Confidence Intervals for Elasticity Estimators in ...

Confidence Intervals for Elasticity Estimators in Translog Models Author(s): Richard G. Anderson and Jerry G. Thursby Source: The Review of Economics and Statistics, Vol. 68, No. 4 (Nov., 1986), pp. 647-656 Published by: The MIT Press Stable URL: http://www.jstor.org/stable/1924524 . Accessed: 28/06/2014 10:35 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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CONFIDENCE INTERVALS FOR ELASTICITY ESTIMATORS IN TRANSLOG MODELS * RichardG. Anderson andJerry G. Thursby Abstract-Thispaperexaminesthedistribution functions of often assume that the normal distribution is an in translog elasticity estimators demandmodels.We consider adequate the conditions for norapproximation, andpresent thenormalandratio-of-normals distributions conthan fortheelasticity Ourresults fidence intervals estimators. sug- malityof the estimatorsare more stringent gestthatonlyelasticity estimators basedon themeansof the the conditions for the estimatorto follow the actualcost sharesare likelyto followeitherthenormalor distribution. We considera wide ratio-of-normals distribution function. Examination of three ratio-of-normals thatinferences re- range of alternativeelasticityestimators-differpublishedempiricalstudiesdemonstrates gardingthevaluesof elasticities cannotbe madefrompoint ing in theirtreatment of factorcost shares-in a alone and suggests a tradeoff estimates betweenthelevelof Monte Carlo experiment, and conclude thatonly and the widthof confidence intervals for the aggregation on the means of the elasticity estimators based elasticity estimators.

actual cost shares are likelyto followeitherthe normal or ratio-of-normals distributionfunction. Methods of constructing confidenceintervalsand E MAND modelsbasedon flexible functional testinghypothesesforotherelasticityestimators, formssuch as thetranslogfunctionare widely includingthe majorityof those whichhave been used by economistsbecause theyplace fewprior used in applied studies,remainunknown. restrictionson substitutionand demand elasticiIn section II we brieflyreview the translog ties. Economic inferencesfromthesemodelsmust model and discuss its use as an approximationto be based upon estimatedsubstitution and demand unknownfunctionalforms.Our analyticalresults elasticities,however, since the estimated funcare presentedin sectionIII, and the Monte Carlo tional formparametershave no economicinterresults are presentedin section IV. Section V pretation. presentsconfidenceintervalestimatorsfor transElasticityestimators in translogmodelsare nonlog elasticityestimators,and demonstratestheir linear transformations of theparameterestimators use in threeapplied studies.A briefsummaryis and the factorcost shares.The observedsample presentedin sectionVI. data do not provide repeatedobservationson a single set of elasticities,and preciseestimatesof the parametersof the translogfunctionneither II. The TranslogModel implynor guaranteepreciseestimatesof the elasticities.This factmakestranslogestimators fundaLet a smooth neoclassicalproductionfunction fromthoseof CES and its dual cost functionbe denotedas y = f(x) mentallydifferent (statistically) functions,whose elasticityestimatorsare solely and C(w, y), respectively, where x and w are functionsof the regressionparameterestimators. vectors of input quantitiesand prices. An apIn thispaper,we examinethestatisticalproper- proximationto C(w, y) is the translogcost functies of translogelasticityestimators.We present tion conditions under which the asymptoticnormal N distributionmay providean acceptable approxilnC(w, y) = a0 + ayln(y) + a ailn(wi) mation to the distributionof the estimatorsin i=l N finitesamples. In addition,we examinethe exact + E 0ayin(y)ln(wJ) finite-sampledistributionfunctionof a ratio-ofi=1 normals random variable.While applied studies

D

I. Introduction

Received forpublicationJuly11, 1985. Revisionacceptedfor publicationJanuary17, 1986. *The Ohio State University. We are indebted to ErnstBerndt,C. A. Knox Lovell, Peter Schmidtand Marie Thursbyforcommentson earlierversions of thispaper.

+ .5ayy ln(y)2 N

+ .5 E

N

E

i=1 j=1

aijIn(wi)ln(wj).

The inputcost sharesand Allen partialelasticities [ 647 ]

