Structural change and demand analysis: a cursory review* · Structural change and demand analysis...

Structural change and demand analysis:a cursory review*

GIANCARLO MOSCHINIIowa State University, USA

DANIELE MOROCatholic University of Piacenza, Italy

(final version received March 1995)

Summary

We address the issue of structural change in demand analysis. After a briefdiscussion of the major problems and questions that arise in this context, themain body of the paper is devoted to a review of analytical methods that arerelevant to investigating structural change in demand. This survey is organisedwithin the broad categories of'nonparametric' and 'parametric' methods. Alongwith the main theoretical contributions, we try to cover a rather large body ofapplied studies. We strive to offer a critical and unified treatment of thisassorted set of contributions, stressing the pros and cons of alternativeapproaches and methods.

Keywords: demand models, flexible functional forms, nonpar ametric demandanalysis, structural change tests.

1. Introduction

'Economists have traditionally been suspicious of changing tastes, and a profession'sintellectual tastes change slowly'. (Robert A. Pollak, 1978)

* Giancarlo Moschini is Professor of Economics, Iowa State University, USA, and DanieleMoro is Researcher in Agricultural Economics, Catholic University of Piacenza, Italy.This paper was prepared while Giancarlo Moschini was visiting the University of Sienaas part of the RAISA research project of Italy's CNR, the support of which is gratefullyacknowledged. Thanks are due to Catherine Kling and Federico Perali for commentingon an earlier draft.

European Review of Agricultural Economics 0165-1587-99603/0023-023923 (1996) 239-261 © Walter de Gruyter, Berlin

240 Giancarlo Moschini and Daniele Moro

The discipline of economics has long endeavoured to be considered a science,but this aspiration often appears frustrated by a fundamental feature ofeconomics as a social science, which sets it apart from physical and naturalsciences. The laws of nature that are studied in physics today are arguablythe same as those studied by Galileo, and this fact has allowed physicists toamass an impressive amount of knowledge. In economics, however, becauseit is human behaviour within social institutions that is studied, the 'world'that economists analyse is constantly changing. For example, productiondecisions rely on ever-changing techniques employing an increasing varietyof constantly improved inputs; consumption choices span an increasingvariety of goods and are carried out by a demographically and culturallychanging population; the ability of economic agents and markets to dealwith fundamental uncertainties of the economic system is constantlyimproved by the emergence of new institutions; and so on. The possibilitythat the very structure that generates economic data may be changing posesserious problems that lie at the heart of the empirical economist's quest.How and what do we learn from empirical analysis if we are in the presenceof such 'structural' changes?

Of course, one could quibble with our opening remarks. Some, likeMcCloskey (1983), would argue that economics is best interpreted as arhetorical discipline rather than a 'scientific' enterprise. Others, perhapsmore to the point, could question the implicit assumption that empiricaleconomics only learns from observing the real world (a presumption ofmuch of econometric theory), and not by running controlled experiments.An in-depth analysis of these and related concepts is not attempted here.Our aim is simply to review analytical methods that are relevant to investi-gating structural change in demand analysis. We open with a brief discussionof the main structural change issues that arise in empirical demand studies.The survey of methods that follows is organised under the broad categoriesof 'nonparametric' and 'parametric' methods. Along with a reference to themain theoretical contributions, we try to cover the rather large body ofapplied studies that pertain to our topic. In doing so, we are mindful of thesubjective bias and unintended omissions that an exercise like this inevitablyentails. The hope is that these limitations will not reduce the usefulness ofour review for the applied researcher.

2. Structural change in demand models

In order to delimit somewhat the scope of this review, it is important todefine what is meant by 'structural change' in demand. Arguing for thelikelihood that economic data typically used by most practitioners may nothave been generated by a constant structure depends, of course, on asomewhat narrow definition of what is the relevant economic structure. The

Structural change and demand analysis 241

relevant structure should be that implied by theory, but this characterisationis not conclusive because economic theories are, by necessity, simplifications.For example, and with a notation that suits the demand focus of whatfollows, a theory may characterise a variable q in terms of a set of variablesp by the function q = /(p) while, at the same time, it may be known that amore general characterisation involves another set of variables z as in thefunction q = F(p, z). However, explicit modelling of the effects of the variablesin z may not be essential because they are only marginally interesting, orquantitatively unimportant, or quite simply unobservable (at least to theanalyst). Under some plausible conditions, the effect of z can be subsumedin the function f(p), which can serve well as a 'theoretical' relation, but inother cases, say associated with considerable changes in z, the concisetheoretical relation q=f(p) will also change. From the point of view of thetheory that explains q in terms of p alone, we would say that we are in thepresence of a structural change.1

In demand analysis, the focus of this paper, q typically represents the percapita aggregate consumption of one or more goods, and p is a vector ofprices deflated by per capita income. Abstracting from considerations ofexactly which quantities and which relative prices are explicitly considered,f(p) can then represent a market demand model such as those in a greatmany studies using time series data. In some cases, however, another set ofvariables z may turn out to be important in explaining demand behaviour.For example, z may include important demographic variables, or condition-ing variables such as advertising, or a state variable capturing systematicallyevolving preferences. The existence of structural change has important impli-cations for demand analysis, generally affecting a variety of welfare andpolicy evaluations that depend on elasticities and other estimates typicallyof interest in empirical demand models.

