A procedure to test for the significance of violations of revealed preference conditions is described. The procedure is simple and hence may especially be appropriate for large data sets. An application to consumption data is presented.Research Methods/ Statistical Methods,

Tsur, Yacov

English

Research Papers in Economics

Staff Papers Series
Staff Paper P89-14  March 1989
ON TESTING FOR
REVEALED PREFERENCE CONDITIONS
March 1989
by
Yacov Tsur
University of Minnesota
Department  of Agricultural and Applied Economics
Staff Papers  are published without formal review within the Department of
Agricultural  and Applied Economics.
The University of Minnesota is  committed to  the policy  that all persons
shall have equal access  to  its programs,  facilities, and employment
without  regard to race,  religion, color, sex, national origin, handicap,
age,  or  veteran  status  or sexual orientation.
University  of Minnesota
Institute of Agriculture,  Forestry and Home Economics
St. Paul, Minnesota  55108On Testing for Revealed Preference Conditions
Yacov Tsur
Department of Agricultural and Applied Economics, University of Minnesota
Abstract
A procedure  to  test  for  the  significance of violations  of revealed
preference  conditions  is  described.  The  procedure  is  simple  and
hence  may  especially  be  appropriate  for  large  data  sets.  An
application  to consumption data is presented.
*
Department of Agricultural  and Applied Economics, University of Minnesota,
1994 Buford Avenue,  St.  Paul, MN 55108.
The helpful comments  of Ted Graham-Tomasi are gratefully acknowledgedIntroduction
Applications  of nonparametric revealed preference theory, while  free of
misspecification  errors, are generally not free of measurement errors.  Data
may be viewed as  being generated from an underlying true, unobserved structure
and some measurement error process. Suppose  the data violate GARP  (Varian,
1982, p. 947).  A question then arises as  to whether  this  is  a significant
indication that  the  true underlying structure violates GARP or  is  it  likely
that the true structure  satisfy GARP and the violations  in the observed data
were caused by the measurement errors?
This note describes  a procedure designed to  test for the  significance of
GARP violation.  Its  implementation is  simple, requiring only a few, fast
computations.  Thus,  it may in particular be useful when dealing with large
data  sets,  possibly as a screening test before a more elaborate method, such
as  that suggested by Varian (1985),  is  considered.  The present approach
provides a natural  interpretation of Afriat's  (1967) efficiency indexes, as
defined in Varian (1987),  in terms of the measurement errors.
Specification of the Data Generating Process
For k goods and n time periods  let Xi and pi be the k by 1 vectors of
observed quantities and prices, respectively, at time period i-l,2,..n.  The
observations  contain errors;  the  corresponding true, unobserved, quantities
and prices are denoted by X *i and Pi. Let  C(i,j)  - Pi  be the observed
expenditure of goods consumed at time j in terms  of period i prices.  In a
similar manner define C (i,j)  - P i.X  . The  revealed preferred relation is
indicated by R  (see Varian 1982, p. 947).  The equivalent concept for  the  true
structure  is  indicated by R ; thus X  R X*X is  interpreted as  "X  is  preferred
to  X  according to  the unobserved quantities and prices  (the term "revealed" to X  according to  the unobserved quantities and prices  (the term "revealed"2
is  dropped since X* j and P*1  are unobserved).  The starred variables P  J, X*j
*
and C  will be referred to as  the  true  structure.
The data  (Xj,p j, j-l,2,..,n)  satisfy GARP if XiRX j implies C(j,j)sC(j,i)
for all  i,j.  The  true quantities  and prices  satisfy GARP if X*iR*Xj  implies
C (j,j)C (j,i)  for all i,j.  Varian  (1987) defines another revealed
preference condition:  the  data (XJ,Pj,eJ,  j-l,2,..,n)  satisfy  GARPe  if XiReXj
