On Speci¯cation and Identi¯cation of Stochastic
Demand Models
Walter Beckert ¤
Department of Economics
University of Florida
January 2003
Abstract
This paper is concerned with stochastic demand systems that arise from structural random
utility models for J continuous choice variables. It examines under which conditions on the structural preference speci¯cation the implied reduced form model is invertible. And it investigates the
conditions under which the structural model can be locally identi¯ed from the reduced form under
moment assumptions on the stochastic components in the structural preference model. The paper shows that analogues to conventional assumptions on preferences, together with some further
conditions which are easily veri¯able in practice, provide enough structure for local identi¯cation
through conditional moments, which can be derived from either the reduced form or, if analytically
intractable, ¯rst-order conditions.
¤
I thank Chunrong Ai, Hide Ichimura, Daniel McFadden, Lea Popovic, Paul Ruud and Chris
Shannon for helpful comments and discussions. All errors are mine.
1
1
Introduction
Demand analysis is an actively researched area, in theoretical and, more prominently, applied work. The task of the applied researcher is to model randomness
in demand data, given prices and income or expenditure. There exist, in principle,
three paradigms to accomplish this task. The ¯rst attributes randomness in demand
data to measurement error, the second interprets randomness as optimization error,
and the third views randomness as arising from unobserved preference heterogeneity.
These potential sources of randomness may contribute simultaneously. This paper
focuses on the third interpretation. Identifying unobserved preference heterogeneity
as the source of interest for randomness in demand data invokes a structural approach for econometric demand analysis. The elementary structural component in
this approach is a random utility or stochastic preference model. This paper addresses two fundamental questions. Under which conditions on the speci¯cation of
structural stochastic preferences over J goods does the implied reduced form induce
choice vectors for the J ¡ 1 inside goods that have a non-degenerate distribution on
the commodity space, given prices and expenditure? And under which conditions is
the structural model locally identi¯able from moment conditions on demand data?
Non-degeneracy of the distribution of demands induced by the reduced form is desirable in applied work if the modeling framework for analysis is not to impose a
priori restrictions on consumers' choice behavior. The question of identi¯ability is
critical in any structural, rather than reduced-form approach.
A structural random utility approach for micro-econometric demand analyses
is of interest for several reasons. First, modeling unobserved preferences heterogeneity frees the empirical analysis from the rigidity of deterministic preferences.
Typically, applied demand analyses maintain the hypothesis that consumers make
rational choices that are consistent with a preference ordering on the commodity
space. Empirically, however, even precisely measured choice data often do not satisfy rationality tests like the Weak Axiom of Revealed Preference.1 The modeling
1
The Weak Axiom is a weak consistency requirement on choices. Let x(p) and x(~
p ) denote
the choice vectors at prices p and p
~ , respectively. Then the Weak Axiom states that if x(~
p) was
feasible when x(p) was chosen, i.e. if x(~
p )0 p · x(p)0p, then it must not be that x(p) was feasible
when x(~
p ) was chosen, i.e. it must be the case that x(~
p )0 p
~ < x(p)0 p
~.
2
assumption of deterministic preferences becomes a liability when there is evidence
that consumers indeed optimize, because then the failure of the data to satisfy the
Weak Axiom suggests that consumers' preferences are inconsistent across choice situations. A structural framework that allows for unobserved preference heterogeneity
is °exible enough to accommodate both, the maintained hypothesis of optimization
in accordance with a consistent preference ordering and the empirically observed
patterns in the data.
Second, next to such conceptual considerations, assessing the entire distribution of preferences is of practical importance when consumers compete for shared
resources. Typical examples are the consumption of services that are delivered
through capacity-constrained networks. 2 If aggregate consumption is close to exhausting capacity, then the service valuation and utilization of a marginal user potentially in°icts a negative consumption externality on all contemporaneous users,
through network congestion and ensuing quality-of-service degradation. Calibrating
consumers' taste heterogeneity is therefore instrumental for e®ective and e±cient
management of capacity-constrained resources.
Quality of service can be enhanced through e±cient capacity allocation. At
least in theory, e±cient capacity allocation can be achieved through nonlinear pricing schemes for such services. The literature on the theory of nonlinear pricing o®ers
preference heterogeneity as the fundamental motivation for the welfare-enhancing
e®ects of nonlinearities in price schedules; see, for example, Wilson (1993). Heterogeneous preferences are a primitive in this theory. The third reason for measuring
preference heterogeneity, then, is the desire to empirically assess the primitives of
the theory of nonlinear pricing, in order implement its prescriptions for optimal price
structures.
Related Literature
The literature on random utility models has primarily focused on discrete choice
models. It goes back to Thurstone (1927) and McFadden (1974, 1981); see also
McFadden and Train (1998) for some recent developments. The literature on sto2
A classical instance of this kind is electricity. More topical examples are Internet access and
wireless networks for voice and data.
3
chastic preferences models for continuous choices is much more sparse. An early
contribution is Barten (1968), expanding a demand theory developed by Barten
(1964) and Theil (1965, 1967). In applied work, Dubin and McFadden (1974) in an
analysis of electricity demand and Fuss, McFadden and Mundlak (1978) in production analysis estimate structural econometric models with random parameters. The
question of identi¯cation in continuous choice models was ¯rst taken up by Brown
(1983). This analysis was extended by Roehrig (1988) to nonparametric and semiparametric implicit nonlinear simultaneous equation models. More recently, Brown
and Matzkin (1995, 1995a, 1998) consider a special class of non-parametric random
utility models, give conditions for identi¯cation and propose a non-parametric estimation methodology for this class; Brown and Wegkamp (2000) outline a weighted
minimum mean-squared distance from independence estimation methodology for
this model. Brown and Walker (1989) and Lewbel (2001) investigate the implications of the utility maximization hypothesis for the reduced form system of stochastic
demand functions if randomness arises from unobserved preference heterogeneity.
Overview
This paper focuses on the econometric analysis of structural, parametric random
utility models for continuous choices. The paper ¯rst examines necessary conditions
that a random preference speci¯cation must satisfy to generate a system of demands
for J ¡ 1 inside goods that has a non-degenerate distribution on its support. Non-
degeneracy is desirable if the model should be capable of statistically explaining
any conceivable observation in the commodity space, given prices and expenditure.