In demand analysis, hypotheses of structural change are often framed interms of 'changing tastes and preferences', although different studies andanalysts may have different notions of what that means. In some cases, onethinks of changes in the shape of individual utility functions of a stablepopulation of consumers, as brought about by health concerns and attentionto quality, possibly related to new information and improved scientificknowledge. In others, one may think of a changing demographic compositionof a heterogeneous collection of consumers who have different preferences.Alternatively, one may postulate changes in the preferences individualsdisplay, induced by firm strategies such as advertising and product innova-tion. Alternatively, individual choices may be based upon their basic charac-teristics (Lancaster, 1966 and 1991), and if the content of those characteristicschanges over time, individual preferences for goods may be affected. Whereasthe differences in these examples may call into question the loose characterof the notion of 'changing tastes and preferences', from an operational point

242 Giancarlo Moschini and Daniele Mow

of view it is sufficient to realise that they can all be modelled under a generalframework such as the one outlined above.

The possibility of variable preferences was considered earlier by Basmann(1956), who postulated that demand models derive from a utility functionexpressed as u = u(q, z), where q represents the consumption vector and zrepresents any other variable affecting preferences over bundles of goods. Insuch a setting, the substitution effects between goods are clearly affected bythe variables z. The notion that preferences underlying demand models maychange over time has found its critics. For example, Stigler and Becker(1977) have argued that various phenomena that may indicate a change intastes can otherwise be explained by stable preferences; often a change inpreferences is invoked to justify the inappropriateness of some economicmodels in interpreting observed data. Others have questioned such a position(see Pollak, 1978), and considered changes in consumer preferences, bothexogenous and endogenous, as an important aspect of demand analysis andempirical work (Pollak, 1976 and 1977).2

Viewed in the light of our earlier discussion, however, much of the debateon whether it is acceptable and/or useful to postulate structural change indemand analysis is a matter of semantics, and depends on whether onethinks in terms of q = f(p) or views the problem from the more generalrelationship q = F(p, z).3 From an operational point of view, more pressingissues are: (i) to determine when a model of the q=f{p) type is appropriateand when a more general model is warranted; (ii) to find which model,among the many possible extensions of the q = F{p, z) type, may be moreappropriate; and (iii) to tie the selected empirical model to a theoreticalframework so that a consistent interpretation of empirical findings may bepossible. Many procedures are available to deal with these specificationproblems. Although these may already be familiar in some other fields ofeconomic analysis, we will review many of them here in the light of theirapplication to demand analysis.

3. The nonparametric approach

There are two basic ways of characterising consumer demand. Classicaldemand theory, moving from basic rationality axioms, assumes that consum-ers have preferences over goods which are described by a utility function.Demand behaviour is then seen as the result of the consumer maximisingher utility (subject to a budget constraint). Alternatively, revealed-preferencetheory takes individual choices as primitive elements, and sets forth restric-tions on these choices that parallel the rationality axioms of classical demandtheory.4 Although much of the intellectual appeal of revealed preferencesderives from the fact that it frees demand theory from the burden of a 'utilityfunction', what is being exploited in modern nonparametric demand analysis


is, in fact, the equivalence of the restrictions that revealed preferences andutility maximisation impose on a finite set of observed consumer choices.Following the seminal contributions of Afriat (1967, 1973), Varian (1982,1983) developed a series of operational procedures to check the consistencyof a body of data with the theoretical properties of revealed preferencesand/or utility maximisation.5 Specifically, these procedures entail testing forconsistency of the data with the weak, strong or generalised axioms ofrevealed preference (WARP, SARP and GARP, respectively). Failure tomeet requirements for these axioms may be interpreted as an indication ofstructural change in demand: if we maintain that consumers are, in fact,maximising utility, then a change in the structure of preferences couldaccount for the observed violations.

To illustrate briefly the consistency implications of revealed preferences,using Varian's notation (Varian, 1992), WARP requires that ifq'R^, thenit is not the case that qJRDq', where RD denotes the directly revealed prefer-ence relation, and q' and qj are any two observed consumption vectors.Testing for consistency of the data with WARP requires searching for viol-ations in pairwise comparisons; a violation is found when both Cu > Cu andCjj > Cp, where C,,- indicates the expenditure with prices in time i of thebundle chosen in time j , and C,, > CtJ is equivalent to q'RDqj.6 A morestringent test for consistency requires ruling out intransitivities, i.e. checkingthe consistency of the data with SARP. This axiom requires that if q'RqJ,then it is not the case that q'Rq\ where R is the transitive closure of therelation RD? Applications of revealed preference procedures to test structuralchange in demand data include Landsburg (1981), Thurman (1987), Chalfantand Alston (1988). Burton and Young (1991), Sakong and Hayes (1993),and Dono and Thompson (1994).

The most important feature of this approach is that it allows one to drawinferences on the nature of a particular data set without making assumptionsabout the functional form of demand functions. This clearly eliminates onesource of arbitrariness present in parametric procedures (see below).Unfortunately, the nonparametric approach has shortcomings of its ownthat make the interpretation of findings from such an approach somewhatdifficult. One of the major concerns is the power and size of such tests,especially in view of their 'exact' nature [the fact that these tests are non-statistical (Epstein and Yatchew, 1985)].8

