a brief overview of nonparametric methods in...

Invited Presentation

A Brief Overview of NonparametricMethods

Arne Hallam

in Economics -

The concept of nonparametric analysis, estima-tion, and inference has along and storied existencein the annals of economic measurement. At leastfour rather distinct types of analysis are lumpedunder the broad heading of nonparametrics. Theoldest, and perhaps most common, is that associ-ated with distribution-free methods and order sta-tistics. Similar in spirit, but different in emphasis,is nonparametric density estimation, such as thecurrently popular kernel estimator for regression.Semi-parametric or semi-nonparametric estimationcombines parametric analysis of portions of theproblem with nonparametric specification for theremainder, such as the specification of a specificfunctional form for a regression function with anonparametric representation of the error distribu-tion. The final type of nonparametrics is that as-sociated with data envelopment analysis and re-vealed preference, although the use of the termnonparametrics for this research is perhaps a mis-nomer. This paper will briefly review each of thefour types of analysis, leaning heavily on otherpublished work for more detailed exposition. Thepaper will then discuss in more detail the applica-tion of the revealed-preference approach to fourspecific economic problems: efficiency, the struc-ture of technology or preferences, technical ortaste change, and risky choice. The paper is notcomplete, exhaustive, or detailed. The primarypurpose is to expose the reader to a variety oftechniques and provide ample reference to the rel-evant literature.

Nonparametric Inference

Much of nonpararnetric inference is based on ranksand order statistics. Early papers include those byHotelling and Pabst, Friedman, and Kendall(1938). The big boost came with the 1945 paper ofWilcoxin, followed by an important theoretical pa-

AmeHahn is an associate professor, Department of Economics, IowaState University. Journal Paper no. J-15075 of the Iowa Agricultural andHome Economics Experiment Station, Ames, IA. Project no, 2894,

per of Mann and Whitney, and the classic mono-graph by Kendall (1948). The current state of thepractice is represented well by the textbooks ofLehmann, Randles and Wolfe, and Gibbons. Acouple of simple examples will give the flavor ofthe analysis.

Much of nonparametrics is based on the use oforder statistics. Consider a random sample X1, Xz,. . . . X. from a population with continuous cu-mulative distribution function F’x. Now let X(~,de-note the smallest element of the sample, and re-arrange the sample in increasing order of magni-tude. The set of variables X(l), X(2), . . . , X(n)aretermed the order statistics of the sample, and X(,) iscalled the rth order statistic. Using the probability-integral transformation, it can be shown that thevariables F(X[l)) s. . . ~F(X(”)) are distributed asthe order statistics from a uniform distribution on(O,1) (Randles and Wolfe, p. 7). Thus, no matterwhat the underlying distribution of X, these statis-tics are uniformly distributed. In this sense, theyare referred to as distribution-free. Joint and mar-ginal distributions, moments, and asymptoticproperties of these statistics are easily found (Gib-bons, pp. 24-29), and analysis can proceed with-out troublesome assumptions about the nature ofthe underlying distributions.

In a similar manner, analysis can be based onthe ranks of the variables in this ordering. For ex-ample consider two populations, one of which issubjected to a treatment. For example, consider thestarting salaries of new professors with training ineconomics as opposed to finance. Rank all the newprofessors by starting salary, with the random vari-able for each individual being his or her rank in thesample. The Wilcoxin rank-sum test is performedby adding the ranks of all those trained in econom-ics and comparing it to some critical value. If thissum is sufficiently large, then one can reject thehypothesis that those trained in finance have highersalaries.

While nonparametric methods avoid many prob-lems associated with specification error, they aregenerally less efficient than parametric methods ifthe form of the underlying distribution is known.

A Brief Ovewiew of Nonparametric Methods in Economics 99

While the true form of underlying distributions isnever known, extensive Monte-Carlo work hasbeen performed comparing order and rank statis-tics to parametric alternatives for many standarddistributions (Lehmann). While the results supportparametric alternatives when the distribution isknown, they also argue for specific Monte Carlowork for individual problems and situations.

Given economists’ preoccupation with modelbuilding and regression analysis, these techniqueshave not had heavy use. There has been increasinginterest in these techniques in recent years, how-ever. Holmes and Hutton have proposed the mul-tiple rank F test for investigating causality betweeneconomic variables as an alternative to the stan-dard Granger tests. Pagan and Schwert proposeseveral nonparametric tests for covariance station-arity and use them in analyzing stock-market data.Such techniques could be a useful precursor tostandard parametric time series analysis. Campbelland Dufour use sign statistics and Wilcoxinsigned-rank statistics to test the independence oftime series that may be subject to the Mankiw-Shapiro criticism due to feedback and rational ex-pectations. They found that these nonparametrictests avoid the too frequent rejection problem as-sociated with standard parametric tests. Whilemore closely related to density estimation dis-cussed in the next section, Han (1987a, b) andMatzkin (1991a) have extended the concept ofrank correlation due to Kendall (1938) to developa maximum rank correlation estimator that gener-alizes in a nonparametric way the BOX-COXtrans-formation. They analyze nonparametric transfor-mations of the linear model of the following form:

(1) yi = D .F(~i’~o,~i i = 1, . . . ,n)>

where the composite transformation D o F is suchthat D: R + R is nondegenerate monotonic and F:R 2 ~ R k strictly monotonic in its arguments.While X’(30is still a linear function, it can affect yin a nonlinear and nonparameterized fashion that isnot separable in the error term. Such models allowfor more general transformations of the variables,but at the cost of computational complexity as theestimation usually involves a grid search. Thegreatest value of these papers is perhaps in remind-ing us that old ideas can often be revived and mod-ified to solve new problems.