If no a priori restrictions on choice behavior are imposed by the model design,
then the distribution of demand for inside goods will be non-degenerate. If such
restrictions were imposed and not valid, then estimates from a restricted model
would be inconsistent.3
Second, the paper presents conditions, easily veri¯able in applications, on random utility so that, as a result, the stochastic demand system is invertible almost
surely. This result is signi¯cant later in the context of identi¯cation.4
3
If an unrestricted model is estimated, then actual constraints on choice behavior can still be
detected, as they manifest themselves in estimates of higher-order moments.
4It is possible to have random preference components of dimensions higher than necessary, but,
4
The second part of the paper turns to local identi¯cation of structural model parameters from moment conditions. Only assumptions on the moments of stochastic
structural model components and other structural model features are maintained;
no further distributional assumptions are required, and no assumptions on the reduced form of the model are made; in particular, separability of the ¯rst-order
conditions or the reduced from in the stochastic structural model components is not
assumed. This distinguishes this work from the analyses by Brown (1983), Roehrig
(1988), Brown and Matzkin (1995, 1995a, 1998) who investigate standard identi¯cation (see, e.g., Newey and McFadden (1994)) under assumptions on the entire
distribution of stochastic model components and/or maintain assumptions on the
reduced form, including separability. In this paper, separability, in the sense of
invertibility of the reduced form as in Roehrig (1988), is implied by conditions on
the structural model. 5 It is shown that the conditions on the stochastic preference
speci¯cation that su±ce for an almost surely invertible model ensure that ¯xed parameters in the random preference model as well as the variance-covariance matrix
of the random preference components are locally identi¯able from demand data.
Due to the nonlinearity typical in the functional relationship between response variables and stochastic preference components, even if the distribution of the stochastic
model component were known, the likelihood function would contain an analytically
intractable Jacobian term. As a consequence, moment assumptions replace distributional assumptions, and estimation and local identi¯cation strategies based on
moment conditions are considered. Potential lack of separability of stochastic components in the reduced form is immaterial for estimation when such moments can
be simulated, so that a method of simulated moments (MSM) approach is feasible.
It is, therefore, quite natural that the identi¯cation question asked in this context
is intimately linked to the question of local identi¯cation in generalized method of
moments (GMM) setups. The proof of the main identi¯cation proposition, Proposition 1, appeals to this connection. It builds on conditional moments derived from
as can be illustrated in easy examples, identi¯cation in such situations is tenuous.
5
Compare, e.g., Brown (1983) who admits that it may be\exceedingly di±cult to determine in
practice whether or not the rank condition [on a matrix with basis vectors for the space generated
by the derivative of the reduced form model with respect to the exogenous variables]" does or
does not hold; Roehrig (1988), Lemma 3.3, Brown and Matzkin (1998), Theorem 1', who assume
separability of the stochastic structural components in the ¯rst-order conditions;
5
the implied reduced form model. A corollary to the proposition demonstrates that
locally identifying moment conditions can be derived alternatively from the ¯rstorder conditions. This strengthens the local identi¯cation result since it makes it
applicable also in cases when the system of stochastic demand functions exists in
theory only, but is analytically intractable in practice.
The ensemble of the speci¯cation and local identi¯cation results show that rather
conventional assumptions about preferences in applied microeconomic consumption
analyses provide enough structure for local identi¯cation. Thus, local identi¯cation
really arises from the economics of the choice problem that generates the observed
consumption data.
The paper proceeds as follows. The second section is devoted to structural
speci¯cations that yield a invertible reduced forms. The third section develops the
results on identi¯cation of the ¯xed parameters of the preference speci¯cation and
the variance-covariance matrix of the stochastic preference components from demand
data. The ¯nal section summarizes and suggests some directions for future research.
2
Speci¯cation
2.1
Synopsis of di®erent approaches
Suppose one observes consumption amounts for J ¡ 1 commodities, x¡J , together
with prices p, including the price (index) of a (composite) outside good J, and
total expenditure m. There are basically three di®erent approaches to designing a
stochastic demand model for the data at hand.
The ¯rst approach starts from a stochastic reduced form, i.e. a system of stochastic demand equations. It is known from the work by Brown and Walker (1989)
that the properties of the demand functions imposed by microeconomic consumer
theory imply that the di®erence between realized demand and its conditional expectation, given p and m, is itself a function of prices and income. It is clear from this
work that any reduced form speci¯cation has the drawback that it is not clear how
6
to impose the restrictions implied by utility maximization. 6
Preferences are of interest not so much because they allow consumption bundles to be ranked, but more fundamentally because they characterize substitution
relationships between goods.7 Once the substitution pattern is determined, at least
in principle ordinal rankings can be inferred by integration. Another approach to
stochastic demand analysis, therefore, might start from a stochastic speci¯cation
of J ¡ 1 marginal rates of substitution. The third section of this paper provides
conditions that yield identi¯cation from ¯rst-order conditions.
A fully structural approach speci¯es stochastic preferences by means of random
utility functions. Under suitable assumptions this approach yields both, stochastic
marginal rates of substitution and stochastic demand functions. While this is the
most comprehensive modeling approach and allows the greatest °exibility in terms
of the stochastic speci¯cation, it comes at the cost that parameters of interest in
the random utility function must be identi¯able from the reduced form or, alternatively, from the ¯rst-order conditions. The third section of the paper shows that the
conditions for identi¯cation of the structural model from the reduced form are the
same as for identi¯cation from the ¯rst-order condition.
2.2
Non-Degeneracy and Invertibility
General speci¯cations must be estimable from the implied system of demand functions or ¯rst-order conditions. For this reason, any sensible stochastic model that
is a candidate for estimation must be representable { at least in principle { as a
system of demand functions that is consistent with the structural model and that
does not a priori limit the space of observables. To formalize the latter requirement,
consider the following
6
A variant of the ¯rst approach is a random coe±cient speci¯cation of the demand functions.
This approach is, however, equally unsatisfying since it is also unclear how to incorporate the
restrictions implied by the utility maximization hypothesis.
7
One might argue { and it has been suggested by some, e.g. Lancaster (1966), p. 132, to
whom the following \tongue in cheek" citation is due { that \determinateness of the sign of the
substitution e®ect [is] the only substantive result of the theory of consumer behavior."
