Let's Take the Con Out of Eco No Metrics

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Let's Take the Con Out of Econometrics

Edward E. Leamer

The American Economic Review, Vol. 73, No. 1. (Mar., 1983), pp. 31-43.

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Let's Take the Con out of Econometrics

Econometricians would like to project theimage of agricultural experimenters who di-vide a farm i nto a set of sm aller plots of landan d w ho select rand om ly the level of fertiliz-er to be used on each plot. If some plots areassigned a certain amount of fertilizer whileothers are assigned n one, then the differencebetween the m ean yield of the fertilized p lotsan d the mea n yield of th e unfertilized plots isa measure of the effect of fertilizer on agri-cultural yields. Th e econometrician's hum blejob is only to determine if that difference islarge enough to suggest a real effect of fertil-izer, or is so small that it is more likely dueto random variation.This image of the applied econometrician'sart is grossly misleading. I would like tosuggest a more accurate one. The appliedeconometrician is like a farmer who noticesthat the yield is somewhat higher under treeswhere birds roost, and he uses thls as evi-dence that bird droppings increase yields.However, when he presents this finding atthe annual meeting of the American Ecologi-cal Association, another farmer in the audi-ence objects that he used the same data butcame up with the conclusion that moderateamounts of shade increase yields. A brightchap in the back of the room then observesthat these two hypotheses are indistinguish-able, given the available data. He mentionsthe phrase "identification problem," which,though no one knows quite what he means,is said with such authority that it is totallyconvincing. The meeting reconvenes in thehalls and in the bars, with heated discussionwhether this is the kind of work that meritspromotion from Associate to Full Farmer:th e Luminists strongly opposed to promo-tion and the Aviophiles equally strong infavor.

"Professor of econo mics , University of California-L osAngeles. This paper was a public lecture presented atthe University of Toront o, January 1082. I acknowledgepartial support by NSF grant SOC78-09479.

One should not jump to the conclusionthat there is necessarily a substantive dif-ference between drawing inferences from ex-perimental as opposed to nonexperimentaldata. The images I have drawn are de-liberately prejudicial. First, we had the ex-perimen tal scientist with hair n eatly combed.wide eyes peering ou t of horn-r imm ed glasses,a wh ite coat, and a n electronic calculator forgenerating the random assignment of fertiliz-er treatment to plots of land. This seems tocontrast sharply with the nonexperimentalfarmer with overalls, unkempt hair, and birddroppings on his boots . Another image,drawn by Orcutt , is even more damaging:"Doing econometrics is like trying to learnthe laws of electricity by playing the radio."However, we need not now submit to thetyranny of images, as many of us have in thepast .

I. Is Random ization Essential?

What is the real difference between thesetwo settings? Rand om ization seems to be theanswer. In the experimental setting, thefertilizer treatm ent 'is "ran dom ly" assignedto plots of land, whereas in the other casenature did the assignment. Now it is thetyranny of words that we must resist. "Ran-dom" does not mean adequately mixed inevery sample. It only means that on theaverage, the fertilizer treatments are ade-quately mixed. Randomization implies thatthe least squares estimator is "unbiased,"but that definitely does not mean that foreach sample the estimate is correct. Some-times the estimate is too high, sometimes toolow. I am reminded of the lawyer who re-marked that "when I was a young m an I lostmany cases that I should have won, butwhen I grew older I won many that I shouldhave lost, so on the average justice was done."In p articular, it is possible for the rand om-ized assignment to lead to exactly the sameallocation as the nonrandom assignment,


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33O L . 7.1 .YO. 1 LE.4 ,MER: T A K E T H E C ON O U T O F E C O N O M E T R I C S

ma trix is just the (prio r) variance of the biascoefficients (a * , P*). If this variance m atrixis small, the least squares bias is likely to besmall. If M is large, it is correspondinglyprobable that ( a* ,P* ) is large.I t would be a remarkable b o o t s t r a~ f wecould determine the extent of the rnisspecifi-cation from the data. The data in fact con-tain no information about the size of thebias. a point which is revealed by studyingthe likelihood function. The misspecificationmatrix M is therefore a pure prior concept.One must decide independent of the datahow good the nonexperiment is.The formal difference between a random-ized experiment and a natural experiment ismeasured bv the matrix M. If the treatmentis randomized, the bias parameters (a *. P*)are exactly zero, or, equivalently, the matrixM is a zero matrix. If M is zero, the leastsauares estimates a re con sistent. If M is notzero, as in the natural experiment, there re-mains a fixed amount of specification uncer-tainty, independent of sample size.There is therefore a sharp difference be-tween inference from randomized experi-ments and inference from natural experi-ments. This seems to draw a sham distinc-tion between economics where randomizedexperiments are rare and "science" whereexperiments are routinely done. But the factof the ma tter is tha t no o ne has ever design-ed an experiment that is free of bias, and noone can. As it turns out, the technician whowas assigning fertilizer levels to plots of land,took his calculator into the fields. and whenhe was out in the sun, the calculator gotheated up and generated large "random"numbers. which the technician took to meanno fertilizer; and when he stood under theshade of the trees, his cool calculator pro-duced small numbers, and these plots re-ceived fertilizer.You may object that this story is ratherfanciful. but I need only m ake you think it ispossible, to force you to set M * 0. Or if youthink a computer can really produce randomnumbers (calculated by a mathematical for-mula and therefore perfectly predictable!), Iwill bring up mismeasurement of the fertiliz-er level, or human error in carrying out thecomputer instructions. Thus, the attempt to

randomize and the attemDt to measure accu-rately ensures that M is small, but not zero,and the difference between scientific experi-ments and natural experiments is differencein degree, but not in kind. Adm ittedly how-ever, the misspecification uncertainty inmany experimental settings may be so smallthat i t is well approximated by zero. T h s canvery rarely be said in nonexperimental set-tings.Examples may be ultimately convincing.There is a great deal of empirical knowledgein th e science of astron om y, yet there ar e noexperiments. Medical knowledge is anothergood example. I was struck by a headline inthe January 5 , 1982 New York Times:"LifeSaving Benefits of Low-Cholesterol Diet Af-firmed in Rigorous Study." The article de-scribes a random ized experiment with a con-trol group and a treated group. "Rigorous"is therefore interpreted as "randomized." Asa matter of fact, there was a great deal ofevidence suggesting a link between heart dis-ease and diet before any experiments wereperformed on humans. There were cross-cultural comparisons and there were animalstudies. Actually, the only reason for perfor-ming the randomized experiment was thatsomeo ne believed there was pretty clear non-experimental evidence to begin with. Theno ne x~ er im en ta l vidence was: of course. in-conclusive, which in my language means thatthe misspecification uncertainty M remaineduncomfortably large. The fact that theJ a ~ a n e s e ave both less incidence of heartdisease and also diets lower in cholesterolcompared to Americans is not convincingevidence. because there are so manv otherfactors that remain unaccounted f ir . Th efact that pigs on a high cholesterol diet de-velop occluded arteries is also not convinc-ing,-because the similarity in physiology inpigs and humans can be questioned.When the sampling uncertainty S getssmall compared to the misspecification un-certainty M,i t is t ime to loo kfo r other formsof evidence, experiments or none xperime nts.Suppose I am interested in measuring thewidth of a coin. and I ~ r o v id e ulers to aroom of volunteers. After each volunteer hasreported a measurement, I comp ute the meanand standard deviation, and I conclude that


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34 T H E A , M E R I C A N E C O N O ,M I C R E V I E W ,MA R C l l I9 8 3

the coin has w idth 1.325 millimeters with astandard error of .013. Since thls amount ofuncertainty is not to my liking, I propose tofind three other rooms full of volunteers,thereby multiplying the sample size by four,and dividing the sta ndard error in half . Thatis a silly way to get a more accurate measure-ment, because I have already reached thepoint where the sampling uncertainty S isvery small compared with the misspecifica-tion uncertainty M. If I want to increase thetrue accuracy of my estimate, it is time forme to consider using a microm eter. So too inthe case of diet and heart disease. Medicalresearchers had more or less exhausted thevein of nonexperimental evidence, and it be-came time to switch to the more expensivebut richer vein of experimental evidence.In economics, too, we are switching toexperimental evidence. There are the labora-tory experiments of C harles Plott and V ernonSmith (1978) and Smith (1980), an d there arethe field experiments such as the Seattle/Denver income maintenance experiment.Another way to limit the rnisspecificationerror M is to gather different kinds of nonex-periments. orm m all^ speaking. we will saytha t experiment 1 is qualitatively differentfrom experimen t 2 if the bias param eters(a:. PT) are distributed independently of thebias parameters ( a ! ,P:). In that ev en t, sim -ple averaging of the data from the twoexperiments yields average bias parameters( a : + a!. /3 + /3:)/2 with rnisspecif icat ionvariance matrix M / 2 , half as large asthe (com mo n) individual variances. MiltonFriedman's study of the permanent incomehypothesis is the best example of t h s that Iknow. Other examples are hard to come by.I believe we need to put much more effortinto identifying qualitatively different andconvincing kinds of evidence.Parenthetically, I note that traditionaleconometric theory, whch does not admitexperimental bias, as a consequence also ad-mits no "hard core" propositions. Demandcurves can be shown to be positively sloped.Utility can be shown not to be maximized.Econometric evidence of a positively slopeddemand curve would, as a matter of fact, beroutinely explained in terms of simultaneitybias. If utility seems not to have been maxi-

mized, it is only that the econometrician hasrnisspecified the utility function. The mis-specification matrix M thus forms ImreLakatos' "protective belt" which protectscertain hard core propositions from falsifi-cation.11. Is C ontrol Essential?