C)1986 Copyright

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648

THE REVIEW OF ECONOMICS AND STATISTICS

the AES estimator is a ratioof a normal randomvariateand a productof lognormals (which is lognormal),and the price-elasticity Mi= ai + E aijln(wj) + ay,ln(y), is a ratioof a normaland a lognormal estimator j=1 normal The adequacyof theasymptotic variate. + a 1 iJ/MiM;, O'ij= in is considered in thiscircumstance distribution ji (ai + Mt2 _-Mi in sectionIV. theMonteCarloexperiment conditional(Hicksian) Considernexttheasymptotic The output-compensated distribution ofthe of inputi withrespectto the estimatorof av based on fittedfactorshares, demandelasticity price of input j is qij= aj Mj. For details, see 6,j = 1 + &iJ/MiMj.Let & denotea BAN MLE of Berndtand Wood(1975). vectorofthetranslog costfunction. theparameter as a Undermildregularity oftenis interpreted The translogfunction conditions, to the approximation Taylor'sseriessecond-order Motivated by an examplein White truefunction. a( - a) ~ N(O, ). charactertheapproximation (1980) thatcriticized isticsof translogmodels,severalrecentpapers Let theexogenousvariablesbe fixedat observed formswhichpro- valuesand let ar denotethevectorofAES based have proposednew functional elas- on fittedsharesat period t. Assumethat the and nearly-unbiased videstrongly-consistent and Gallant(1985) partialderivatives (see Chalfant ticityestimators existandletVa, denote Byronand Bera the gradientof daa/da However, therein). and references a,. Then a BAN MLE of a, iS A, exam- and numerical thatWhite's (1983)demonstrated providedan excelthetranslog ple was incorrect; in the toWhite'struefunction lentapproximation ~a,) EvaA); N(Av, aT( example.In thispaper,we assumethat corrected providesan acceptableap- see Zacks(1971). the translogfunction of based on byByron The finitesampledistribution withsmallbiasas suggested proximation on fixed Conditional fitted sharesis moredifficult. and Bera. valuesof the exogenousvariables(w, y), & and (AES) are of substitution N

III.

StatisticalResults

in finite Mi (i = 1, . . ., N) are normallydistributed

disturbance is normal. samplesifthecostfunction If certainconditions aresatisfied, thenormaldisof A. Asymptoticand Finite-SampleDistributions an acceptableapproximatribution may furnish ElasticityEstimators of ij in moderate-size tion to the distribution suggests in translog modelsarenon- samples,and a weakersetof conditions estimators Elasticity function ofa ratio-of-normals of the randomvariables thatthedistribution linear transformations aj ei M,, and Mj. In empiricalstudies,re- randomvariablemaybe an acceptableapproxiof belowtheimportance factor shares mation.We investigate the" fitted" haveusedeither searchers and the adequacyof both apa=6 + F>li& ln(w1)+ &yiln(y) or the ob- theseconditions of vi. in finite to the distribution served (actual) shares Mi (i = 1,..., N) in esti- proximations samples. matorsof theAES andpriceelasticities. A numberof studieshave proposedmodels of aij based on Considerfirstthe estimator stochastic exogenousinputpricevecactual factorshares, 61]= 1 + aij/MiM1. If dis- containing in opti- tors(e.g.,Fuss,McFadden,and Mundlak(1978); in production arisefromerrors turbances variablesarenonsto- Guilkeyand Lovell(1980); Guilkey,Lovell,and mizationand theexogenous of these consideration assumedin Sickles(1983)). Analytical chastic(bothof whichare frequently the is not but we investigate possible, models behaviorof appliedstudies),thenthestochastic in the Monte of estimators their behavior elasticity of M, follows from the stochasticbehavior in sectionIV. reported C(w, y)-1 since Mi = (wixi) - C(w, y)-1. If Carloexperiment estimators of two additional that Finally, note will and then y)-l C(w, lnC(w, y) is normal, Mi if the means will one uses be sample obtained ai distribution -a inverse of the lognormal follow or actualsharesinsteadof Ml or Mi. In thiscase, of thefitted whichis itselflognormal. function,

