Chaos Theory and Microeconomics: An Application to Model Specification and Hedonic
Estimation
Author(s): Steven G. Craig, Janet E. Kohlhase, David H. Papell
Source: The Review of Economics and Statistics, Vol. 73, No. 2 (May, 1991), pp. 208-215
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/2109510
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CHAOS THEORY AND MICROECONOMICS: AN
APPLICATION TO MODEL SPECIFICATION AND
HEDONIC ESTIMATION
StevenG. Craig,JanetE. Kohlhase,andDavidH. Papell*
Abstract-This paperis the firstto applythe theoryof deterministicchaos to a microeconomicproblem.Previousapplications of chaos theoryto time series data, while successfulin
uncoveringnonlinearities,have not providedguidelines for
problems.In contrast,we
resolvinguncoveredmisspecification
show that a modifiedtest statistic from chaos theory is an
extremelyvaluabletool in microeconomicmodel specification
becauseit showswhen excludedinformationis correlatedwith
includedinformation.This test, appliedto hedonicestimation
of marginalhousingprices,is able to distinguishamongalternative regressionspecificationsand assists in discoveringa
parsimoniousspecificationdevoidof nonlineareffects.
I. Introduction
JNTERESTin the theoryof deterministicchaos
has recently exploded. Although the theory
originated in the hard sciences, applicationsof
chaos theory to economics, as described by
Grandmont (1985), Brock (1988), and Baumol
and Benhabib (1989), are becoming more common. All of the empirical applicationsto date,
however, have used macroeconomicor financial
time series data. Even though researchershave
been successful in detecting evidence of nonlinear structure,they have not yet used chaos theory
to provide guidelines for model specificationto
purge nonlinearitiesfrom the data. Our application of chaos theoryto microeconomicdata offers
a naturalway to attempt to resolve the presence
of nonlinearstructurethroughan examinationof
model specification.
In this paper we present an innovativestatistical test, based on the BDS test of Brock,Dechert,
and Scheinkman(1987),to detect the presence of
nonlinear structurein microeconomicdata. The
essence of our test is to examinethe residualsof
a regression for systematic patterns. In a time
series context, the data follow a natural order.
With cross section data, the choice by which
variable to order the data becomes imnportant.
We demonstratethat our modified BDS test examineswhetherthere is informationomittedfrom
the regressionthat is correlatedwith the variable
by which the residuals are ordered. Omitted informationcorrelatedwith includedvariablescan
either be truly omitted variables, or can be a
nonlinearcombinationof the informationalready
included in the regression. While there are no
systematic methods for finding excluded variables, it is possibleto explorenonlinearcombinations of included variablesas alternativeregression specifications.
To illustratethe importanceof chaos theory to
microeconomicresearch, we apply the test for
nonlinear structure to a hedonic housing price
regression.This applicationis particularlyimportant since model specificationis a crucial issue
without clear theoreticalguidancein the housing
literature.' We show that the residualsfrom the
standardspecificationtypicallyexhibitnonlinearities; that is, the residuals are not independently
and identicallydistributed(iid). For some of the
cases that we examine,we are able to purge the
nonlinearitiesfrom the system by examiningalternativespecificationsof the regressionmodel.
There are two separateaspects of specification
in applied economics that are highlightedin the
hedonichousingliterature.One is to find the best
overallfit of a regressionequation.The second is
to determine whether the central coefficient of
the model is correctly estimated. While the
econometric literature has recently proliferated
with new specificationtests that addressthe first
of these problems,no test exists that is focused
Received for publicationSeptember1, 1989. Revision accepted for publicationAugust9, 1990.
* Universityof Houston and Hunter College and Graduate
Center, CUNY; Universityof Houston and Hunter College
and Graduate Center, CUNY; and Universityof Houston,
respectively.
We would like to thank Dee Dechert for unravellingsome
of the complexitiesof chaos theory and the BDS test for us.