The use of standard nonparametric techniques isdue for a resurgence in economics. These tech-niques are particularly useful for comparing theproperties of two populations. Comparing yielddistributions over time and across space is one pos-sible application. The techniques are also usefulfor analyzing the residuals from estimated models

such as the standard K-S test for normality. Theycould also prove useful in investigating the prop-erties of random variables generated using optimi-zation techniques or parameters estimated by sim-ulation. An example is the work of Tolley andPope on stochastic dominance tests.

Nonparametric Regression Models

Two approaches to nonparametric curve fittinghave become popular in the last decade. The first isthe use of series approximations popularized byGallant (Gallant 1981, 1982; Gallant and Nychka;Andrews). Spline approximations would also fit inthis category (Engle et al.). The second is kernelestimation, or smoothing, as discussed by Bierens,Ullah, and in the recent monograph by Hardle.White and Wooldridge argue that both of thesetechniques are special cases of the method ofsieves proposed by Grenander. These and otherrelated techniques are discussed in the recent sur-vey paper of Pagan and Wickens.

Consider first the advantages of series estima-tors. The economist is often interested in estimat-ing the moments of a variable y conditional on therealization of some random vector x = -f. Denotethis conditional moment as O(xt) = E(y~xt = it).The regression model is then written as

(2) y, = m(xt) + %

where y,, Et f R, Xt < D C Rd, m(”) C M, rmdivl isthe class of continuous functions from D to R, Theset D is the set of d-element vectors considered asthe sample space for the variable x. The objectiveis to estimate various functions of m(”), such asm(l) and the derivatives of m for arbitrary valuesof x. A parametric approach would then approxi-mate this unknown function with a specific func-tional form, such as a quadratic, translog, or CES.Significant work has been done on the various ad-vantages and disadvantages of various functionalforms (Barnett and Lee; Diewert and Wales). Onedifficulty is that these approximations may not beglobally valid. Another is that they do not improveas the sample size increases. The nonparametricapproach is to approximate m(”) by a finite seriesexpansion of the following form:

(3) m(,) = ~ z,(.)y,j~=1

where {z~:s = 1, 2, . . .} is a family of functions(such as the trigonometric ones) from D to R, y =

(?’1> ~ . . , Ykn) is an unknown parameter vector,

100 October 1992 NJARE

and k. is the number of summands in the seriesexpansion when the sample size is n. For example,Gallant proposed that z include linear and qua-dratic terms in x plus the trigonometric termscos(b’x) and sin(b’x) for some integer vector b 6Rd The number of terms included in the trigono-metric series depends explicitly on the sample size,and thus both the number of parameters and, hope-fully, accuracy increase as the sample size in-creases. Andrews has given conditions on the rateat which the number of terms in the series mustapproach infinity, as the sample size increases toensure consistency and asymptotic normality of theestimator. These series estimators can be also beviewed as a type of GMM estimator, where theorthogonality condition holds in the limit as thenumber of terms in the series increases (Pagan andWickens). Since these estimators almost alwaysinclude parametric as well as series-expansionterms and parametrize the series expansion, theyare often called semi-nonparametric. The primaryadvantage of this type of estimator is its ability toapproximate any underlying functional relation-ship to a high degree of accuracy as the sample sizeincreases. Disadvantages include a possibly largenumber of parameters and the distinctly periodicbehavior of the estimated relationship for serieswith small numbers of terms (Pope). For a discus-sion of these and other issues, see the symposiumpapers contained in the 1984 American Journal ofAgricultural Economics (Gallant 1984). Recentpapers using this approach include Chalfant, Paganand Hong, and Barnett and Yue.

Kernel estimation is based on the work ofRosenblatt, Nadaraya (1964a, b), and Watson.Two standard references are Prakasa Rao and Sil-verman. The use of kernel estimators for time se-ries is discussed by Robinson (1983), while a re-cent text with economic examples is Hardle. Thekernel estimator is basically a method of smooth-ing. A smoothing for m(x) in (2) is given by aweighted average of values of Ytin the neighbor-hood of x,. Specifically, for the case of a single xvariable, a smoothing estimator would be

5 wlIAx)Yt~=1(4) lb(x) = n ,

where Wnt is a weight that may depend on theentire set of x variables.

As the kernel estimator is a smoothing estimatorof the conditional expectation of y in (2), considerthe joint density of y and a k X 1x vector. Becausem(”) is the expected value of y given x, its value is

(5) TTZ(X)= j’yf(ylX) dy

= ~y f(y,x)— dy, if f(x) >0,f(x)

wheref(ylx) is the conditional density of y given x,f(x,y) is the joint density, and f(x) = Jf(y,x) dy isthe marginal density of x. The function m can beestimated by obtaining estimates off (y,x)andf (x).The kernel estimator off(x) is given as

[1x—x,()nK—

(6) it(x) = : x $ ~t= 1

where K(”) is a chosen real function on Rk thatsatisfies

(7) .([K(x)I b < m .fK(x) ah = 1.

