunction, and because the approximation is perfectwhen H = oo, trigonometric polynomials have anumber of fruitful applications in physics and theengineering sciences. The following additional in-formation is needed to interpret equation (A.1): his an n-dimensional vector of integers (..., -1, 0,1, ... ) with "length" I hl* = hi ; H is exoge-

nously chosen by the researcher and serves to limitthe number of terms in (A. 1). Each coefficient Yh iscomplex-valued, Yh = "h + i y , where i2 = -1 and

'h and Yh are the real and complex components ofY,, respectively. Thus, each 7Y really represents twoseparate parameters.

If y(x) is known and the terms Yh are chosen ina certain manner (see Tolstov, pp. 13-14, 174; orRees, Shah, and Stanojevic, p. 225), then these termsare Fourier coefficients, and equation (A.1) identi-fies a Fourier series approximating y(x). For all pos-sible trigonometric polynomials of limited order ap-proximating y(x), the Fourier series provides thebest approximation as measured by least sum-of-square differences across the domain (Rudin, pp.172-73; Tolstov, p. 174).

In production analysis we are not interested inperiodic or complex-valued functions. Because pro-duction functions are not expected to be periodic,the exogenous factors, the xi, must be scaled so thatthey lie in an interval of length less than 2ir. Thisis typically accomplished by scaling or normalizingxi into [0, 27r] (Gallant 1981). In order to guarantee

that y(x) in equation (A. 1) is real-valued, it is suf-ficient to impose

(A.2) , = =0, Yh= Y'-h, and yr = -'y .

Equation (A. 1), conditions (A.2), and the identity

exp(i hix,) = cos(2 hix,) + i sin(2 hx,i)

can be used to obtain

(A.3) y(x) = 1 ['ycos(2 hx,)Ihl-<H

- 'ysin(2 hix)].

While (A. 1) is a more compact representation of thegeneral trigonometric polynomial, application toproduction function analysis begins with equation(A.3). Note that equation (A.3) and its derivativesare obviously real-valued, and conditions (A.2) muststill be econometrically imposed.

Gallant (1981) is responsible for combining thequadratic form and equation (A.3) to produce theso-called Fourier form or, by our nomenclature, theaugmented Fourier form:

(A.4) y= fx, + Ox dxji i j

+ Z [-y Ycos( hx) - y, sin(2 hxI)].Ihl*-H

The constant term, a, is omitted to avoid redun-dancy since it is contained in the Fourier expression(h = 0). The number of parameters in the augmentedFourier form depends on n and H, but we havebeen unable to obtain a completely general formulafor the number of parameters in the Fourier portionof (A.4). The following formula applies for n = 1,2, 3, or 4.

n(n + 3)m = ml + m2= + m2,

2

where m is the total number of coefficients in theaugmented Fourier form, ml is the number of termsin the quadratic portion (excluding the constantterm), and m2 is the number of terms (including theconstant term) in the Fourier portion; m2 is givenby an nth degree polynomial in H:

m2 = ao + alH1 + a2H2 + ... + anHn

with coefficients identified in the following table forn up to 4.

Table A.1. Polynomial Coefficients for m2

ao al

a2 a3 a4

1 1 2 0 0 02 1 2 2 0 03 1 %/3 2 4/3 04 3/1 53 -2/ 164/ 24 11 6/63 2/3 3 /99 /3

The number of terms in the Fourier portion can

226 December 1987


be quite large if either n or H is large. To illustratethe effect of increasing n when H is small: if H = 2and n = 2, ml = 5 and m2 = 13; ifH = 2 and n =4, ml = 14 and m2 = 41.

Sufficient (but not necessary) conditions for con-


cavity of the augmented Fourier form can be de-rived by following the procedure of Gallant (1981).To do so requires substantial additional notationand will not be pursued here.

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