Th e experimental scientist who notices thatthe fertilizer treatment is correlated with thelight level can correct his experimental de-sign. He ca n control the light level, or he canallocate the fertilizer treatment in such a waythat the fertilizer level and the light level arenot perfectly correlated.The nonexperimental scientist by defini-tion cannot control the levels of extraneousinfluences such as light. But he can controlfor the variable light level by including lightin the estimating equation. Provided naturedoes not select values for light and values forfertilizer levels that are perfectly correlated,the effect of fertilizer on yields can be esti-ma ted w ith a m ultiple regression. The collin-earity in naturally selected treatment vari-ables may mean that the data evidence isweak, but it does not invalidate in any waythe usual least squares estimates. Here, again,there is no essential difference between ex-perimental and nonexperimental inference.

111. Are the Degr ees of Freedom Inadequatewith Nonexperimental Data?

As a substitute for experimental control,the nonexperimental researcher is obligatedto include in the regression equ ation all vari-ables that might have an important effect.T h e NBER data banks contain time-seriesdata on 2,000 macroeconomic variables. Amodel explaining gross national product interms of all these variables would face asevere degrees-of-freedom deficit since thenumber of annual observations is less thanthirty. Thoug h the numbe r of observations ofany phenomenon is clearly limited, the num-ber of explanatory variables is logically un-limited. If a polynom ial could have a de greeas hlgh as k, it would usually be admittedthat the degree could be k + 1 as well. Atheory that allows k lagged explanatory vari-


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VOL-. 73 NO. I 1.EA-MER: TAKE THE C0.V OUT O F C(-'O!VOZ.IETRICS

FERTILIZER PER ACRE

ables would ordinarily allow k + 1. If thelevel of money might affect GNP, then whynot the number of presidential sneezes, orthe size of the polar ice cap?The number of explanatory variables isunlimited in a nonexperimental setting, butit is also unlimited in an experimental set-ting. Consider again the fertilizer example inwhch the farmer randomly decides either toapply F , pounds of fertilizer per acre or zeropounds, and obtains the data illustrated inFigure 1. These dat a adm it the inference thatfertilizer level F, produces higher yields thanno fertilizer. But the farmer is interested inselecting the fertilizer level that maximizesprofits. If it is hypothesized that yield is alinear function of the fertilizer intensity Y =a + P F + U , then profits are

where A is total acreage, p is the productprice. and p , is the price per pound of fertil-izer. This profit function is linear in F withslope A ( /3p - p,). The farmer maximizesprofits therefo re by using n o fertilizer if theprice of fertilizer is high, j3p < p,, and usingan unlimited amount of fertilizer if the priceis low. /3p > p,. It is to be expected that youwill find this answer unacceptable for one of

I l l I0 Fl F, F",

FERTILIZER PER ACRE

several reasons:1) When the farmer tries to buy anunlimited amount of fertilizer, he will driveup its price, and the problem should bereformulated to make p , a function of F.2) Uncertainty in the fertilizer effect j3causes uncertainty in profits , Variunce( p r o f i t s )= p 2 A 2 F 2V c t r ( P ) ,and risk aversionwill limit the level of fertilizer applied.3) The yield function is nonlinear.Economic theorists doubtless find reasons1) and 2) compelling, but I suspect that thereal reason farmers don't use huge amountsof fertilizer is that the marginal increase inthe yield eventually decreases. Plants don'tgrow in fertilizer alone.So let us suppose that yield is a quadraticfunction of fertilizer intensity, Y = a + P IF+ p, F~ + L7, and suppose we have only thedata illustrated in Figure 1. Unfortunately.there are an infinite number of quadratic

functions all of which fit the data equallywell, three of which are drawn. If there wereno other information available, we couldconclude only that the yield is hgher at F ,than at zero. Formally speaking, there is anidentification problem, whch can be solvedby altering the experim ental design. T he yieldmust be observed at a third point, as inFigure 2, where I have draw n the least squaresestimated quad ratic function and have indi-cated the fertilizer intensity F,, that maxi-mizes the yield. I expect that most peoplewould question whether these data admit the


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37OL 7 1 ' vO I LEA W F R T A KF THF COY OL T OF ECOVOZfLTR1C.Y

in fact, all the concepts of traditional theory ,utterly lose their meaning by the time anapplied researcher pulls from the bramble ofcomputer out pu t the one thorn of a model helikes best, the one he chooses to portray as arose. The consuming public is hardly fooledby this chicanery. The econometrician'sshabby art is humorously and disparaginglylabelled "da ta mining," "fishing," "grub -bing," "nu mb er crunching." A joke evokesthe Inquisit ion: "If you torture the data longenough, Nature wil l confess" (Coase) .Another suggests methodological fickleness:"Econometricians, like artists, tend to fall inlove with their models" (wag unknown). Orhow about: "There are two things you arebetter off not watching in the making:sausages and econometric estimates."This is a sad and decidedly unscientificstate of affairs we find ourselves in. Hardlyanyone takes data analyses seriously. Or per-haps more accurately, hardly anyone takesanyone else's data analyses seriously. Likeelaborately plumed birds w ho have long sincelost the ability to procreate but not the de-sire, we preen and strut and display ourt-values.

If we want to make progress, the first stepwe must take is to discard the counterpro-ductive goal of objective inference. The dic-tionary defines an inference as a logical con-clusion based on a set of facts. Th e "facts"used for statistical inference about 0 are firstthe data, symbolized by x, second a condi-tional probabili ty density, known as a sam-pling distribution, f( x l0 ), a nd , third, ex-plicitly for a Bayesian and implicitly for "allothers," a marginal or prior probabili ty d en-sity function f (0 ) . Because both the sam-pling distribution and the prior distributionare actually oplnions and not facts , a s tatis-tical inference is and must forever remain anopinlon. W hat is a fact? A fact is merely an op inionheld by all, or at least held by a set of p eopleyou regard to be a close approximation toall . ' For some that set includes only one

'This notion of "truth bq consen.\usn is espoused bqThomas Kuhn ( 1962) and Michael Polanqi ( 1964). OscarWilde agrees bq dissent: "A truth ceases to be true whenrnore than one person believes it."

person. I myself have the opinion thatAndrew Jackson was the sixteenth presidentof the Unite d States. If many of my friendsagree, I may take it to be a fact. Actually, Iam m ost likely to regard it to be a fa ct if theauthor s of one or more books say it is so.The difference between a fact and an op in-ion for purposes of decision making andinference is that when I use opinions, I getuncomfortable. I am not too uncomfortablewith the opinion that error terms are nor-mally distributed because most econometri-cians make use of that assumption. Thisobservation has deluded me into thinkingthat the opinion that error terms are normalmay be a fact, when I know deep inside thatnormal distributions are actually used onlyfor convenience. In contrast, I am quite un-comfortable using a prior distribution, mostlyI suspect because hardly anyone uses them.If c o nv en ie n t ~ r i o r istributions were used asoften as convenient sampling distributions, Isuspect that I co uld be as easily deluded in tothinking that prior distributions ar e facts as Ihave been into thinking that sam pling distri-butions are facts .To emphasize this hierarchy of statements,I display them in order: truths; facts; opin-ions ; conventions. No te that I have added tothe top of the or der, the category tru ths. Thiswill appeal to those of you who feel com-pelled to believe in such things. At the bot-tom are conventions. In practice, i t may bedifficult to distinguish a fact from a conven-tion, but when facts are clearly unavailable,we must strongly resist the deceit or delusionthat con ventions can represent.What troubles me about using opinions istheir whimsical nature. Some niornings whenI arise, I have the opinion that R aisin Bran isbetter than eggs. By the time I get to thekitchen, I may well decide on eggs, oroatmeal. I usuallv do recall that the sixteenthpresident distinguished himself. Sometimes Ithink he was Jackson; often I think he wasLincoln.A data analysis is similar. Sometimes Itake the error terms to be correlated, some-times uncorrelated; som etimes normal andsometimes nonnormal; sometimes I includeobservations from the decade of the fifties,sometimes I exclude them; sometimes the


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38 THI- 4 t f 1 5 R l C AY F C O Z 'O M I C R E V I L W M4 R C H I983

equation is linear and sometimes nonlinear;sometimes I control for variable z , some-times I don't. Does it depend on what I hadfor breakfast?As I see i t , the fundamental problem fac-ing econometrics is how adequately to con-trol the whimsical character of inference, howsensibly to base inferences on opinions whenfacts are unavailable. At least a partial solu-tion to this problem has already been formedby practicing econometricians. A commonreporting style is to record the inferencesimplied by alternative sets of opinions. It isnot unusual to find tables that show how aninference changes as variables are added toor deleted from the equation. This kind ofsensitivity analysis reports special features ofthe mapping from the space of assumptionsto the space of inferences. The defect of thisstyle is that the coverage of assumptions isinfinitesimal. in fact a zero volume set in thespace of assumptions. What is needed in-stead is a more complete, but still economi-cal way to report the mapping of assump-tions into inferences. What I propose to do isto develop a correspondence between regionsin the assumption space and regions in theinference space. I will report that all assump-tions in a certain set lead to essentially thesame inference. Or I will report that thereare assumptions within the set under consid-eration that lead to radically different in-ferences. In the latter case, I will suspendinference and decision, or I will work harderto narrow the set of assumptions.Thus what I am asserting is that the choiceof a particular sampling distribution, or aparticular prior distribution, is inherentlywhimsical. But statements such as "The sam-pling distribution is symmetric and uni-modal" and "My prior is located at theorigin" are not necessarily whimsical, and incertain circumstances do not make me un-comfortable.To put this somewhat differently, an in-ference is not believable if it is fragile, if itcan be reversed by minor changes in assump-tions. As consumers of research, we correctlyreserve judgment on an inference until itstands up to a study of fragility. usually byother researchers advocating opposite opin-ions. It is. however. much more efficient for