The parameter h is called a window width param-eter and satisfies

(8) lim h. = O lim nh~ = Q.-X -m

A kernel estimator of f(y,x) consistent with thederived estimator of the marginal density f (x) is

[

~ K*(Y-Yr) (x–%)— —

()(9) }@,X) = ; ~ ‘nhk+; ‘n1

,=1 nwhere K* satisfies

(10) .f yK*(y,x) dy = O ~ K*(y,x) dy = K(x).

Clearly, ~(x) is the marginal density since

(11)

k) =~.f(.hx)@

[“K*(Y-YJ (x–%)

1$ (-) E

h. ‘ h. 1.k+ 1

hady

n,= 1

[

K* b – Yt) (x -%)

oxIn 1=—n ()

S “j ‘n d;

n n~= 1

[1

~ K* (x – x,)

ox1 hn=_n h;

~=1

Hallam A Brief Overview of Nonparametric Methods in Economics 101

The estimator of m(x) is then given by

(12)

~(x) = J Y?OJ) ~Y

k)

[

~K*(Y-Yt) (x–-k)

() 1$Y ; ~ “’h,+: ‘n dy

/=1 n

j(x)

This estimator is a weighted mean of the depen-dent variable y,. The closer x is to x,, the moreweight is put on yz. There is a large literature onthe choice of the appropriate kernel. Commonchoices include the standard normal (K(x) =(2Ir - ‘1’)exp(– VU*)) and various polynomials.The properties of the estimators are not particularlysensitive to the choice of the kernel function. Bie-rens, Epanechinikov, and Hardle (chapter 4) eachcontain a more detailed discussion of this issue. Ofperhaps more serious concern is the choice of thebandwidth, h. The bandwidth is chosen so that itsrate of decrease with the sample size will givedesirable asymptotic properties for the estimator.If h is chosen to be too large, the estimate will betoo smooth and obscure the shape of the densityfunction. If h is too small, the estimate of m(x) willfollow the data closely and may result in overfit-ting. The literature suggests several way to chooseh, most giving forms similar to h = cn -b, whereb is a fraction. While several suggestions are givenfor choosing h in an “optimal” fashion, in practicethe choice has tended to be arbitr~.

There has been an explosion of papers usingkernel estimators. Some of these combine para-metric with kernel specifications. Vinod and Ullahconsider the estimation of production functions us-ing this technique. Moschini combines parametricand nonparametric kernel estimation in investigat-ing preference change in meat demand. Stoker

uses a kernel estimator to test additive constraintson the first and second derivatives of a regressionmodel. Pagan and Ullah discuss the use of kernelestimators for incorporating risk as a regressor.Hardle, Hildenbrand, and Jensen use a nonpara-metric kernel estimator to estimate the mean in-come effect matrix in a cross section of demands.Rilstone examines the use of kernel estimators foraverage, as opposed to point, estimates of deriva-tives. Robinson (199 1) uses kernel estimators toestimate the Kullback-Liebler information crite-rion in forming a test of the independence of timeseries data. McCurdy and Stengos compare non-parametric kernel estimators to traditional para-metric estimators in studying the time pattern ofthe risk premium, while Hong and Pagan comparekernel and series estimators using Monte Carlotechniques,

Kernel estimators avoid many of the specifica-tion problems associated with parametric func-tional forms. They also have a number of limita-tions. Kernel estimators may be sensitive to thechosen bandwidth. To the extent that the band-width influences the results, there is arbitrarinessin any reported estimate. Nonparametric estima-tors have strong asymptotic properties. They maynot be particularly useful in small samples sincesmoothing tends to be more useful with largeamounts of data, Given the sample sizes used inmany studies employing annual data, kernel esti-mators may see limited use in agricultural econom-ics.

Semiparametric Econometrics

Semiparametric models combine a parametriccomponent of known form with a nonparametric orinfinite dimensional one. Consider a simple linearregression model of the form

(13) y, = X1’(3+ E,,

where the error Ethas density function ~(.). If theerrors are identically and independently distrib-uted, ordinary least squares estimates of (3 areGauss-Markov efficient. If the errors are alsoGaussian and satisfy standard regularity condi-tions, the variance of the least squares estimatorachieves the Cramer-Rao lower bound. If the er-rors are heteroskedastic or serially dependent withknown covariance variance, the generalized leastsquares (GLS) estimator is efficient; but, if thiscovariance matrix is not known, the GLS estimatoris not defined. Feasible GLS provides a way toobtain consistent estimates of the parameters of themodel but requires some assumptions about theproperties of ~. In this model, a semiparametric


estimator would be one that used a nonparametrictechnique to obtain estimates of the unknown co-variance matrix. Robinson (1988), in his survey ofsemiparametric methods, suggests that semipara-metric techniques might be viewed as a way toavoid paxameterizing nuisance parameters in econo-metric models. In the above model, the propertiesof c as regards to heteroskedasticity, serial corre-lation, or form of the underlying distribution (nor-mal, log-normal, beta, etc,) are really nuisanceparameters as far as obtaining estimates of ~. Ifthese properties can be parameterized (such as nor-mality with first-order serial correlation and ho-moskedasticity), feasible generalized least squares(in this case, for example, use of the CochraneOrcutt procedure) is consistent and achieves theCramer-Rao lower bound asymptotically. Semi-parametric estimation may be viewed then as theparametric estimation of (13), without parametricspecification of the error structure ~(cf).

Consider the log-likelihood function for estima-tion of (13):

n

(14) L = > logf(y[ – X,’~).