3.1. Power and size of nonparametric tests

Under the view that nonparametric tests are exact, their size is zero; thus,even a single violation should be enough to invalidate the maintainedhypothesis. Hence, in this setting, the only admissible concern is the powerof such tests. In our case, the power of a nonparametric test relates to itscapability of detecting violations of consistency when, in fact, there is


structural change. There are legitimate reasons for believing that such apower may not be great. For example, Chalfant and Alston's (1988) resultsbased on a quarterly Australian data set, imply, among other things, theabsence of seasonality in consumption, although seasonality is a fairlyaccepted feature of quarterly meat demand. The failure to detect at leastsome seasonality may lend support to Varian's (1982) and Landsburg's(1981) observation that nonparametric tests based on revealed preferencesare likely to have little power with aggregate time series consumption data.This may be due to the fact that, as initially pointed out by Varian (1982),when income (or expenditure) growth is large compared to changes inrelative prices, then it may be difficult to find comparable points: at thelimit, every observed bundle is revealed preferred to all the previous ones,and there is no possibility of finding any violation, even if a strong shift inpreferences may have occurred.9

A possible solution to this problem is that of adjusting the data. Ideally,in such cases, one would like to work with income-compensated demandbundles, but unfortunately they are not observable. Chalfant and Alston(1988) try to recover them by adjusting their data for expenditure growthunder the assumption that all expenditure elasticities are equal to one.Clearly, such a procedure is admissible only when preferences are homo-thetic, a hypothesis that is systematically rejected by virtually any data set.Sakong and Hayes (1993), extending Chavas and Cox's (1990) linear pro-gramming technique to consumer demand analysis, adjust their data in away that allows for non-unitary income elasticities. However, this approachessentially recovers Hicksian demand by a first-order approximation and,strictly speaking, there are no theoretical restrictions one should expect tobe satisfied by such adjusted data. The bottom line is that compensateddemands cannot be recovered from observed Marshallian consumption bun-dles unless one resorts to functional form assumptions of a nature typicalof parametric models, thereby undermining the main distinguishing featureof nonparametric analysis.

An alternative tactic is to relax the 'exact' view of nonparametric tests.The proposed solution is to postulate that observed consumption patternsare measured with error. As it stands, this has to do both with the powerand the size of the test. The approach is described in Varian (1985) and, inits original form, it is basically related to the size of the test: the idea is tominimise a 'distance' function of the observed data from the closest pointthat satisfies the null hypothesis.10 As common in the parametric approach,it is assumed that observed quantities are random variables, and acceptance(rejection) of the null hypothesis may simply be due to the impossibility ofobserving 'true' quantities.11 Thus data may be perturbed in order to elimi-nate (or induce) violations of consistency with revealed preference axioms,and some measure of this adjustment gives the level of significance of thenonparametric test.


3.2. Limitations of the nonparametric approach

The appealing features of the nonparametric approach, and its extension totesting for structural change, are challenged by the issues of power and sizeof these tests. The proposed solutions are not completely satisfactory. Forinstance, an example in Gross (1991) strongly undermines the usual test ofWARP. Using a Cobb-Douglas utility function with two goods, he showsthat a very large shift in preferences between two points, combined with nochange in expenditure and a small change in prices, produces a smallviolation of WARP when measured in terms of C,,. In particular, a slightmodification of the example shows that, under large preference change, asmall change in income (of less than 1 per cent) will make it impossible todetect the preference shift.12 Thus, simply showing that postulating a smallmeasurement error can restore theoretical consistency in observed quantitieshas the potential of being quite misleading.

Gross (1991) proposed to test for consistency of the data by using themoney metric utility or the direct compensation function, instead of theexpenditure index (i.e. testing for WARP), by extending an idea of Varian(1990): under the assumption that preferences are stable, the direct compen-sation function provides a measure of the minimum expenditure at time j ,given that preferences are unchanged with respect to time i, and that theindividual is willing to attain the level of utility that his/her observed bundlewould provide. Thus, if preferences are unchanged and the individual ismaximising utility, the minimum expenditure will be the same as the actualexpenditure. If preferences are unstable, then the difference between mini-mum and actual expenditures gives an indication of the hypothetical wastein expenditure when maximising utility in time j with preferences of time i:the larger the difference, the more serious the shift in preferences betweenthe two periods.13 Although, in principle, the adoption of the direct compen-sation function can increase the power of the test, its computation requiresknowledge of the particular structure of preferences. However, that is typi-cally not known, and use of the approximations of the direct compensationfunction discussed by Varian (1982) does not necessarily improve the powerof the test.14

Some final points should be kept in mind in evaluating the ability of theseprocedures to detect structural change. First, insofar as applications pertainto a subset of the goods comprising consumer's choice (say, three or fourmeat goods), there are virtually no restrictions that one can place on sucha subset of data, unless the assumption of separability is invoked. Varian(1988) concludes that consistency of observed data must be interpreted only'... as tests for separability of the observed choices from other variables inthe utility function rather than as a test of maximisation per se\ Hence,nonparametric tests on subsets of data are conditional on a correct assump-tion of separability. Put another way, the null hypothesis of a consistency

246 Gicmcarlo Moschini and Daniele Moro

check is composite, and the problem remains of how to interpret consistency.To be sure, this is a specification problem that affects parametric andnonparametric applications alike. Still, it should suggest more caution inclaiming that structural change has not occurred when the analysis is basedon a handful of goods only.15 Second, the whole revealed theory apparatuspertains to one individual's choices, but empirical applications typically relyon aggregate (market) data. The validity of this procedure would seem todepend on the exact aggregation conditions to be satisfied. Such conditionsare known to be rather stringent (Deaton and Muellbauer, 1980a: Chapt. 6),but they are typically ignored in empirical applications of nonparametricmethods.

