--- Page 1 ---
The Power of Nonparametric Tests of Preference Maximization 
Author(s): Stephen G. Bronars 
Source: Econometrica , May, 1987, Vol. 55, No. 3 (May, 1987), pp. 693-698 
Published by: The Econometric Society 
Stable URL: https://www.jstor.org/stable/1913608
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--- Page 2 ---
 Econometrica, Vol. 55, No. 3 (May, 1987), 693-698
 THE POWER OF NONPARAMETRIC TESTS OF
 PREFERENCE MAXIMIZATION'
 BY STEPHEN G. BRONARS
 1. NONPARAMETRIC TESTS OF PREFERENCE MAXIMIZATION
 IN A RECENT PAPER in this journal, Varian (1982) presented algorithms which test a
 finite body of data for consistency with the Generalized Axiom of Revealed Preference
 (GARP). Despite the fact that he used broadly defined aggregate data, Varian was unable
 to uncover any violations of GARP over the period 1947-1978. Varian attributes the
 inability of his test to reject preference maximization to the postwar aggregate data which
 he uses:2
 Most existing sets of aggregate consumption data are post-war data, and this period
 has been characterized by small changes in relative prices and large changes in income.
 Hence each year has been revealed preferred to the previous years in the sense that it has
 been typically possible in a given year to purchase the consumption bundles of each of
 the previous years. Hence no "revealed preference" cycles can occur and the data are
 consistent with the maximization hypothesis.
 In each year the "representative consumer" has been typically able to purchase the observed
 consumption bundles of previous years, but since consumption bundles are potentially the
 outcome of preference maximizing decisions, this is not a meaningful criticism of postwar
 aggregate data for tests of preference maximization. Varian is incorrect in asserting that
 relative price changes are unimportant in postwar aggregate data, for the budget sets of
 the "representative consumer" frequently intersect in the positive orthant. Varian finds
 that the observed consumption bundles do not provide much of an improvement on the
 "classical" bounds for true cost of living indexes, but this does not imply that tests of
 preference maximization using aggregate data have low power. Any budget set intersec-
 tions, whether or not they improve the "classical" bounds for cost of living indexes,
 increase the power of tests of preference maximization. A data set contains no useful
 information about preference maximization only if budget sets do not intersect (the
 "representative consumer" is able to purchase all feasible consumption bundles of previous
 years).
 This point is made clearly in Figures 1 and 2, which contain budget sets for years I,
 II, and III. The budget sets are identical across figures, but consumer 1 in Figure 1 chooses
 A, B, and C, while consumer 2 in Figure 2 chooses D, E, and F. Since budget line III
 does not intersect with the other two budget sets, it provides no useful information about
 preference maximization in either figure: GARP cannot be violated by any possible choice
 in year III. A violation of GARP only occurs if a consumer chooses consumption along
 GH in year I and along HI in year II, which is possible for either consumer. The fact
 that consumer 2 was "typically possible in a given year to purchase the consumption
 bundles of each of the previous years" is dependent on consumer 2's choice of bundles
 across years, and does not diminish the power of this test relative to Figure 1 (consumer
 1 is unable to afford year I's consumption bundle in year II).
 l I would like to thank the editor and referees for comments which have improved the pap
 exposition. Michael Baye, Donald Deere, Chris Fawson, Martyn Houtman, John Lott, and Bob Reed
 provided insightful comments. James Van Beek and Julie Holleman provided excellent research
 assistance. Any remaining errors are my responsibility.
 2 See Varian (1982, p. 965). The problem to which Varian refers is price collinearity; since nominal
 prices tend to be highly correlated across commodities, relative prices are characterized by "small"
 changes. This note argues that even these "small" changes are enough to detect violations of GARP
 against certain alternatives.
 693
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--- Page 3 ---
 694 STEPHEN G. BRONARS
 C2
 J
 Year I
 G A C
 B
 Year III
 Year II
 I K C1
 FIGURE 1.
 The observed consumption bundles in Figure 2 reveal more information about the set
 of consumption bundles "not revealed worse" than year I's consumption bundle (in
 Varian's notation), since E is revealed preferred to D, but neither A is revealed preferred
 to B, nor is B revealed preferred to A. Varian discusses why the set of consumption
 bundles in Figure 2 provide tighter approximate bounds for the true costs of living index
 for year I compared to the observed choices in Figure 1. Improvements in cost of living
 bounds depend on observed consumption choices; the power of the test of preference
 maximization depends on the way in which these consumption choices are made.
 The power of this test (the probability of rejecting the null hypothesis) is zero when
 the null hypothesis is true, given the absence of an error term in the theory of preference
 maximization upon which GARP is based. One would like to know the entire power
 C2
 YearI
 F
 E
 D Year III
 Year II
 C1
 FIGURE 2.
 3 See Varian (1985) for a discussion of tests of preference maximization with measurement error.
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--- Page 4 ---
 NONPARAMETRIC TESTS 695
 function, for any arbitrary alternative hypothesis. Since any type of irrational behavior is
 an admissable alternative hypothesis, some restrictions must be placed on the alternatives
 to preference maximization in order to make power calculations.
