Structural change and demand analysis:
a cursory review*
GIANCARLO MOSCHINI
Iowa State University, USA
DANIELE MORO
Catholic University of Piacenza, Italy
(final version received March 1995)
Summary
We address the issue of structural change in demand analysis. After a brief
discussion of the major problems and questions that arise in this context, the
main body of the paper is devoted to a review of analytical methods that are
relevant to investigating structural change in demand. This survey is organised
within the broad categories of'nonparametric' and 'parametric' methods. Along
with the main theoretical contributions, we try to cover a rather large body of
applied studies. We strive to offer a critical and unified treatment of this
assorted set of contributions, stressing the pros and cons of alternative
approaches and methods.
Keywords: demand models, flexible functional forms, nonparametric demand
analysis, structural change tests.
1.
Introduction
'Economists have traditionally been suspicious of changing tastes, and a profession's
intellectual tastes change slowly'. (Robert A. Pollak, 1978)
*
Giancarlo Moschini is Professor of Economics, Iowa State University, USA, and Daniele
Moro is Researcher in Agricultural Economics, Catholic University of Piacenza, Italy.
This paper was prepared while Giancarlo Moschini was visiting the University of Siena
as part of the RAISA research project of Italy's CNR, the support of which is gratefully
acknowledged. Thanks are due to Catherine Kling and Federico Perali for commenting
on an earlier draft.
European Review of Agricultural Economics
23 (1996) 239-261
0165-1587-99603/0023-0239
© Walter de Gruyter, Berlin
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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 of
economics as a social science, which sets it apart from physical and natural
sciences. The laws of nature that are studied in physics today are arguably
the same as those studied by Galileo, and this fact has allowed physicists to
amass an impressive amount of knowledge. In economics, however, because
it is human behaviour within social institutions that is studied, the 'world'
that economists analyse is constantly changing. For example, production
decisions rely on ever-changing techniques employing an increasing variety
of constantly improved inputs; consumption choices span an increasing
variety of goods and are carried out by a demographically and culturally
changing population; the ability of economic agents and markets to deal
with fundamental uncertainties of the economic system is constantly
improved by the emergence of new institutions; and so on. The possibility
that the very structure that generates economic data may be changing poses
serious 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 presence
of such 'structural' changes?
Of course, one could quibble with our opening remarks. Some, like
McCloskey (1983), would argue that economics is best interpreted as a
rhetorical discipline rather than a 'scientific' enterprise. Others, perhaps
more to the point, could question the implicit assumption that empirical
economics only learns from observing the real world (a presumption of
much 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 investigating structural change in demand analysis. We open with a brief discussion
of the main structural change issues that arise in empirical demand studies.
The survey of methods that follows is organised under the broad categories
of 'nonparametric' and 'parametric' methods. Along with a reference to the
main theoretical contributions, we try to cover the rather large body of
applied studies that pertain to our topic. In doing so, we are mindful of the
subjective bias and unintended omissions that an exercise like this inevitably
entails. The hope is that these limitations will not reduce the usefulness of
our review for the applied researcher.
2.
Structural change in demand models
In order to delimit somewhat the scope of this review, it is important to
define what is meant by 'structural change' in demand. Arguing for the
likelihood that economic data typically used by most practitioners may not
have been generated by a constant structure depends, of course, on a
somewhat 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 characterisation
is not conclusive because economic theories are, by necessity, simplifications.
For example, and with a notation that suits the demand focus of what
follows, a theory may characterise a variable q in terms of a set of variables
p by the function q = /(p) while, at the same time, it may be known that a
more general characterisation involves another set of variables z as in the
function q = F(p, z). However, explicit modelling of the effects of the variables
in z may not be essential because they are only marginally interesting, or
quantitatively unimportant, or quite simply unobservable (at least to the
analyst). Under some plausible conditions, the effect of z can be subsumed
in the function f(p), which can serve well as a 'theoretical' relation, but in
other cases, say associated with considerable changes in z, the concise
theoretical relation q=f(p) will also change. From the point of view of the
theory that explains q in terms of p alone, we would say that we are in the
presence of a structural change.1
In demand analysis, the focus of this paper, q typically represents the per
capita aggregate consumption of one or more goods, and p is a vector of
prices deflated by per capita income. Abstracting from considerations of
exactly which quantities and which relative prices are explicitly considered,
f(p) can then represent a market demand model such as those in a great
many studies using time series data. In some cases, however, another set of
variables z may turn out to be important in explaining demand behaviour.
For example, z may include important demographic variables, or conditioning variables such as advertising, or a state variable capturing systematically
evolving preferences. The existence of structural change has important implications for demand analysis, generally affecting a variety of welfare and
policy evaluations that depend on elasticities and other estimates typically
of interest in empirical demand models.