4. The parametric approach

In this approach an explicit model to represent demand functions is specified,either directly or in terms of a primitive utility (direct or indirect) or expendi-ture function. Relative to the nonparametric approach just described, theanalyst is required to make an assumption about the functional form relatingthe variables of interest. Whereas this concession to arbitrariness may bedeemed undesirable, there is really no particular novelty here, as this stepis required in virtually every econometric application. It is common to tryand ensure that the arbitrariness of this step does not prejudge importantaspects of the analysis by choosing a 'general enough' specification(see below). Having chosen a functional representation of preferences, thereare at least three alternative (but related) ways of examining structuralchange issues: consistency analysis, parameter instability analysis and explicitmodelling of structural change by a trend or other economic variables.

4.1. Consistency analysis

Given the chosen functional representation of preferences, the analysis couldproceed as in the nonparametric approach. Specifically, one could testwhether a body of data satisfies the theoretical restrictions of demand theory,such as homogeneity of degree zero in prices and income, and symmetryand negative semidefiniteness of the matrix of Slutsky substitution terms.Testing for these properties is, indeed, the parametric counterpart of checkingthe satisfaction of WARP, SARP, or GARP. This set of equivalences hasbeen made explicit in the work of Kihlstrom et al. (1976), Hildenbrand andJerison (1989) and John (1995), who show the relation between WARP andhomogeneity, and the equivalence between a weaker version of WARP andthe semidefinite negativeness (but not the symmetry) of the Slutsky substitu-tion matrix. The relation of the nonparametric approach with integrability


of demand functions is also reviewed in Mas-Colell (1978), and relateddevelopments can be found in Chiappori and Rochet (1987).

Curiously, although the relationship between common nonparametricconsistency checks and tests of homogeneity, symmetry, and negativity isobvious, few studies seem to have taken that approach. Perhaps it is due tothe recognition that other sources of misspecification can affect the outcomeof such tests, including separability assumptions, aggregation conditions,omitted variables, distributional assumptions, and functional form choice.16

Although true, it is important to emphasise that all these limitations (withthe exception of the functional form selection) also affect the nonparametricprocedures described above. Perhaps more important, the fact that theparametric approach has followed other routes may be due to the relativerichness of options that it offers. Thus, analysts have typically preferred tomaintain the theoretical properties of demand theory as much as possible,and look for evidence of structural change in other ways.17

4.2. Tests of parameter stability

A more common approach to detecting structural change originates fromthe observation that such changes will alter parameter values, given anunchanged functional form. Thus, testing for parameter instability has beena common way to test for structural change. Tests that can be used includethe classical Chow test (Chow, 1960; Fisher, 1970) and the CUSUM test(Brown et al., 1975), with their modifications and generalisations (Dufour,1982; Davidson and MacKinnon, 1993: Chapt. 11), the Farley-Hinich test(Farley and Hinich, 1970; Farley et al., 1975), the fluctuation test (Plobergeret al., 1989) and other tests (Andrews, 1993). Applications in demand analysisinclude Moschini and Meilke (1984), Martin and Porter (1985), Atkins et al.(1989), Chang and Kinnucan (1991), and Chen and Veeman (1991).

The relevant issues here are the size and power of the proposed tests. Thesize of the test is crucial, because the null hypothesis that we are typicallytesting is actually a set of hypotheses that are best grouped generically as anull hypothesis for misspecification (type-I errors). Again, the main problemis that parameter instability may arise from different sources, with structuralchange being only one of the possibilities. Hence, these tests are properlyinterpreted as tests for misspecification arising from any source of invalidconditioning. The problem has been widely recognised in the econometricliterature, and it has received a great deal of attention in applied demandanalysis (see Alston and Chalfant, 1991).

As for power, the performance of the tests for parameter instability pro-posed in literature often depends on the nature of the alternative hypothe-sis.18 Testing the null hypothesis that a parameter vector is stable over thesample period implies a comparison with an alternative hypothesis that cantake different forms, like a one-time 'structural' change, with a known or


unknown change point, or smooth patterns of change, or other cases whereno information is available on the timing of structural change (see Andrews,1993). Thus, the particular nature of the structural change may be important,and therefore it is convenient to resort to tests that have high power againstdifferent alternatives of structural change: a priori information may be usefulin choosing the appropriate test.19

4.3. Explicit modelling of structural change

A related procedure consists of explicitly modelling the effect of structuralchange by introducing, in the estimating model, variables indexing tasteshifters. The simplest version of this approach is to include a time trend ora (set of) dummy variable(s). One basic application of this approach is theuse of dummy variables to account for seasonality in consumption whenusing frequent time series data (say, quarterly). Alternatively, under theassumption that preferences have changed smoothly over the observationperiod, a trend variable can provide a useful proxy for these effects, andfollowing Stone (1954), Deaton (1975), and Jorgenson and Lau (1975), thisapproach has been adopted in a number of studies. Furthermore, a distinc-tion can be made between systematic and random variations in parameters:systematic variations imply that a certain deterministic pattern of changefor the parameter vector can be modelled, even the most general, whilestochastic variations allow for a more flexible specification of the pattern ofchange.20 The statistical problem then reduces to that of the significance ofthe added variable(s). The relationship with parameter instability tests isobvious. For example, one can restate the Chow tests in terms of thesignificance of a set of dummy variables. Thus, our econometric discussionof parameter instability tests applies here as well.