implies  ejPijXJ  x  P .Xi;  where  Ro  is defined  as  XJR:X iff eJPjXJ2PjX,  Re  is
the  transitive closure of Ro and  the  e- are n scalars  satisfying OseJ<l.  The
set  (Xj,PJ,ej,  j-l,2,..,n) will be  called the  perturbed data set  and e  -
1  2  n (e ,e ,..,e n ) is  the perturbation vector.
It is assumed that the observed quantities  and prices are  related to  the
true structure according to X  - V.X*j and P  W  ,  where V. and W. are
scalar random variables such that  v. - log V. and w. - log W. are
*J  J  J
independently and identically distributed (iid)  normal variates with zero
means and variances given by a2  and a', respectively.  Letting c - log C and ~~~~*  *W
c  - log C ,  it  follows that
c(ij) - c(i,j) + v+  ;  vi+ wj  - NO, 1  j  i  id  v  w+a),
which characterizes the data  generating process.  In particular, for  i-j,
c(j,j)  - c  (j,j) +  j;  e  -lid N(O,  ),  j-1,2,...,n,  (1)
where  c.  - vj+ w  and a2-  2a+  a  . The  n  by  1  vector  which  consists  of  the
diagonal  elements  of c(.,.)  [resp.  C(.,.)]  will  be  indicated  by  c  [resp.  C]
and  will  be  refer  to  as  the  expenditure  vector.
The  Test  Procedure
Our  test  concerns  observations  which  violate  GARP.  We  make  use  of  the
idea  set  forth  by  Varian  (1985)  and  perturb  the  data  until  GARPe  is  satisfied.
We  then  inquire  whether  it  is  plausible  that  the  perturbed  data  represent  a3
feasible  true structure  from which the actual data were generated.  Following
Afriat  (1967) and Varian (1987),  perturbations are allowed only on the
diagonal  elements  of the expenditure matrix, i.e.  on the expenditure vector C.
Let the distance between any two vectors  in Rn be defined as  the square
of the Euclidean norm of  the difference vector divided by n.  For p > 0, let
T (p) - (uERn: Zi[uj-cj)] /n < p) be the  set  of all n by 1 vectors  that are at
most distance p away from  the expenditure vector c.  The  set T (p) is  such
that T  (P')  C  T  (p")  whenever p'sp", with T (0) - (c) and Tn(O)  - Rn. n  n  n  n
Any uERn has a perturbation vector attached to  it defined by
e -exp(uj-ci)  such that Uj-exp(ui)  is  derived from the expenditure vector C
according to U -eJC,  j-1,2,..,n.  In this way one can attach a perturbed data
set to  any vector ueRn .
Definition:  T (p) satisfies GARPe if there exists u E  T (p) such  that
the perturbed data generated by ej - exp[u.-c  ], j-1,2,..,n, satisfies GARPe.
J j
Clearly T (0) does not satisfy GARP. (since  it entails  ej - 1 and GARP is
violated by the data) whereas T (c) vacuously satisfies GARPe  (setting u.--C n  c
implies  eJ-O).  Moreover, if T  (p')  satisfies  (does not satisfy) GARPe  then
the  same holds  true for any T (p)  with p > ()  p'.
The true expenditure vector c  is  of distance  [cf.  equation (1)]
2  2
s  - .e./n n  j-li 
away  from c.  Define p  - Min(p:  T (p) satisfies GARPe) and suppose p  n  s . n  n  n  n
Then it must be that T (s ) satisfies GARPe  as well.  Obviously T (s ) n  n  n  n
contains c ,  which means that the possibility that the true expenditure vector
satisfies GARP cannot be ruled out.  Under such circumstances we shall say
that the violation of GARP by the observed data  is not sufficiently severe to
imply that the  true structure violates  GARP as well.  On the other hand, if Pn
, tn  ay  st  T  ()  tt  s  G  c  p  c 
> s2 n  then any set T (.) that satisfies GARPe  cannot possibly contain c  and n'  n4
we  shall  interpret this  as  evidence that the true structure violates GARP.
In view of  the above, the null hypothesis, maintaining that the true
structure satisfies GARP, is  specified as:
2
Ho:  p  s n  n
Consider  the  simple  hypothesis
2
Hoo:  p  - s n  n
and note  that Hoo  is  contained  in Ho  and rejecting Hoo  in  favor  of the
alternative p >s 2 implies rejecting any other simple hypothesis  contained in
n
2  2  2  2
Ho.  Now s2  Z  ej/n  - (a  /n)X  [cf.  equation (1)].  Hence, under Hoo,
2  2
np  -a  Xn.