In demand analysis, hypotheses of structural change are often framed in
terms of 'changing tastes and preferences', although different studies and
analysts may have different notions of what that means. In some cases, one
thinks of changes in the shape of individual utility functions of a stable
population of consumers, as brought about by health concerns and attention
to quality, possibly related to new information and improved scientific
knowledge. In others, one may think of a changing demographic composition
of a heterogeneous collection of consumers who have different preferences.
Alternatively, one may postulate changes in the preferences individuals
display, induced by firm strategies such as advertising and product innovation. Alternatively, individual choices may be based upon their basic characteristics (Lancaster, 1966 and 1991), and if the content of those characteristics
changes over time, individual preferences for goods may be affected. Whereas
the differences in these examples may call into question the loose character
of the notion of 'changing tastes and preferences', from an operational point
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Giancarlo Moschini and Daniele Mow
of view it is sufficient to realise that they can all be modelled under a general
framework 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 function
expressed as u = u(q, z), where q represents the consumption vector and z
represents any other variable affecting preferences over bundles of goods. In
such a setting, the substitution effects between goods are clearly affected by
the variables z. The notion that preferences underlying demand models may
change over time has found its critics. For example, Stigler and Becker
(1977) have argued that various phenomena that may indicate a change in
tastes can otherwise be explained by stable preferences; often a change in
preferences is invoked to justify the inappropriateness of some economic
models in interpreting observed data. Others have questioned such a position
(see Pollak, 1978), and considered changes in consumer preferences, both
exogenous and endogenous, as an important aspect of demand analysis and
empirical work (Pollak, 1976 and 1977).2
Viewed in the light of our earlier discussion, however, much of the debate
on whether it is acceptable and/or useful to postulate structural change in
demand analysis is a matter of semantics, and depends on whether one
thinks in terms of q = f(p) or views the problem from the more general
relationship q = F(p, z).3 From an operational point of view, more pressing
issues are: (i) to determine when a model of the q=f{p) type is appropriate
and 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 more
appropriate; and (iii) to tie the selected empirical model to a theoretical
framework so that a consistent interpretation of empirical findings may be
possible. Many procedures are available to deal with these specification
problems. Although these may already be familiar in some other fields of
economic analysis, we will review many of them here in the light of their
application to demand analysis.
3.
The nonparametric approach
There are two basic ways of characterising consumer demand. Classical
demand theory, moving from basic rationality axioms, assumes that consumers have preferences over goods which are described by a utility function.
Demand behaviour is then seen as the result of the consumer maximising
her utility (subject to a budget constraint). Alternatively, revealed-preference
theory takes individual choices as primitive elements, and sets forth restrictions on these choices that parallel the rationality axioms of classical demand
theory.4 Although much of the intellectual appeal of revealed preferences
derives from the fact that it frees demand theory from the burden of a 'utility
function', what is being exploited in modern nonparametric demand analysis
Structural change and demand analysis
243
is, in fact, the equivalence of the restrictions that revealed preferences and
utility 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 consistency
of a body of data with the theoretical properties of revealed preferences
and/or utility maximisation.5 Specifically, these procedures entail testing for
consistency of the data with the weak, strong or generalised axioms of
revealed preference (WARP, SARP and GARP, respectively). Failure to
meet requirements for these axioms may be interpreted as an indication of
structural change in demand: if we maintain that consumers are, in fact,
maximising utility, then a change in the structure of preferences could
account for the observed violations.
To illustrate briefly the consistency implications of revealed preferences,
using Varian's notation (Varian, 1992), WARP requires that ifq'R^,
then
it is not the case that qJRDq', where RD denotes the directly revealed preference relation, and q' and qj are any two observed consumption vectors.
Testing for consistency of the data with WARP requires searching for violations in pairwise comparisons; a violation is found when both Cu > C u and
Cjj > Cp, where C,,- indicates the expenditure with prices in time i of the
bundle chosen in time j , and C,, > CtJ is equivalent to q'RDqj.6 A more
stringent test for consistency requires ruling out intransitivities, i.e. checking
the 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 the
relation RD? Applications of revealed preference procedures to test structural
change in demand data include Landsburg (1981), Thurman (1987), Chalfant
and 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 draw
inferences on the nature of a particular data set without making assumptions
about the functional form of demand functions. This clearly eliminates one
source of arbitrariness present in parametric procedures (see below).