Of course, modelling structural change by a time trend is a somewhatunpleasing shortcut, and the use of such a trend is often interpreted as anadmission of ignorance of the true form of structural change. The alternativeis to be more specific about what sort of structural change one has in mind,and model it accordingly. For example, under the assumption that USconsumers' egg preferences were changing due to an increased concern aboutthe health hazards of large intakes of cholesterol and saturated fats, Brownand Schrader (1990) constructed a 'cholesterol information index' from thenumber of articles in medical journals linking cholesterol in diet to heartdisease. Whereas, in principle, this approach is attractive, it still involvesrather arbitrary choices,21 or may turn out to be econometrically equivalentto a trend (for example, the correlation coefficient between Brown andSchrader's cholesterol index and a linear time trend is 0.986).

In some cases, direct modelling of preference shifters is clearly the preferredalternative, especially when dealing with issues of demographic or dynamiceffects in demand. That individuals may have different preferences (and


therefore demands) is widely accepted. This diversity is supported by anumber of cross-sectional studies showing the significance of demographiceffects (such as household size, age of household members, region of resi-dence, employment status, and others) in demand models.22 Insofar as rele-vant demographic variables change over time, market demand functionstypically investigated in demand studies using time series will be affectedand may display structural change.

Similarly, the importance of dynamic components in aggregate demandhas long been recognised. One explanation for such dynamics postulatesthat individuals' consumption is sluggish due to their dependence on a statevariable that can be interpreted as a 'stock of habits' (Houthakker andTaylor, 1970; Pollak, 1970). This 'habit formation' hypothesis is usuallyimplemented in demand models by specifying preferences as depending onpast consumption levels. The econometric problem of testing for this specificinstance of structural change then reduces to that of the significance of theadded lagged consumption variable(s).23 Alternatively, as in Anderson andBlundell (1983), one can evaluate how accounting for dynamics affectsconsistency tests (homogeneity and symmetry). Of course, dynamic consider-ations can be nested in other structural change issues. For example, onecould consider testing for parameter instability within a dynamic model(Kramer et al, 1988).24

Other variables that may play an important role at one time or anotherin explaining consumption are advertising, product innovation, quality,information and environmental changes, and many studies have tried toaccount for these effects in demand systems.25 However, some of them aremeasured (or proxied) by variables that display a strong time trend, especi-ally when some index is used; thus, their effect is almost indistinguishablefrom a trend, and the interpretation of results from this model may still bemisleading in attributing a role for some effects (see above).

4.4. The issue of functional form

Consistency, parameter instability and explicit modelling of structuralchange are all concerned with the issue of model specification. As such, theyaddress the basic problem of structural change, which relates to both theparametric and nonparametric methods we have discussed. On the otherhand, the parametric approach requires a few additional assumptions, whichexposes it to some additional problems. First, one often needs to make aprior choice of endogeneity and/or exogeneity of prices, quantities andincome in standard demand studies. Overlooking the endogeneity of somevariables has the potential of introducing considerable biases in inference(LaFrance, 1991). Whereas preliminary tests may help in adopting a correctspecification, it is clear that wrong assumptions may lead to spurious identi-fication of structural change (Wahl and Hayes, 1990; Eales and Unnevehr,


1993). For instance, structural change in supply, such as technologicalchange, can appear as a demand shift.26 However, the most obvious short-coming of parametric relative to nonparametric methods is that one typicallyneeds to assume a specific functional form. Hence, inference in parametricmodels is usually conditional on the choice of the functional form. This hasled to considerable controversy over the analysis of structural change withparametric models, typically centring on the interpretation of the resultsand questioning the possibility of ever concluding anything useful from aparametric analysis of structural change (Alston and Chalfant, 1991).

One can try to take some precautions against the limitations of anarbitrary functional form by selecting a sufficiently general functional form.For example, in a single equation framework, Moschini and Meilke (1984)and Martin and Porter (1985) test for structural change in flexible parame-terisations based on the Box-Cox transformation. More generally, in asystem's framework, it has become common to rely on the concept of flexiblefunctional forms (FFFs), pioneered by Diewert (1971, 1974) and byChristensen et al. (1971), which centre on the approximation properties ofa Taylor series expansion of the unknown true function. Systems of equationscommonly used in demand analysis, such as the translog (Christensen et al.,1975), the almost ideal (Deaton and Muellbauer, 1980b), and the normalisedquadratic (Diewert and Wales, 1988) satisfy this notion of flexibility; anothercommonly used system, the Rotterdam model (Barten, 1966; Theil, 1975and 1976), satisfies an equivalent definition of flexibility.

Whereas this notion of flexibility is useful, it does not eliminate thearbitrariness of the functional form chosen. FFFs provide an accurateapproximation only at a point, but not much can be said about theirapproximation properties away from that point; thus, Diewert's notion offlexibility is not sufficient to ensure that these second-order forms canapproximate the underlying (true) function over the entire region of theprice-income space. Therefore, if the approximation properties are only validaround part of a point, and there is great variability in the price-incomespace over the sample, then parameter instability can be detected.27 Becauseof this, it has long been recognised that estimation and hypothesis testingare sensitive to this choice (Gallant, 1981). The open issue, specifically intesting for structural change, concerns the extent of this sensitivity. Inparticular, work by Alston and Chalfant (1991) has shown that the perfor-mance of the test statistic for structural change can be heavily affected bythe choice of the functional form. Specifically, in a simulation setting wherethe true function is known by construction, it is shown that the adoptionof an incorrect functional form enormously increases the proportion ofrejections of the (true) null hypothesis of stable preferences.