*=]

One approach (Gallant and Nychka) would be toreplace the unknown ~(”) with the product of aN(0,u2) variable denoted by ~(~) and a polynomialin ~, P(6). The approximate log-likelihood func-tion L* is then given as

n

(15) L* = ~ log 4(y1 – X,’(3)

t=1

n

+ ~ log P@t – X,’(3).f=1

Gallant, Hsieh, and Tauchen have used this for-mulation in the study of exchange rates.

An alternative to approximation is to solve thefirst-order conditions that come from the scoresassociated with (13). The score, by definition, isthe derivative of the log-likelihood function withrespect to the parameters. For (14), this gives

(16)[1

- i xtf(d-’ ~aq -t=1

Nonparametric estimates of the density and its de-rivatives could then be used to find the values of ~

that optimize the scores. Stone was the first to usethis method, and Manski (1984, 1985, 1987) pro-vides several extensions. A recent paper byHorowitz uses the kernel estimator to smooth thescores for a binary response model. Lee proposes anonlinear least squares estimator similar to a kernelestimator for the truncated regression model. Sev-eral of the suggested approaches involve OLS es-timation and then use of the residuals as estimatesof the underlying empirical density function.Given some structure on the underlying problem,semiparametric estimation may reduce to findingan optimal nonparametric estimate of some partic-ular moment of the unknown distribution (Cham-berlain 1987; Newey 1988). Pagan and Wickensgive a GMM interpretation of semiparametric es-timators.

An important concept in semiparametric estima-tion is the property of adaptation. A semiparamet-ric estimate is said to be “adaptive” if it is asefficient over a broad class of assumptions con-cerning ~ as a correct finite parameterization(Stein). Another important concept in semipara-metric estimation is the efficiency bound. This isan extension to the semiparametric problem of theCramer-Rao bound and can be viewed as repre-senting the information lost from using a nonpara-metric estimator as opposed to a correctly param-etrized one. This topic is currently being inten-sively investigated. The survey by Robinson(1988) on semiparametric methods analyzes thesebounds in some detail. Coslett discusses boundsfor binary choice and censored regression models.Chamberlain (1992) discusses bounds using con-ditional moment restrictions, while Newey (1990)provides a lucid comprehensive survey, Recent ap-plication papers using semiparametric methods areSentana and Wadhwani using stock-market data,Deaton on income distribution in Thailand, Mos-chini on Ontario dairy farms, and Holt and Mos-chini on sow farrowing decisions.

Semiparametric estimators may have a signifi-cant role in future econometric work. They allowfor flexible specification of nuisance and other un-known parameters. They are particularly useful inanalyzing large data sets with small numbers ofvariables. They have some drawbacks, however.While they have good asymptotic properties, theymay do poorly in small samples compared to in-correctly specified parametric estimators sincetheir rate of convergence is typically slower thanusual maximum-likelihood estimators. They arenot really practical for large numbers of parametersor moment restrictions. Two-step implementationsexploiting the parametric portions may alleviatesome of these problems.

Hallum A Brief Overview of Nonparametric Methods in Economics 103

Revealed Preference, Behavior, and“Nonparametric” Analysis

The work of Farrell on efficiency and Afriat (1967,1972, 1973) on revealed preference has given riseto another type of analysis that is sometimes callednonparametric. This research attempts to under-stand technology and behavior without imposingany functional form on the data. However, it dif-fers from the nonparametric analysis previouslydiscussed in that statistical inference is not explic-itly or implicitly used. In some ways, it is a kind ofreduced-form analysis. Consider, for example, aconsumer who maximizes utility subject to a bud-get constraint. Certain assumptions about the util-ity function lead to restrictions on the derived de-mand functions. The nonparametric approachwould examine the actual choices of a consumerand ask if there exists a utility function consistentwith the observed choices and the assumptions. Ifone exists, the researcher concludes that the utilitymaximization hypothesis and the assumptions arevalid. The answer to the question, however, canonly be yes or no. The methods do not give prob-abilities of the hypothesis being true. These meth-ods are nonparametric in the sense that no func-tional form is assumed. They are not nonparamet-ric in the sense of nonparametric statisticalanalysis. Better terms for this analysis might berevealed behavior analysis (RBA) or revealedstructure analysis (RSA). Revealed-behavior ap-proaches have been used in both consumption andproduction analysis. The basic idea is the same,but the emphasis has typically been different.Work in production has tended to concentrate oncharacterization of the isoquant and efficiency,while work in consumption has emphasized testsof optimizing behavior and the structure of prefer-ences. The discussion will proceed by first discuss-ing the efficiency problem in some detail and thenmoving to more general revealed-preference argu-ments. The discussion of efficiency is presented asa basis for illustrating the nonparametric approachand is not a comprehensive survey.

A Digression on Eflciency Analysis

Modem efficiency analysis originated with the pi-oneering work of Farrell. In essence, he developeda way to measure how far a given input vector isfrom the boundary of the input requirement set. Inorder to consider this measure in more detail, con-sider a set of assumptions on the technology. As-sume that the firm produces m outputs y using ninputs x. The input correspondence y - L(y) CR:is the subset of all input vectors capable of

producing at least output vector y. The correspon-dence is assumed to have the following properties:

(17)L.1 O~L(y) L(O) = R:.

L.3 Ifx 6 L(y), Ax f L(y) for h > 1.

L.4 L is a closed correspondence.