7
De¯nition: The random variable x 2 RJ has dimension J, denoted throughout
by dim(x) = J, if it has a non-degenerate distribution on RJ .8
Denote by U the collection of all random (direct) utility functions U : RJ+ £
£ £ RK ! R, de¯ned up to a ¯nite dimensional parameter vector µ 2 £ and a K
dimensional random component ² 2 RK .
The following properties of U are henceforth assumed.
Assumption A1: For each ², U 2 U is continuous in its arguments, continuously
di®erentiable in ²; x; µ, strongly monotone, concave and strictly quasi-concave in x.
Assumption A1 is more than su±cient to guarantee that a system of Marshallian
demand functions
hM (p; m; µ; ²) = (h(p; m; µ; ²)0 ; hJ (p; m; µ; ²))0 = arg max
fU(x; µ; ²) : p0x = mg
K
x2R
exists. For measurable representatives of equivalence classes9 in the space of random
utility functions, they essentially mimic the ones given in Mas-Colell (1977) for each
² and guarantee that, for each ², a particular demand function cannot be generated
by representatives of di®erent equivalence classes in U . Di®erentiability is added so
that marginal rates of substitution exist.
De¯nition: A stochastic model for demand data on consumption, prices and
income, (x0¡J ; p0 ; m)0 2 RJ¡1
£ RJ++ £ R+, is non-singular if its reduced form
+
constitutes a system of stochastic Marshallian demand functions hM (p; m; µ; ²) =
(h(p; m; µ; ²)0 ; hJ (p; m; µ; ²))0 such that x¡J = h(p; m; µ; ²) has dimension J ¡ 1 for
all p and m.
The requirement of the de¯nition is essential if, given prices and expenditure
0
(p ; m)0 > 0, the model should havehthe capability
every conceivable
i
hto explain
i
m
m
observation in the commodity space 0; p1 £ : : : £ 0; pJ , subject to the linear
budget constraint p0 x = m.
10
8
For simplicity, there is no notational distinction between the random variable and its realiza-
tion.
9
For conditions under which measurable utility functions exist, see Aumann (1969), Wesley
(1976) and Wieczorek (1980).
10For applied demand analyses, it seems preferable to ¯rst consider a structural model without
8
Since, given p and m, the response variables x ¡J are in a J ¡1 dimensional subset
of RJ¡1
+ , it is at least plausible, though only under certain conditions necessary,
that the stochastic components ² of non-singular stochastic models should have
dim(²) = J ¡ 1. These conditions are spelled out in the following
Lemma 1: Consider the map x : E ! X , where E ½ RK and X ½ RJ , for the
vector-valued random variable ² with dim(²) = K. If the map x is non-constant and
either (i) uniformly Lipschitz continuous or (ii) C 1, then a necessary condition for
x(²) to have a non-degenerate distribution on RJ is that K ¸ J.
The proof of Lemma 1 is given in an appendix; it is essentially an analysis result. Its implication for structural random utility models with J continuous choice variables is that their speci¯cation has to involve ² of at least stochastic dimension J ¡ 1 if the implied reduced form stochastic demand model
hM (p; m; ²) = (h(p; m; ²)0 ; hJ (p; m; ²))0 is either continuously di®erentiable or Lipschitz continuous in ².11
Continuous di®erentiability of h with respect to ² is convenient for several reasons. First, as Lemma 2 will show, it can be justi¯ed from assumptions on structural
preference speci¯cations which are analogues to conventional assumptions on utility in microeconomic consumer theory. Second, under conditions on the structural
model that can be easily checked in applications, the Jacobian of h with respect
to ² has full column rank J ¡ 1 almost surely. Together with ² having stochastic
dimension J ¡ 1, this implies that the J ¡ 1-component choice vector of the J ¡ 1
inside goods, x ¡J , has a non-degenerate distribution. Third, Lemma 2 also shows
implicit restriction on choice behavior. If the restriction were in fact true, then it would manifest itself in estimates of higher-order moments. If it is wrong, yet imposed by the mo del, then
model estimates are inconsistent; on the other hand, if the restriction is true, then imposing the
restriction yields more e±cient estimates. Therefore, as a modeling strategy, one would estimate
an unrestricted model ¯rst, test for the validity of restrictions, and if, they are not rejected, one
might proceed by estimate the restricted model.
11
It is well known that, in the absence of such continuity, ² of lower stochastic dimension su±ces
for non-degeneracy of the distribution of x. For example, to generate two independently uniformly
distributed random variables x0 = (x1 ; x 2 ), it is enough to generate a single uniformly distributed
random variable and assign the odd digits to x1 and the even digits to x2 . The map corresponding
to this assignment is not continuous.
9
that, under such assumptions, x ¡J = h(p; m; ²) and ² are one-to-one. This theoretical property of the reduced form is used later to argue that structural parameters of
random utility models that imply such reduced form models are locally identi¯able
under moment assumptions on ² from moments of implied demans.
While Lemma 1 provides necessary conditions for non-singularity of the reduced
form, Lemma 2 gives conditions on the structural random utility model that are
su±cient for non-singularity and invertibility of the reduced form. To eliminate the
ambiguity arising from di®erent members in equivalence classes in U, all of which
yield the same marginal rates of substitution, Lemma 2 directly makes the following
assumption on marginal rates of substitution:
Assumption A2: Let MRS(x; µ; ²) =
h
i
@
@
@xj U(x; µ; ²)= @xJ U (x; µ; ²)
j=1;::: ;J¡1
.
Suppose that the (J ¡ 1) £ (J ¡ 1) matrix r²MRS(x; µ; ²) has full rank J ¡ 1 for
all µ for all ².
In addition, Lemma 2 requires:
Assumption A3: Suppose that
¯
¯ r 0 U (x; µ; ²) r U(x; µ; ²)
w
¯ ww
¯
¯ rw0 U(x; µ; ²)
0
for all µ and w 0 = (x 0 ; ²0 ).
¯
¯
¯ 6= 0
¯
¯
The counterpart in classical microeconomic theory to assumption A3, the nonzero determinant of the bordered Hessian, is usually referred to as smoothness in
the sense of Debreu (1972). Tow further assumptions are used to guarantee that the
image of the stochastic Marshallian demand functions is entire consumption space.
Assumption A4: dim(²) = J ¡ 1.