The point of Alston and Chalfant (1991) is well taken, but the results theyactually present are too extreme and pessimistic because their Monte Carlosimulations do not account for the (arbitrary) choice of the signal-to-noise


ratio (which turns out to be important when the explanatory variables aretrended). To illustrate, consider their experiment using the General Electric(GE) data reported in Theil (1971: 296), which shows that when the dataare generated by a linear model without structural change, a Chow test forstructural change at the 5 per cent significance level will give the wronganswer 100 per cent of the time if carried out with the wrong functionalform (double-log, semi-log, or square root functions). Crucial to this finding,however, is the fact that simulated samples are obtained by adding randomterms with mean zero and unit variance, but a variance of one, given thedesign matrix at hand, corresponds to a population R2 in excess of 0.999.Because a perfect fit is rarely a feature of typical empirical problems, it maybe of interest to consider the same experiment with a more reasonableassumption on the error term. For example, one can repeat the experimentby fixing the variance at the level required to replicate the R2 obtained byfitting the linear model with the GE data (which, by assumption, is the 'true'model), i.e., R2 = 0.705. The results in this case are quite different: the wrongrejection of the hypothesis of no structural change (at the 5 per cent level)falls from 100 per cent to, respectively, 21 per cent (semi-log), 23 per cent(double-log), and 11 per cent (square root). It is clear that the fraction ofwrong rejections of the true hypothesis of no structural change is dramati-cally affected by the signal-to-noise assumption, although for incorrect func-tions it is still greater than the nominal size.

In order to reduce the potential bias of the arbitrary choice of a functionalform, one can test the validity of the maintained form, say by using Ramsey's(1969) RESET test and its generalisations (Davidson and MacKinnon, 1993:Chapt. 6). An alternative strategy is that of specifying larger models thatcan encompass different functional specifications: structural change testsconducted on these larger specifications should reduce the initial condition-ing. Along the same line is the idea of reducing misspecification effects bycomparing and selecting different models. FFFs can be compared, butusually these models are not nested, and hence this strategy requires the useof appropriate procedures to test non-nested models.28

5. The 'econometric' nonparametric approach

Whereas FFFs can offer a workable solution to the arbitrariness of specifyingan explicit parametric model, and have become widely adopted in empiricalapplications, it is apparent from the foregoing that their use has come undersome rightful criticism. In particular, because most FFFs are essentiallyTaylor series expansions of the unknown true function, these forms have theundesirable feature that their approximation properties are invariant to thesample size. An improved approximation was obtained by Gallant (1981)by adding trigonometric terms to the leading terms of a Taylor series


expansion, which produces the Fourier functional form. The increased flexi-bility of this approach hinges on the requirement that the number of trigono-metric terms tends to infinity as the sample size grows. An alternativeapproximation strategy relies on the use of nonparametric methods. Here,the term 'nonparametric' applies to the statistical method of density estima-tion (Silverman, 1986). Using this approach, one can estimate the regressionfunction, or other function of interest, without making any parametricassumption about the true functional form (Hardle, 1990; Ullah, 1988). Aparametric form can sometimes be retained for a portion of the model, andthis leads to semiparametric specifications (Robinson, 1988).

Applications of these nonparametric methods in the study of structuralchange issues are clearly possible. The simplest application, again, would beto test for the consistency of a body of data with the properties of demandtheory. For example, one could test the homogeneity property following theprocedure used by Moschini (1990) to test for constant returns to scale.One could also test symmetry along the lines discussed by Stoker (1989).Alternatively, one could use the nonparametric approach to test for thesignificance of added variables meant to index structural change. Particularlyuseful here may be Robinson's semiparametric approach, as applied, forexample, by Moschini (1991).

A distinctive feature of the nonparametric approach based on densityestimation, as compared with other nonparametric methods discussed earlier,is that it has a solid statistical foundation which is appealing for empiricalapplications requiring formal tests of hypotheses. In a sense, therefore,nonparametric estimation of the regression function represents the obviousevolution of the framework of analysis based on parametric flexible func-tional forms. The main advantage is that it allows econometric modellingand hypothesis testing that do not depend on arbitrary (and possiblyrestrictive) parametric assumptions. However, there are also a number oflimitations to this approach. For example, an important issue at the applica-tion stage is the choice of smoothing parameter (window width). This iscrucial to estimation results as it corresponds to a sort of implicit parameteri-sation. Unfortunately, there is no widely accepted method for the choice ofwindow width, and this introduces an element of arbitrariness in the estima-tion process.

Another issue concerns the statistical properties of the nonparametricestimators. Under certain assumptions, one can prove consistency andasymptotic normality for most nonparametric estimators. However, the rateof convergence in distribution is typically slower than that of parametricmodels. Also, this rate can be very slow if the number of regressors is large(the 'curse of dimensionality'). A workable solution may sometimes be foundby using the semiparametric model, where the parametric portion can beestimated consistently and efficiently, or by using the method of averaging(Hardle and Stoker, 1989). In general, however, large samples and simple


models seem necessary for a reasonable application of nonparametricmethods of estimation.

6. Concluding remarks

A number of explanations of certain features of market demand may fallunder the heading of 'structural change'. Several methods are used to detectthese situations in applied analysis, and to improve demand models typicallyused for welfare and policy analysis. In this paper, we have reviewed mostof these procedures, grouping them under the broad categories of 'nonpara-metric' and 'parametric' methods. Nonparametric methods based onrevealed-preference theory can be used to analyse the consistency of a bodyof consumption data with the assumption of utility maximisation. Becausethis can be accomplished with few accessory assumptions, nonparametricmethods have the potential of yielding conclusions needing minimal qualifi-cations. However, such methods are difficult to extend over the most basicframework of standard demand analysis, and are incapable of producingaccurate quantitative predictions and response functions typically necessaryfor applied problems.