individual researchers to perform their ownsensitivity analyses, and we ought to be de-manding much more complete and morehonest reporting of the fragility of claimedinferences.The job of a researcher is then to reporteconomically and informatively the mappingfrom assumptions into inferences. In a slogan,"The mapping is the message." The mappingdoes not depend on opinions (assumptions),but reporting the mapping economically andinformatively does. A researcher has to de-cide which assumptions or which sets of al-ternative assumptions are worth reporting. Aresearcher is therefore forced either to antic-ipate the opinions of his consuming public,or to recommend his own opinions. It isactually a good idea to do both, and a seri-ous defect of current practice is that it con-centrates excessively on convincing one's selfand, as a consequence, fails to convince thegeneral professional audience.The whimsical character of econometricinference has been partially controlled in thepast by an incomplete sensitivity analysis. Ithas also been controlled by the use of con-ventions. The normal distribution is now socommon that there is nothing at all whimsi-cal in its use. In some areas of study, the listof variables is partially conventional, oftenbased on whatever list the first researcherhappened to select. Even conventional priordistributions have been proposed and areused with nonnegligible frequency. I am re-ferring to Robert Shiller's (1973) smoothnessprior for distributed lag analysis and toArthur Hoerl and Robert Kennard's (1970)ridge regression prior. It used to aggravateme that these methods seem to find publicfavor whereas overt and complete Bayesianmethods such as my own proposals (1972)for distributed lag priors are generallyignored. However, there is a very good rea-son for this: the attempt to form a priordistribution from scratch involves an untoldnumber of partly arbitrary decisions. Thepublic is rightfully resistant to the whimsicalinferences which result, but at the same timeis receptive to the use of priors in ways thatcontrol the whimsy. Though the use of con-ventions does control the whimsy, it can doso at the cost of relevance. Inferences based


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39OI.. ' 3 T O . I I ,E A ,b fE R : T A K E T H E C 0 .V O U T O F E C 0 , V O M E T R I C S

on Hoerl and Kenn ard's conventional "ridgeregression" prior are usually irrelevant, be-cause it is rarely sensible to tak e the prio r tobe spherical and located at the origin, andbecause a closer approximation to prior be-lief can be suspected to lead to substantiallydifferent inferences. In contrast, the conven-tional assump tion of n ormality a t least uses adistribution which usually cannot be ruledout altogether. Still , we may properly de-mand a demonstrat ion that the inferencesare insensitive to this distributional assump-tion.A. The Horizon Problem: Sherlock

Holrvles InferenceConventions are not to be ruled out al to-gether, however. One can go mad trying toreport completely the m apping from assump-tions into inferences since the space of as-sumptions is infinite dimensional. A formalstatistical analysis therefore has to be donewithin the limits of a reasonable horizon. Aninformed convention can usefully limit thishorizon. If it turned out that sensible neigh-borhoods of distributions around the normal

distribution 99 times out of 100 producedthe same inference, then we could all agreethat there are other more important things toworry abou t , and we may properly adop t theconvention of normality. The consistency ofleast squares estimates under wide sets ofassumptions is used improperly as supportfor this convention, since the inferences froma given finite sample may nonetheless bequi te sensit ive to the normali ty a s ~ u m p t i o n . ~The truly sharp distinction between in-ference from experimental and inferencefrom nonexperimental data is that experi-mental inference sensibly admits a conven-tional horizon in a critical dime nsion, namelythe choice of explanatory variables. If fertil-izer is randomly assigned to plots of land, itis conventional to restrict attention to therelationship between yield an d fertilizer, and' In particular. least squar es estima tes are com pletel)sensitive to the indepen dence a.\sumption, since b) choice

of sarnple covariance matrix a generalized least squaresest imate can be made to assume an! value w ha tso e~ er( s ee m! 1981 p ap e r )

to proceed as if the model were perfectlyspecified, which in my notation means thatthe misspecification matrix M is the zeromatrix. There is only a small risk that whenyou present your findings, someone will ob-ject that fertilizer and light level are corre-lated, and there is an even smaller risk thatthe conventional zero value for M will leadto inappropriate inferences. In contrast , i twould be foolhardy to adopt such a limitedhorizon with nonexperimental data. But ifyou decide to include light level in yourhorizon, then why not rainfall; and if rain-fall, then why not temperature; and if tem-pera ture, then why no t soil depth, an d if soildepth, then why not the soil grade; ad in -finitum. Though this list is never ending, itcan be made so long that a nonexperimentalresearcher can feel as comfortable as an ex-perimental researcher that the risk of havinghis findings upset by an extension of thehorizon is very low. The exact point wherethe list is terminated must be whimsical, butthe inferences can be expected not to besensitive to the termination point if thehorizon is wide enough.Still, the horizon within which we all doour statistical analyses has to be ultimatelytroublesome, since there is no formal way toknow what inferential monsters lurk beyondour immediate field of vision. "Diagnostic"tests with explicit alternative hypothese s suchas the Durbin-Watson test for first-order au-tocorrelation do not truly ask if the horizonshould be extended, since first-order au-tocorrelation is explicitly identified andclearly in our field of vision. Diagnostic testssuch as goodness-of-fit tests, without explicitalternative hypotheses, are useless since, ifthe sample size is large enough, any main-tained hypothesis will be rejected (for exam-ple, no observed distribution is exactly nor-mal). Such tests therefore degenerate intoelaborate rituals for measuring the effectivesample size.The only way I know to ask the questionwhether the horizon is wide enough is tostudy the anom alies of the da ta. In the wordsof the physiologist, C. Bernard:

A great surgeon performs operationsfor stones by a single method; later he


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40 T H E A M E R I C A N E C O N OM I C R E V I E W M A R C H 1983

makes a statistical summary of deathsand recoveries, and he concludes fromthese statistics that the mortality lawfor this operation is two out of five.Well, I say that this ratio means liter-ally n o th n g scientifically, and gives nocertainty in performing the next opera-tion. What really should be done, in-stead of gathering facts empirically, isto study them more accurately, each inits special d eterm inism .. .b y statistics,we get a conjecture of greater or lessprobability about a given case, butnever any certainty, never any ab solutedeterminism. . .only basing itself on ex-perimental determinism can medicinebecome a true science.11927, pp . 137-381A study of the anomalies of the data iswhat I have called "Sherlock Holmes" in-ference. since Holmes turns statistical in-ference on its head: "It is a capital mistaketo theorize before you have all the evidence.It biases the judgem ents." Statistical theorycounsels us to begin with an elicitation ofopinions about the sampling process and itsparam eters; the theory, in other words. After

that, data may be studied in a purely me-chanical way. Holmes warns that this biasesthe judgements, m eaning that a theory con-structed before seeing the facts can be disas-trously inappropriate and psychologicallydifficult to discard. But if theories are con-structed after having studied the data, it isdifficult to establish by how m uch, if at all,the d ata favor the d ata-instigated hypothesis.For example, suppose I think that a certaincoefficient ought t o be positive, an d m y reac-tion to the anomalous result of a negativeestimate is to find another variable to in-clude in the equation so that the estimate ispositive. Have I found evidence that thecoefficient is positive? It would seem that weshould reau ire evidence that is mo re convinc-ing than 'the traditional stan dard . I haveproposed a method for discounting such evi-den ce (1974). Initially, when you regress yieldon fertilizer as in equation (2), you are re-quired to assess a prior distribution for theexperimental bias parameter P*; that is, youmust select the misspecification matrix M.Then , when the le as t squares estimate of P

turns out to be negative, and you decide toinclud e in t he equ ation the Light level as wellas the fertilizer level, you are obligated toform a prior for the light coefficient y corl-sistent with the prior for p*, given tha t p* =y r , , where r , is the regression coefficient oflight on f e r t i l i ~ e r . ~T h s method for discounting the output ofexploratory data analysis requires a disci-pline that is lacking even in its author. It isconsequently important that we reduce therisk of Holmesian discoveries by extendingthe horizon reasonably far. The degree of apolynomial or the order of a distributed lagneed n ot be da ta instigated, since the horizonis easily extended to include l g h degreesand high orders. It is similarly wise to askyourself before examining the dat a what youwould do if the estim ate of yo ur favoritecoefficient ha d th e wron g sign. If that m akesyou t l n k of a specific left-out variable, it isbetter to include it from the beginning.Though it is wise to select a wide horizonto reduLe the risk of Holmesian discoveries,it is mistaken then to analyze a data set as ifthe horizon were wide enough. Within thelimits of a horizon, no revolutionary in-ference can be made, since all possible infer-ences are predicted in advance (admittedly,some with low probabilities). Within thehorizon, inference and decision can be turn edover completely to a com puter. But the greathuman revolutionary discoveries are madewhen the horizon is extended for reasonsthat cannot be predicted in advance andcan not be com puterized. If you w ish to makesuch discoveries, you will have to poke at thehorizon, and poke again.

V. An ExampleT h s rhetoric is understandably t iring.Methodology, like sex, is better demon-strated than discussed, though often betteranticipated than experienced. Accordingly,let me give you an example of what all this' ~ n r an d omi ze d e xp e rimen t wi th r ,= 0, the con-stra int p* = y r , is irrelevant, and you are free to playthese exploratory games without penalty. This is a very

critical difference between randomized experiments andnonrandomized nonexperiments.