©
ª
¡1
Assumption 5: Im (MRS(x; µ; ²)) = MRS(x; µ; ²) : ² 2 RJ ¡1 = RJ++
.
Lemma 2: Suppose A1-A5 hold. Then, x¡J = h(p; m; µ; ²) and ² are one-to-one
a.s., for all (p0 ; m)0 > 0 and µ.
The proof of Lemma 2 is in the appendix. Its logic is straightforward. A1
guarantees the existence of a system of stochastic demand functions, rather than
10
correspondences. A3 ensures that this system is continuously di®erentiable in ². A2
implies that its Jacobian with respect to ² has full rank a.s., which in conjunction
with A4 then implies non-degeneracy of the reduced form, and together with A5
invertibility.
Within the class U of random utility functions with K = J ¡ 1, one may wish
to consider those for which xj is non-separable from ²j only, for j = 1; : : : ; J ¡ 1. A
special case of random utility functions of this form is given by U(x) + x 0², ² J = 1
w.p.1, a nonparametric model adopted in Brown and Matzkin (1995). The following
result is a corollary to Lemma 2 and shows that this class of models is nested within
the framework of this paper and yields systems of stochastic demand functions with
non-degenerate distributions of the endogenous demand variables.
Corollary 1: Suppose that U (x; ²) 2 U satis¯es U(x; ²) = u(x) + x0 ², with
²J = 1 w.p.1 and dim(²¡J ) = J ¡1, and that u(x) is smooth in the sense of Debreu.
Then, dim(x¡J ) = J ¡ 1.
Proof: Smoothness yields that the system of stochastic demand functions is
continuously di®erentiable in it arguments, so that the Jacobian r²¡J hM (p; m; ²)
exists. The vector of stochastic marginal utilities is
rx U(x; ²) = rx u(x) + ²;
so that rx¡J ²0¡J U(x; ²) = IJ¡1 and hence r²MRS(x; µ; ²) = IJ¡1 , the identity
matrix of dimension J ¡ 1. The result now follows by Lemma 2.
¤
Notice that the ¯rst-order conditions in this model are separable in ². Lemma
2 shows that such separability is not necessary as long as the rank assumption is
satis¯ed.
3
Identi¯cation
Building on Lemmas 1 and 2, the question of identi¯cation of model parameters
from demand data can be addressed. Identi¯cation is critical in any structural
econometric approach.
11
Consider random utility models in the class U. In light of standard axiomatic as-
sumption about preference in economic theory, the stochastic preference components
are assumed throughout to be independent of prices and income; see assumption A7
below. This axiomatic assumption is empirically not uncontroversial: Recent work
by Calvet and Comon (2000) establishes strong empirical correlation between tastes
and income. It is nonetheless retained here to illuminate the close link between
standard assumptions in microeconomic consumer choice theory and identi¯cation
of structural microeconomic choice models. In fact, this section shows that standard
microeconomic theory, suitably interpreted in the random utility setup, is itself rich
enough to yield identi¯cation.
Proposition 1 below provides su±cient conditions under which model parameters
µ0 and §0 in random-parameter utility models are locally identi¯ed12 from demand
data on consumption choices, prices and income (x0¡J ; p0; m)0 under moment con-
ditions on ²; no assumptions on their joint distribution are made. The proposition
considers the class of models U 2 U satisfying U(x; µ; ²) = U(x; µ1 + ²; µ 2), where
x 2 X ½ RJ+, p 2 RJ++ , m > 0, µ = (µ01; µ20 )0 , µ1 2 £1 ½ RJ¡1 ,µ2 2 £2 ½ RK¡(J¡1),
where £1 and £2 are open sets and J ¡ 1 · K · 2(J ¡ 1).13 Denote the true
parameter vector by µ0. The following moment assumptions on ² will be invoked:
Assumption A6: E[²] = 0 and E[²²0 ] = § 0 positive de¯nite.
Moreover, it is assumed that variation in preference, parameterized by ², is independent of prices and income, whose distribution is non-degenerate:
Assumption A7: The distribution Fp¡J ;m of (p0¡J ; m) 0 is non-degenerate on
RJ++, independent of ².
Finally, an assumption akin to A2 is maintained with respect to µ2 :
Assumption A8: rµ2 MRS(x; µ; ²) has full column rank.
12
For a de¯nition of local identi¯cation, see Rothenberg (1971). For a random variable U with
parametric p.f. fU (u; µ); µ 2 £ ½ RK ; µ is locally identi¯ed if for any other parameter vector µ~ 6= µ
~ is strictly positive.
in a neighborhood of µ the probability Pr(fU (u; µ) 6= f U (u; µ))
13
An alternative parameterization would have U (x; µ2 ; ²), i.e. U parameterized by µ2 and ², with
E[²] = µ1 . The parameterization adopted in assumption A2 of the proposition was chosen because
it is more parsimonious in terms of notation.
12
Proposition 1: Consider U 2 U satisfying U(x; µ; ²) = U(x; µ1 + ²; µ2), where
x 2 X ½ RJ+, p 2 RJ++ , m > 0, µ = (µ01; µ20 )0 , µ1 2 £1 ½ RJ¡1 ,µ2 2 £2 ½ RK¡(J¡1),
where £ 1 and £2 are open sets and J ¡ 1 · K · 2(J ¡ 1). Suppose that A1-A8
hold. Then, the parameters µ 0 and § 0 are locally identi¯ed from data (x0¡J ; p0 ; m)0 .
The signi¯cance of the proposition lies in demonstrating that microeconomic
consumer theory, suitably recast in a structural econometric framework, provides
enough structure for identi¯cation. Its proof is outlined in an appendix.
Proposition 1 derives from the following logic. Suppose one wanted to estimate
µ0 and §0. Proposition 1 suggests that this could be done using a GMM estimator.14 Speci¯cally, assumptions A1-A5 imply that g(p; m; µ; x¡J ) exists, so that, by
assumption A6,
E[g(p; m; µ0 ; x¡J )] = 0
E[g(p; m; µ0 ; x¡J )g(p; m; µ 0; x ¡J )0 ] = §0 ;
(3-1)
(3-2)
where the expectation is taken with respect to the joint distribution of the vector
², which, by A7, is independent of p and m. This implies that §0 is identi¯ed
by the second moment condition, whether or not µ 0 is identi¯ed, while µ0 must be
identi¯able from the ¯rst moment condition.15 Therefore, if the dimension of µ is
higher than J ¡ 1, this is only possible through the choice of suitable instruments.