Parametric methods usually require one to specify a functional form, andthus conclusions are conditional on a degree of arbitrariness. However, suchmethods are typically easy to adapt and generalise in order to handle agreater variety of problems and questions, and the careful analyst can takeprecautions against the danger of being misled by a grossly incorrect func-tional form. Most of the parametric approaches that we have reviewed are,in fact, model specification tests. Whereas this is quite appropriate for anumber of hypotheses concerning structural change in demand analysis, itdoes raise the issue of interpreting the results one obtains, and emphasisesthe fact that a carefully constructed theoretical model should always precedeany empirical application.

We have, in addition, reviewed what we called the 'econometric' nonpara-metric approach. This is similar in spirit to the nonparametric methodsdiscussed earlier, because it attempts to do away with arbitrary functionalspecification choices, but is also akin to the parametric methods in incorpo-rating the explicit stochastic framework that constitutes the hallmark ofeconometrics. If this approach is taken as a good-faith effort to reconcilethe two classes of methods discussed earlier, it offers a sobering message. Inmaking explicit the large information requirements for its implementation,it suggests that overcoming the main limitations of parametric methods maybe difficult, and that the potential advantages of nonparametric methodsmay prove hard to achieve in applied research.


Notes

1. One should not be misled into concluding that structural change issues can be assumedaway by working only with models of the type q = F(p, z). Because the vector z potentiallyincludes a great many things, virtually all of the economic models used for empiricalanalysis are of the concise type q=f(p), and which component of z will turn out to beimportant as a vehicle of 'structural change' may depend on circumstances or specificapplications.

2. Endogenous tastes are defined as a preference ordering that depends on other people'sconsumption and/or on prices, such as in the 'snob' effect (see also Pollak and Wales, 1992).

3. Somewhat related to this, Bessler and Covey (1993) distinguish between associational andstructural inference, and claim that results from observed data often show just prima faciecausal relations, and drawing associational inference on the economic structure is incorrectand misleading.

4. Revealed-preference theory was developed by Samuelson (1938, 1948) and Houthakker(1950). Other contributions include Koo (1963,1971) and Richter (1966).

5. Related results can be found in Diewert and Parkan (1985) and Chavas and Cox (1993).6. As in Chalfant and Alston (1988), a test for WARP may be implemented by constructing

the matrix <D whose elements are defined as Oy = Cij/Cu. We have a violation of WARP ifboth <t>ij and <!>,•,• are less than one.

7. See Varian (1992) for more details. One important difference between WARP, SARP, andGARP is that the first two axioms assume that a unique bundle is chosen with each budgetline (as implied by strictly convex preferences in utility theory), while GARP is consistentwith multiple optimal bundles for any given price-income combination.

8. The power of the test is related to the probability of type-II errors, i.e. the probability ofaccepting the null hypothesis when it is false; the size of the test is the probability of type-Ierrors, i.e. the probability of rejecting the null hypothesis when it is true. Alston andChalfant (1992) provide a discussion on the issue of power and size of nonparametric tests.

9. Bronars (1987) stresses the fact that useful information may be obtained from the observeddata when budget lines cross, even if there are no comparable bundles. An application ofthis approach is found in Burton (1994).

10. See also discussion in Epstein and Yatchew (1985). This procedure is applied by Burtonand Young (1991). Tsur (1989), working under some special assumptions, proposes a fastprocedure to test for the significance of GARP violations. A discussion about the size andthe power of the test under the provision that data can be measured with error can befound in Alston and Chalfant (1992).

11. In empirical work, it is common to assume that only quantities are observed with error,while prices and expenditure are taken as exact (a parallel assumption is usually made inparametric models). Of course, measurement errors are not the only possible source ofrandomness. Errors may arise in the budgeting process, and observed quantities may notbe the maximising ones because of errors in expenditure allocation. Finally, as in Varian(1990), individuals may have a 'nearly' optimising behaviour: thus, the usual approach ofanalysing randomness in quantities, both in parametric and nonparametric models, maybe misleading, since relatively large errors in quantities may be consistent with smalldepartures from optimising behaviour.

12. Gross (1991) considers a Cobb-Douglas utility function of the form u = q\q\~", where a =0.1 at time i and a = 0.9 at time j . Expenditure in $10,000 in both periods, and prices arep, = $99andp2 = S>101 at time i, andp t =p2 = $100 at t ime/Given optimal budget shares,consumption at time i is 10.1 for <j, and 89.11 for q2, whereas at timej it is 90 for q, and 10for q2- This extremely large shift in preferences produces <J>(J = 0.992 and <bJt = 0.992, whichare very close to unity: thus, small adjustment in quantities could restore consistency.Alternatively, if expenditure at timey were equal to $9,915, then <P0 = 0.984 and Oj-, = 1.001.


In this case there would be no violation, and thus a small change in income would implythat the exact test does not detect structural change.

13. Gross (1991) shows that by computing the direct compensation at time j by solvingmin,{ piq:ui(q) = ui{qi)}, one obtains a minimum expenditure of $1,724.27, implying thatunder the hypothesis of no structural change the consumer would have wasted about83 per cent of her income.