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ELASTICITY ESTIMATORS IN TRANSLOG MODELS

649

establishes thecondiof the samplemeansmore The secondproposition If the distributions estimaoftheAllenelasticity thando tionsfora function closelyresemblethenormaldistribution normaldistribution: of individualM, and Mi, then torto followthestandard the distributions of aij maymorecloselyfolloweither PROPOSITION 2: Normalityof theAllen Elasticity theestimator We distribution. the normalor ratio-of-normals Estimator. in the Monte Carlo this possibility investigate oftheAES as Denotethestochastic portion in sectionIV. reported experiment ~~~~~~~~A betweenafj Let yl denotethesimplecorrelation the In this section,we examineanalytically and MiMj where of distribution and asymptotic finite-sample thata, Mi and Mj on theassumption conditional N(aij,a2) aij we willnot For brevity, distributed. are normally above.Then and all othertermsareas defined thenormalaboveregarding repeatourdiscussion ity of fittedand actualfactorsharesexceptto - ai )/(a2 + o2a2- 2yia af )1/2 (ijA oftheactualinputshares thatnormality emphasize N(O,1) as (Pij/la) -* c. correctsincethesharesare cannotbe rigorously boundedby 0 and 1. fromthereProof:The prooffollowsdirectly result.Let sults of Fieller (1932), Marsaglia(1965), and use thefollowing Our propositions the randomvariablesM, and Mj followa joint Hinkley(1969). distributionwith means (,ui,yj), variances These propositions place conditions upon the coefficient 2), andcorrelation (ai2, yij.ThenMiMJ asymptoticand finite-sample behaviorof the has mean factorshares ofindividual coefficients ofvariation of and the shares product (lAi/ai) (Pi,/a.). In PijE E[M,MJ] = iL j + YJij on the valuesof the large samples,conditional and variance exogenousvariables,the conditionsof propositions1 and 2 willbe satisfied byMi and Mj since = 2 + r2_Var(M M1) p3+ta0Jr 2+a to zeroas thevaritheirvarianceswillconverge + (1 + yij)722j to of thevectora converge ancesof theelements of the condiIn satisfaction zero. finite samples, (Mood,Graybilland Boes(1974),p. 180). stateall whichmaybe said tionsdependsupon thevaluesof theexogenous Two propositions of a (and thereby in variablesand thedistribution oftheAES estimator distribution the regarding of the cost function disfor upon the distribution theconditions thiscase. The firstestablishes productsof normalrandomvariatesto followa turbance). A oftheseconditions byactualfactor and hencefor . to follow Satisfaction normaldistribution, oftheshares of a ratio of normal sharesdependsuponthedistribution the distribution function theadequacy and thevaluesof ui,a,. Empirically, randomvariables: normalapproximation of theasymptotic maydifof Productsof Factor ferfordifferent PROPOSITION 1: Normality pairsofinputssincethesharesof Cost Shares. withrelatively some factorinputsare distributed Let the inputsharesMi and Mj followa bi- less variationand largervalues of ni/a, (e.g., labor) thanotherinputs(e.g.,capital).Note also Let variatenormaldistribution. ofmeansharesbasedon thatthesmallervariances z=(MMifM -Pi )/P actual or fittedsharesmayallowthemto fulfill function of these conditionsmore readilythan individual and let 4Dz(z) denotethedistribution oftheproduct Z. Then 4z(z) -* N(0, 1) as eithernul/ai-* oo or shares.Finally,notethatnormality is a butnotsuffi(proposition 1) necessary MiMj o0. Ai(yJ J/MMJ the to follow cient condition for ratio ai fromthe immediately Proof:The prooffollows the the normal distribution Thus, (proposition 2). resultsofCraig(1936),Aroian(1947),andAroian, to follow the ratio-of-normals aj. condition for (1978). Taneja,and Cornwell B. AnalyticalResults

A

-*

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650

THE REVIEW OF ECONOMICS AND STATISTICS

is weakerthantheconditionsexogenousvariablesw, x, and y are fixedat function distribution in thecostfuncof ty. requiredforthenormality observedvalues.A disturbance tionarisesfromuncorrelated in minimizing errors productioncost foreach (w1,x,,yt) tripletand IV. MonteCarloResults costs are givenby C(w, y) = (Y2=jwjxj)e' where

that E - N(O, a,). In this specification,C(w, y) is In this section,we test the hypotheses distributed and strictly positive, and in translogmodelsfollow lognormally elasticityestimators lnC(w, y) is normally distributed. Thetruefactor or theratio-of-noreithera normaldistribution in finitesamples.We cost shares Mi = (wixi) - [(V jwjxj)e']-' are function mals distribution fortheAES and sevenesti- distributed as the inverseof a lognormaland testsevenestimators the estimatorstherefore are lognormal.The fittedfactorcost matorsfor the priceelasticities; of factorcost sharesMi are normally withrespectto thetreatment differ since aij distributed are N(a1ij, d2). estimators shares(see table 1). All fourteen whether thenormaldistribution The second,whichwe label Case II, assumes testedto ascertain ofthetruedistribu-thatfactorquantities is an adequateapproximation and thelevelof outputare and 8-12 and estimators also aretested nonstochastic. 1-5 The onlydisturbances tion, are uncorfunctionrelatederrorsin inputpricesw whicharisewhen distribution againsttheratio-of-normals 6, 7, 13, and 14 are neverdistributeddecisionsregarding (estimators thelevelsofinputsin periodt as a ratio-of-normals). mustbe madein periodt - 1 basedon forecasts of the elasticity estimatorsof w whichdiffer The distributions fromobservedw; see,forexamand factorcostsharesdependuponthestochasticple, Fuss, McFadden, and Mundlak (1978), of thetranslog model.We examine Guilkeyand Lovell(1980),and Guilkey,Lovell, specifications whichhave and Sickles(1983). In thiscase, ln wI = ln w* + E two different specifications stochastic function.wherew1 is theobservedpriceof inputi, w* is beenusedin priorstudiesofthetranslog whichwe labelCase I, assumesthatthe the latentprice whichproducersrespondto in The first, TABLE 1.-DEFINITIONS OF ALLEN- AND DEMAND-ELASTICITYESTIMATORS Expression

Description

A. Allen SubstitutionElasticities 1. (&i1/M.M.) + 1 2. (&,i/&a, &J) + 1

AES usingmeansof actualcost shares AES usingcostsharesevaluatedat mean of ytand w,: ln(y), ln(Tv) 3. (&a,/M1RMj)+ 1 AES usingcost sharesevaluatedat mean of ln(y) and ln(w): ln(y), ln(w). 4. AES value at midpointof time-series ofAES estimatesusingfittedfactorcost sharesM, 5. AES value at midpointof time-series ofAES estimatesusingactual factorcost sharesM, T

6. T-1 ? ((&Ij/MItMjt) + t= 1 T

7. T-1 ? ((&1J/MIMJA ) + t=1

1) 1)