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V O L . 73 .YO. I LE.4 M E R : T A K E T I f E C O N O U T O F E C O ~ V O M E T R I C S ill

ranting and raving is about. I trust you willfind it even better in the experience than inthe anticipation. A problem of considerablepolicy importance is whether or not to havecapital punishment. If capital punishmenthad no deterrent value, most of us wouldprefer not to impose such an irreversiblepunishment, though, for a significant rninor-ity, the pure joy of vengeance is reasonenough. The deterrent value of capitalpunishment is, of course, an empirical issue.The unresolved debate over its effectivenessbegan wh en evolution was judging thesurvival value of the vengeance gene. Naturewas unab le to m ake a decisive judgm ent.Possibly econometricians can.In Table 1, you will find a list of variablesthat are hypothesized to influence the mu rderrate.4 The d ata to be examined are state-by-state murder rates in 1950. The variables aredivided into three sets. There are four deter-rent variables that characterize the criminaljustice system, or in economic parlance, theexpected out-of-pocket cost of crime. Thereare four economic variables that measurethe opportunity cost of crime. And thereare four social/environm ental variables thatpossibly condition the taste for crime. Thisleaves unmeasured only the expected re-wards for criminal behavior, though theseare possibly related to the economic andsocial variables and are otherwise assumednot to vary from state to state.A simple regression of the murder rate onall these variables leads to the conclusionthat each additional execution d eters thirteenmurders, uith a standard error of seven.Th at seems like such a healthy rate of retu rn,we might want just to randomly draft ex-ecutees from the population at large. Thisproposal would be unlikely to withstandthe scrutiny of any macroeconomists whoare slulled at finding rational expectationsequlibria.Th e issue I would like to addres s instead iswhether this conclusion is fragile or not.Do es it hold up if the list of variable s in themodel is changed? Individuals with differentexperiences and different training will find4 ~ h saterial is taken from a study by a student ofmine. Walter McManus ( 1982).

TABLL-VARIABLESUSED N THE ANALYSIS

a. Dependent VariableM =Murder rate per 100,000, FBI estimate.b. Independent Deterrent VariablesP C = (Conditional) Probability of conviction for

murder given commisaion. Defined by PC =C / Q . where C = convictions for murder, Q = M~ V S ,l'S = state population. This is to correctfor the fact that M is an estimate based on asample from each state.P X = (Conditional) Probability of execution givenconviction (average number of executions1946-50 divided by C).

T= Median time served in months for murder byprisoners released in 195 1.X P O S =A dummy equal to 1 if P X > 0.

c. Independent Economic VariablesW =Median income of families in 1949.X = Percent of families in 1949 with less than one-

half Mi. U =Unemployment rate. L F = Labor force participation rate. d. Independent Social and Environmental Variables

N W = Percent nonwhite.A G E = Percent 15-24 years old.U R B = Percent urban.M A L E =Percent male.F A M H O = Percent of families that are husband andwife both present families.SOC'Tl I= A dummy equal to 1 for southern states(Alabama, Arkansas, Delaware. Florida,Kentucky, Louisiana, Macland, Missis-sippi, North Carolina. Oklahoma. SouthCarolina. Tennessee. Texas, Virginia, WestVirginia).

e. Weighting VariableS Q R T ? I F = Square root of the population of theFBI-reporting region. Note that weight-ing is done by multiplying variables bySQRT.VF.

f . Level of ObservationObservations are for 44 states, 35 executing and 9nonexecuting. The executing states are: Alabama,Arizona, Arkansas. California, Colorado, Connecti-cut, Delaware, Florida, Illinois. Indiana, Kansas,Kentucky, Louisiana, Maryland, Massachusetts, Mis-sissippi. Missouri. Nebraska, Nevada, New Jersey,New Mexico, New York. North Carolina. Oho,Oklahoma, Oregon, Pennsylvania, South Carolina,South Dakota. Tennessee, Texas, Virginia, Washng-ton. West Virginia.

The nonexecuting states are: Idaho, Maine, Min-nesota. Montana, New Hampshre, Rhode Island,Utah, Wisconsin, Wyoming.

different subsets of the variables to becandidates for omission from the equation.Five different lists of doubtful variables arereported in Table 2. A right winger expects
http:///reader/full/SQRT.VFhttp:///reader/full/SQRT.VF


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42 T H C 4 M E R I C 4 I F C O I O W I C R E V I E W M A R C I I 1 98 3

Prior PC PX' T X P O S M/ X ti L F .VU' A G E C R B M.4 LE F 4 W H O S O t i T I fRight Winger 1 1 1 * D D D D D D D D D DRational hlaximizer I I I + I I I I D D D D 1) DE3e-for-an-Eye I I D * D D D D D D D D D DBleeding Heart D D D + 1 1 1 1 D D D D D DCrim e of Passion D I) D + I I I I I I I I I INore,r: 1) I indicates variables considered important by a researcher with the respecti\,e prior. Thus, every modelconsidered by the researcher will include these variables. D indicates variables considered doubtful by the researcher.* indicates X P O S , the dumm y equal to 1 for executing states . Each pr ior was pooled with the dat a two ways: onewith X P O S t reated as imp ortant , and on e with i t as doubtful .2) With five basic priors and X P O S treated as doubtf ul or imp ortan t by ea ch, we get ten alternative priorspecifications.

~ - E X T R E L L E OF THE EFFECTON MURDERSthe punishment variables to have an effect, TABLE E S T I . ~ T E S OFbu t treats al l other variables as doub tful . He EXECUTIONSwants to know whether the data still favorthe large deterr ent effect, if h e om its some of Minimum MaximumPrior Estimate Estimatethese doubtful variables. The rational maxi-mizer takes the variables that measure the Right Winger - 22.56 - .86expected economic return of crime as im- Rational Maximizer - 15.91 - 10.24portant, but treats the taste variables as Eye-for-an-E!e - 28.66 1.91Bleeding Hea rt -25.59 12.37doubtful. The eye-for-an-eye prior treats all Crime of Passion - 17 32 4.10variables as doubtful except the probabilityof execution. An individual with the bleeding Nort: Least squares is - 13.22 with a standard error of

heart prior sees murder as the result of eco- 7.2.nomic impoverishment. Finally, if murder isthought to be a crime of passion then thepunishment variables are doubtful. I come away from a study of Table 3 withIn Table 3. I have listed the extreme esti- the feeling that an y inference from these dat amates that could be found by each of these about the deterrent effect of capital punish-groups of researchers. The right-winger min- ment is too fragile to be believed. It is possi-im um of -22.56 me ans that a regression of ble credibly to narrow the set of assump-the murder rate data on the three punish- tions, but I d o not think tha t a credibly largement variables and a suitably selected linear set of alternative assumptions will lead to acombination of the other variables yields an sharp set of estimates. In another paperestim ate of the deterren t effect equal to 22.56 (1982), I found a narrower set of priors stilllives per execution. It is possible also to find leads to inconclusive inferences. And I havean estimate of - .86. Any thing between these ignored the important simultaneity issue (thetwo extremes can be similarly obtained; but death penalty may have been imposed inno estimate outside this interval can be gen- crime ridden states to dete r murder) which iserated no matter how the doubtful variables often a so urce of great inferential fragility.are manipulated (linearly). Thus the rightwinger can report that the inference from VI. Conclusionsthis data set that executions deter murders isnot fragile. The rational maximizer similarly After three decades of churning out esti-finds that conclusion insensitive to choice of mates, the econometrics club finds itself un-model, but the other three priors allow ex- der critical scrutiny and faces incredulity asecution actually to encourage murder, possi- never before. Fischer Black writes of "Thebly by a brutalizing effect on society. Troub le with Econom etric Models." David


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You have printed the following article:

Let's Take the Con Out of Econometrics

Edward E. Leamer

The American Economic Review, Vol. 73, No. 1. (Mar., 1983), pp. 31-43.

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Econometrics-Alchemy or Science?

David F. Hendry

Economica, New Series, Vol. 47, No. 188. (Nov., 1980), pp. 387-406.

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Ridge Regression: Biased Estimation for Nonorthogonal Problems

Arthur E. Hoerl; Robert W. Kennard

Technometrics, Vol. 12, No. 1. (Feb., 1970), pp. 55-67.

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A Class of Informative Priors and Distributed Lag Analysis

Edward E. Leamer

Econometrica, Vol. 40, No. 6. (Nov., 1972), pp. 1059-1081.

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Sets of Posterior Means with Bounded Variance Priors

Edward E. Leamer

Econometrica, Vol. 50, No. 3. (May, 1982), pp. 725-736.

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An Experimental Examination of Two Exchange Institutions

Charles R. Plott; Vernon L. Smith

The Review of Economic Studies, Vol. 45, No. 1. (Feb., 1978), pp. 133-153.

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A Distributed Lag Estimator Derived from Smoothness Priors

Robert J. ShillerEconometrica, Vol. 41, No. 4. (Jul., 1973), pp. 775-788.

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Macroeconomics and Reality

Christopher A. Sims

Econometrica, Vol. 48, No. 1. (Jan., 1980), pp. 1-48.

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JournalofAgriculturaland ResourceEconomics, 17(2): 294-302Copyright 1992 Western Agricultural Economics Association

A Linear Inverse Demand SystemGiancarlo Moschini and Anuradha Vissa

We present an inverse demand system that can be estimated in a linear form.The model is derived from a specification of the distance function which isparametrically similar to the cost function underlying the Almost Ideal DemandSystem. Simulation results suggest that this linear inverse demand system hasgood approximation properties.Key words: Almost Ideal Demand System, demand analysis, distance func-tion, duality.