14. One can analyse further the example in Gross (1991) by computing the overcompensationfunction m + {p, q) as in Varian (1982), where m + (p, q) is denned as the solution tomin:{pz:zRq}- In our example m*(p',qJ) = Cij, and the procedure proposed by Grosscollapses to the one used by Chalfant and Alston (1988).

15. For example, one may find that a separable group is not affected by structural change.However, if the change of preferences takes place in the (typically untested) first stage, thenthe consumption of the goods in the separable group is biased if the group is not homothetic(Moschini, 1991).

16. For this reason, McGuirk et al. (1993, 1995) insist on the notion that the model must be'statistically adequate' before one can meaningfully test hypotheses.

17. Using Japanese data, Maki (1992) finds that homogeneity and symmetry are satisfied whentaste change is allowed. A similar approach is used in Chen and Veeman (1991).

18. See Kramer et al. (1988), for a discussion on the power of the CUSUM test and itsextension to dynamic models; this test is usually affected by the local power problem, thatis, the power is dependent on the form of the alternatives. The types of tests proposed byAndrews (1993) do not suffer from this problem, and may have power against severalalternatives. However, much of the work on the power of tests for structural change isdevoted to models where parameter instability is originated by structural change, thus theissue of a composite null hypothesis may still be crucial in empirical applications. See alsoAndrews and Fair (1988) for testing for structural change in nonlinear models.

19. The sample size may be a critical aspect here, since most of these tests have asymptoticproperties, and do not perform well in small samples (see Fachin, 1994, for a discussion ofthis issue in the context of variable additional tests). For example, Wales (1984) has shownthe tendency towards over-rejection of tests for structural change based on the likelihoodratio: this bias is more serious when the sample size is small.

20. Several applications of these techniques can be found. Moschini and Meilke (1989) adopteda gradual switching regression model for structural change in a demand system (see alsoReynolds and Goddard, 1991; Burton and Young, 1992; Eales and Unnevehr, 1988; andBjorndtil et al., 1992). A more complex gradual switching model, with stochastic cross-equational constraints, has been proposed by Tsurumi et al. (1986), and adapted to meatdemand analysis by Goodwin (1992), using a Bayesian approach. Poirier (1976) studiedthe use of spline functions in modelling structural change. Chavas (1983) estimated arandom coefficient model, and Leybourne (1993) a time-varying coefficient AIDS model,both based on the Kalman filter. Other applications of systematic and stochastic changesin parameters are in Kinnucan and Venkateswaran (1994). A slightly different applicationis in Choi and Sosin (1990), who test for structural change using time-varying multiplicativeterms. Finally, for other applications, see Buse (1989).

21. For example, the cholesterol index developed by McGuirk et al. (1995) is quite differentfrom the cholesterol index constructed by Brown and Schrader (1990), and which of thetwo is more appropriate is not clear.

22. Demographic effects have been widely incorporated in demand analysis [Pollak and Wales(1981); Ray (1984); Lewbel (1985); Rossi (1988); Alessie and Kapteyn (1991); see alsoPollak and Wales (1992, Chapt. 3)]. Applications in food demand analysis include Kokoski(1986), Heien and Wessells (1988), Heien and Pompelli (1988), Gould et al. (1991), Hortonand Campbell (1991), Gao and Spreen (1994), and Gould et al. (1994).

23. A general dynamic specification of demand systems, that includes habit formation as aspecial case, can be found in Anderson and Blundell (1982, 1983).


24. This approach is found in Burton and Young (1992). Long-run dynamics are accountedfor in Kesavan et a!. (1993).

25. Scaling and translating, typically used for demographic effects, can also be applied for othereffects, following Lewbel's (1985) generalisation. For a treatment of advertising in demandsystems, see also Selvanathan (1989) and Duffy (1987). Applications can be found inKinnucan (1987), Chang and Kinnucan (1991), Green et al. (1991), Brown and Lee (1993),Kinnucan and Venkateswaran (1994). Increasing attention has been devoted to the role ofinformation and health concerns: see applications in Brown and Schrader (1990), Cappsand Schmitz (1991), Chang and Kinnucan (1991), Jensen et al. (1992), Gould and Lin(1994). The treatment of quality variation is considered in the pioneering work byHouthakker (1952) and Theil (1952); see also Hanemann (1981), Cox and Wohlgenant(1986), Deaton (1989), and Nelson (1991). Other factors that have been accounted for areconvenience (Capps et al., 1985), retailing (Bjorndal et al., 1992), government intervention(Gould et al., 1991), and product innovation (Gould et al., 1994).

26. Eales and Unnevehr (1993) and Dahlgram (1988) use an inverse demand system to test forstructural change in meat demand. Moschini and Vissa (1993) develop the intermediateassumption of a mixed demand specification.

27. A possible solution to the problem may be to resort to FFFs with larger approximationproperties. Gallant (1982) proposed a form, based on a multivariate Fourier series expan-sion, where a better approximation is obtained at the expense of an increased number ofparameters. This approach has been applied in food demand estimation and testing forstructural change by Wohlgenant (1985).

28. For an early review of non-nested testing procedures, see MacKinnon (1983). Severalapplications can be found in demand analysis since Deaton (1978); a recent application isBrester and Wohlgenant (1991). An alternative interesting approach, rooted in the modelselection literature, is the likelihood dominance criterion proposed by Pollak and Wales(1991).

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Daniele MoroIstituto di Economia Agro-alimentareUniversita Cattolica del Sacro CuoreVia Emilia Parmense, 8429100 PiacenzaItaly

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