Unfortunately, the nonparametric approach has shortcomings of its own
that make the interpretation of findings from such an approach somewhat
difficult. 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 nonstatistical (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 maintained
hypothesis. Hence, in this setting, the only admissible concern is the power
of such tests. In our case, the power of a nonparametric test relates to its
capability of detecting violations of consistency when, in fact, there is
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Giancarlo Moschini and Daniele Moro
structural change. There are legitimate reasons for believing that such a
power may not be great. For example, Chalfant and Alston's (1988) results
based on a quarterly Australian data set, imply, among other things, the
absence of seasonality in consumption, although seasonality is a fairly
accepted feature of quarterly meat demand. The failure to detect at least
some seasonality may lend support to Varian's (1982) and Landsburg's
(1981) observation that nonparametric tests based on revealed preferences
are 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 in
relative prices, then it may be difficult to find comparable points: at the
limit, 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 in
preferences 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 demand
bundles, but unfortunately they are not observable. Chalfant and Alston
(1988) try to recover them by adjusting their data for expenditure growth
under the assumption that all expenditure elasticities are equal to one.
Clearly, such a procedure is admissible only when preferences are homothetic, a hypothesis that is systematically rejected by virtually any data set.
Sakong and Hayes (1993), extending Chavas and Cox's (1990) linear programming technique to consumer demand analysis, adjust their data in a
way that allows for non-unitary income elasticities. However, this approach
essentially recovers Hicksian demand by a first-order approximation and,
strictly speaking, there are no theoretical restrictions one should expect to
be satisfied by such adjusted data. The bottom line is that compensated
demands cannot be recovered from observed Marshallian consumption bundles unless one resorts to functional form assumptions of a nature typical
of parametric models, thereby undermining the main distinguishing feature
of nonparametric analysis.
An alternative tactic is to relax the 'exact' view of nonparametric tests.
The proposed solution is to postulate that observed consumption patterns
are measured with error. As it stands, this has to do both with the power
and the size of the test. The approach is described in Varian (1985) and, in
its original form, it is basically related to the size of the test: the idea is to
minimise a 'distance' function of the observed data from the closest point
that 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 of
observing 'true' quantities.11 Thus data may be perturbed in order to eliminate (or induce) violations of consistency with revealed preference axioms,
and some measure of this adjustment gives the level of significance of the
nonparametric test.
Structural change and demand analysis
245
3.2. Limitations of the nonparametric approach
The appealing features of the nonparametric approach, and its extension to
testing for structural change, are challenged by the issues of power and size
of these tests. The proposed solutions are not completely satisfactory. For
instance, an example in Gross (1991) strongly undermines the usual test of
WARP. Using a Cobb-Douglas utility function with two goods, he shows
that a very large shift in preferences between two points, combined with no
change in expenditure and a small change in prices, produces a small
violation of WARP when measured in terms of C,,. In particular, a slight
modification of the example shows that, under large preference change, a
small change in income (of less than 1 per cent) will make it impossible to
detect the preference shift.12 Thus, simply showing that postulating a small
measurement error can restore theoretical consistency in observed quantities
has the potential of being quite misleading.
Gross (1991) proposed to test for consistency of the data by using the
money metric utility or the direct compensation function, instead of the
expenditure index (i.e. testing for WARP), by extending an idea of Varian
(1990): under the assumption that preferences are stable, the direct compensation function provides a measure of the minimum expenditure at time j ,
given that preferences are unchanged with respect to time i, and that the
individual is willing to attain the level of utility that his/her observed bundle
would provide. Thus, if preferences are unchanged and the individual is
maximising utility, the minimum expenditure will be the same as the actual
expenditure. If preferences are unstable, then the difference between minimum and actual expenditures gives an indication of the hypothetical waste
in expenditure when maximising utility in time j with preferences of time i:
the larger the difference, the more serious the shift in preferences between
the two periods.13 Although, in principle, the adoption of the direct compensation function can increase the power of the test, its computation requires
knowledge of the particular structure of preferences. However, that is typically not known, and use of the approximations of the direct compensation
function discussed by Varian (1982) does not necessarily improve the power
of the test.14
Some final points should be kept in mind in evaluating the ability of these
procedures to detect structural change. First, insofar as applications pertain
to a subset of the goods comprising consumer's choice (say, three or four
meat goods), there are virtually no restrictions that one can place on such
a 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 in
the utility function rather than as a test of maximisation per se\ Hence,
nonparametric tests on subsets of data are conditional on a correct assumption of separability. Put another way, the null hypothesis of a consistency
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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 and
nonparametric applications alike. Still, it should suggest more caution in
claiming that structural change has not occurred when the analysis is based
on a handful of goods only.15 Second, the whole revealed theory apparatus
pertains to one individual's choices, but empirical applications typically rely
on aggregate (market) data. The validity of this procedure would seem to
depend on the exact aggregation conditions to be satisfied. Such conditions
are known to be rather stringent (Deaton and Muellbauer, 1980a: Chapt. 6),
but they are typically ignored in empirical applications of nonparametric
methods.