By A7, for any M ¡ (J ¡ 1) £ (J ¡ 1) array of instruments z(p; m; µ), with M ¸ K
and for µ 2 £, it follows that also
E[z(p; m; µ)g(p; m; µ 0; x¡J )] = 0;
where the expectation is taken with respect to the joint distribution of (p0 ; m; x 0¡J )0 .
The vector µ0 is identi¯ed from the set of unconditional ¯rst moment conditions
14
In fact, due to the nonlinearity of the function h in ² and of the inverse function g in x¡ J , the
procedure would employ simulation to replace analytically intractable theoretical expectations by
their simulated counterparts; in applications, the appropriate estimator therefore is a Method-ofSimulated-Moments estimator; compare, for example, Beckert (2001) for an application. The issue
of simulation does not impede the interpretation given here.
15
This is analogous to identi¯cation of the variance-covariance matrix of the disturbances in the
ordinary linear regression model. This matrix is identi¯ed even if the covariate matrix does not
have full column rank since the conditional mean of the response variables is identi¯ed.
13
E[Z(p; m; µ)g(p; m; µ 0; x¡J )] = 0, where Z(p; m; µ) = [IJ ¡1 ; z(p; m; µ)0 ]0 is an M £
(J ¡ 1) dimensional array of instruments.
In particular, evaluating instruments at µ = µ 0, Z(p; m; µ0), µ0 is locally identi¯ed
if and only if
~ x¡J );
E[Z(p; m; µ0 )g(p; m; µ 0; x¡J )] 6= E[Z(p; m; µ0)g(p; m; µ;
whenever µ0 6= µ~ for any µ~ in a neighborhood of µ0. Thus, for local identi¯cation of
µ0, it is su±cient that
~ x¡J )]
E[Z(p; m; µ 0)g(p; m; µ0; x ¡J )] ¡ E[Z(p; m; µ0)g(p; m; µ;
¼ E[Z(p; m; µ 0)rµ g(p; m; µ; x¡J )jµ=µ0 ]dµ
6= 0;
(3-3)
for dµ = µ~ ¡ µ0 and any µ~ in a neighborhood of µ0. Hence, local identi¯cation of µ0
follows if the classical rank condition su±cient for local identi¯cation in Generalized
Method of Moments setups is met, namely that E[Z(p; m; µ0)r µ g(p; m; µ; x ¡J )jµ=µ 0 ]
have constant, full column rank K in a neighborhood of µ0; see, for example, Newey
and McFadden (1994), Lemma 2.3.16
Notice that the GMM estimator for µ0 has the interpretation of a nonlinear instrumental variable estimator. Nonlinearity arises to the extent that rµ g(p; m; µ; x¡J )
may still depend on µ, unlike in the linear case. The strategy of proof of Proposition 1
then is straightforward. It needs to be deduced that E[Z(p; m; µ0)r µ g(p; m; µ0; x ¡J )]
has full rank; this is analogous to the usual condition in the linear case that the expectation of the product of instruments and covariates be nonsingular. And it needs
to be ensured that the variance-covariance matrix of the array of instruments is
non-singular in a neighborhood of µ0. These conditions are shown in the appendix
for theoretically e±cient, though practically infeasible instruments. The following
Lemma identi¯es these instruments.
Lemma 3: Under the assumptions of Proposition 1,
16
£
¤
D(p; m; µ0) = E (r² h(p; m; µ 0; ²))¡1r µ2 h(p; m; µ0; ²)jp; m
See also Rothenberg (1971), Theorem 1, and, for identi¯cation conditions that imply the ones
required by this theorem and cover a general class of reduced form models, Roehrig (1988).
14
constitutes an array of e±cient instruments for the estimation of µ0 .
The proof of this Lemma is given in the appendix.
The identi¯cation argument for µ bene¯ts from being closely related to the
(simulation-assisted) GMM estimation methodology that one would use to uncover
the structural model parameters from choice data. Conversely, the identi¯cation
argument can guide estimation and enhance its e±ciency since it points to e±cient
instruments. This distinguishes this line of reasoning from the one o®ered by Brown
and Matzkin (1998)17 whose identi¯cation and estimation approaches appear to be
mutually uninformative. Brown and Wegkamp (2000) o®er a weighted mean-squared
distance from independence estimation methodology but require an assumption on
the inverted reduced form model at a single point in the space of covariates and ². 18
The proof of Proposition 1 is cast in terms of the system of inverse demand functions g(p; m; µ; x ¡J ) which may be analytically intractable in applications; it could
have been equally derived in terms of the system of demand functions h(p; m; µ; ²).
One might therefore be led to believe that, for the purpose of GMM estimation of µ,
it is necessary to be able to explicitly derive the system of stochastic demamd functions, as in the example above. This would limit the applicability of Proposition 1,
since in many interesting cases U 2 U , this system may be analytically intractable,
even though in theory the existence of this system is guaranteed. In applications,
all that is analytically tractable may be the ¯rst-order conditions.
19
It follows,
however, as a corollary to Proposition 1 that the system of ¯rst-order conditions is
17
Brown and Matzkin (1998), Theorem 3 and Corollary 1.
and Wegkamp (2000), p.6. A footnote points to an extension of their identi¯cation
18Brown
result for nonlinear simultaneous equation models with multiple equilibria. As a comment, if the
model does not yield unique predictions, then identi¯cation appears tenuous and it is unclear what
their result then means.
19
A variant of this approach, focusing primarily on the substitution pattern between commodities, has been pursued in the literature on intertemporal consumption decisions and essentially
works with an intertemporal Euler condition, which amounts to the equality of an expected marginal rate of substitution and the relative price of consumption between periods, where both analyst
and decision maker face the same uncertainty. Compare, for example, Attanasio and Weber (1993).
The stochastic component is often interpreted as arising from permanent and transitory income
shocks, rather than from taste uncertainty.
15
absolutely su±cient for local identi¯cation of µ.
Corollary 2: Suppose the assumptions of Proposition 1 hold. Then µ is locally
identi¯able from the ¯rst-order conditions.
The reasoning behind the proof mimics the one of Proposition 1 and is given in
an appendix.