We have also benefitedfrom discussionswith Chera Sayers,
the commentsof two excellentreferees,and fromseminarsat
the Dallas FederalReserveBank,the Universityof Houston,
1Applications in labor economics,such as the specification
HunterCollege,and WayneState University.Partialresearch
supportfor Craigand Kohlhasewas providedby Texas Ad- of earningsequations(Murphyand Welch, 1990), may also
benefitfromour approach.
vancedResearchProgramGrant003652-113.
[ 208 ]
Copyright? 1991
CHAOS THEORY AND MICROECONOMICS
upon a single variable in order to address the
second of these problems.The modifiedBDS test
is appliedwith respect to a single variablein the
equation and does not rely on the overall goodness of fit of the regressionmodel.
Complicatedfunctionalforms are often needed
to obtain the highest possible goodness of fit
measures.Achievingthe best overall goodness of
fit sacrifices,however, the parsimoniousspecification needed to make precise statements about
single regression coefficients. The problem, as
discussed in Cassel and Mendelsohn (1985), is
that complicated functional forms can make it
difficultto calculatemarginalprices and standard
errorsof specifichousingcharacteristics.We find
that the modified BDS test gives valuable informationwhen appliedto the analysisof individual
variablesin hedonic housingregressions.
We applythe modifiedBDS test to the specification of a hedonic housing price equation used
in Kohlhase (1991). We search for the existence
of systematicnonlinearitiesin the errorterm of a
regression.If systematicnonlinearitiesare found,
we can conclude that the error term is not independentlyand identicallydistributedand that the
regressionspecificationneeds to be exploredfurther. We applythe test and show how it leads to
a parsimoniousspecificationwith ease of interpretationof the coefficientsof a single variable.
The test is described in section II. We show
that orderingby one of the right hand side variables of the regression allows the BDS test to
detect correlationsbetween the includedvariable
and informationexcluded from the regression.
Section III discussesthe applicationof the modified BDS test to the hedonic housing equation.
This applicationshowsthat a parsimoniousspecification,which allows straightforwardanalysisof
the effects of a single variable and relativelyuncomplicatedcalculationof the resultingstandard
errors, retains some of the desirable features of
more complexfunctionalforms. Section IV offers
conclusionsand suggestionsfor furtherresearch.
II. A One Variable Specification Test
The problem we analyze is to determine the
correct specificationof a hedonic housing price
equationsuch as:
g(P)=f(X)+e
(1)
209
where P is the price of a residential dwelling,
and X representsa vectorof characteristicsof the
house, the neighborhoodcharacteristicsof the
area, and its location. Goodness of fit tests, such
as Box and Cox (1964),examinethe overallspecification of (1) and maximize an appropriately
defined likelihood function. As pointed out in
Cassel and Mendelsohn (1985), however, if the
true specificationof (1) is
ln(P)
=
Eai ln(Xi) + bjXj + e
(2)
ri
then goodness of fit tests will prefer a log-log2
specificationover all others.This is in spite of the
fact that Xj is not specified correctly in the
log-log form. Such a misspecificationmay be crucial if Xj is the variableof interest.The modified
BDS test developed here is the first diagnostic
test capableof examiningthe errorstructurewith
respect to a single variable.
A. Outlineof the Basic BDS Test
The BDS test developed in Brock et al. (1987)
examinesthe patternof a series of data. Compare
the ith observationto the (i + 1)th observation.
There is some positiveprobabilitythat this pair of
observationsis within a known distance (d) of
each other. Now consider a third observation.
Again there is a probabilitythat it is within the
same distanceof the second. If these three observations are from an iid series, the probabilitythat
each pair is within d is well defined. The BDS
test compareshow often a series of data points
are actually within d of each other to the expected value if the series were distributediid.
Define an Indicatorfunction I, where I = 1 if
the absolutevalue of the differencebetween two
points is less than or equal to d, and I = 0
otherwise. The embeddingdimension m defines
the number of points that are compared in a
series.3The function C depends on the sum over
the productof the indicatorfunctionsfor each m
2 Here we define log-linearto mean ln(P) is a functionof
linearX's, linearto meana linearspecification,and log-logto
indicatelogs of both the left and righthand side variables.