Introduction

The Almost Ideal Demand System (ALIDS) of Deaton and Muellbauer is one of the mostcommonly used in applied demand analysis. While the ideal connotation of this modelstems from its aggregation properties, it is arguable that one of the main reasons for itspopularity is the availability of an approximate version of this system that is linear inthe parameters; in fact, it is this linear version of the ALIDS model that is typicallyestimated (Heien and Wessells; Gould, Cox, and Perali; Moschini and Meilke). Thepurpose of this article is to illustrate how a linear system for inverse demand equationsthat resembles the ALIDS model can be derived, and we term this system the LinearInverse Demand System (LIDS).'Inverse demand functions, where prices are functions of quantities, provide an alter-native and fully dual approach to the standard analysis of consumer demand (Anderson),and may be more appropriate when quantities are exogenously given and it is the pricethat must adjust to clear the market (Barten and Bettendorf). This situation is likely tobe of relevance to modeling agricultural demand using data based on frequent time seriesobservations (say monthly or quarterly). The chief advantage of using LIDS to modelinverse demands is linearity, which may be useful for some applications (say large demandsystems or systems involving dynamic adjustment). Although the parametric structure ofthe model that we present is similar to that of ALIDS, it does not claim the sameaggregation properties. Nonetheless, its simplicity and its approximation abilities, doc-umented in this article, are likely to make the LIDS model suitable for empirical studies.

Duality and the Linear Inverse Demand SystemCommonly used demand systems typically are derived from parameterizations of dualrepresentations of preferences through the derivative properties. This approach ensuresintegrability of the resulting demand equations by construction. To derive an inversedemand system, one can start either from the direct utility function and exploit Wold'sidentity (which yields ordinary inverse demands), or start from the distance (transfor-mation) function and exploit Shephard's theorem (which yields compensated inverse


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A Linear Inverse Demand System 295demand functions) (Weymark). As will be clear in what follows, for ou r purposes it isbetter to start with the distance function, an alternative representation of preferenceswhich has proved convenient in related contexts (Deaton).If U(q) is the direct utility function, where q denotes the vector of quantities, thetransformation or distance function F(u, q) is implicitly defined by U[q/F(u, q)] - u,where u is the reference utility level. Under standard regularity conditions, F(u, q) iscontinuous in (u , q), decreasing in u, and nondecreasing, concave, and homogeneous ofdegree one in q. These properties establish a useful parallel between the distance functionand the cost function C(u, p) derived from the utility-constrained expenditure minimi-zation problem (where p is the price vector corresponding to q). As Blackorby, Primont,and Russell put it (p. 27), ". .. except for the direction of monotonicity of the utilityvariable, these conditions suggest that C could be interpreted as a transformation functionand F as a cost function."The parallel features of cost and distance functions are useful because, as emphasizedby Hanoch, they imply that any standard functional form for the cost function can beapplied also to the distance function. 2 The preceding discussion is pertinent to the problemat hand because the useful linear form of the approximate ALIDS model is made possibleby the specific functional form chosen for the cost function. Exchanging the role of thevariables (u , p) in the PIGLOG cost function of the ALIDS model with the variables (-u,q) of the distance function, where the negative sign on u emphasizes the opposite monoto-nicity direction ofFand C relative to the utility index, one obtains the following parametricspecification for F(u, q):(1) ln(F) = a(q) - ub(q),where a(q) and b(q) are quantity aggregator functions defined as:(2) a(q) = ao + , aoln(q) + - C yjln(q3)ln(q),

i i j

(3) b(q)= fo0 qfi.Because F(u, q) is homogeneous of degree one in q, the following restrictions apply: Zi a,= 1, j yi= 2;i yi = 0, and ,i i = 0. Also, without loss of generality, yi = ji (the symmetryproperty).From Shephard's theorem, the first derivatives of the distance function yield compen-sated inverse demands as iri = OF/Oq, = h(u, q), where r, = p,/x is the normalized priceof the ith good (the nominal price divided by total expenditure x). Because at F = 1 thedistance function is an implicit form of the direct utility function, then (1) implies theutility function U(q) = a(q)/b(q). This, together with the derivative property, implies thatthe uncompensated inverse demand functions associated with (1)-(3) can be written inshare form as:(4) w = a, + iln(qi) fln(Q),where w 7riqi is the ith budget share, and ln(Q) is a quantity index defined as ln(Q)a(q).The distance function in (1)-(3) has the same parametric structure of the PIGLOG costfunction of the ALIDS model. It should be clear, however, that this distance function isnot dual to the PIGLOG cost function of the ALIDS model. It follows that the aggregationproperties of the ALIDS are not shared by the inverse demand system in (4). Hence, theattribute "Almost Ideal," used by Eales and Unnevehr and by Barten an d Bettendorf to

Moschini and Vissa


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JournalofAgriculturaland Resource EconomicsIn practice, however, ln(Q) can be replaced by an index ln(Q*) constructed prior toestimation of the share system to yield:(5) wi =a + yln(qj) - ln(Q*).

The resulting set of equations (5) is a linear system of inverse demands, the LIDS model.Many index formulae for ln(Q*) may be considered here. Similar to the original suggestionof Deaton and Muellbauer, one may use the geometric aggregator ln(Q*) = Zi wiln(qi),although other indices (say Diewert's superlative indices) may have better approximationproperties. It should be understood, however, that in general quantities must be properlyscaled for the geometric aggregator to be admissible. This point also applies to the equiv-alent price aggregator of direct ALIDS models, typically referred to as the Stone priceindex.3The inverse demand system presented here satisfies standard flexibility properties. Itcan be verified that the distance function (2)-(4) has enough parameters to be a flexiblefunctional form for an arbitrary distance function once it is realized that the ordinalityof utility always allows one to put d21n(F)/du2 = 0 at a point.4The notion of flexible functional form in demand perhaps is defined more usefully interms of demand functions (which are ultimately estimated) rather than in terms of thefunction representing preferences (which are unobservable). Hence, a flexible inversedemand system must have enough parameters to approximate, at a point, an arbitraryset of quantity elasticities and of normalized price levels (i.e., it must provide a first orderlocal approximation to an arbitrary inverse demand system). If n is the number of goods,it is verified that (after imposing homogeneity, adding-up, and symmetry) the demandsystem (5) has 1/2(n - 1)(n + 4) free parameters [(n - 1) parameters a,, (n - 1) parametersf, and /2n(n - 1) parameters ij]. These constants could be chosen to represent at apoint an arbitrary set of quantity elasticities [of which 1/2n(n + 1) - 1 are independentafter accounting for homogeneity, adding-up, and symmetry] and an arbitrary set of left-hand-side shares [of which (n - 1) are independent after accounting for adding-up].

Simulation ResultsTo illustrate the approximation properties of the LIDS model, we report the results of asmall simulation exercise. Specifically, we generate repeated stochastic realizations froma known structure and then look at how close the elasticity estimates from LIDS are tothe true ones. Following similar studies by Kiefer and MacKinnon, and Wales, the datagenerating model chosen is a Linear Expenditure System (LES). Specifically, shares for athree-good system are generated using the inverse share equations of LES; that is,

ai[qif(qi - 7.)](6) wi ~ aj[q/(q- - )

where i,a,= 1.The quantity data that we use for qj, q2, and q3 are U.S. per-capita demandof beef, pork, and chicken, respectively, for the period 1960-89. These data, normalizedto equal one at the mean of the sample period, are reported in the appendix. 5 The pa-rameters used are: a1 = .5, a 2 = .3, a 3 = .2, Y1= .2, 72 = .3, and 73 = -. 3. From thisstructure we generated 250 samples, each with 30 observations, by appending multinormaldisturbances to the shares. The (full) covariance matrix used to generate the multinormalerrors is the same as that used by Kiefer and MacKinnon, and Wales; that is,

296 December 1992


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A Linear InverseDemandSystem 297With these data, we estimate five different models 250 times. First, we estimate thenonlinear inverse demand system of equation (4), and we label this system NLIDS. Similarto the case of ALIDS discussed by Deaton and Muellbauer, the parameter a0 is virtuallyimpossible to estimate, so we set a0 = 0.6 Second, we estimate the LIDS model, that isequation (5) with the geometric index ln(Q*) = zj wjln(qj). Third, as a benchmark, we

estimate the true LES model of equation (6). Note that while LES has five free parameters,both NLIDS and LIDS have seven free parameters. Finally, for comparison, we estimatetwo versions of the inverse translog (ITL) system introduced by Christensen, Jorgenson,and Lau, and applied by Christensen and Manser, which, after an arbitrary normalizationof parameters, can be written as:ai + fPijln(q)

(8) J(8) 1 + Z f3jln(qj)'j i

where 2i ai = 1 and fi3 = fji-It can be verified that the ITL system has eight parameters, one more parameter thanthe LIDS model. Hence, ITL has one more parameter than is needed to make it a flexible(local) approximation to an arbitrary utility, which means that (8) could be suitablyrestricted without affecting its flexibility properties. Specifically, one can always find amonotonic transformation of utility such that i Zj d2U/dln(qi)dln(qj) = 0 at a point. Tomake this argument more explicit, let U(q) denote an arbitrary utility function for which,at a point qO , U/Cdln(q,) = ai and 02 U/dln(qi)dln(qj)= ai. Because U(q) is ordinal, one canput i a, = 1 without loss of generality. Now consider the monotonic transformation U= G(U(q)). Then, at the point qO , 2 U/0ln(qi)dln(qj)= (G"aia,+ G'aij),where the derivativesG' and G" are evaluated at U(q). Hence, at the point qO , Zi j 02U/dln(q)dln(qj) = G" +G' (2Si 2 ai). If one chooses the transformation G(.) such that, at the point qO, G' = 1 and