This section concludes with some comments on estimation which are closely
related to the insights developed up to this point. The above arguments underlying propositions 1 and 2 implicitly point to the fact that model parameters can be
identi¯ed and therefore estimated from conditional and unconditional moments. Alternatively, they can in principle also be estimated by maximizing the log-likelihood
function if one is willing to make further assumptions on the joint distribution of ².
Assuming that ² has a density f², the density of x¡J , given p; m and parameterized
by µ and §, is given by
³ 1
´¯ 1¯¯
¯
¯
¯
fx¡J (x ¡J jp; m; µ; §)dx¡J = f² §¡ 2 g(p; m; µ; x¡J ) ¯§¡ 2 ¯ ¯ rx¡J g (p; m; µ; x ¡J )¯ dx ¡J :
From the de¯nition of the inverse g, the Jacobian is given by jrx¡J g(p; m; µ; x ¡J )j =
jr²h(p; m; µ; ²)j ¡1 . Notice that the Jacobian term jr x¡J gj depends on x¡J and jr²hj
on ², respectively. This must be the case, since the assumption of E[²] = 0 does
not allow h(p; m; µ; ²) to be linear in ² because in this case P (h(p; m; µ; ²) < 0) > 0
since the support of ² is RJ¡1, while the support of x is the hyper-rectangle X =
[0; pm1 ] £ : : : £ [0; pmJ ].
20
Estimation from moment conditions may be more practical
and robust in applications, since this approach does not require the assumption of
the stochastic utility components having a density. Also, compared to an approach
based on the likelihood, moments of the response variables can be obtained directly,
without recourse to the inverted model. The inverted model, however, is necessary
to evaluate the density of the stochastic utility components. On the other hand,
the likelihood-based approach yields an important by-product. Recall that Lemma
1 implies that model invertibility and a non-degenerate distribution of the J ¡ 1
vector ² are su±cient for a non-degenerate distribution of the vector of observed
response variables x¡J . In comparison to nonparametric models, parametric models
20
This, in part, distinguishes this work from Newey, Powell and Vella (1999) who consider
triangular simultaneous equations models with additive, conditional mean-zero errors.
16
therefore require that the Jacobian r²h(p; m; µ; ²) have full rank for all µ in the parameter space, with probability one.21 The maximum likelihood estimator imposes
this requirement automatically, since the logarithm of the Jacobian term approaches
negative in¯nity for values of µ that render the Jacobian singular. 22
4
Conclusions
This paper presents a framework for speci¯cations of structural stochastic demand
models in which randomness in demand data arises from preference heterogeneity.
It provides necessary and su±cient conditions on the speci¯cation of random preference under which the vector of demands, conditional on prices and income, has a
non-degenerate distribution and which are straightforward to check in applications.
Furthermore, an identi¯cation theorem for natural speci¯cations of parametric random utility models is derived, based on assumptions about the ¯rst two moments
of the stochastic model components and analogues to conventional assumptions on
structural microeconomic utility models. The theorem and a corollary demonstrate
that identi¯cation and estimation of structural model parameters is possible from
generalized moment conditions that can be derived from the implied stochastic demand system or, alternatively, from the ¯rst-order conditions. The latter option
renders the identi¯cation result more robust, as it makes it applicable to structural
preference models whose implied stochastic demand system exists in theory, yet is
analytically intractable. The identi¯cation results show that preferences themselves
provide enough structure for identi¯cation. The economics of the consumer choice
problem that generates the observed consumption data, thus, is seen to provide
su±cient structure for identi¯cation.
Future work might attempt to unify the framework adopted here { which is
21
The Implicit Function Theorem then implies that rx ¡J g(p; m; µ; x¡J ) has full rank for all µ
and almost all x¡J in the opportunity set.
22
This property of the ML estimator in this context is analogous to the fact that the ML estimator in autoregressive models, when based on the entire data set, restricts the coe±cients of the
innovation process to stationary. It is an open question at this point how to impose non-singularity
of the Jacobian for µ in method-of-moments estimation.
17
parametric in terms of the speci¯cation of random preference models, but only
maintains moment assumptions on their stochastic model components { with the
non-parametric work of Brown and Matzkin { which is non-parametric in terms of
the speci¯cation of random utility functions, but operates under the assumption that
the stochastic components have an absolutely continuous distribution. A second line
of future research might re-examine the propositions of this paper when the demand
variables are not continuously distributed, but only take on, e.g., integer values.
A
Proofs
A.1
Proof of Lemma 1
Since dim(²) = K, rk(§) = K as well. Suppose K < J.
Case (i): Since x is uniformly Lipschitz, there exists a ¯nite scalar ·, such that
kx(²) ¡ x(² 0)k · ·k² ¡ ²0 k, except on a set of measure zero, where k ¢ k denotes
Euclidean distance; · is a uniform Lipschitz constant. In particular,
kx(²) ¡ x(0)k · ·k²k:
This is equivalent to (x(²) ¡ x(0))0 (x(²) ¡ x(0)) · ·2²0 ², so that
tr((x(²) ¡ x(0))0 (x(²) ¡ x(0))) · ·2tr(²0 ²)
() tr(var(x(²)) + (E[x(²)] ¡ x(0)) (E[x(²)] ¡ x(0))0 ) · ·2 tr (§) ;
where the last inequality follows from commuting tr(¢) and E[¢]. The last inequality
then implies that var(x(²)) exists.
Furthermore, Lipschitz continuity of x(²) implies that jxj(²k ek ) ¡ x(0)j · ·j²k j,
for j = 1; : : : ; J and any k, where ek is the K dimensional unit vector with 1 in
position k; similarly, jxj (² + ®ek ) ¡ x(²)j · ·j®j for any scalar ®. The triangle
P
inequality, then, implies that jxj(²) ¡ x(0)j · · K
k=1 j²k j for all j. Hence, there
exists a J £ K matrix K, for which jx(²) ¡ x(0)j · Kj²j. Again by the triangle
inequality, it follows that jx(²) ¡ E[x(²)]j is stochastically dominated by a constant
and the random variable Kj²j. If K has J > K, this random variable has a singular
18
distribution, because its covariance matrix has rank at most K. Its support is some
subspace of RJ with dimension no larger than K. Since jx(²) ¡ E[x(²)]j is bounded
by this random variable, it must also lie in this subspace, and so its distribution is
degenerate in RJ .