The properties are standard and are discussed inmore detail by Fiire (1988), and Knox Lovell andSchmidt. Three important subsets of L(y) are usedto discuss efficiency. They are the isoquant and theweak-efficient subset and the efficient subset.They are defined as

(18) lsoq L(y) = [x:x < L(y),Ax @ L(y), A < [0,1)], y = O,

W%ffU’y) + [x:x < L(j),u<*x~u EZL(y)], y20,

E&L(y) = [X:X < ~(y),usx+ueFL(y)], y= (),

where the notation u <* x implies Ui< xi or Ui =xi = Ofor all i.It is obvious the E&L ~ WEffL cIsoq L, Consider Figure 1 with two variable inputs.The isoquant is given by ABCD, the weak efficientset by BCD, and the efficient set by CD. Consider

xl

Figure 1. Efficient Subsets of L(Y)


now the Farrell measure of technical efficiency. Itis given by

(19) F(x;y) = min [h: Ax { L(y), h a O].

The Farrell measure computes the ratio of thesmallest feasible radial contraction of an observedinput vector to the input vector itself. Call thisminimum value of h, A“. If ho < 1, the firm canproduce y with a radially smaller input vector andthe observed input vector is inefficient. Farrell alsoconsiders cost efficiency. Consider the cost-minimization problem for the firm:

(20) C(y,w) = min [w’x: x f L(y)].x

A measure of cost or overall efficiency (CE) isgiven by the ratio of actual cost to the minimumcost of producing output vector y:

(21)ay,w)

CE(x;y,w) = —W’x “

Farrell defines allocative efficiency as the ratio ofcost efficiency to technical efficiency or

Clqx;y, w)(22) AE(x;y,w) =

F(x;y) “

These can all be illustrated graphically using Fig-ure 2. Consider an input combination P. Technicalefficiency is given by the ratio of OQ to QP. Costefficiency is given by the ratio of OS to OP andallocative efficiency by the ratio OS to OQ. In asimilar way, a weak input measure of technicalefficiency can be defined (Fiire, Grosskopf, and

Knox Lovell). The two measures are equal onlywith strong disposability of inputs. A variety ofnonradial measures of efficiency can also be de-fined (Russell). Efficiency can also discussed interms of cost and profit functions (Kopp andDiewert).

Consider now nonparametric measurement ofthe technology and various efficiency measures.Let there be K firms each using n inputs to producem outputs. Let M be a K X m matrix of the outputlevels, N a K X n matrix of input levels, y theoutput vector containing the total of each output,and x the vector of total inputs. Shephard; Fiire,Grosskopf, and Knox Lovell; and Banker,Charnes, and Cooper have shown how to constructan input requirement set satisfying weak dispos-ability of inputs that is consistent with observeddata. This is done by constructing the convexweak-disposal hull of the observed data. The inputset so constructed is the smallest set the includesall K observations in the sample and satisfies theproperties L. 1–L.5. For the case of constant re-turns to scale, this set is given by

(23) L(y) = [x: z ( R~, ~Z’ikf = y,8X = z’N, p,8 [ (0,1)],

where z is an intensity vector that specifies com-binations of the various firms’ technologies used toconstruct the reference technology, and p, and 8scale output and input respectively. Modificationsto the set for alternative assumptions concerningreturns to scale and disposability are discussed byKnox Lovell and Schmidt, as well as by F&-e,Grosskopf, and Knox Lovell. The isoquant is con-structed in essence by enveloping the data. If allthe firm’s outputs were the same, this could bevisualized as finding the set of convex facets clos-est to the origin that allow production of the out-put y.

The Farrell efficiency measure is computed foreach firm in the sample by solving the followingprogramming problem.

(24) F(#,y9 = min[h: z < Rf, IJ,Z’M = yk,

M& = z’;, p,a < (0,1)]

= min[k: M < L(yk)].

As written, the problem is not well defined due tothe scalars p and 8, but can be written in a more

ostandard programming format by using a change of

xl variables. Define s = z/8, y = (l/@) and then

Figure 2. Farrell Eftlciency write the problem as

A Brief Overview of Nonparametric Methods in Economics 105Hallam

(25) min h

S,?,h

St.

S’M – @ = os’N– A&=O

S20, y=l.

The solution finds the shortest radial distance fromthe input-output combination of firm k to the iso-quant constructed from the technologies of all thefirms. Minimum cost can be obtained by solvingthe program in (20) given the technology in (23).Cost and allocative efficiency are then easily com-puted using these solutions.

Several points about the nonparametric ap-proach can now be made. The approach is dataenveloping in the sense that it finds the “minimal”set of points that satisfy some hypothesis. In thiscase, it finds the smallest input requirement set thatsatisfies the axioms and is consistent with the data.In a different context, it might find the set of pointsconsistent with utility or profit maximization. Insome cases, this set might be empty (no feasiblesolution), and the researcher would conclude thatthere is no set of points that is consistent with thehypothesized axioms and the data. The particularconstraints that give rise to infeasibility may beuseful in analyzing the problem, however. The ref-erence technology is then used to construct variousother useful measures such as efficiency and thecost-minimal input function, For example, the ef-ficiency measures could be related to firm size orcountry of origin.

Revealed Preference andNonparametric Analysis

While the work of Farrell is the precursor to thenonparametric revealed-behavior approach in pro-duction, the work of Afriat (1967, 1973) is thefoundation of analysis for consumers. In his orig-inal paper, Afriat (1967) developed a set of in-equalities that must be satisfied by observed priceand individual demand data if they were generatedby utility maximization. He termed these cyclical,multiplier, and level consistency. These basic in-equalities have been slightly modified and exten-sively discussed (Diewert 1973a, 1978; Diewertand Parkin 1985; Varian 1982, 1983a). Similar tothe construction of the isoquant in efficiency anal-ysis, they provide a set of conditions that data gen-erated by a utility-maximizing consumer must sat-isfy, In order to discuss these, three definitions areneeded (Varian 1982, 1983a).