G" = -(2,i j ai), then at this point ,i ij d2U/dln(q,)dln(qj) = 0. Because in the translogutility underlying (8), iv= a2 U/ln(qi)dln(qj), it follows that we can set iZj 3 = 0 and stillhave a local approximation to an arbitrary utility function.7 Given that the translog model(8) with the normalization Zi Zj fl = 0 achieves what Barnett and Lee called the "mini-mality" property, the resulting model is termed here the minimal inverse translog (MITL).Like LIDS and NLIDS (with ao = 0), MITL has seven free parameters.The approximation properties of the models considered are illustrated in terms of "howclose" the estimated elasticities are to the true elasticities. We consider uncompensatedquantity elasticities (flexibilities) and scale elasticities (in inverse demand analysis theconcept of scale effect, discussed by Anderson, plays a role similar to that of the incomeeffect of direct demands). Quantity elasticities are defined asij dln(p,)/dln(qj), and scaleelasticities are defined as ei dln[pj(Oq)]/0ln(O). Quantity elasticities for LES are com-puted as:(9) = Wi(q - qiwhereas for LIDS and NLIDS they are computed as:(10) Ei ~ Faj + 2 ykln(qk)) -bWi W.i k -and for ITL and MITL they are computed as:

Moschini and Vissa


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JournalofAgriculturalandResource Economicswhere bi is the Kronecker delta (6 = 1 for i = j and 6b= 0 otherwise). Scale elasticitiesare readily computed using (9)-(11) because Ei = j ijiA possible issue, in light of the arguments presented in Green and Alston, is whether(10) is an appropriate formula for LIDS. It is verified easily that under procedures wehave followed (that is, scaling the right-hand-side variables, and setting a0 = 0), eachparameter of LIDS will approximate the corresponding parameter ofNLIDS. Thus, for-mula (10), which is derived from NLIDS, is appropriate for LIDS as well. An alternativefor LIDS, which is consistent with taking ln(Q*) as given in estimation, is to use:(12) w- -wi Wi

To investigate what may be called the "local" approximation properties of the model,elasticities were computed at the mean point (at which qi = 1 V i), and summary statisticsare reported in table 1.8 The first column of table 1 reports the elasticities, at the meanpoint, of the true LES model used to generate the data. Then, for each ofLES, ITL, MITL,NLIDS, and LIDS we report the mean, computed over the 250 replications, of the esti-mated elasticities at the mean point, and the root mean square error (RMSE) of each ofthese estimated elasticities. Also, for each model we report the average RMSE for the 12elasticities involved.All models seem to provide a reasonable approximation. As expected, the best resultsare obtained by estimating the true LES model. The performances ofLIDS, NLIDS, andMITL are similar, with an average RMSE roughly double that of the true model. The factthat MITL does better than ITL perhaps may seem surprising. The reason is that therestriction (i 2,j j = 0) is not rejected; when estimating ITL, the empirical distributionof the quantity (,i 2j ri) over the 250 replications had a mean of .4 and a standarddeviation of 1.7. Maintaining the restriction (,i Zj i, = 0) in MITL results in a considerableefficiency gain (the average absolute t-ratio for the five independent fli in MITL over the250 replications was about 3, whereas the average absolute t-ratio for the six independentfijs in ITL was about 1.4).Table 1 makes it clear that the linear approximation made possible by the use ofln(Q*)instead of ln(Q) is very good, as LIDS and NLIDS produce virtually identical results. Inthe context of ALIDS for direct demands, it is believed that the use of the Stone indexis likely to produce good approximations because prices typically are highly correlated(Deaton and Muellbauer). In our application, however, the data are not very correlated:the coefficient of correlation between q, and q2 is -. 25, between q, and q3 is .05, andbetween q2 and q3 is .27. Yet the approximation made possible by the use ofln(Q*) appearsquite good, suggesting that it is robust to the design matrix of the exogenous variables.Although the results of table 1 are encouraging as to the approximation properties ofLIDS, and consistent with the notion that all the models considered (apart from the truemodel) are capable of providing a local approximation to an arbitrary demand system,the question arises as to "how local" these results are. If the inverse demand system isto be used for forecasting or welfare analysis, one would want to be reassured that theapproximation abilities of the model extend to a reasonably wide range of the data. Toinvestigate this issue, we consider what we term the "extended" approximation propertiesof the models. Specifically, we evaluate true and estimated elasticities at each of the 30sample points, and for each of the 12 elasticities we compute the mean square error overthe resulting 7,500 estimates (30 sample points for 250 replications).The square roots of such mean square errors, and their average over all 12 elasticities,are reported in table 2.9 The approximation abilities of MITL, NLIDS, and LIDS holdup very well in this extended analysis, with the average RMSE increasing only by .004relative to the approximation at the mean point (up 6.6%). For these models the average

298 December 1992


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23/33

JournalofAgriculturalandResource EconomicsTable 2. Extended Approximation Properties

Elasticity LES IT L MITL NLIDS LIDS---------- -----------------------------------------MSE -------------------------------------------------

El1 .024 .045 .029 .030 .030E12 .030 .043 .036 .036 .036E13 .005 .013 .009 .009 .009E21 .026 .051 .040 .041 .042E22 .053 .086 .071 .069 .069E23 .005 .017 .015 .016 .016E31 .026 .140 .096 .101 .106E32 .030 .132 .104 .107 .110E33 .033 .062 .043 .048 .049El .041 .069 .050 .051 .052E2 .063 .104 .087 .087 .089E3 .047 .277 .169 .178 .186Avg. .032 .087 .062 .064 .066

Note: Entries are RMSEs over all 30 sample points.ConclusionIn this article we have illustrated a linear specification for an inverse demand system.This specification is based on a distance function which has a parametric structure similarto the PIGLOG cost function underlying the ALIDS model commonly used for directdemand models. The approximation properties of the new model were illustrated with asimulation exercise. Of course, although the results presented are useful in terms of rankingthe models used relative to the performance of the true model, the actual size of theapproximation error (say, the average RMSE) cannot be generalized because it depends,among other things, on the design matrix of right-hand-side variables, on the structureand parameters of the true model, and on the signal-to-noise ratio of the stochastic terms.The simulation results show that the new (nonlinear) inverse demand system derivedfrom the chosen parametric specification of the distance function performs well relativeto the true model, and very similar to that of an (appropriately restricted) inverse translogdemand system. Moreover, the linear version of the new inverse demand system, whichwe have termed LIDS, results in a good approximation to the nonlinear model. Thesimplicity of LIDS is likely to make it a useful specification for empirical analysis, es-pecially in applications where linearity is appealing (for example, in dynamic demandsystems). Because the derivation of LIDS parallels that of ALIDS for direct demandsystems, the simulation results reported in this article are somewhat more general andcould be interpreted, with minor modifications, as evidence of the approximation prop-erties of ALIDS models, and as supporting the linear version of ALIDS as a good ap-proximation to the nonlinear ALIDS.

[Received July 1991; final revision receivedApril 1992.]

Notes1After the first draft of this article was completed, a paper by Eales and Unnevehr, giving a similar derivationof the linear inverse demand system, came to our attention. They call this system the "Inverse Almost IdealDemand System," and use it to model U.S. quarterly meat demand. Barten and Bettendorf also allude to an"Almost Ideal Inverse Demand System," but they do no t provide an explicit derivation.2 Hanoch formalizes this parallel further by developing the concept of"symmetric" duality, which in our case

300 December 1992


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A LinearInverse Demand System 301It is clear that the Stone index, or equivalently the geometric aggregator ln(Q*) = Zi wiln(q,), is not invariant tothe choice of units of measurement. This problem arises when one uses natural units (i.e., pounds or metrictons). In such a situation, an easy way to get around the problem is to scale prices (or quantities for LIDS) bydividing through by the mean. When one aggregate indices with a common base, such as in Deaton andMuellbauer, the problem clearly does not arise.

4 A similar argument applies to the ALIDS model (Deaton and Muellbauer, p. 313).5These data are from U.S. Department of Agriculture sources. The sample means were 78.417 lbs./capita(retail cut equivalent) for beef, 60.037 lbs./capita (retail cut equivalent) for pork, and 44.283 lbs./capita (ready-to-cook weight) for chicken.6 Fixing a0 basically entails a local normalization of the utility function at the point q, = 1 (the mean pointin ou r case).7 The direct demand system derived from an indirect translog utility function also has one more parameterthan the linear ALIDS model (o r the nonlinear ALIDS with ao set to some constant). In this context, imposingthe restriction ,i ,Z j , = 0 reduces the indirect translog utility function to be a member of the PIGLOG family

of preferences, thereby giving it desirable aggregation properties (Lewbel).8 Given that we are evaluating elasticities at the point q, = 1, the distinction between formulae (10) and (12)for LIDS is immaterial, as the two formulae reduce to the same expression at this point.9When evaluating elasticities away from the point q, = 1, formulae (10) and (12) for LIDS are not identical.However, for the three-digit rounding reported in table 2, formulae (10) and (12) give the same results, whereasat a five-digit rounding level formula (10) gives slightly smaller RMSEs.

