Alternative stochastic specification of conditional demand models are considered. The results of LaFrance concerning the inconsistency of least squares are supported, but the class of models that allow standard instrumental variable estimation is broadened considerably.Demand and Price Analysis,

Edgerton, David L.

English

Research Papers in Economics

Journal ofAgricultural  and Resource Economics, 18(2):  141-146
Copyright  1993 Western Agricultural  Economics Association
On the Estimation of Separable Demand Models
David L. Edgerton
Alternative  stochastic  specifications  of conditional  demand  models  are con-
sidered.  The results of LaFrance concerning the inconsistency of least squares
are supported, but the class of models that allow standard instrumental variable
estimation is broadened  considerably.
Key words:  conditional demand models, instrumental  variables,  weak sepa-
rability.
Introduction
The problems connected  with the  estimation  of conditional  demand  models have  been
addressed by a number of authors,  with  somewhat contradictory  results.  The block re-
cursivity of separable  systems  has sometimes been cited as an argument  for using least
squares (LS)'  (see,  for example, Bieri and de Janvry, p.  21).  Other authors (e.g.,  Deaton,
p.  167) have stressed the interrelations between the conditional  and unconditional error
terms as  a reason for expecting LS to be inconsistent.2
In an important paper, LaFrance clarified many of the issues that had previously caused
confusion  in the  literature.  Starting from  a  plausible  stochastic  generalization  of usual
demand theory, he showed, among other things, (a) that LS estimation of the conditional
demand model will in general be inconsistent, and (b) that standard instrumental variable
(IV) techniques will also yield inconsistent results in nonlinear models. He suggested using
an iterative estimation  method proposed by Anderson to obtain consistency.
In this article  we will show that the stochastic generalization  suggested by LaFrance is
not unique.  While  it seems unlikely  that  an alternative  specification  would  invalidate
LaFrance's  results concerning  LS, it is easy to  show that another plausible specification
will enable standard IV methods to yield consistent estimates in certain nonlinear models.
It is thus important  to formally  incorporate  the stochastic  specification  and estimation
methodology into the modeling process.
Conditional and Unconditional Demand  Systems
Consider a vector of commodities of interest x with corresponding  prices p. The goods x
are assumed to be weakly  separable from all  other goods z, whose  prices are given by q.
The unconditional  demand for x will be a function of all prices and total expenditure (y),
which can  be expressed in nonstochastic form as
(1)  x = h(p,  q, y).
Defining group expenditure as
(2)  x = p'x =  (p,  q, y)
The author is an associate professor in the Department of Economics,  Lund University,  Lund,  Sweden.
This research is a part of the Nordic Food Demand Study, supported by the Swedish Agricultural and Forestry
Research  Council.
The  comments  of an  anonymous  reviewer  are  greatly  appreciated.  Any  errors  or  omissions  remain  the
responsibility of the author.
141Journal  of Agricultural  and  Resource Economics
we can utilize  separability,  and write the conditional  demand function as
(3)  x = a(p, Yx).
Separability thus implies, and is implied by, the relation h(p, q, y) = h(p, k(p, q, y)).
Since  economic  theory  tells  us little  about the  stochastic nature  of the  econometric
model, LaFrance  suggests  expressing a stochastic demand model in terms of the condi-
tional expectations of x and yx,  i.e., replacing  (1) through (3) by
(4)  x = E(x Ip,  q, y) = h(p, q, y),
(5)  Y. =p'x  =  q(p,  q, y),
and
(6)  = h(p, yx),
where (6)  follows from (4) and (5).  If we write x = x +  Ex and y, = :y  +  vx, then it follows