Definition 1: A utility function, U(x), rational-izes the data (p’, x’), i = 1, . . . ? n, if U(x’) >U(x) for all x such that pix~ > p!x, for i = 1,. . . . n.Definition 2: An observation xi is directl~ re-vealed preferred to a bundle x,, written xi R x, ifpixi a pix. An observation x’ is revealed pre-ferred to a bundle x, written x’ Rx, if there issome sequence of bundles (i, ~, . . . , Xe)suchthat xiRO~, ~RO#, . . . . A?ROxe.Definition 3: The data satisfies the GeneralizedAxiom of,Revealed Preference (GARP) if x’Ra?implies #x’ > #x’.

Afriat’s theorem then says that the following fourconditions are equivalent:

1. There exists a nonsatiated utility function thatrationalizes the data.2. The data satisfies GARP.3. There exist numbers ~i, hi >0, i = 1, . . . ,n that satisfy the Afriat inequalities: p,i s # +)/#(xi – .@ fori,j = 1, . . . ,n.

4. There exists a concave, monotonic, continu-ous nonsatiated utility function that rationalizesthe data.

The point is that if there exists a set of numbersthat rationalize the data, there exists a well-behaved utility function and vice versa. Tests ofthe utility-maximization hypothesis can then beconstructed by considering whether there exists asolution to a set of linear inequalities. This prob-lem was initially solved by Afriat using linear pro-gramming. Diewert and Parkin (1985) provide ex-tensions of this approach. A major contribution ofVarian was the development of solution techniquesfor some types of problems that do not require thesolution of a programming problem. In fact, muchof the current popularity of the approach is the easeof use of Varian’s canned procedures,

While the initial paper of Afriat (1967) concen-trated on the existence of a utility function ratio-nalizing the data, more recent work (Varian 1983a;Diewert and Parkin 1985) has emphasized testingfor the existence of a utility function satisfyingcertain properties and also rationalizing the data.Thus, the approach can be used to test for homo-theticity, separability, and even technical change.

Contemporaneous with the development of therevealed-preference approach to consumer prob-lems, work proceeded on nonparametric ap-proaches to production analysis (Afriat 1972; Ha-noch and Rothschild). Again the approach was todevelop a set of conditions that must be satisfiedby the data if they were generated by a produceroptimizing some objective function subject tosome technology, Varian (1984) again providedextensions of the original work along with sugges-


tions for easy computation for many of the simplertests. An excellent discussion of testing using pro-gramming techniques is Diewert and Parkin(1983). Banker and Maindiratta have extendedVarian’s approach to cases when all the firms maynot satisfy the assumption of profit maximizationand link Varian’s results to those suggested by dataenvelopment. As in the consumer case, the testscan be used to test for specific structures as well asfor consistency with optimization hypotheses andan underlying technology. The early work of Far-rell is implicit in most of this research.

The revealed-behavior approach has many ad-vantages over the traditional parametric approachto production and consumption. It allows for the4‘testing” of economic hypotheses independent offunctional form. This is important since a separa-bility test using a translog form is a simultaneoustest for separability and the translog form. Theapproach also has several drawbacks. The primaryone is that the tests as currently implemented haveno statistical properties. The data either satisfy theaxioms or they do not. Consider, for example, thecase when the appropriate linear-programmingproblem has an infeasible solution. Suppose thatthe constraints (inequalities) are only violated atone point and that the violation is of a small orderof magnitude. Does this mean that the data is notconsistent with the maintained hypothesis, or doesit mean that a very small error occurred somewhere(measurement, optimization, model structure,etc.)? Varian (1985, 1990) has addressed the mea-surement error issue in some detail. His approachis to consider the degree of measurement error nec-essary for the data to satisfy the axioms. Tsur hasdeveloped a simpler way to implement the ap-proach for large data sets. Epstein and Yatchewhave proposed an alternative way of analyzing theproblem using a nonparametric approach to regres-sion analysis. While this approach shows promise,it requires that the variance of the error term beknown. Yatchew (1992) has extended the ap-proach in ways that are less restrictive, but someprior information is still required. Work by Matz-kin (1991b) on the semiparametric estimation ofresponse subject to concavity and monotonicityconstraints may also prove relevant, Bronars hasproposed a way to compute the power of nonpara-metric tests by comparing the actual pairs withthose that would occur if the data points were ran-domly generated. Aizcorbe has developed a lowerbound based on the same approach that is not asnumerically intensive. Nonparametric statisticalapproaches might prove to be a valuable tool inthis area. Since the revealed-behavior approachconsiders all data points in the analysis, outliers

may seriously bias the results. Whereas regressionweights the various points so that a single outlierhas minimal significance, this approach has noway to weight effectively the various observations.

Applications Using theRevealed-Behavior Approach

Analysis of Economic Efficiency

The work of Farrell was rapidly applied by re-searchers in agricultural economics. The earlywork was almost all associated with the Universityof California at Berkeley (Boles 1966; Bressler;Seitz). Most research concentrated on identifica-tion of the production frontier (efficiency locus orisoquant) and measurement of deviations from it.Boles (197 1) developed linear-programming soft-ware allowing for efficient estimation of efficiencyfrontiers.