Mean of time-series of estimatedAES usingfittedfactorcost shares of estimatedAES Mean of time-series usingactual factorcost shares

B. Factor PriceElasticitiesof Demand 8. AES estimator1 multipliedby Mj 9. AES estimator2 multipliedby &, 10. AES estimator3 multipliedby Mj 11. AES estimator4 multipliedby Mj 12. AES estimator5 multiplied by Mj T

13. T-1 ? ?(&X/Ml M1M+ t=1

)

Mean of time-series of price elasticitiesusingfittedfactorshares

+ Mit)

of price Mean of time-series elasticitiesusingactual factorshares

T

14. T'

F (Oulmit) t= 1

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ELASTICITY ESTIMATORS IN TRANSLOG MODELS

651

elasticitiesare and i is a random The Allen- and demand-price choosinginputquantities, costfuncmultiply each from the estimated translog calculated thismodel,we error.To implement a disturbance the formulas in section II above. below) by tion according to (defined inputpricewi pairs of (6, y) exp(,E) wherethe vectore (E1,C2, E3) iS inde- We examinedthreedifferent and an almost-homogeneous N(0,ac2I). values, generating pendentand identicallydistributed respectively: distributed in two almost-homothetic technologies, Onlylnwi (i = 1,2,3) arenormally The cost function(6, y) = (0.0, 1.0), (0.1, 1.2),(1.0, 1.2). We comthis stochasticspecification. C(w, y) = (E3=lwixi) is a sum of lognormalran- binedeach (6, y) pairwitheach of 18 combinadom variablesand hencelnC(w, y) is not nor- tionsof (P, Pl, P2, P3) (fromGuilkeyand Lovell The technologies. and the factorsharesMi = (1980)),a totalof54 production mally distributed; of and "difficult" include cases "easy" models of a ranare ratios lognormal (w xi)/(y3=lwixi) in nonand complementarity random factorsubstitution dom variableand a sum of lognormal andalmost-homoalmost-homothetic, In homothetic, distributed. variables,and are not normally In all models,weset81 = 82 of translog cost geneoustechnologies. estimates addition,least-squares in thiserrors- = 0.3 and 83 = 0.4. areinconsistent function parameters in in-variables model.Theseresultssuggestthatin- We used twosetsof factorinputquantities For the first,we willbe dif- the Monte Carlo experiment. estimators ferences regarding elasticity random generatedthreevectorsof 25 lognormal a conjecture confirmed by ourexperiments ficult, Forthesecond, nev- variatesas factorinputquantities. below.We includethisspecification reported indexesof capital, in previouswe used the Divisia quantity erthelessbecause of its popularity fromBerndtand Wood(1975). labor,and energy function. studiesof thetranslog Sample sizes of 50 and 100 wereobtainedby Output,total repetition of the 25 observations. A. Setup of theExperiment inputprice cost,and theimpliedcost-minimizing We chooseas an experimental techproduction vectorw weregenerated above. from theequations funcnologyHanoch's1971 CRESH production byfive 1,396models,eachdefined We examined tion. This function, popular in Monte Carlo thesize oneofthe54 technologies; characteristics: studies,has been usedin previousstudiesof the of thedisturbance variance,ae2= 0.1 or 0.5; the functional formmodtranslogand otherflexible size of 25, 50, or 100; the set of factor sample els, e.g.,Guilkeyand Lovell(1980) and Guilkey, input quantities(dataset); and the stochastic Lovell,and Sickles(1983). eitherCase I or Case II. We made is conductedas follows.On specification, The experiment thetranslog 300 trialsforeach model,estimating fromthequantities each trial,outputis calculated and thevaluesof the14 elasticity cost function of threefactorinputsaccordingto the CRESH in table1 on eachtrial.After300trials estimators function production of a model (for fixedvalues of the exogenous variables),we testedeach elasticityestimator -(Y/P) 3 distribuagainstthenormaland ratio-of-normals t = 1,..., T. s3ix0 yte0t= x2 testat tion functions a using goodness-of-fit i=l the5% level. Given the vectorof inputquantities xt and the which B. Resultsof theExperiment conditions levelofoutputyt,thefirst-order are solvedto yield definethedual costfunction forCase I are The resultsof the experiment thenormalized inputpricevector in table2, whichshowstheproporsummarized tion of modelsforwhichthe two hypothesized Theresults forCase arenotrejected. distributions wit= [eYt(1 + yt) II are notpresented: of a normal thehypotheses +P12 P) -0( wererejectedin 3 distribution or ratio-of-normals morethan95% ofthemodels.As notedin section i=l given IV, theresultsforCase II arenotsurprising estiof the the inconsistency translog parameter 2,3). (=1,

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652

THE REVIEW OF ECONOMICS AND STATISTICS TABLE 2.-RESULTS OF MONTE CARLO EXPERIMENTS(ENTRIES ARE PROPORTIONOF MODELS ACCEPTED)