that E(x Ip  ,  ,  ) = O, E(vx Ip,  q, y)  = 0, and p'x = vx.  The conditional  demand function
can now be expressed with the observable level of group expenditure  on the right-hand
side, that is
(7)  x = h(p, y,)  +  Zx,
where
(8)  x=  E  +  h(p, yx)  - i(p, Yx  + vx).
LaFrance uses these relations to prove,  among other things, the following:
Lemma  1: Given that E(ex)  =  0 and E(vx) = 0,  then E(Zx)  = 0 if and  only if either
p'Ex = 0 or A is linear in yx (LaFrance, lemmas  1 and 2).
The first condition is equivalent to  I  2x I = 0, where 2x is the covariance  matrix of the
unconditional  errors cx.  Note that this matrix is in general nonsingular,  since  adding-up
merely implies that p'x + q'fz = 0 and p'x = 0.  A second result is that
(9)  Cov(yx,  Zx)  =  xxp  - E[h(p, Yx  + P'ex)ex],
from which the following can be obtained:
Lemma 2: Cov(yx,  -x)  =  0 if and only if (a) I  Z  I =  0;  (b)  2xx is constant  and A is
linear in yx, with its coefficient equal to a specific function ofp and 2xx; or (c)  i is
linear in  yx,  and  the unconditional  errors  satisfy  the generalized  rational  random
errors hypothesis (LaFrance, lemmas  1, 3, and 4).
The above equality holds approximately  in cases (b) or (c) when A is nonlinear in yx, as
long as the first-order Taylor approximation of A satisfies the given restrictions.
These results are very strong. LaFrance  interprets the second result as showing that LS
will only yield consistent estimates of (7) under very restrictive conditions. The first result
is  interpreted  as  implying  that  standard  IV  methods  (see  Amemiya,  or Bowden  and
Turkington) will only be consistent when A  is linear in yx. The estimation method suggested
by LaFrance  is an iterated generalized least squares  method proposed by Anderson.
All the above results  are dependent,  however,  on the stochastic  generalization  of the
deterministic  demand system given by (4) and (5).  The definitions given there seem both
natural  and  plausible,  but  an obvious  question  is  whether  there  exist  other plausible
definitions  that lead to different  results.
Consider,  for example,  Deaton and Muellbauer's  Almost Ideal Demand System. The
conditional  demand function for this model explains the within group budget shares3 (wi
= xpi/yx)  as a linear function of prices and the logarithm of group expenditure  (7x = In
x).  It is thus reasonable to assume that w, and  x  are the variables of primary interest in
the model, rather than x and yx.  In this case it is more convenient to assume a stochastic
specification  based on the conditional  expectations  R = E(w IP, q, y) and  7x  = E(nx IP, q,
y), instead of (4) and (5).
142  December 1993Separable  Demand Models  143
An Alternative Stochastic  Specification
Let us now consider quite general transformations of x and yx,
(10)  w  = f(x, P. Y),
and
(11)  x =  g(Yx)
where the inverse functions of g and f  (for given p and yx) both exist.  Using the deter-
ministic demand functions  (1) through (3),  we can write
(12)  w=  f(h(p,  q, y), p,  I(p, q, y)) = k(p,  q, y),
(13)  x = g(p'x) = g(p(p,  , y))  =  (p, q, y),
and
(14)  w  = f(h(p,  yx), P, Y)  = f(h(p, g-'(rx)), P, g-(rx)) = k(p,  7x),
where k(p,  q,  y) = k(p,  '(p, q, y)), and4p'x = p'f-(w, p,  g-
1 (x))  = g-l(x). Note that we
can also write  w = f(x, p, p'x) = f(x, p), but that f  need not be invertable for given p.
A  stochastic  generalization  of the  deterministic  model,  in terms  of the conditional
expectations  of w and  77,  can now be constructed.  Equations (4) through (6)  will become
(15)  · = E(w Ip,  q,  y) = k(p,  q,  y),
(16)  (,  q, Y),
and
(17)  = k(p,  x)
If we now write w = w +  ew and 77 =  xi +  vw,  then  ,w and vw have conditional expectations
equal to zero.  Defining x = f-  l(  , p, g-'(i))  implies that  ,x = g(p'x), but x is no longer
the conditional  expectation  of x if  f - is nonlinear in w or 7x.  Note that there no longer