Case (ii): Since the map x is C 1, the rank of the map x in a neighborhood of
any ² is rk(r²x(²)) · minfK; J g. Since K < J by hypothesis, the Rank Theorem23
implies that x in a neighborhood of ² is at most a K-dimensional submanifold of
RJ . This implies that the distribution of x(²) is degenerate. Note that the moments
¤
of ² are not needed in this case.
A.2
Proof of Lemma 2
This proof is essentially an application of the Implicit Function Theorem. Consider
the system of equations
"
# "
#
1
p
MRS(x;
µ;
²)
¡J
pJ
¡
= 0 =: M (x; p; m; µ; ²):
m
p0 x
(A-4)
Given any ² and µ, the system M can be solved for x as a function of p and m,
since U 2 U implies that the solution for x, h(p; m; µ; ²), exists and is continuous
in p and m. The condition on the bordered Hessian implies in addition that it is
di®erentiable. In particular, the J £ J matrix rx M (p; m; µ; ²) has full rank for all
²; p and m. Then, for all ², given p and m,
0 = rx M (x; p; m; µ; ²)r²h(p; m; µ; ²)d² + r²M (x; p; m; µ; ²)d²:
From this, it follows that
r²h = ¡ (r xM (p; m; µ; ²))¡1
"
r²MRS(x; µ; ²)
00J¡1
#
d² =
"
r²hM (p; m; µ; ²)d²
dxJ
#
;
where r²h(p; m; µ; ²) is decomposed into two matrices; the ¯rst has rank J ¡1 since
rxM has rank J because U 2 U , while the second has full rank J ¡ 1 by A2, so
that the Jacobian inherits its rank.
23
See, e.g, Boothby, Theorem 7.1, p.47
19
The proof
by³contradiction.
Given (p0 ; m)0 > 0 and µ 2 £, consider
³ proceeds
´
´
any ¹x 2 0; pm1 £ : : : £ 0; pmJ . Since U 2 U , x
¹ uniquely solves the ¯rst-order
conditions
r x¡J U(¹
x ; µ; ¹² )
1
p¡J =
pJ
rxJ U(¹
x ; µ; ¹² )
0
px
¹ = m;
for some ¹², since A5 implies that rx U(x; µ; ¢) is surjective.24 Suppose there exists
some ² such that hM (p; m; µ; ²) = hM (p; m; µ; ¹²) = x
¹. This implies that
rxU (¹
x; µ; ²) / r xU (¹x; µ; ²):
Therefore, for ¹², the vector ² ¡ ¹² is orthogonal to the associated change in the
gradient rxU (¹
x; µ; ²) ¡ rxU (¹
x; µ; ²), i.e. rx²0 U(x; µ; ²)(² ¡ ¹²) = 0: This contradicts
A2. Therefore, h(p; m; µ; ²) is injective, 25 for all ² and any(p0 ; m) 0 > 0 and µ. This
¤
completes the argument.
A.3
Proof of Lemma 3
Consider the identity
x ¡J = h(p; m; µ; g(p; m; µ; x ¡J ));
which holds for all x¡J ; p; m and µ. Di®erentiating with respect to µ,
r µ h(p; m; µ; ²) + r²h(p; m; µ; ²)rµ g(p; m; µ; x ¡J ) = 0;
for all x; p; m. Therefore,
rµ g(p; m; µ; x ¡J ) = ¡(r ²h(p; m; µ; ²))¡1rµ h(p; m; µ; ²)
= ¡[IJ ¡1 ; (r²h(p; m; µ; ²))¡1rµ 2 h(p; m; µ; ²)]
24
= ¡[IJ ¡1 ; d(p; m; µ; ²)]
A function f : x ! y, x 2 X, y 2 Y is surjective, or \onto", if for any y 2 Y there exists an
x 2 X such that y = f (x).
25
A function f : x ! y, x 2 X; y 2 Y , is injective, or one-to-one, if f (x) = f (y) implies x = y.
20
where d(p; m; µ; ²) = (r²h(p; m; µ; ²))¡1rµ2 h(p; m; µ; ²) and the partition on the
right-hand side of the last equalities, w.l.o.g., can be achieved by appropriate labeling
of the elements of µ. Considering the rank condition 3 ¡ 3, it can be restated as at
µ0,
E[Z(p; m; µ0)rµ g(p; m; µ; x ¡J )jµ=µ 0 jp; m] = ¡
"
IJ ¡1
D(p; m; µ0)
z(p; m; µ0 ) z(p; m; µ0)D(p; m; µ 0)
#
;
for D(p; m; µ) = E²[d(p; m; µ; ²)]. Then, choosing instruments z(p; m; µ 0) = D(p; m; µ0 )0
and taking also expectation with respect to the joint distribution of (p0 ; m)0, the last
equality implies
E[Z(p; m; µ0 )rµ g(p; m; µ0 ; x¡J )]
"
#
IJ¡1
E[D(p; m; µ0)]
= ¡
E[D(p; m; µ0)] 0 E[D(p; m; µ0)0 D(p; m; µ0 )]
= ¡E [E[r µ g(p; m; µ0; x ¡J )jp; m] 0 E[r µ g(p; m; µ0; x ¡J )jp; m]] :
Note that
Z(p; m; µ0) = E[rµ g(p; m; µ0 ; x¡J )jp; m]
= E[[rµ 1 h(p; m; µ 0; ²)] ¡1rµ h(p; m; µ0 ; ²)]jp; m]:
(A-5)
As such, these instruments minimize the asymptotic variance-covariance matrix of
the nonlinear IV estimator for µ0, since they minimize the mean squared error of
the linear prediction of rµ g(p; m; µ 0; x¡J ) on the basis of predictors Z which satisfy
E[Zrµ g] nonsingular and E[Z 0 Z] nonsingular; compare, e.g., Ruud (1999), Lemma
¤
20.4 and example 21.3. Thus, they are an e±cient array of instruments.