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 expenditure function. Relative to the nonparametric approach just described, the
analyst is required to make an assumption about the functional form relating
the variables of interest. Whereas this concession to arbitrariness may be
deemed undesirable, there is really no particular novelty here, as this step
is required in virtually every econometric application. It is common to try
and ensure that the arbitrariness of this step does not prejudge important
aspects of the analysis by choosing a 'general enough' specification
(see below). Having chosen a functional representation of preferences, there
are at least three alternative (but related) ways of examining structural
change issues: consistency analysis, parameter instability analysis and explicit
modelling of structural change by a trend or other economic variables.
4.1. Consistency analysis
Given the chosen functional representation of preferences, the analysis could
proceed as in the nonparametric approach. Specifically, one could test
whether a body of data satisfies the theoretical restrictions of demand theory,
such as homogeneity of degree zero in prices and income, and symmetry
and negative semidefiniteness of the matrix of Slutsky substitution terms.
Testing for these properties is, indeed, the parametric counterpart of checking
the satisfaction of WARP, SARP, or GARP. This set of equivalences has
been made explicit in the work of Kihlstrom et al. (1976), Hildenbrand and
Jerison (1989) and John (1995), who show the relation between WARP and
homogeneity, and the equivalence between a weaker version of WARP and
the semidefinite negativeness (but not the symmetry) of the Slutsky substitution matrix. The relation of the nonparametric approach with integrability
Structural change and demand analysis
247
of demand functions is also reviewed in Mas-Colell (1978), and related
developments can be found in Chiappori and Rochet (1987).
Curiously, although the relationship between common nonparametric
consistency checks and tests of homogeneity, symmetry, and negativity is
obvious, few studies seem to have taken that approach. Perhaps it is due to
the recognition that other sources of misspecification can affect the outcome
of 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 (with
the exception of the functional form selection) also affect the nonparametric
procedures described above. Perhaps more important, the fact that the
parametric approach has followed other routes may be due to the relative
richness of options that it offers. Thus, analysts have typically preferred to
maintain 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 from
the observation that such changes will alter parameter values, given an
unchanged functional form. Thus, testing for parameter instability has been
a common way to test for structural change. Tests that can be used include
the 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 (Ploberger
et al., 1989) and other tests (Andrews, 1993). Applications in demand analysis
include 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. The
size of the test is crucial, because the null hypothesis that we are typically
testing is actually a set of hypotheses that are best grouped generically as a
null hypothesis for misspecification (type-I errors). Again, the main problem
is that parameter instability may arise from different sources, with structural
change being only one of the possibilities. Hence, these tests are properly
interpreted as tests for misspecification arising from any source of invalid
conditioning. The problem has been widely recognised in the econometric
literature, and it has received a great deal of attention in applied demand
analysis (see Alston and Chalfant, 1991).
As for power, the performance of the tests for parameter instability proposed in literature often depends on the nature of the alternative hypothesis.18 Testing the null hypothesis that a parameter vector is stable over the
sample period implies a comparison with an alternative hypothesis that can
take different forms, like a one-time 'structural' change, with a known or
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Giancarlo Moschini and Daniele Moro
unknown change point, or smooth patterns of change, or other cases where
no 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 against
different alternatives of structural change: a priori information may be useful
in choosing the appropriate test.19
4.3.
Explicit modelling of structural change
A related procedure consists of explicitly modelling the effect of structural
change by introducing, in the estimating model, variables indexing taste
shifters. The simplest version of this approach is to include a time trend or
a (set of) dummy variable(s). One basic application of this approach is the
use of dummy variables to account for seasonality in consumption when
using frequent time series data (say, quarterly). Alternatively, under the
assumption that preferences have changed smoothly over the observation
period, a trend variable can provide a useful proxy for these effects, and
following Stone (1954), Deaton (1975), and Jorgenson and Lau (1975), this
approach has been adopted in a number of studies. Furthermore, a distinction can be made between systematic and random variations in parameters:
systematic variations imply that a certain deterministic pattern of change
for the parameter vector can be modelled, even the most general, while
stochastic variations allow for a more flexible specification of the pattern of
change.20 The statistical problem then reduces to that of the significance of
the added variable(s). The relationship with parameter instability tests is
obvious. For example, one can restate the Chow tests in terms of the
significance of a set of dummy variables. Thus, our econometric discussion
of parameter instability tests applies here as well.