exists any simple relationship between  vw and  Ew.
Equations  (15)  through  (17) and  (4) through  (6),  respectively,  comprise mutually  ex-
clusive  alternatives  when either  g or f  is  nonlinear.  They  thus  represent  two  distinct
stochastic  models.  Both are  quite plausible  generalizations  of the deterministic  model,
but they have somewhat different implications. Estimation of LaFrance's specification (if
it is correct)  yields unbiased  predictions  of x and y,,  while  the  specification  given here
yields (if  correct) unbiased predictions of w and 7,. This is quite a normal situation when
comparing budget-share  and expenditure  models, and is not usually a cause for concern.
We  can continue  our analysis by developing  the conditional  demand  function  as  for
(7), namely
(18)  w =  k(p,  7x)  +  Ew,
where
(19)  =  Ew  +  k(p, I)  - k,  7x  +  VW).
The following result is now directly comparable  to lemma  1:
Lemma 3: Given  that E(Ew  = 0  and E(vw)  = 0,  then E(w) =  0 if and only if either
p'(x - x) = 0 or k  is linear in 7x.
Thus if k is linear in 7x, then usual nonlinear IV estimation will yield consistent estimates.5
The first condition in lemma  3 has a similar interpretation to that in lemma  1, namely
that the vector of unconditional demand  errors,  ex  = x - X, has a singular distribution
(although ,x  does not have zero conditional expectation here). Note also that the conditional
budget  share errors,  Ew,  has a singular distribution,  due to adding up, whether or not p'(x
- )  =  0.6
As previously, LS  will not be consistent,  since
EdgertonJournal  of Agricultural  and  Resource Economics
(20)  Cov(nx,  ,w)  = E(vx,)  - E[k(p,  x +  vx)vx]
can only be zero under very special conditions,  analogous to those in lemma 2.
An  Example and Some  Implications
The Almost Ideal Demand System is usually written in determinstic form as
(21)  wi = ai  +  ,,  In pj +  fi(ln Y, - In P,),
where w = xipi/yx, and In PI is a price index to be estimated (as a nonlinear function of
p and the a and y parameters)  or approximated  (e.g.,  as Stone's price  index, 
2 kWk In Pk).
The group expenditure  function for yx can be given by
(22)  x .=  ,(P,  Y;  0),
which may,  or may not, be in Almost Ideal  form, and where p  and q may,  or may not,
form group price indices.
Using  LaFrance's  specification  of the stochastic  Almost  Ideal  model,  we  obtain  the
following conditional  demand system expressed in terms of observed  expenditures:
(23)  xi= ai(yp  1) + 2j(yxp- 1 )n  pj + fi(yxp, - )(ln yx  - In P,) +  xi,
Since E(Ex)  0 in general,  the model to estimate  would be
(24)  xi =  a,(yxp; -1)  +  ,  (yxp- 1 )ln  p  +  (yp)(ln  - In PI) +  ex)
(25)  Yx=  (P, q, y; 0) +  vx,
and Anderson's  method (or double regression  nonlinear TSLS) will yield  consistent  es-
timates. Note that the function r has to be specified.
The alternative  stochastic  specification  used in this article yields, for  x  = In yx,
(26)  W i = ai +  yij In p  +  fi(7 - In Pi) +  wi,
which, being linear in nx, implies that E(ew) = 0. Using y or In y in the instrument set for
7x will yield consistent estimates. Note that we do not need to specify the group expenditure
function when applying this method. If, however,  we estimate
(27)  1x=  '(P, q,  y; 0) +  v,,
then this can be used explicitly to form instruments  (this is called the method of internal
instruments  by  Bowden  and  Turkington,  p.  166).  LS  will not be  consistent,  except  in
exceptional circumstances.  The bias introduced by using LS may be small, however, and
the usual Hausman-Wu test could be used to detect this.
The specification  (26)  seems more convenient  than (24)-(25),  but the choice  between
them must be an empirical matter.  Some form of nonnested test seems to be called for,
although  the  necessity  of different  estimation  methods  in the  two  models  will  cause
complications.  An alternative  approach could be to test for specification  error in each of