ReferencesAnderson, R. W. "Some Theory of Inverse Demand for Applied Demand Analysis." Eur.Econ. Rev. 14(1980):281-90.Barnett, W. A., and Y. W. Lee. "The Global Properties of the Minflex-Laurent, Generalized Leontief, andTranslog Flexible Functional Forms." Econometrica53(1985):1421-37.Barten, A. P., and L. J. Bettendorf. "Price Formation of Fish: An Application of an Inverse Demand System."Eur. Econ. Rev. 33(1989):1509-25.Blackorby, C., D. Primont, and R. R. Russell. Duality, Separability, and FunctionalStructure: Theory andEconomic Applications. New York: North-Holland, 1978.Christensen, L. R., D. W. Jorgenson, and L. J. Lau. "Transcendental Logarithmic Utility Functions." Amer.Econ. Rev. 65(1975):367-83.Christensen, L. R., and M. E. Manser. "Estimating U.S. Consumer Preferences for Meat with a Flexible UtilityFunction." J. Econometrics 5(1977):37-53.Deaton, A. "The Distance Function in Consumer Behaviour with an Application to Index Numbers and OptimalTaxation." Rev. Econ. Stud. 46(1979):391-405.Deaton, A., and J. Muellbauer. "A n Almost Ideal Demand System." Amer. Econ. Rev. 70(1980):312-26.Diewert, W. E. "Index Numbers." In The New Palgrave-ADictionaryof Economics, Vol. 2, eds., J. Eatwell,M. Milgate, and P. Newman, pp. 767-80. Ne w York: The Stockton Press, 1987.Eales, J., and L. Unnevehr. "The Inverse Almost Ideal Demand System." Paper presented at the NCR-134Conference on Applied Price Analysis, Forecasting, and Market Risk Management, Chicago, April 1991.Gould, B. W., T. L. Cox, and F. Perali. "The Demand for Fluid Milk Products in the U.S.: A Demand SystemsApproach." West. J. Agr. Econ. 15(1990):1-12.Green, R., and J. M. Alston. "Elasticities in AIDS Models." Amer. J. Agr. Econ. 72(1990):442-45.Hanoch, G. "Symmetric Duality and Polar Production Functions." In ProductionEconomics: A DualApproachto Theory and Applications, Vol. 1, eds., M. Fuss and D. McFadden, pp . 111-31. Amsterdam: North-Holland, 1978.Heien, D. M., and C. R. Wessells. "The Demand for Dairy Products: Structure, Prediction, and Decomposition."Amer. J. Agr. Econ. 70(1988):219-28.Kiefer, N. M., and J. G. MacKinnon. "Small Sample Properties ofDemand System Estimates." In Studies inNonlinearEstimation, eds., S. M. Goldfeld and R. E. Quandt. Cambridge MA: Ballinger Publishing Co.,1976.Lewbel; A. "AIDS, Translog, and the Gorman Polar Form." Econ. Letters 24(1987):161-63.Moschini, G., and K. D. Meilke. "Modeling the Pattern of Structural Change in U.S. Meat Demand." Amer. J.Agr. Econ. 71(1989):253-61.Wales, T. J. "A Note on Likelihood Ratio Tests of Functional Form and Structural Change in Demand Systems."Econ. Letters. 14(1984):213-20.Weymark, J. A. "Duality Results in Demand Theory." Eur. Econ.Rev. 14(1980):377-95.

IMoschini antd Vissa


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JournalofAgriculturaland ResourceEconomicsAppendix

Table Al. Normalized U.S. Per-CapitaConsumption of Beef, Pork,and ChickenYear ql q2 q31960 .81870 1.00439 .627781961 .83911 .96108 .675201962 .84421 .98440 .672941963 .89139 1.01605 .695521964 .94240 1.01605 .704551965 .93858 .91111 .751981966 .99596 .90944 .801661967 1.01764 .99939 .819721968 1.04570 1.02937 .824241969 1.05207 1.00938 .860371970 1.07630 1.03104 .905531971 1.06738 1.13098 .905531972 1.09033 1.03936 .937151973 1.02657 .94942 .907791974 1.08905 1.02271 .914571975 1.12221 .84115 .901021976 1.20255 .89279 .959731977 1.16557 .92943 .989091978 1.11201 .92943 1.047801979 .99469 1.06102 1.135871980 .97428 1.13431 1.124581981 .98321 1.08101 1.158451982 .97938 .97440 1.190061983 .99724 1.03104 1.205871984 .99596 1.02437 1.246521985 1.00489 1.03270 1.300721986 .99979 .97607 1.325561987 .93603 .98440 1.415881988 .91945 1.05102 1.456531989 .87736 1.04270 1.53782

302 December 1992


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Dermot J. Hayes

Pioneer Chair in Agribusiness

Iowa State University

Ames, IA 50010-9802

Work: (515) 294-6185

Home: (515) 233-4309

EDUCATION:

UNIVERSITY OF CALIFORNIA, BERKELEY

Ph.D., 1986Major Field: International Trade

Minor: Comparative Economic Systems

UNIVERSITY OF CALIFORNIA, BERKELEY

Masters in Agricultural Economics, June 1982

Fields: Econometrics, Economic Theory

UNIVERSITY COLLEGE, DUBLIN

First Class Honors in Agricultural Economics, June 1981

Fields: Agricultural Economics

Awards:

2007 Fellow of the American Agricultural Economics AssociationRecipient of 2006 AAEA Publication of Enduring Quality Award for Valuing

Food Safety in Experimental Auction MarketsAmerican Journal of

Agricultural Economics, 1995 by Dermot Hayes, Jason Shogren,; Seung-

Youll Shin, andJames Kliebenstein

Recipient of the 2005 ISU J. H. Ellis Award for Excellence in UndergraduateIntroductory Teaching, a University level award for excellence in teaching

undergraduate introductory classes

Listed as an outstanding Faculty Member for 2005, 2004, 2003, 2002, 2001,2000 by the ISU Pan-Hellenic Council and the Interfraternity Council

2003 VEISHA Faculty Member of the Year for the College of Agriculture

Listed in the 4th Edition ofWhos Whoin Economics as one of top 6% of themost cited economists for work published from 1990 to 2000

Awarded Pioneer Chair in Agribusiness in 1999

AAEA Distinguished Extension Program Group for Managing Risk and

Profits, 20001999 College of Agriculture Team award for Dollars and Cents

1990 ISU Livestock Service Award, sponsored by Walnut Grove

Scholarship for 1st place in class of 80 in 1978, 25 in 1979, and 25 in 1980

WORK HISTORY:

2001-present Leader Policy Task Force, Plant Science Institute, ISU1999-present Pioneer Chair in Agribusiness, Iowa State University


27/33

1999-present Professor of Finance, Iowa State University1996-present Professor of Economics, Iowa State University

1991-1996 Associate Professor of Economics, Iowa State University

1986-1991 Assistant Professor of Economics, Iowa State University1990-1998 Leader, Trade and Agricultural Policy Division,

Center for Agricultural and Rural Development1988-2000 Assistant Director, Meat Export Research Center1981-1985 Research Assistant, University of California, Berkeley

PERSONAL: Date of Birth: September 1, 1959

Married: 4 children

PAPERS PUBLISHED IN PEER-REVIEWED JOURNALS

Lence, Sergio H., and Dermot J. Hayes (2008). Welfare Impacts of Cross-CountrySpillovers in Agricultural Research. Forthcoming in theAmerican Journal of Agricultural

Economics.

Lence, Sergio, Stphan Marette, Dermot Hayes, and William Foster. Collective Marketing

Arrangements for Geographically Differentiated Agricultural Products: Welfare Impacts and

Policy Implications.American Journal of Agricultural Economics, 89(4) November 2007:947-963.

Mennecke, Brian, Anthony Townsend, Dermot J. Hayes, and Steven Lonergan. A Study ofthe Factors that Influence Consumer Attitudes Toward Beef Products Using the Conjoint

Market Analysis Tool. Journal of Animal Science, 85(10) 2007: 2639-2659.

Elobeid, Amani, Simla Tokgoz, Dermot J. Hayes, Bruce A. Babcock, and Chad E. Hart. The

Long-Run Impact of Corn-Based Ethanol on the Grain, Oilseed, and Livestock Sectors: A

Preliminary Assessment.AgBioForum,10(1) 2007: 11-18.

Monchuk, D. C., J. A. Miranowski, D. Hayes, and B. Babcock. An Analysis of Regional

Economic Growth in the U.S. Midwest.Review of Agricultural Economics, 29(1) Spring

2007: 17-39.

Lence, Sergio H., and Dermot J. Hayes. EU and US Regulations for Handling and

Transporting Genetically Modified Grains: Are Both Positions Correct?EuroChoices, 5(2)August 2006: 20-27.

Hart, Chad E., Dermot J. Hayes, and Bruce A. Babcock. Insuring Eggs in Baskets Shouldthe Government Insure Against Individual Risks, Canadian Journal of Agricultural

Economics 54(1) March 2006: 121-137.


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Appleby, Michael C., Kathy Chinn, Lawrence Douglas, Lawrence D. Firkins, David Fraser,Gail C. Golab, Dermot Hayes, Katherine A. Houpt, Christa Irwin, John J. McGlone, R. Tracy

Rhodes, Paul Sundberg, Lisa Tokach, and Robert W. Wills. Housing Pregnant SowsA

Comprehensive Review,Journal of the American Veterinary Medical Association. 227(10)November 2005: 1580-1590.

Lence, Sergio H., and Dermot J. Hayes. Genetically Modified Crops: Their Market andWelfare Impacts,American Journal of Agricultural Economics. 87(4) November 2005: 935-

950.