Of course, modelling structural change by a time trend is a somewhat
unpleasing shortcut, and the use of such a trend is often interpreted as an
admission of ignorance of the true form of structural change. The alternative
is to be more specific about what sort of structural change one has in mind,
and model it accordingly. For example, under the assumption that US
consumers' egg preferences were changing due to an increased concern about
the health hazards of large intakes of cholesterol and saturated fats, Brown
and Schrader (1990) constructed a 'cholesterol information index' from the
number of articles in medical journals linking cholesterol in diet to heart
disease. Whereas, in principle, this approach is attractive, it still involves
rather arbitrary choices,21 or may turn out to be econometrically equivalent
to a trend (for example, the correlation coefficient between Brown and
Schrader's cholesterol index and a linear time trend is 0.986).
In some cases, direct modelling of preference shifters is clearly the preferred
alternative, especially when dealing with issues of demographic or dynamic
effects in demand. That individuals may have different preferences (and
Structural change and demand analysis 249
therefore demands) is widely accepted. This diversity is supported by a
number of cross-sectional studies showing the significance of demographic
effects (such as household size, age of household members, region of residence, employment status, and others) in demand models.22 Insofar as relevant demographic variables change over time, market demand functions
typically investigated in demand studies using time series will be affected
and may display structural change.
Similarly, the importance of dynamic components in aggregate demand
has long been recognised. One explanation for such dynamics postulates
that individuals' consumption is sluggish due to their dependence on a state
variable that can be interpreted as a 'stock of habits' (Houthakker and
Taylor, 1970; Pollak, 1970). This 'habit formation' hypothesis is usually
implemented in demand models by specifying preferences as depending on
past consumption levels. The econometric problem of testing for this specific
instance of structural change then reduces to that of the significance of the
added lagged consumption variable(s).23 Alternatively, as in Anderson and
Blundell (1983), one can evaluate how accounting for dynamics affects
consistency tests (homogeneity and symmetry). Of course, dynamic considerations can be nested in other structural change issues. For example, one
could 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 another
in explaining consumption are advertising, product innovation, quality,
information and environmental changes, and many studies have tried to
account for these effects in demand systems.25 However, some of them are
measured (or proxied) by variables that display a strong time trend, especially when some index is used; thus, their effect is almost indistinguishable
from a trend, and the interpretation of results from this model may still be
misleading in attributing a role for some effects (see above).
4.4. The issue of functional form
Consistency, parameter instability and explicit modelling of structural
change are all concerned with the issue of model specification. As such, they
address the basic problem of structural change, which relates to both the
parametric and nonparametric methods we have discussed. On the other
hand, the parametric approach requires a few additional assumptions, which
exposes it to some additional problems. First, one often needs to make a
prior choice of endogeneity and/or exogeneity of prices, quantities and
income in standard demand studies. Overlooking the endogeneity of some
variables has the potential of introducing considerable biases in inference
(LaFrance, 1991). Whereas preliminary tests may help in adopting a correct
specification, it is clear that wrong assumptions may lead to spurious identification of structural change (Wahl and Hayes, 1990; Eales and Unnevehr,
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1993). For instance, structural change in supply, such as technological
change, can appear as a demand shift.26 However, the most obvious shortcoming of parametric relative to nonparametric methods is that one typically
needs to assume a specific functional form. Hence, inference in parametric
models is usually conditional on the choice of the functional form. This has
led to considerable controversy over the analysis of structural change with
parametric models, typically centring on the interpretation of the results
and questioning the possibility of ever concluding anything useful from a
parametric analysis of structural change (Alston and Chalfant, 1991).
One can try to take some precautions against the limitations of an
arbitrary 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 inflexibleparameterisations based on the Box-Cox transformation. More generally, in a
system's framework, it has become common to rely on the concept of flexible
functional forms (FFFs), pioneered by Diewert (1971, 1974) and by
Christensen et al. (1971), which centre on the approximation properties of
a Taylor series expansion of the unknown true function. Systems of equations
commonly used in demand analysis, such as the translog (Christensen et al.,
1975), the almost ideal (Deaton and Muellbauer, 1980b), and the normalised
quadratic (Diewert and Wales, 1988) satisfy this notion offlexibility;another
commonly used system, the Rotterdam model (Barten, 1966; Theil, 1975
and 1976), satisfies an equivalent definition of flexibility.
Whereas this notion of flexibility is useful, it does not eliminate the
arbitrariness of the functional form chosen. FFFs provide an accurate
approximation only at a point, but not much can be said about their
approximation properties away from that point; thus, Diewert's notion of
flexibility is not sufficient to ensure that these second-order forms can
approximate the underlying (true) function over the entire region of the
price-income space. Therefore, if the approximation properties are only valid
around part of a point, and there is great variability in the price-income
space over the sample, then parameter instability can be detected.27 Because
of this, it has long been recognised that estimation and hypothesis testing
are sensitive to this choice (Gallant, 1981). The open issue, specifically in
testing for structural change, concerns the extent of this sensitivity. In
particular, work by Alston and Chalfant (1991) has shown that the performance of the test statistic for structural change can be heavily affected by
the choice of the functional form. Specifically, in a simulation setting where
the true function is known by construction, it is shown that the adoption
of an incorrect functional form enormously increases the proportion of
rejections of the (true) null hypothesis of stable preferences.