3 The embedding dimension defines the "m dimensional
cube" that would be filled by the data if the data are truly
random.If the data are deterministic,then a functionof the
data would show a relationshipin m dimensionalspace. For
our purposes,the embeddingdimensionsimplyrepresentsa
parameterto be specified,where m can be specified larger
the largeris the data set.
THE REVIEW OF ECONOMICS AND STATISTICS
210
as
C(m,n,d)
=
cl
E
-1<i<j<n
( 1iId(Xi+k,
Xj+k))j
k=O
(3)
where m is the embedding dimension, n is the
number of observationsminus m, d is the distance between two points that causes the indicator function to equal one, and X represents a
data point. The expression in parentheses will
equal one if all m - 1 pairs of points are within
d of each other, and it will equal zero if even one
of the pairs of points is fartherapartthan d. The
summationshows how many times in an entire
data series there is a succession of m - 1 pairs
that are close (within d).
The BDS test relies on a proof that C(m, n, d)
asymptoticallyapproachesCmin probabilitywhen
the data series is independent and identically
distributed.Brock et al. (1987) use this result to
derive a statistic, called a C statistic, that depends on comparing the actual calculation of
C(m, n, d) to the calculationof [C(1, n, d)]m.If
the data are independent and identically distributed, this difference should approach zero.
Asymptotically,the difference between the two
functions divided by the standard error approachesa standardnormaldistribution.Dechert
(1987) has written a computerprogramto calculate the BDS statisticdefined as
BDS =
nl/2(C(m,
n, d) - [C(1, n, d)]m}/bm
(4)
where the numeratoris called the C statisticand
bmis the standarddeviationof the C statistic.4If
there are more (or less) points close together
than would be expected from iid data, where
[C(, n, d)]mis the numberof times a series of iid
points would be expected to be close, then the
BDS statistic indicates the likelihood that those
points could have been generatedby an iid series.
B. Applicationof the BDS test to MicroData
The BDS test describedin (4) has been applied
to macroeconomictime series data to search for
4 This is the C test based on Corollary2 of Theorem2 of
Brocket al. (1987),ratherthan the S test whichrelies on the
ratio of C functions.
occurrencesof systematicnonlinearities(for example Brock and Sayers, 1988). In a time series
context, the data have a natural order. In the
microeconomiccontextwe must impose sequencing on the data by sorting on the variable of
interest. What further differentiatesour application of the BDS test from the time series context
is that applicationto micro data allowsconsideration of why a series of residuals might or might
not be "close"to one another.If the observations
are ordered randomly,it would be expected that
the BDS statisticwouldbe zero, as the errorterm
should be iid. On the other hand, assume the
data are ordered by one of the right hand side
variables.If there is a misspecificationerror, either through incorrect functional form or an
omitted variable that is correlated with the included variables, then the BDS statistic will be
significantlydifferentfrom zero.
Brockand Dechert (1988) have proventhat the
BDS test is a good specification test for time
series data provided the residuals of the estimated regressioncan be written as a function of
the currentand lagged values of the residualsof
the true relation.In the case of omittedvariables
for micro data, the intuitionworks the same way
because the estimatedresidualsare a functionof
the omittedvariables.This can be seen by assuming the true model is
(5)
g(P) = f (X) + h(Z) + u
where u is independently and identically distributed.Assume the equation is estimatedomitting the Z terms:
(6)
g(P) = f(X) + v
so that the errorterm of the estimatedregression
is
(7)
v=h(Z)+u.
If the excluded information(Z) from the true
regression(5) is correlatedwith the included information(X) in the actual regression(6), then
equation (7) can be rewritten to depend upon
only the includedinformation:
(8)
v = w(X) + e + u = g(P) - f(X)
where e is an iid error term independent of u
and w(X) describes how the excluded information is correlatedwith the included information.