Lence, Sergio H., Dermot J. Hayes, Alan McCunn, Stephen Smith, and William S. Niebur.Welfare Impacts of Intellectual Property Protection in the Seed Industry,American Journalof Agricultural Economics. 87(4) November 2005: 951-968.

Lence, Sergio H., and Dermot J. Hayes. Technology Fees Versus GURTs in the Presence Of

Spillovers: World Welfare Impacts.AgBioForum, 8(2&3) September 2005: 172-186.

Hayes, Dermot, Sergio Lence, and Bruce Babcock. Geographic Indications and Farmer-

Owned Brands: Why Do the U.S. and E.U. Disagree?EuroChoices, 4(2) August 2005: 28-

35.

Hayes, Dermot J., and Helen H. Jensen. Lessons from the Danish Ban on Feed-Grade

Antibiotics, Choices, Fall 2004.

Hayes, D. J., S. H. Lence, and A. Stoppa. Farmer-Owned Brands?Agribusiness: AnInternational Journal, 20(3) July 2004: 269-285.

Babcock, B.A., C. E. Hart, and D. J. Hayes. Actuarial Fairness of Crop Insurance Rates with

Constant Rate Relativities,American Journal of Agricultural Economics, 86, 2004: 563-575.

Hurd, H. Scott, Stephanie Doores, Dermot Hayes, Alan Mathew, John Maurer, Peter Silley,Randall S. Singer, and Ronald Jones. Public Health Consequences of Macrolide Use in

Food Animals: A Deterministic Risk Assessment,Journal of Food Protection, 67(5) 2004:

980992.

Hayes, D. J., S. H. Lence, and C. Mason. Could the Government Manage its Exposure to

Crop Reinsurance Risk?Agricultural Finance Review, 63, Fall 2003: 127-142.

Mason, C., D. J. Hayes, and S. H. Lence. Systemic Risk in U.S. Crop Reinsurance

Programs,Agricultural Finance Review, 63, Spring 2003: 23-39.

Lence, S. H., and D. J. Hayes. Impact of Biotech Grains on Market Structure and Societal

Welfare,AgBioForum, 5(3) 2002: 85-89.


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Hayes, Dermot, and Sergio Lence. Farmer-Owned Brands? Choices, Fall 2002.

Lence, Sergio H., and Dermot J. Hayes. U.S. Farm Policy and the Volatility of Commodity

Prices and Farm Revenues,American Journal of Agricultural Economics, 84(2) May 2002:350-351.

Shogren, Jason, Dermot Hayes, John Fox, and Todd Cherry. Auctions 101: Lessons from aDecade in the Lab. What Am I Bid for Safer Food? Choices, Spring 2002: 16-21.

Hayes, Dermot, Helen Jensen, and Jay Fabiosa. Technology Choice and the Economic

Effects of a Ban on the Use of Antimicrobial Feed Additives in Swine Rations, Journal of

Food Control, 13(2) March 2002: 97-101.

Hayes, Dermot, John Fox, and Jason Shogren. Experts and Activists: How InformationAffects the Demand for Food Irradiation, Food Policy, 27(2) 2002: 185-193.

Fox, John, Dermot Hayes, and Jay Shogren. Consumer Preferences for Food Irradiation:How Favorable and Unfavorable Descriptions Affect Preferences for Irradiated Pork in

Experimental Auctions,Journal of Risk and Uncertainty, 24(1) 2002: 75-95.

Hayes, Dermot, Helen Jensen, Lennart Backstrom, and Jay Fabiosa. Economic Impact of aBan on the Use of Over the Counter Antibiotics in U.S. Swine Rations,International Foodand Agribusiness Management Review, 3, 2001.

Hart, Chad, Bruce Babcock, and Dermot Hayes. Livestock Revenue Insurance,Journal ofFutures Markets, 21(6) 2001: 553-580.

Shogren, Jason F., John A. List, and Dermot J. Hayes. Preference Learning in Consecutive

Experimental Auctions,American Journal of Agricultural Economics, 82(4) November2000: 1016-1021.

Fuller, Frank H., and Dermot J. Hayes. Reconciling Chinese Meat Production andConsumption Data,Economic Development and Cultural Change, 49(1) October 2000: 23-

45.

Hennessy, David A., and Dermot Hayes. Competition and Tying in Agri Chemical and SeedMarkets,Review of Agricultural Economics, 22(2) June 2000: 389-406.

Kyun, Kim Tae, and Dermot Hayes. Input Demand and Producer Welfare Effects of CropRevenue Insurance Schemes,Journal of Rural Development, 22, Summer 1999: 81-95.

Hayes, Dermot J. Chinas Role in World Livestock and Feed-Grain Markets, Choices, FirstQuarter, 1999.

Shogren, Jason F., John Fox, Dermot Hayes, and Jutta Roosen. Observed Choices for FoodSafety in Retail, Survey and Auction Markets,American Journal of Agricultural


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Economics, 81(5) 1999: 1192-1199.

Hayes, Dermot J. Costs of Eliminating Subtherapeutic Use of Antibiotics, in The Use of

Drugs in Food Animals: Benefits and Risks, National Academy Press, Washington, D.C.,1999.

Fox, John A., Dermot J. Hayes, Jason Shogren, and James B. Kliebenstein. CVM-X:Calibrating Contingent Values with Experimental Auction Markets,American Journal of

Agricultural Economics, 80(3) August 1998: 455-465.

Wang, Qingbin, Frank Fuller, Catherine Halbrendt, and Dermot J. Hayes. ChineseConsumer Demand for Animal Products and Implications for U.S. Pork and Poultry

Exports,Journal of Agricultural and Applied Economics, 30(1) July 1998: 127-140.

Lence, Sergio, and Dermot J. Hayes. The Forward-Looking Competitive Firm under

Uncertainty,American Journal of Agricultural Economics, 80(2) May 1998: 303-312.

Hennessy, David A., Bruce A. Babcock, and Dermot J. Hayes. Budgetary and Producer

Welfare Effects of Revenue Assurance,American Journal of Agricultural Economics, 79(3)

August 1997: 1024-1034.

Shogren, Jason F., and Dermot J. Hayes. Resolving Differences in Willingness to Pay and

Willingness To Accept: Reply,American Economic Review, March 1997: 241-244.

Baumel, C. Phillip, and Dermot J. Hayes. Risk Management Strategies That Can Increase

Risk, Wallaces Farmer, 122(3) February 1997.

Fox, John A., Dermot J. Hayes, Jason F. Shogren, and James B. Kliebenstein. Experimental

Methods in Consumer Preference Studies,Journal of Food Distribution Research, 27, July

1996: 1-7.

Hayes, Dermot J., Jason F. Shogren, John A. Fox, and James B. Kliebenstein. Test

Marketing New Food Products Using a Multi-Trial Nonhypothetical Experimental Auctionwith Market Discipline,Journal of Psychology and Marketing, 13(4) July 1996: 365-379.

Hayes, Dermot J., Marvin L. Hayenga, and B. Melton. The Impact of Grade Equivalency onBeef and Cattle Trade Between the United States and Canada, U.S. Meat Export Analysis

and Trade News, 4(2) 1996: 3-15.

Hayes, Dermot J., and Sergio H. Lence. Optimal Hedging Under Forward-LookingBehavior,Economic Record, 71(215) December 1995: 329-342.

Lence, Sergio H., Dermot J. Hayes, and William H. Meyers. The Behavior of Forward-Looking Firms in the Very Short Run,American Journal of Agricultural Economics, 77(4)

November 1995: 922-934.


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Sakong, Yong, and Dermot J. Hayes. Expected Utility Maximization with TruncatedDistributions: An Application of the Mean-Value Theorem,International EconomicJournal, 9(3) Autumn 1995.

Lence, Sergio H., and Dermot J. Hayes. Land Allocation in the Presence of Estimation

Risk,Journal of Agricultural and Resource Economics, 20(1) July 1995: 49-62.

Hayes, Dermot J., Alexander Kumi, and Stanley R. Johnson. Trade Impacts of Soviet

Reform: A Heckscher-Ohlin-Vanek Approach,Review of Agricultural Economics, 17, May

1995: 131-145.

Hayes, Dermot J., Jason F. Shogren, Seung Youll Shin, and James B. Kliebenstein. Valuing

Food Safety in Experimental Auction Markets,American Journal of Agricultural

Economics, 77(1) February 1995: 40-53.

Fox, John A., Brian L. Buhr, Jason F. Shogren, James B. Kliebenstein, and Dermot J. Hayes.

A Comparison of Preferences for Pork Sandwiches Produced from Animals With andWithout Somatotropin Administration,Journal of Animal Science, 73, 1995: 1048-1054.

Jensen, Helen J., Steven W. Voight, and Dermot J. Hayes. Measuring International

Competitiveness in the Pork Sector,Agribusiness: An International Journal, 11(2) 1995:169-177.

Shogren, Jason F., John A. Fox, Dermot J. Hayes, and James B. Kliebenstein. BidSensitivity and the Structure of the Vickrey Auction,American Journal of AgriculturalEconomics, 76, December 1994: 1089-1095.

Lence, Sergio H., Yong Sakong, and Dermot J. Hayes. Multiperiod Production with

Forward and Option Markets,American Journal of Agricultural Economics, 76(2) May1994: 286-95.

Shogren, Jason F., Seung Y. Shin, Dermot J. Hayes, and James B. Kliebenstein. ResolvingDifferences in Willingness to Pay and Willingness to Accept,American Economic Review,

84(1) March 1994: 255-70.

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