The point of Alston and Chalfant (1991) is well taken, but the results they
actually present are too extreme and pessimistic because their Monte Carlo
simulations do not account for the (arbitrary) choice of the signal-to-noise
Structural change and demand analysis
251
ratio (which turns out to be important when the explanatory variables are
trended). To illustrate, consider their experiment using the General Electric
(GE) data reported in Theil (1971: 296), which shows that when the data
are generated by a linear model without structural change, a Chow test for
structural change at the 5 per cent significance level will give the wrong
answer 100 per cent of the time if carried out with the wrong functional
form (double-log, semi-log, or square root functions). Crucial to this finding,
however, is the fact that simulated samples are obtained by adding random
terms with mean zero and unit variance, but a variance of one, given the
design 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 may
be of interest to consider the same experiment with a more reasonable
assumption on the error term. For example, one can repeat the experiment
by fixing the variance at the level required to replicate the R2 obtained by
fitting 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 wrong
rejection 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 of
wrong rejections of the true hypothesis of no structural change is dramatically affected by the signal-to-noise assumption, although for incorrect functions it is still greater than the nominal size.
In order to reduce the potential bias of the arbitrary choice of a functional
form, 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 that
can encompass different functional specifications: structural change tests
conducted on these larger specifications should reduce the initial conditioning. Along the same line is the idea of reducing misspecification effects by
comparing and selecting different models. FFFs can be compared, but
usually these models are not nested, and hence this strategy requires the use
of appropriate procedures to test non-nested models.28
5. The 'econometric' nonparametric approach
Whereas FFFs can offer a workable solution to the arbitrariness of specifying
an explicit parametric model, and have become widely adopted in empirical
applications, it is apparent from the foregoing that their use has come under
some rightful criticism. In particular, because most FFFs are essentially
Taylor series expansions of the unknown true function, these forms have the
undesirable feature that their approximation properties are invariant to the
sample size. An improved approximation was obtained by Gallant (1981)
by adding trigonometric terms to the leading terms of a Taylor series
252
Giancarlo Moschini and Daniele Mow
expansion, which produces the Fourier functional form. The increased flexibility of this approach hinges on the requirement that the number of trigonometric terms tends to infinity as the sample size grows. An alternative
approximation strategy relies on the use of nonparametric methods. Here,
the term 'nonparametric' applies to the statistical method of density estimation (Silverman, 1986). Using this approach, one can estimate the regression
function, or other function of interest, without making any parametric
assumption about the true functional form (Hardle, 1990; Ullah, 1988). A
parametric form can sometimes be retained for a portion of the model, and
this leads to semiparametric specifications (Robinson, 1988).
Applications of these nonparametric methods in the study of structural
change issues are clearly possible. The simplest application, again, would be
to test for the consistency of a body of data with the properties of demand
theory. For example, one could test the homogeneity property following the
procedure 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 the
significance of added variables meant to index structural change. Particularly
useful here may be Robinson's semiparametric approach, as applied, for
example, by Moschini (1991).
A distinctive feature of the nonparametric approach based on density
estimation, as compared with other nonparametric methods discussed earlier,
is that it has a solid statistical foundation which is appealing for empirical
applications requiring formal tests of hypotheses. In a sense, therefore,
nonparametric estimation of the regression function represents the obvious
evolution of the framework of analysis based on parametric flexible functional forms. The main advantage is that it allows econometric modelling
and hypothesis testing that do not depend on arbitrary (and possibly
restrictive) parametric assumptions. However, there are also a number of
limitations to this approach. For example, an important issue at the application stage is the choice of smoothing parameter (window width). This is
crucial to estimation results as it corresponds to a sort of implicit parameterisation. Unfortunately, there is no widely accepted method for the choice of
window width, and this introduces an element of arbitrariness in the estimation process.
Another issue concerns the statistical properties of the nonparametric
estimators. Under certain assumptions, one can prove consistency and
asymptotic normality for most nonparametric estimators. However, the rate
of convergence in distribution is typically slower than that of parametric
models. Also, this rate can be very slow if the number of regressors is large
(the 'curse of dimensionality'). A workable solution may sometimes be found
by using the semiparametric model, where the parametric portion can be
estimated consistently and efficiently, or by using the method of averaging
(Hardle and Stoker, 1989). In general, however, large samples and simple
Structural change and demand analysis
253
models seem necessary for a reasonable application of nonparametric
methods of estimation.