 In this note I adopt Becker's (1962) notion of irrational behavior. The representative
 consumer is assumed to choose consumption bundles randomly from his budget hyper-
 plane. In Becker's example the consumer chose consumption from a uniform distribution
 across all bundles in the budget hyperplane.4 For example, in Figure 1 a violation of
 GARP occurs if agents choose bundles along GH and HI, which occurs with the following
 probability:
 Power = Pr (Violating GARP) = (11 GH 1 * 11HI 1 /(11 GK 11 * I!JI 1)
 where lid I denotes the length of the budget line denoted by d. Since 11HK 1 I HI and
 IIJH 1 11GH 1, this probability is bounded between 0 and .25, given only one budget se
 intersection.' The power of this test can increase substantially as the number of budget
 set intersections increases.6 In general, the probability of rejecting the null hypothesis
 increases with the number of budget set intersections in the data, although for more than
 two commodities, and with multiple budget set intersections, no simple formula for the
 power of this test can be obtained (even for the uniform distribution case). This suggests
 that a fruitful approach to obtaining the approximate power of this test is to perform a
 Monte-Carlo type study. Consumption data sets based on random choice behavior, given
 the constraints implied by actual post-war U.S. price and total expenditure data, are
 generated and Varian's algorithm is used to check the random data for consistency with
 GARP.
 2. CALCULATING THE APPROXIMATE POWER OF VARIAN'S NONPARAMETRIC
 TEST USING RANDOM CONSUMPTION DATA
 The first step in calculating the approximate power of Varian's test is to construct
 algorithms that generate random consumption data which exhaust the budget set in each
 year. Given N commodities, in each year we seek N random variables, SI,, . . ., S,, that
 sum to one and represent budget shares of each commodity in year t. Algorithm 1 generates
 budget shares in such a way as to induce a uniform distribution of consumption choices
 across the budget hyperplane, following Becker's example.7 The generated budget shares
 in year t are multiplied by either aggregate or per capita total expenditure in t and divided
 by the actual price of the corresponding commodity in t, to obtain the random consumption
 quantities for year t.
 Algorithm 2 draws N i.i.d. uniform random variables in each year, Z1, . . ., Zn, and Si,
 is determined by the following equation:
 N
 4Becker (1962) showed
 the alternative of individual random behavior, when aggregate data are used (if individual randomness
 is independent across agents). Houtman and Maks (1985) verify this by generating random individual
 data, aggregating the data, and finding that the aggregate data are rarely inconsistent with GARP.
 This paper concerns another issue: are there enough budget set intersections in the data for a powerful
 test of GARP even if one had individual consumption data?
 5 The power is highest when 1I GH II -I JH II and II HI II - HK 11, or the two budget sets are "nearly
 parallel. The power is lowest when 11 GH II/ IIJH 11 and 11 HI / HK 11 are nearly zero. This occurs wh
 the budget sets intersect at close to a 90? angle.
 6 Two sets of mutually exclusive budget set intersections could result in a power of the test as hi
 as .4375. Each budget set intersection would result in a rejection probability of at most .25. The
 probability of a rejection due to either budget set intersection equals at most .25 + .25 - (.25)2 =.4375,
 given independence. Using similar arguments the power of a test given three mutually exclusive pairs
 of intersections is at most .25 +.25 +.25 - 3(.25)2- (.25)3 .547.
 7A copy of this algorithm is available from the author.
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--- Page 5 ---
 696 STEPHEN G. BRONARS
 Since this algorithm imposes an expected budget share of 1/ N, algorithm 3 determines
 Si, by:
 N \
 Sit-=K.ZZ / E KJj JZj
 j=l
 where Ki is the mean budget share of good i in the actual data across all
 3 mimics the actual consumption data in the sense that mean budget shares 
 and actual data are equal.
 In all of these algorithms I imposed independence of choices over time. The approximate
 power of the test given each alternative hypothesis (algorithm for generating consumption
 choices) is the fraction of data sets in which violations of GARP occurred.
 Following Varian, I use aggregate U.S. consumption data by nine categories, over the
 period 1947-1978, for the nonparametric test of preference maximization.8 My analysis
 differs from Varian's in that I use both aggregate and per capita consumption data in
 testing for violations of GARP. Working with per capita rather than aggregate data
 significantly alters the nature and frequency of budget set intersections in the sample period.
 Using Varian's algorithms for the nonparametric tests of GARP, I verified that neither
 the actual aggregate or the actual per capita consumption data violated GARP over this
 sample period. Using aggregate data, budget hyperplanes in 7 of the 32 years did not
 intersect (in the positive orthant) with any other budget hyperplanes. For the purposes of
 nonparametric tests of GARP, one can view the years 1947-1961 and 1968-1977 as two
 separate subsamples. Using per capita data, the budget hyperplane in each year intersected
 (in the positive orthant) with budget hyperplanes from at least two other years.
 Using each of the algorithms described above, I generated 200 ranuom consumption
 time series, using actual prices and both per capita and aggregate total expenditure
 measures. For each of the six sets of 200 time series (32 years in each series), I calculated
 the percentage of times that GARP was rejected. These results are reported in Table I.