TestsforNormalDistribution LognormalData Estimator(see table 1) A. Allen SubstitutionElasticity 1. mean actual shares 2. sharesat mean of y, w 3. mean of [ln y], [ln w] AES at midpointof sample, 4. usingestimatedshares 5. usingactual shares Mean of AES estimates, 6. usingestimatedshares 7. usingactual shares B. PriceElasticityEstimators 8. mean actual shares 9. sharesat mean of y, w 10. mean of [ln y], [ln w] At midpointof sample, 11. usingestimatedshares 12. usingactual shares Mean of elasticityestimates, 13. usingestimatedshares 14. usingactual shares

Berndt-WoodData

Tests forRatio-of-Normals Distribution LognormalData

Berndt-WoodData

q,2 = 0.1

aq2 = 0.5

aq2 = 0.1

aq2 = 0.5

aq2 = 0.1

aq2 = 0.5

aq2 = 0.1

aq2 = 0.5

.963 .148 .142

.883 .031 .025

.969 .000 .000

.963 .000 .000

.852 .568 .691

.858 .358 .512

.914 .006 .019

.901 .000 .000

.049 .037

.012 .000

.000 .000

.000 .000

.420 .074

.222 .000

.000 .265

.000 .000

.000 .889

.000 .475

.000 .957

.000 .617

.988 .191 .420

.914 .043 .105

.938 .000 .000

.944 .000 .000

.957 .858 .889

.864 .741 .907

.858 .840 .938

.846 .691 .932

.247 .506

.068 .025

.000 .790

.000 .000

.802 .531

.475 .080

.735 .852

.519 .191

.056 .957

.000 .889

.000 .932

.000 .944

-

-

-

-

-

-

-

-

Onlyestimators8 and 14 matorsand the natureof theprobabilitydistribu- of-normalsdistributions. using actual cost sharesare acceptedas normally tions. Our resultsfor seven alternativeestimatorsof distributed(forboth values of a,2 and both datathe AES are shownin partA of table 2. Only the sets),whileestimators8, 9, 10, and 11 are accepted distribution. firstestimator,using the mean of the actual cost as following the ratio-of-normals distribution shares, is consistentlyaccepted as followingthe Acceptance of the ratio-of-normals normaldistributionfunctionforboth sizes of the functionafterrejectionof thenormaldistribution disturbancevariance q,2 and both datasets.Esti- is consistentwiththe analyticalresultspresented followtheratio-of- in sectionIII above. mators2 and 3 approximately forthe We conclude thata normaldistribution normals distributiononly for the smaller disturbance variance and the lognormal dataset. AES estimatoris appropriateif theestimatoruses of AES the means of the actual factorshares.Second,we Estimator7, the mean of the time-series estimatorsusing actual cost sharesat each sample conclude thatestimatorsof thepriceelasticitiesof thanare observation, appears nearly-normalfor both demand are "betterbehaved"statistically more are significantly AES: they datasetsbut only forthe smallerdisturbancevari- estimatorsof the or ratio-of-northe normal either follow ance (q, = 0.1). Estimator6, similarto estimator likelyto 7 except using fittedfactor shares, fails to be mals distribution.Thus, confidenceintervalsconnormallydistributed(note that the estimatorsin structedfor price elasticityestimatesare more likelyto be robustwithrespectto departuresfrom the sum are not independent). and policyconclusions Results for seven alternativeestimatorsof the the assumed distributions, estimatesare more from elasticity drawn price elasticity of demand price Hicksian (conditional) thosedrawnfrom than (statistically) judged easily B in table 2. shown of Estimators 8-12 are part of AES estimators AES estimatesalone. are nonlineartransformations These conclusionsare robustwithrespectto the 1-5. Estimators13 and 14 are mean values of the time-seriesof price elasticityestimatorsat each structureof the technology.The completeresults sample observationusing estimatedand actual of our experiment(omittedforbrevity)show that of factorsubstitution We finda striking the "ease" or "difficulty" factorcost shares,respectively. the difference betweentestsforthenormaland ratio- (including cases of factorcomplementarity),

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ELASTICITY ESTIMATORS IN TRANSLOG MODELS

653

homogeneous orhomothetic natureofthetechnol- popular econometriccomputerprogramsexcept ogy,and thedifference betweenCES and CRES rl, forwhicha value of zero appears reasonable. The conditional (output-compensated)price technologies are all unimportant. elasticityestimatoris

V. ConfidenceIntervalEstimatorsand Applications

ij =(a= J

=

+

and its confidenceintervalfor the normal-distriIn this section,we presentconfidenceintervals bution case is

estimators forelasticity basedon boththeratio-of normalsand normaldistribution functions. Due ri; ? zo 1[T 1ij 5- 2rjiijsi to its complexity, we obtainconfidence intervals + V2) 1/2 + S2 + V2 1/2 X(T-1(s2 fortheratio-of-normals distribution bynumerical integration ofthefiducial thelatteris distribution; where a sumof twobivariate normaldistribution funcT tions(Marsaglia,1965)wherein unknown parame= T3 [ r2 ((Mit -)(mitmit ij tershave been replacedby theirestimators. The confidence intervals basedon thenormaldistribution are givenin partA. In partB, we present + T(T - )isi(Si + riisi))] intervalsfor threeapplied elasticityconfidence studies.In thesestudies, confidence intervals from IL= MiMJ+ ri1s1si the ratio-of-normals distribution were almost identicalto intervals calculatedfromtheexpres- and all othertermsare as previouslydefined.Note sionsforthenormaldistribution (thelargestdif- that r2 is the estimatedcorrelationbetweenthe conditionalon ference was in thethirddecimalplace).Hence,we numeratorand denominatorof zero correlationbetweena. and M presentonlythelatterintervals. A