6.
Concluding remarks
A number of explanations of certain features of market demand may fall
under the heading of 'structural change'. Several methods are used to detect
these situations in applied analysis, and to improve demand models typically
used for welfare and policy analysis. In this paper, we have reviewed most
of these procedures, grouping them under the broad categories of 'nonparametric' and 'parametric' methods. Nonparametric methods based on
revealed-preference theory can be used to analyse the consistency of a body
of consumption data with the assumption of utility maximisation. Because
this can be accomplished with few accessory assumptions, nonparametric
methods have the potential of yielding conclusions needing minimal qualifications. However, such methods are difficult to extend over the most basic
framework of standard demand analysis, and are incapable of producing
accurate quantitative predictions and response functions typically necessary
for applied problems.
Parametric methods usually require one to specify a functional form, and
thus conclusions are conditional on a degree of arbitrariness. However, such
methods are typically easy to adapt and generalise in order to handle a
greater variety of problems and questions, and the careful analyst can take
precautions against the danger of being misled by a grossly incorrect functional form. Most of the parametric approaches that we have reviewed are,
in fact, model specification tests. Whereas this is quite appropriate for a
number of hypotheses concerning structural change in demand analysis, it
does raise the issue of interpreting the results one obtains, and emphasises
the fact that a carefully constructed theoretical model should always precede
any empirical application.
We have, in addition, reviewed what we called the 'econometric' nonparametric approach. This is similar in spirit to the nonparametric methods
discussed earlier, because it attempts to do away with arbitrary functional
specification choices, but is also akin to the parametric methods in incorporating the explicit stochastic framework that constitutes the hallmark of
econometrics. If this approach is taken as a good-faith effort to reconcile
the two classes of methods discussed earlier, it offers a sobering message. In
making explicit the large information requirements for its implementation,
it suggests that overcoming the main limitations of parametric methods may
be difficult, and that the potential advantages of nonparametric methods
may prove hard to achieve in applied research.
254
Giancarlo Moschini and Daniele Moro
Notes
1. One should not be misled into concluding that structural change issues can be assumed
away by working only with models of the type q = F(p, z). Because the vector z potentially
includes a great many things, virtually all of the economic models used for empirical
analysis are of the concise type q=f(p), and which component of z will turn out to be
important as a vehicle of 'structural change' may depend on circumstances or specific
applications.
2. Endogenous tastes are defined as a preference ordering that depends on other people's
consumption 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 and
structural inference, and claim that results from observed data often show just prima facie
causal relations, and drawing associational inference on the economic structure is incorrect
and 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 O y = Cij/Cu. We have a violation of WARP if
both <t>ij and <!>,•,• are less than one.
7. See Varian (1992) for more details. One important difference between WARP, SARP, and
GARP is that the first two axioms assume that a unique bundle is chosen with each budget
line (as implied by strictly convex preferences in utility theory), while GARP is consistent
with 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 of
accepting the null hypothesis when it is false; the size of the test is the probability of type-I
errors, i.e. the probability of rejecting the null hypothesis when it is true. Alston and
Chalfant (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 observed
data when budget lines cross, even if there are no comparable bundles. An application of
this approach is found in Burton (1994).
10. See also discussion in Epstein and Yatchew (1985). This procedure is applied by Burton
and Young (1991). Tsur (1989), working under some special assumptions, proposes a fast
procedure to test for the significance of GARP violations. A discussion about the size and
the power of the test under the provision that data can be measured with error can be
found 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 in
parametric models). Of course, measurement errors are not the only possible source of
randomness. Errors may arise in the budgeting process, and observed quantities may not
be 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 of
analysing randomness in quantities, both in parametric and nonparametric models, may
be misleading, since relatively large errors in quantities may be consistent with small
departures 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 are
p, = $99andp 2 = S>101 at time i, a n d p t =p2 = $100 at time/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 10
for q2- This extremely large shift in preferences produces <J>(J = 0.992 and <bJt = 0.992, which
are 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.
Structural change and demand analysis
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
255
In this case there would be no violation, and thus a small change in income would imply
that the exact test does not detect structural change.
Gross (1991) shows that by computing the direct compensation at time j by solving
min,{ piq:ui(q) = ui{qi)}, one obtains a minimum expenditure of $1,724.27, implying that
under the hypothesis of no structural change the consumer would have wasted about
83 per cent of her income.
One can analyse further the example in Gross (1991) by computing the overcompensation
function m + {p, q) as in Varian (1982), where m + (p, q) is denned as the solution to
min:{pz:zRq}- In our example m*(p',qJ) = Cij, and the procedure proposed by Gross
collapses to the one used by Chalfant and Alston (1988).