 Table I indicates that U.S. per capita consumption data provide a surprisingly powerful
 test of preference maximization against the alternative hypothesis of random behavior by
 the "representative consumer." Rejection probabilities exceed 90 per cent for all three per
 capita versions of the tests, but are considerably lower for aggregate versions of the tests.
 The pattern of rejection probabilities presented in Table I is consistent with the notion
 that in aggregate versions of the tests, most budget set intersections in post-war U.S. data
 are similar to those in Figure 3. Algorithms 2 and 3 make it unlikely that a consumer
 randomly chooses a bundle near a corner solution, and since violations of GARP can
 only be detected if consumers choose a bundle between points A and B in Figure 3, the
 test has low power against this alternative. Algorithm 1 results in a considerably higher
 probability of rejecting preference maximization because it induces a much higher probabil-
 ity of observing consumption bundles in the neighborhood of a corner solution.9
 The third and fifth columns of Table I report the mean number of violations of GARP
 across time series (conditional on the occurrence of at least one violation of GARP). For
 example, using algorithm 1 and aggregate total expenditure, 67.5 per cent of the time
 series had at least one violation of GARP, and among these series, the average number
 of violations per series was 3.58. The use of per capita data substantially increases the
 8The nine consumption categories include Motor Vehicles, Furniture, Other Durables, Food,
 Clothing, Fuel, Housing, Transportation, and Other Services. The price of each good is given by the
 GNP deflator for each commodity. One abstracts from savings decisions by assuming that total
 disposable income in each period equals total expNenditures on these 9 commodities.
 9 As N gets large, the probability that Zi_j= Zj is "close to" one approaches zero. With 9
 commodities it is relatively rare that a consumer would be devoting a "large" share of his income
 to just one or two commodities, given algorithms 2 or 3. A uniform distribution of choices implies
 that this choice "near" a corner solution is just as likely as equal budget shares across commodities.
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--- Page 6 ---
 NONPARAMETRIC TESTS 697
 TABLE I
 PROBABILITY OF REJECTING GARP AND AVERAGE NUMBER OF VIOLATIONS PER TIME SERIES
 Aggregate Version Per Capita Version
 Algorithm Probability Avg. Violations Probability Avg. Violations
 1 .675 3.58 .985 14.52
 2 .305 2.80 .910 6.34
 3 .220 2.18 .915 6.19
 C2
 Year I
 Year II
 A~~~~~
 B C.
 FIGURE 3.
 average number of violations of GARP relative to aggregate data. The variation in real
 per capita total expenditure and relative prices over the period 1947-1978 has been sufficient
 to provide a powerful test of preference maximization against the alternative of random
 behavior, although aggregate versions of the tests appear much less powerful than per
 capita versions of the tests against this alternative.
 III. CONCLUSION
 This note outlines a simple and inexpensive method for calculating the approximate
 power of a nonparametric test of preference maximization for a simple alternative
 hypothesis. Before one estimates a system of demand equations, it is important and
 instructive to first examine whether or not the data are consistent with GARP, as Varian
 has noted. Before one accepts the results of a nonparametric test of preference maximization
 based on a set of observed consumption data, it is important and instructive to examine
 the nature and frequency of budget set intersections. The power calculations in this note
 suggest that Varian's test of preference maximization using per capita data is quite powerful
 against the alternative of random behavior, but tests using aggregate data appear much
 less powerful. One should be wary of tests that have low power against the rather naive
 alternative of random behavior, for failure to reject the null hypothesis may convey little
 useful information.
 As always, one must be cautious in drawing conclusions about a "representative
 consumer" based on observed aggregate or per capita data. As Becker (1962) noted,
 random behavior by individuals may still result in aggregate or per capita consumption
 data that are consistent with revealed preference theory. The results presented here indicate
 that U.S. post-war consumption data are inconsistent with an irrational "representative
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--- Page 7 ---
 698 STEPHEN G. BRONARS
 consumer," but i.i.d. randomness in behavior across individuals would be difficult to detect
 with aggregate or per capita data. This is a limitation of the use of aggregate rather than
 individual data, and not a criticism of nonparametric tests in general.
 Department of Economics, University of California-Santa Barbara, Santa Barbar
 93106, U.S.A.
 AManuscript received November, 1985; final revision received July, 1986.
 REFERENCES
 BECKER, G. S. (1962): "Irrational Behavior and Economic Theory," Journal of Political Economy,
 70, 1-13.
 HOUTMAN, M., AND J. A. H. MAKS (1985): "The Consistency of Aggregate Random Data with the
 Hypotheses of Cost Minimization and Utility Maximization," Paper presented at 1985 World
 Congress of the Econometric Society, Department of Economics, University of Groningen, Gronin-
 gen, The Netherlands.
 LANDSBURG, S. E. (1981): "Taste Change in the United Kingdom, 1900-1955," Journal of Political
 Economy, 89, 92-104.
 VARIAN, H. (1982): "The Nonparametric Approach to Demand Analysis," Econometrica, 50,945-973.
 (1985): "Non-parametric Analysis of Optimizing Behavior With Measurement Error," Journal
 of Econometrics, 30, 445-458.
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