Note thattheboundsof theconfidenceintervals presentedabove are functionsof therandomvariables themselves,a resultwhich followsdirectly The confidence interval estimator fortheAllen fromproposition2 and long has been recognized in thenormal-distribution case is 6,,? in studiescomparingthenormaland ratio-of-norelasticity A/B where mals distributionfunctions(e.g., Hinkley,1969). A. The NormalDistribution Case

ij=

1 + aiJ/Mimj

2 vsara + B=M~~M1+r,r1as /T + B = MiMj + rijsisj/T

A =z0

( V2,2 -

21

B. EmpiricalApplications

A frequentquestionin applicationsis whether inputs i and j are substitutesor complements. v2= (Mi2s2 + MH2s + 2M1M1jsisjri The most common "test" statisticfor this hypothesis is the sign of the AES or conditional 2 + (1 + rij)Si2 )IT price elasticityof demand. Confidenceinterval zo is thecriticalvaluefromthestandardnor- estimateshighlighttwo serious shortcomingsof maldistribution; this test,however.First,a confidenceintervalfor MT and sT are the sample mean and standard the Allen elasticitymay span both positiveand negative values, simultaneouslysupportingthe deviationof MT (T = i, j); mutually-exclusive hypothesesthatinputsi and j standard errorof aij; sa is theestimated are substitutesand complements. Second,theconfidence interval for the price elasticity may span r1 is the samplecorrelation betweenaij and and values while the interval for positive negative MiMj; theAllen elasticitydoes not,and viceversa.Hence betweenMi and ri. is the samplecorrelation tests based on confidenceintervalsneed not supMj; and T is thesamplesize. port the same inferencesas testsbased solelyon All requiredstatisticsmay be calculatedfrom the signsof point-estimates of the elasticities.

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654

THE REVIEW OF ECONOMICS AND STATISTICS TABLE 3.-CONFIDENCE

AllenElasticities Inputs K, L K, E K, M L, E L, M E, M

Estimated Value 1.01 -3.25 0.54 0.65 0.60 0.76

0.40 -6.67 -0.14 0.17 0.46 -0.14

Estimated Value

STUDIES

A. Bemdtand Wood (1975)a

Interval Lower Upper

AllenElasticities Inputs

INTERVALS FOR APPLIED

1.62 0.16 1.23 1.13 0.73 1.66

Inputs K, L K, E K, M L, K L, E L, M E, K E, L E, M M, K M, L M, E

B. Griffenand Gregory(1976)b

Interval Lower Upper

Inputs

(95% Intervals) Cross-PriceElasticities

Estimated Value

Lower

0.28 -0.15 0.34 0.05 0.03 0.37 -0.17 0.18 0.48 0.03 0.16 0.03

0.11 -0.30 -0.09 0.02 0.01 0.29 -0.36 0.05 -0.09 -0.01 0.13 -0.01

Interval Upper 0.44 0.01 0.77 0.09 0.05 0.46 0.01 0.31 1.04 0.07 0.20 0.07

Cross-PriceElasticities Estimated Value

Lower

Interval Upper

K, L K, E L, E

0.06 1.06 0.88

UnitedStates -0.38 -0.70 0.65

0.49 2.82 1.11

K, L K, E L, K L, E E, K E, L

0.04 0.14 0.01 0.12 0.15 0.64

UnitedStates -0.27 -0.09 -0.05 0.08 -0.10 0.45

0.36 0.37 0.07 0.15 0.41 0.82

K, L K, E L, E

WestGermany 0.50 0.26 1.03 0.29 0.79 0.40

0.73 1.76 1.18

K, L K, E L, E

0.50 1.04 0.82

K, L K, E L, K L, E E, K E, L K, L K, E L, K L, E E, K E, L

WestGermany 0.24 0.12 0.12 0.03 0.10 0.20 0.09 0.04 0.42 0.11 0.38 France 0.18 0.27 0.10 0.11 -0.01 0.13 0.05 0.08 0.05 -0.04 0.31 0.49 0.27

0.35 0.20 0.30 0.14 0.72 0.57 0.44 0.23 0.21 0.12 0.65 0.71

France

0.19 -0.12 0.46

0.71 2.20 1.16

C. Four-InputCapital and Labor Model (thispaper) AllenElasticitiesc Four-InputModel Three-InputModel Inputs B, W B, E B, S W, E WI $ E, S

Estimated Value

Lower

1.64 1.73 1.12 -1.67 1.04 0.20

0.91 1.05 0.31 -2.92 -0.44 -1.41

Interval Upper 2.38 2.41 1.94 -0.42 2.52 1.81

Inputs B, W B, K W, K

Estimated Value

Lower

2.39 1.47 -1.13

1.56 0.79 -2.18

Interval Upper

aElasticity values are calculated frommeans of input cost shares and the translogcost functionestimated by Bemdt and Wood (1975). frommean factorcost shares foreach countryand the translogcost functionof Griffenand Gregory (1976), table 1. bCalculated cBased on mean factorcost shares.