Brockand Dechert (1988) show that if g(P) f(X) is independent of X, then v is iid. The
CHAOS THEORY AND MICROECONOMICS
modificationto the BDS test that we propose
involves ordering the data by X, the included
variableof interest. If there is a relation such as
w(X), then sorting by X will uncover the fact
that g(P) - f(X) will depend on the size of X.
Thus the indicatorfunction in the BDS test will
capturethe relationshipin w(X) by showingthat
the error terms do not have equal probabilityof
being close to each other. We therefore are able
to use the BDS test as a specificationtest to look
for the importanceof excludedinformationfrom
the regression.
To see why the patternof residualsreflectsthe
information omitted from the regression, consider a set of residuals sorted by X and the
probabilitythat a given pair of residualswill be
within, in absolute value terms, d of each other.
If the series of residuals are iid, the probability
that one pair of residualsis close to each other is
unaffectedby whether the previouspair of residuals is close or not. If the residuals reflect the
omitted information w(X), however, then the
probabilitythat they are within d of each other
can be predictedbased on knowledgeof whether
the previous residuals are close to each other.
The predictionis possible only because the previous pair reflects the impact of the omitted information. That is, the BDS test will show a significantly greater probabilitythan independence of
finding residuals close to each other when the
residualshave been orderedby an includedvariable. Therefore the probability that the BDS
statistic is significantlydifferent from zero depends on the probabilitythat there is an omitted
variablethat is correlatedwith the includedvariable.
The key to the test we propose is how the
excluded informationimpacts the included variables.When the functionalformof a regressionis
not correct, it is quite likely that the excluded
informationis related to the included information. Even in the case of truly omitted variables,
however,the BDS test will detect the influenceof
omitted informationif it is correlated with the
included information. This poses a potential
problem for the applied researcher, in that a
significantvalue of BDS (showing a lack of iid)
may mean more than one problemexists. On the
other hand, a BDS value insignificantlydifferent
from zero shows that excluded informationdoes
not have any significanteffect on the estimated
211
parametersof interest. Thus our modified BDS
test has the advantage of providinga stopping
rule in the search for an appropriatespecification.
III.
Empirical Application of the Modified
BDS Test
We applythe modifiedBDS test to the residuals from the housing price equation used in
Kohlhase (1991). Hedonic estimation to find
marginalprices of varioushousingattributesprovides a particularlyuseful application because
there is no theoreticallypreferredfunctionalform
and the correct functional form is likely to be
rathercomplicated.For manypurposes,however,
what is of interest is the marginal price of a
specific housing attribute.We focus here on the
marginal price of square feet of living area
(SQFT) for illustrativepurposes.
The data set consists of 1366 cross sectional
observations on housing prices in Houston in
1985. There are three types of variablesdifferentiated by geographiclevel.5 The first type differs
by observationand includes housing characteristics such as SQFT. The second and third types
vary by geographic area, Keymap number and
census tract level.6 The second and third type of
variablesmay have systematipmeasurementerror
due to geographicaggregationbias, so we concentrate on discussing the specification of a
house-specificvariable SQFT, and report other
specificationsfor comparison.Variablesthat are
truly constant over geographic area, however,
should not pose a problemfor the modifiedBDS
test.
Our goal is to find an estimate of the marginal
price of SQFT where we are confident that we
have capturedany nonlinearitiesin the specification of this variable. We proceed in two steps.
5 Variables included in the regression are of 4 types:
(1) house-specific-square feet, lotsize, parking,bedrooms,
fireplaces, baths, condition, age, central air, range, dishwasher,(2) Keymapnumberspecific-distance to toxicwaste
site, (3) census tract specific-distance to employment,%
owneroccupied,% with high school education,income,poor,
blue collar, black, hispanic,% under 19 years, and (4) time
period-dummies for quarter.A data appendixis available
from the authors.
6 There are 286 keymapletter areas and 155 census tract
areas in the data set. A Keymapletter area is about 0.56
squaremiles and the averagecensustractis about3.44 square
miles.