A.4
Proof of Proposition 1
Using the e±cient instruments identi¯ed by Lemma 4, µ0 is locally identi¯ed if
D(p; m; µ) has a nonsingular variance-covariance matrix for µ in a neighborhood of
µ0, i.e. if for any nonzero ® 2 RK ¡(J¡1),
0 < ®0Ep;m [D(p; m; µ)0 D(p; m; µ)]® ¡ ®0 Ep;m [D(p; m; µ)]0 Ep;m [D(p; m; µ)]®
= Ep;m [D(p; m; µ; ®)0D(p; m; µ; ®)] ¡ Ep;m [D(p; m; µ; ®)]0 Ep;m [D(p; m; µ; ®)]
= Var (D(p; m; µ; ®))
(A-6)
21
for the vector D(p; m; µ; ®) = D(p; m; µ)® and µ in a neighborhood of µ 0. Thus, it is
su±cient to show that for any nonzero vector ® 2 RK¡(J¡1) and µ in a neighborhood
of µ 0, D(p; m; µ; ®) has a non-degenerate distribution on RJ ¡1 .
Lemma 2 implies that [r ²h(p; m; µ; ²)] ¡1 exists a.s. The following properties of
d(p; m; µ; ²) then easily follow for any µ 2 £:
(i) kd(p; m; µ; ²)k < 1 a.s. (by A1);
(ii) d(p; m; µ; ²)® 6= 0 a.s. for all nonzero ® 2 RK¡(J¡1); d(p; m; µ; ²)® = 0
a.s. would be equivalent to rµ2 h(p; m; µ; ²)® = 0 a.s., for any nonzero ® 2
RK¡(J¡1), which contradicts assumption A8, following the line of argument in
Lemma 2; note that this is akin to the full rank assumption on the design
matrix in ordinary least squares regression;
(iii) d(p; m; µ; ²) is homogeneous of degree zero in p and m; this follows from the
homogeneity of degree zero of h(p; m; µ; ²) in (p0 ; m)0 , for any µ; ².
These properties imply that
rp;mD(p; m; µ; ®)
"
p
m
#
=0
for all p; m and ®. Since d(p; m; µ; ²) depends on all p and m,
"
#
p¡J
rp ¡J ;m D(p; m; µ; ®)
6= 0;
m
for all p; m; ®. By A7, therefore, D(p; m; µ; ®) has a non-degenerate distribution on
RJ¡1, which implies ( A ¡ 6). This completes the proof.
A.5
Proof of Corollary 2
From the ¯rst-order conditions,
¯
·
¸
¯
1
0
0
E
p ¡ MRS((h(p; m; µ; ²) ; hJ (p; m; µ; ²)) ; µ; ²)¯¯ p; m = 0:
pJ ¡J
22
¤
Therefore, µ is locally identi¯ed if
E [Z(p; m; µ0 )rµ MRS((h(p; m; µ; ²)0; hJ (p; m; µ; ²))0 ; µ; ²)] dµ 6= 0;
for some choice of instruments Z(p; m; µ0), which, as before, can be any function of
prices, outlays and optionally the parameters of interest.
Consider the following identities:
1
p¡J = MRS(x; µ; g(p; m; µ; x ¡J ))
pJ
x¡J = h(p; m; µ; g(p; m; µ; x ¡J ))
f orall ²; p; m; µ;
8 x ¡J ; p; m; µ:
From the ¯rst, it follows that, for any dµ 6= 0,
(r µ MRS(x; µ; ²) + r²MRS(x; µ; ²)r µ g) dµ = 0;
where rµ g = rµ g(p; m; µ; x¡J ), for any p; m; µ and x satisfying the budget constraint. Using Lemma 3, this implies that
rµ MRS(x; µ; ²) = ¡r²MRS(x; µ; ²)rµ g(p; m; µ; x ¡J )
= r ²MRS(x; µ; ²) [IJ ¡1 ; d(p; m; µ; ²)] :
The remainder of the proof proceeds in the same logic as the proof of Proposition
1. That is, it is shown that E [r² MRS(x; µ 0; ²)] and E [r ²MRS(x; µ0; ²)d(p; m; µ; ²)]
have nonsingular variance-covariance matrices. Then, these can be used to form optimal instruments Z(p; m; µ0), so that the su±cient condition for local identi¯cation
of µ 0 is met.
By A2, rk (r ²MRS(x; µ0; ²)) = J ¡ 1 a.s. Hence, r²MRS(x; µ; ²)d(p; m; µ; ²) 6=
0 w.p.1. Strict monotonicity implies that r²MRS(x; µ; ²), when evaluated at x =
(h(p; m; µ; ²)0 ; hJ (p; m; µ; ²))0, depends on all prices, is homogeneous of degree zero
in p and m, and (h(p; m; µ; ²)0 ; hJ (p; m; µ; ²))0 > 0, and so kr ²MRS(x; µ; ² j )k < 1
a.s., for j = 1; : : : ; J ¡ 1. Therefore, it follows that E[r²MRS(x; µ; ²)], evaluated
at x = (h(p; m; µ; ²)0 ; hJ (p; m; µ; ²))0 , has a nonsingular variance-covariance matrix.
Similarly, from these properties and the properties of d(p; m; µ; ²), one can derive
the analogously that, for any µ,
23
(i) r²MRS(x; µ; ²)d(p; m; µ; ²)® 6= 0 a.s. for all nonzero ® 2 RK¡(J¡1) (by A1
and U 2 U);
(ii) kr²MRS(x; µ; ²)d(p; m; µ; ²)k < 1 a.s. (by A1);
(iii) r²MRS(x; µ; ²)d(p; m; µ; ²) 6= 0 is homogeneous of degree zero in p and m.
From these properties, dim (r² MRS(x; µ; ²)d(p; m; µ; ²)®) = J ¡ 1 for any nonzero
® 2 RK¡(J¡1). Thus, E [r ²MRS(x; µ; ²)D(p; m; µ; ²)jp; m] has a nonsingular variancecovariance matrix for any µ.
Therefore, choose the array of optimal, though in practice infeasible instruments
£
¤0
Z(p; m; µ0 ) = E [r²MRS(x; µ0 ; ²)jp; m]0 ; D(p; m; µ0) 0 ;
where x = (h(p; m; µ0; ²)0 ; hJ (p; m; µ0 ; ²))0 and D(p; m; µ 0) = E [d(p; m; µ0; ²)jp; m].
The two preceding paragraphs establish that this array of instruments has a nonsingular variance-covariance matrix. Therefore, µ is locally identi¯ed from the ¯rst¤
order conditions.
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