An early digression was the work of Aigner andChu, and later Timmer on deterministic parametricfrontiers. They avoided some of the restrictive as-sumptions of Farrell’s work by postulating a func-tional form for the production function and thenestimating it subject to the constraint that the re-siduals all be nonpositive. Thus, they moved theanalysis from input space to input-output space andconverted a parameterless technique of envelopingthe input requirement set to the estimation of para-metric production functions. This approach isprobably what led to differentiating the Farrell ap-proach as nonparametric, Once the problem wasformulated as a type of least squares problem, thepossibility of stochastic assumptions and statisticalinference became obvious. A whole new approachand growth industry was spawned by the stochasticfrontier method proposed by Aigner, Knox Lovell,and Schmidt, and simultaneously by Meeusen andvan den Broeck. They proposed estimating a pro-duction function with a composed error consistingof a symmetric part reflecting measurement errorand a one-sided part reflecting inefficiency on thepart of producers not on the frontier. This approachdominated efficiency measurement for manyyears. Excellent surveys are Schmidt, Knox Lovelland Schmidt, and Bauer.

Work has continued on the ‘‘nonparametric”approach, spurred primarily by F&e and his asso-ciates, and a group in management science(Banker, Charnes, and Cooper; Chames et al.)who call the approach data envelopment analysis,A recent survey of new developments is Seifordand Thrall. The two approaches are different inmany ways. Banker, Conrad and Strauss; Banker,

Hallam A Brief Overview of Nonparametric Methods inEconomics 107

Charnes, Cooper, and Maindiratta; and Gong andSickles provide empirical comparisons betweenthe methods. A recent issue of the Journal ofEconometrics (Lewin and Knox Lovell) is also de-voted to both approaches. One disadvantage of theparametric approach is the difficulty of estimatingmultiproduct production functions. This has ledseveral researchers to propose the use of multi-product cost structures to evaluate efficiency(Schmidt and Knox Lovell; Kopp and Diewert; Zi-eschang). While this approach has significantmerit, it requires price data and a specific objectivefunction to measure a purely technical phenome-non. An alternative is to estimate the multiproductdistance function (Grosskopf and Hayes) using acomposed error term. The nonparametric approachgeneralizes easily to multiple products and has astraightforward interpretation in terms of the inputcorrespondence. It also allows wide variety interms of technological assumptions.

The nonparametric approach also allows deter-mination of the inefficiency of any individual firmin the sample. The stochastic frontier approachcomputes the frontier and an average level of in-efficiency. Jondrow et al. have proposed a way tocompute the expected value of the technical-inefficiency component of the error given the totalvalue, while Battese and Coelli obtain a best pre-dictor of the inefficiency component for panel datasets. For determination of individual firm effi-ciency, the nonparametric approach has the upperhand at the moment.

The nonparametric approach does not allow formeasurement error or outliers in any meaningfulsense. While the work of Varian (1985), and Ep-stein and Yatchew offers some hope, this is still amajor shortcoming of the approach. The stochasticfrontier approach is very flexible in this regard andallows an approximate frontier that best fits thedata.

The other disadvantage of the nonparametric ap-proach is the lack of any statistical inference.Given the types of measurement and modeler errorinherent in economics, this is a flaw that needs tobe rectified. Traditional nonparametric statisticalmethods may provide a way to solve some of theseproblems. For single-output firms, kernel or semi-parametric estimators may provide a way to esti-mate stochastic frontiers without imposition offunctional form. Nonparametric tests could also bedeveloped to consider the results of programmingor optimization models used in constructing effi-cient sets.

There have been numerous applications of thenonparametric approach to efficiency. The earlypaper by Timrner considered U.S. agriculture us-

ing state data as individual observations. Hall andLeveen used the Farrell approach to consider largeand small farms in California. Bymes, F&e, andGrosskopf analyze the efficiency of Illinois stripmines and relate it to optimal scale. Fare,Grabowski, and Grosskopf analyze aggregate Phil-ippine agriculture. Rangan, Grabowski, Aly, andPasurka apply the technique to U.S, banks, whileGrabowski and Pasurka study U.S. farms in 1860.Byrnes, Fiire, Grosskopf, and Kraft analyze therelative performance of Illinois grain farms anddecompose the efficiency measures to determinethat size is a major explanation for differences inefficiency. Fare, Grosskopf, and Kokkelenbergconsider plant capacity and technical change in astudy of Illinois electric utilities.

Testing for Functional Structure

Tests for functional structure have become popularsince the advent of flexible functional forms(Blackorby, Primont, and Russell). Flexible func-tional forms were an attempt to avoid the main-tained hypotheses inherent in traditional forms,such as the Cobb-Douglas (Diewert 1973b; Lau).While these forms have developed immense pop-ularity, they still are not completely general, asexemplified by their separability-inflexibility(Blackorby, Primont, and Russell). Various ap-proaches have been proposed to solve this problem(Pope and Hallam). Nonparametric tests provideone plausible alternative. The techniques ofVarian, and Diewert and Parkin allow a nonstatis-tical test of a variety of functional hypotheses.Chavas and Cox (1988) use aggregate U.S. dataand test hypotheses regarding separability andtechnical change. Fawson and Shumway test sep-arability and jointness hypotheses for ten U. S. ag-ricultural production regions. Barnhart and Whit-ney provide some comparisons between ap-proaches.