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3.21 2.15 -0.09

ELASTICITY ESTIMATORS IN TRANSLOG MODELS

655

we estimatetwo As a finalempiricalillustration, In an importantpaper,Berndtand Wood (1975) estimatedsubstitutionelasticitiesin U.S. manu- models of capital-labor substitutionin U.S. based upon thedata of Berndtand facturingamong capital (K), labor (L), energy manufacturing materials(M). We display Christensen(1973, 1974). These models demon(E), and intermediate content"of a elasticity estimates and confidence intervals stratethat the overall"information -based upon mean factorcost sharesand Berndt translogdemand model is governedby a tradeoff of a and Wood's estimatedtranslogcost function-in between the higher information-content model and the generally-narpart A of table 3. Note the small widthof the more-disaggregate intervalfor8LM wherebothinputshave largecost rower confidenceintervalsobtained in models shares,and the largewidthof theintervalfora'KE based on moreaggregatedata. Whileflexiblefuncwhereboth inputshave small cost shares.Larger tional forms-such as the translog-have their factorcost sharesdo not guaranteenarrowercon- greatestvalue in disaggregatemodels with large fidenceintervals,however:compare the intervals numbersof factorinputs(and small cost shares for8KL and 8KM wherethecost shareof M is for each input), elasticityestimatorsbased on approximatelytwice that of L. This tradeoffbe- smaller cost shares generallydisplay wider contweenthe size of factorcost sharesand thewidth fidenceintervals,certeris paribus.The functional of elasticityconfidenceintervalsis exploredin our relationshipbetween disaggregation,cost share thirdempiricalstudy,below. size, and the widthof confidenceintervalsis not Our confidenceintervalssupport Berndt and monotone,however,as the estimatesfor a7KL and Our Wood's conclusion that(i) capital and labor, (ii) cJKM in the Berndt-Woodstudydemonstrate. labor and energy,and (iii) labor and materials,are models show that thistradeoff is pronouncedfor substitutes: 95% confidence intervals for the as fewas threeor fourfactorinputs. posiAllen- and cross-priceelasticitiesare strictly translogcost funcWe estimateda four-input tive.We findthattheirresultsprovidelittleinfor- tion model using blue- and white-collarlabor mation regardingthe otherthreepairs of factor (B, W), equipment(E), and structures (S), and a inputs,however,sinceconfidenceintervalsforthe three-inputmodel using B, W, and aggregate elasticitiesincludeboth positiveand negativeval- capital (K). For brevity, we presentonlytheAES ues. In particular,the oft-citedcomplementarityestimatesin part C of table 3. Blue-collarlabor of capital and energyis not supportedby a 95% and capital (E, S, or aggregateK) are conconfidenceintervalabout the estimatedelasticity sistently substitutes: 95% confidence intervals value. for the AES contain no negativevalues. In both In a later paper, Griffenand Gregory(1976) models, white-collarlabor is complementaryto amongthree equipment (E), but in the four-inputmodel its estimatedlong-runfactorsubstitution inputs(capital,labor,and energy)in theindustrial relationshipto structures(S) is ambiguous: the sectors of nine Westerncountries.We display confidenceintervalfor the AES contains both elasticityestimatesand confidenceintervalsfor positive and negativevalues. In the more-aggrelabor model,however,white-collar threecountries-based on mean factorcost shares gate three-input with for each countryand Griffenand Gregory'sesti- and aggregatecapital (K) are complements, mated translogcost function-in part B of table the confidenceintervalcontainingonly negative modelprovidesa narrower 3. While Griffenand Gregoryconcluded that values. The three-input the confidenceintervaland (perhaps) a sharperpiccapital and energywere generallysubstitutes, evidence for that conclusionis weak. Confidence tureof the relationshipbetweenW and K. intervalsforthe Allen-and cross-priceelasticities Finally, note that elasticityestimatorsfor the model may be biased and inconfor the United States and France include both more-aggregate are uninsistent-even though displayingnarrowerconpositiveand negativevalues,and hence formativeregardingfactorsubstitution.Further, fidence intervals-if the necessary separability of (aKE are similar restrictionsfor the consistentaggregationof capnote thatwhile point-estimates 1.04 fortheUnited ital are not satisfied.This tradeoffis similarto and countries across (1.06,1.03, States, West Germany,and France,respectively), the well-knownrestricted-least-squares-regression see the three8KE displaythewidestconfidenceinter- tradeoffbetweenbias and mean-square-error; Toro-Vizcorrondoand Wallace (1968). vals of the estimatedelasticities.

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656

THE REVIEW OF ECONOMICS AND STATISTICS VI.