The  test  is performed, utilizing aspects of Varian's  (1985) procedure,  in
the following fashion:  i) calculate p  > - p  [see procedure below];  ii)
-2  2  2  2  2
calculate  2a  np  /X  (a),  where X  (a) is defined from Pr(XnXn(a)) - a;  iii)
2  2
reject Ho,  at a percent significance level,  if  it  is  believed that  a  <  a2.
29  2  2
When a  is  known a priori, Ho  is rejected if npn/a  > Xn(a).
A 
It remains to  calculate  p  ,  which is  done by the following algorithm.
Input:  the n by n expenditure matrix C(-,.)
Output:  a perturbed data set satisfying GARPe and a distance  index pn > Pn
1) set M(j)-l and Ce(i,j)-C(i,j)  for i,j-l,2,..,n;
2) set Ce(j,j)-M(j)C(j,j),  j-1,2,..,n;
3) for  i,j-l,2,..,n set Re(i,j)-l  or  0 as  Ce(i,i) >  or <  Ce(j,i),
respectively, and calculate its  transitive closure Re  [for  an algorithm to
calculate the transitive closure  of a matrix see Varian (1982, p. 972)];
4) set Ge-{j:  Re(i,j)-l and Ce(j,j)  > Ce(j,i)  for at  least one case  i);
5) if Ge  - 0  then go to 7, else go to 6;
6) calculate the n by 1 vector M as5
min  (C(j,i)/C(j,j))  ;jeGe
M. - {  x  Rex
J  M  j  ;j6Ge ;jeG'
(Ge  indicating the complement of Ge)  and go  to  2;
7) calculate
n
2
Pn -jl  [log M]  /n  (2)
and stop.
This procedure  can take  at  most n iterations,  since  once any case j has
been corrected  it can never violate GARPe again.  Clearly,  the perturbed data
generated by the perturbation vector e - M satisfy GARPe.  The j'th element of
the associated perturbed expenditure vector is  Ce(j,j)-eJC(j,j).  It is  easy
to verify that  the distance of  the log of this vector from the log of the
actual  expenditure vector,  c, is  given by Pn of equation (2).  Thus pn<Pn.
(It may be possible  to  design a procedure that calculates pn  itself;  in the
spirit of simplicity, however, we  shall not pursue this task here.)  With
pn>Pn,  the  test is more conservative in the  sense  that if Ho  cannot be
rejected with Pn it obviously will not be rejected with pn'
Application
Our data set contains  consumption and price  data of four  (k-4) major meat
types  in Spain for the  150 months  (n-150) of the period January 1970  through
July 1982.  The mean and variance of the  expenditure sample  (cj  - log C(j,j),
j-l,2,..,n)  are 16.634 and 0.0243, respectively.  A computer code realization
(fortran) of a GARP test, based on the algorithms described in Varian (1982),
detected 33  cases  (months) in which at  least one GARP violation occurs.  A
perturbation vector e - M that  satisfies GARPe was constructed according to
the  procedure described in the previous  section (it took one  iteration to
n
satisfy CARP.)  and provided the distance  index Pn - [log M.  /n - .000046.6
Choosing a - .01 and  .1 yields X2  (.01)  - 192.431 and x2(.1) - 172.482 n·p  I  (1)-(.1
) - 172.482
which  imply a2 n/Xn(a)  - 7.9x10  6 and 8.8xl06, respectively.
As noted above,  the variance of the  sample  (c,  j-l,2,..,n)  is  0.0243.
*
Even if c;,  j-1,2,..,n,  explain  99 percent of this variance we are still  left
Eof  1,-4 with an error variance which  is  of order of magnitude of 10 4 . It  therefore
appears unreasonable  to  suppose  that  the variance of the  (log of) actual
expenditures  is  less  than a2.  Thus Ho  cannot be  rejected.  We concluded that
the violation of GARP by the data  is not sufficiently large  to  imply that the
true structure violate GARP as  well.7
References
Afriat,  Sydney N.,  1967, The  construction of utility function from expenditure data, International Economic Review 8, 67-77.
Varian, Hal  R.,  1982, The nonparametric approach  to demand analysis, Econometrica 50,  945-972.
Varian,  Hal  R.,  1985, Nonparametric analysis of optimizing behavior with measurement error, Journal of Econometrics  30,  445-458.
Varian,  Hal R.,  1987,  On goodness-of-fit  of revealed preference conditions, CREST Working Paper, Department of Economics, University of Michigan.

ON TESTING FOR REVEALED PREFERENCE CONDITIONS

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