Nonparametric tests on the Varian type are analternative to traditional parametric tests of func-tional structure. They have several shortcomings,however. The most critical is the lack of a statis-tical basis. A comment by Hanoch and Rothschildis useful in this regard:

We suggest that researchers run these tests ondata before they use the data to estimate specificcost or production functions. Doing so will helpidentify outlying observations and gross incon-sistencies, and also provide more insight into thepotential usefulness of the data.

The greatest value of the methods may be in per-forming preliminary data analysis and not in anyfinal hypothesis testing. A second criticism is the


inability to deal with outliers or “strange” obser-vations. Points that violate the maintained hypoth-eses may be isolated cases, but there is no way toweight their significance. The third problem is theease of use of the tests. Given the simplicity of theprocedures, there is a danger that significant re-search resources will be devoted to applying themindiscriminately to a wide range of possibly inter-esting, but not terribly important, problems,

Changes in Taste and Technology

A recent application of the nonparametric ap-proach is the changes in taste or technology. Asdiscussed by Varian and others, the reason that agiven set of time-series data may not satisfy theaxioms of utility maximization is that tastes maychange over time. Chalfant and Alston have uti-lized this point to construct a test for taste change,They analyze data on meat demand and concludethat since it satisfies the revealed-preference axi-oms, taste change did not occur. They further an-alyze this issue in a later paper comparing para-metric and nonparametric procedures (Alston andChalfant). Browning has used a similar approachto test aggregate data for consistency with the ra-tional expectations hypothesis. The general rise inincome, however, may bias such tests. Specifi-cally, as budget lines shift out over time as incomerises, there will be few bundles not consumed atcurrent income that were affordable at previousincome levels. Thus, the power of these tests maybe low. Chalfant and Alston suggest that incomeelasticities could be used to investigate the powerof this test. This idea was pursued by Sakong, whoconstructed a compensated test of taste change thattested for consistency of the income-compensateddemand curves. In this work, violations of the Af-riat axioms were attributed to taste change, onlyafter allowing for changes in income. In effect, hisalgorithm computes the income elasticities neces-sary for the data to satisfy the axioms along com-pensated demand curves.

Chavas and Cox (1990) have used a similartechnique to study technical change in the U.S.and Japan. They postulate a particular form of in-put augmenting technical progress and then test forit nonparametrically. Their testis based on the ideathat if the data do not satisfy the postulates of aregular technology and cost minimization, it mustbe due to technical change. The augmentation co-efficients allow for this modification of the tech-nology in order to satisfy the axioms. Thus, theprocedure allows for a shifter (technical change) tosatisfy the postulates.

Both of these nonparametric approaches are sub-

ject to a number of criticisms. The most importantis that both explicitly or implicitly attribute all vi-olations of the behavioral and technological postu-lates to a particular factor, taste or technicalchange. Given the amount of measurement error inthe data and the models, and the presence of otherfactors, this seems undesirable. This is perhapseven more serious than in the efficiency approach,where no explicit reason is usually given for devi-ations from the frontier. This approach also suffersfrom the criticisms of no statistical properties andoutliers discussed earlier.

Applications to Finance

A promising area of application of nonparametricsis finance. Varian (1983b) suggested this applica-tion long ago. It may be particularly fruitful giventhe large data series available in finance. It mayalso be a way to supplement traditional stochasticdominance analysis. Sengupta has provided a wayto estimate the portfolio efficiency frontier usingthis method. It may also be useful in analyzingmodels with risk. For example, one could deter-mine the level of risk aversion consistent with aparticular set of choices. Another applicationmight be to measure the risk premium in financialmarkets. While this literature is just starting, itmay be an important application in the future.

Conclusions

This paper has discussed a variety of nonparamet-ric approaches to economic analysis. A few sum-mary points may be useful.

1. Traditional nonparametric methods, such asorder statistics, are underutilized in much ofeconomic research and could be fruitfully ex-ploited in the future, particularly for preliminarydata analysis and in analyzing the results of eco-nomic models.2. Series and kernel estimators provide an im-portant way to analyze large data sets. Theyavoid unnecessary functional specifications butmay be of limited use for many data sets andtime periods.3. Semiparametric estimators are a compromisethat show great promise. They will require im-proved computing skills on the part of research-ers. They will also require more emphasis on thestatistical basis of estimates. Most researcherstend to be spoiled by the assumptions and strongproperties of least squares, and have forgottenmost of the probability and inference they everlearned.4, The nonparametric procedures cataloged un-der the terms data envelopment analysis and re-


vealed preference suffer from a lack of statisticalbasis. They are nevertheless extremely powerfulways to analyze data in a general framework.They may be preferred to stochastic methodsusing restrictive functional forms. The searchfor a statistical basis for this work is a worth-while one that could have enormous payoff.5. There is some danger in attributing violationsof axioms or hypotheses to any particular factorwhen several are possible. This is not just acriticism of the nonparametric approach, but ofmodel building in general.6. There is some danger in the indiscriminateuse of nonparametric techniques. It would beunfortunate if they became the latest “new tech-nique” in search of a problem.Perhaps the greatest advantage of nonparametric

techniques is that they are data-centered. There isa tendency among economists to be obsessed withmodels and model building, and to be less con-cerned with the nature of the underlying data, Non-parametric techniques encourage the active explo-ration of the data and its central tendencies. To theextent that the use of this approach encouragesmore attention to data, economics will be a richerscience.

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