Summaryand Conclusions

Aroian, Leo A., V. S. Taneja, and L. W. Cornwell,"Mathematical Forms of the Distributionof the Product of Two Normal Variables," Communications in StatisIn thispaper we have examinedthe finitesamtics-Theory and MethodA7 (1978), 165-172. ple adequacy of the normaland ratio-of-normalsBerndt, Ernst R., and LauritsR. Christensen, "The Translog distributionfunctionsfor translogelasticityestiFunction and the Substitutionof Equipment,Structures,and Labor in U.S. Manufacturing1929-1968," mators,and presentedconfidenceintervalsforthe 1 (Mar. 1973), 81-114. Journalof Econometrics elasticityestimators.Our analyticresultssuggest , "Testing for the Existenceof a ConsistentAggregate that the elasticityestimatorsmay be normally Index of Labor Inputs,"AmericanEconomicReview64 (June1974), 391-404. distributedif all stochasticelementsin the model Berndt,Ernst R., and David 0. Wood, "Technology,Prices, are normal and additive.Our Monte Carlo eviand the Derived Demand forEnergy,"thisREVIEW 57 based estimators dence suggeststhatonlyelasticity (Aug. 1975), 259-268. on the meansof the actualcost sharesare likelyto Byron,R. P., and A. K. Bera,"Least Squares Approximations to Unknown RegressionFunctions:A Comment,"Indistrifolloweitherthenormalor ratio-of-normals ternational EconomicReview24 (Feb. 1983), 255-260. bution function. Chalfant,JamesA., and A. Ronald Gallant,"EstimatingSubstitutionElasticitieswith the Fourier Cost Function: Our empirical results demonstratethat inSome Monte Carlo Results,"JournalofEconometrics 28 ferencesregardingthe values of factordemand (Feb. 1985), 205-222. and substitutionelasticitiescannotbe made from Craig, Cecil C., "On the FrequencyFunctionof XY," Annals ofMathematicalStatistics7 (1936), 1-15. point-estimatesalone. Examinationof confidence Fieller, E. C., "The Distributionof the Index in a Normal of the that demonstrates point-estimates intervals Bivariate Population," Biometrika 24 (Nov. 1932), regarding elasticitiesoftenprovideno information 429-440. the structureof technologyor factordemand: the Fuss, Melvyn, Daniel McFadden, and Yair Mundlak, "A Surveyof FunctionalFormsin the EconomicAnalysis confidenceintervalsspan both positiveand negaof Production,"in M. Fuss and D. McFadden (eds.), tive values. Our applicationssuggesta tradeoff The EconometricAnalysisof Production(Amsterdam: North-Holland,1978), 219-268. data and between the use of highly-disaggregate "An Intercountry James,and PeterR. Gregory, Transthe widthof the confidenceintervalssurrounding Griffen, log Model of EnergySubstitutionResponses,"Amerimay reelasticityestimates.Finer disaggregation can EconomicReview66 (Dec. 1976), 845-857. duce the informationcontentof the model if it Guilkey,David, and C. A. Knox Lovell,"On theFlexibilityof the Translog Approximation,"InternationalEconomic reduces the precisionof the elasticityestimators Review21 (Feb. 1980), 137-147. and allows a largernumberof confidenceintervals Guilkey,David, C. A. Knox Lovell, and Robin Sickles,"A Comparison of the Performanceof Three Flexible to includeboth positiveand negativevalues. Functional Forms," International EconomicReview 24 Uzawa's well-known"ImpossibilityTheorem" (Oct. 1983), 591-616. demonstratesthattheredoes not exista function- Hanoch, Giora, " CRESH ProductionFunctions,"Econometrica 39 (Sept. 1971), 695-712. al form which admits both an arbitrary(unreHinkley,David V., "On the Ratio of Two CorrelatedNorstricted)matrixof AES and elasticitieswhichare mal Random Variables," Biometrika56 (Dec. 1969), constantacross sampleobservations(such thatthe 635-639. sample data providerepeatedobservationson the Marsaglia,George,"Ratios of NormalVariablesand Ratios of Sums of UniformVariables,"Journalof theAmerican elasticities).Our resultsextendUzawa's theorem StatisticalAssociation60 (Mar. 1965), 193-204. by demonstratingthat statisticalinferencecon- Mood, Alexander M., FranklinA. Graybill,and David C. Boes, Introduction totheTheoryofStatistics(New York: cerning the estimated matrix of elasticitiesin McGraw-Hill,1974). of the the mean at models is feasible only translog Toro-Vizcarrondo,Carlos,and T. Dudley Wallace, "A Test of observedfactorcost shares. the Mean Square Error Criterionfor Restrictionsin Linear Regression,"Journalof theAmericanStatistical Association63 (June1968), 558-572. White, Halbert, "Using Least Squares to ApproximateUnREFERENCES known RegressionFunctions,"International Economic Aroian, Leo A., "The ProbabilityFunctionof a Product of Review21 (Feb. 1980), 149-170. Two NormallyDistributedVariables,"AnnalsofMath- Zacks, Shelemyahu,The Theoryof StatisticalInference(New ematicalStatistics18 (1947), 265-271. York: JohnWileyand Sons, 1971).

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