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, then
the consumption of the goods in the separable group is biased if the group is not homothetic
(Moschini, 1991).
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.
Using Japanese data, Maki (1992) finds that homogeneity and symmetry are satisfied when
taste change is allowed. A similar approach is used in Chen and Veeman (1991).
See Kramer et al. (1988), for a discussion on the power of the CUSUM test and its
extension to dynamic models; this test is usually affected by the local power problem, that
is, the power is dependent on the form of the alternatives. The types of tests proposed by
Andrews (1993) do not suffer from this problem, and may have power against several
alternatives. However, much of the work on the power of tests for structural change is
devoted to models where parameter instability is originated by structural change, thus the
issue of a composite null hypothesis may still be crucial in empirical applications. See also
Andrews and Fair (1988) for testing for structural change in nonlinear models.
The sample size may be a critical aspect here, since most of these tests have asymptotic
properties, and do not perform well in small samples (see Fachin, 1994, for a discussion of
this issue in the context of variable additional tests). For example, Wales (1984) has shown
the tendency towards over-rejection of tests for structural change based on the likelihood
ratio: this bias is more serious when the sample size is small.
Several applications of these techniques can be found. Moschini and Meilke (1989) adopted
a gradual switching regression model for structural change in a demand system (see also
Reynolds and Goddard, 1991; Burton and Young, 1992; Eales and Unnevehr, 1988; and
Bjorndtil et al., 1992). A more complex gradual switching model, with stochastic crossequational constraints, has been proposed by Tsurumi et al. (1986), and adapted to meat
demand analysis by Goodwin (1992), using a Bayesian approach. Poirier (1976) studied
the use of spline functions in modelling structural change. Chavas (1983) estimated a
random coefficient model, and Leybourne (1993) a time-varying coefficient AIDS model,
both based on the Kalman filter. Other applications of systematic and stochastic changes
in parameters are in Kinnucan and Venkateswaran (1994). A slightly different application
is in Choi and Sosin (1990), who test for structural change using time-varying multiplicative
terms. Finally, for other applications, see Buse (1989).
For example, the cholesterol index developed by McGuirk et al. (1995) is quite different
from the cholesterol index constructed by Brown and Schrader (1990), and which of the
two is more appropriate is not clear.
Demographic effects have been widely incorporated in demand analysis [Pollak and Wales
(1981); Ray (1984); Lewbel (1985); Rossi (1988); Alessie and Kapteyn (1991); see also
Pollak 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), Horton
and Campbell (1991), Gao and Spreen (1994), and Gould et al. (1994).
A general dynamic specification of demand systems, that includes habit formation as a
special case, can be found in Anderson and Blundell (1982, 1983).
256
24.
25.
26.
27.
28.
Giancarlo Moschini and Daniele Moro
This approach is found in Burton and Young (1992). Long-run dynamics are accounted
for in Kesavan et a!. (1993).
Scaling and translating, typically used for demographic effects, can also be applied for other
effects, following Lewbel's (1985) generalisation. For a treatment of advertising in demand
systems, see also Selvanathan (1989) and Duffy (1987). Applications can be found in
Kinnucan (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 of
information and health concerns: see applications in Brown and Schrader (1990), Capps
and 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 by
Houthakker (1952) and Theil (1952); see also Hanemann (1981), Cox and Wohlgenant
(1986), Deaton (1989), and Nelson (1991). Other factors that have been accounted for are
convenience (Capps et al., 1985), retailing (Bjorndal et al., 1992), government intervention
(Gould et al., 1991), and product innovation (Gould et al., 1994).
Eales and Unnevehr (1993) and Dahlgram (1988) use an inverse demand system to test for
structural change in meat demand. Moschini and Vissa (1993) develop the intermediate
assumption of a mixed demand specification.
A possible solution to the problem may be to resort to FFFs with larger approximation
properties. Gallant (1982) proposed a form, based on a multivariate Fourier series expansion, where a better approximation is obtained at the expense of an increased number of
parameters. This approach has been applied in food demand estimation and testing for
structural change by Wohlgenant (1985).
For an early review of non-nested testing procedures, see MacKinnon (1983). Several
applications can be found in demand analysis since Deaton (1978); a recent application is
Brester and Wohlgenant (1991). An alternative interesting approach, rooted in the model
selection literature, is the likelihood dominance criterion proposed by Pollak and Wales
(1991).
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Daniele Moro
Istituto di Economia Agro-alimentare
Universita Cattolica del Sacro Cuore
Via Emilia Parmense, 84
29100 Piacenza
Italy