the two models,  using some IV generalization  of Ramsey's RESET  test.  Note  also that
the two  models given above  are not the  only possible  stochastic  generalizations  of the
Almost  Ideal Demand System.  The following  model,
(28)  wi =  a  +  Zi j In p, +  Ai(ln  x - In P,) +  E,
together with (25), will have to be estimated by Anderson's method, but will yield different
estimates than those given by (24).  Note that reformulating (28),  so that the right-hand
side is expressed in terms of yx, will imply
(29)  w  = ai +  y  In p  +  fi  (ln  y  - In P)  +  Zwi,
which only differs from  (26) as regards the formulation of the error terms. That is,
144  December 1993Separable  Demand Models  145
(30)  w = E, +  ((ln  - ln(yx  +  Vx)),
while
(31)  =  Ew +  #vW.
Conclusions
The importance  of the  stochastic  specification  of demand  systems is  often overlooked.
The implications of separability on the specification  of conditional  demand models have
only recently been  investigated,  and the  article by LaFrance  concerning  exogeneity is of
crucial  importance  in empirical analysis.  Some  of the conclusions  drawn in that article
are,  however,  dependent  on the specific  stochastic  form that was assumed.  Other speci-
fications,  which are at least equally plausible,  lead to other conclusions.
In particular,  conditional demand systems that are linear in some (linear or nonlinear)
function of the group expenditures can be specified so that usual IV methods are consistent.
Other specifications,  which demand a double regression  interpretation  of TSLS, are also
always  possible.  The choice  of which  specification  to  use will  be an empirical  matter,
which generally can be difficult to separate from other misspecification problems such as
choice of funcational form,  etc.
The automatic  use of LS in conditional demand systems cannot be justified. The matter
of finding a consistent estimation method to use as a yardstick,  seems, unfortunately,  still
to be unresolved.
[Received April 1993;final revision received July 1993.]
Notes
' We use the notation LS here to denote all estimation methods that are based on a theoretical zero correlation
between regressors  and errors, e.g.,  ordinary LS, nonlinear  LS,  seemingly  unrelated regressions (SUR), iterated
SUR, etc.
2 Deaton argues that the asymptotic bias  in using  LS should be small, however.
3 LaFrance  mentions  budget-share  models  in his footnote 2,  but in his case,  he considers  the unconditional
budget  shares xipi/y.
4 The notation f-'(w, p, yx)  is used loosely here to mean the inverse  of w = fix, p, yx)  for given p and yx.
5 Anderson's  method  is  equivalent  to  iterated  generalized  two-stage  least  squares performed  as  a double
regression  method.  Both Kelejian  and Edgerton  (1972) have  shown that  in nonlinear  sitautions,  the double
regression and  IV interpretations  of TSLS  cannot both be  consistent for the  same stochastic specification.  In
linear models the two forms  are, of course, numerically  identical.  Edgerton (1973) has  shown how a nonlinear
form of Wold's fix-point method (which is equivalent to iterated  double regression TSLS)  can be applied if,  in
the present  notation, E(Qw)  =  0.
6 In the case of simple functions, such as expenditures, budget shares, or uniform transformations of quantities
demanded, the results of Lau are of  interest. Note also that if we consider uniform simple functions of quantities,
expenditures,- or budget  shares, then lemmas  1 and 3 restrict us to rank 2 conditional  demand models in order
that the error terms should have  the requisite properties.
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When Is Expenditure 'Exogenous' in Separable Demand Models?&quot;

ON THE ESTIMATION OF SEPARABLE DEMAND MODELS

ON THE ESTIMATION OF SEPARABLE DEMAND MODELS

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