212
THE REVIEWOF ECONOMICSAND STATISTICS
First we examine three primarysimple specifications, linear, log-log, and log-linear to find a
preferredbasic functionalform. Because the preferred log-linear specificationdoes not pass the
modifiedBDS test, we examine alternativeinteraction and higher order terms. This allows us to
find a relativelyparsimoniousspecificationwhere
the marginal price of SQFT and its standard
error are straightforwardto calculate,but which
still meets the requirementsof statisticalreliability as identifiedby an iid error structure.
TABLE 1.-MODIFIED
FOR ALTERNATIVE
Basic Modelb
Linear
Log-Log
Log-Linear
a
BDS TEST RESULTS
FUNCrIONAL FORMS
ca
(Standard
Error)
1.57
(0.15)
0.43
(0.09)
0.29
(0.07)
BDS
Statisticc
10.61
4.84
4.06
Calculated from the regression residuals sorted by SQFT.
See text.
Significance test from zero.
A.
Evidence for Systematic Nonlinearities
Table 1 presentsthe C statistics,their standard
errors, and the resulting BDS statistics for the
three alternativehedonic housing price equation
specificationswith the data ordered by SQFT.7
There are two basic results presented in table 1.
First, there is considerableevidence of nonlinear
structure,as the BDS statistics are all over 4V8
Second, table 1 shows that the log-linear functional form is clearlypreferredto the linear, and
is weakly preferred to the log-log. This can be
seen because the log-linearspecificationhas a C
statistic significantlybelow that of the linear
model, and a C statistic lower than that of the
log-log model.9We will thereforeuse the log-linear specificationin the remainderof the paper.10
We find the BDS test to have considerably
more powerfor detectingcomplexpatternsin the
error structurethan do commonlyused tests for
functionalform. We performtwo standardfunctional form tests: MacKinnon,White and David-
son's (1983) PE test and Kmenta's (1986)
Durbin-Watson(DW) statistic test. Neither provides any clear guidance about which functional
form of the regressionis preferredand the inconclusive results in the DW test do not even warn
the researcherof the nonlinearityproblem.
We also estimate an autoregressivemovingaverage (ARMA) model on the basic residualsordered by SQFT for the log-linearregression,and
find that, based on the Ljung-BoxQ Statistic,the
appropriate specification for our data is an
ARMA (1, 1). We then apply the modified BDS
test to the residuals from the ARMA model.
Purgingthe linear structurefrom the residuals,
however, appears to have little effect. The C
statistic falls insignificantlyfrom 0.29 (0.07) for
the basic model residuals to 0.27 (0.07) on the
ARMA residuals. Further, the resulting BDS
statistic of 3.82 indicates that significantnonlinear structureremains.
7 We use an embeddingdimension(m) of 5 and a distance
(d) of the standarddeviationof the data dividedby the range. B. Respecification of the Basic Model
The resultsare robustfor values of m between 3 and 10 and
While the preferencefor the log-linearspecififor valuesof d between 0.5 and 1.5 times that chosen.
8 We use standardnormaltables to calculateprobabilities. cation is consistent with results elsewhere, the
Hsieh and LeBaron(1988) show that smallsampleproperties BDS statistics in table 1 indicate that further
of the BDS statisticonly become importantfor sample sizes
model specificationis needed. There is statistical
less than 500.
9 The test of significanceis approximateonly and is based evidence that the regression residuals vary syson comparing the statistic z = (Cl
-
C2)/(Var(C1 - C2))5
to a standardnormaldistribution,where Cl is the C-statistic
in model 1. This calculationis approximatebecause the covariancebetween Cl and C2 has not yet been theoretically
derivedand is assumedhere to be zero. z equals7.73 comparing the linear model to the log-linear,and z equals 1.21
comparingthe log-log to the log-linearmodel.
10Both the linear and the log-linearspecificationinclude
SQFT squared as in the basic Kohlhase model. The BDS
statisticwithoutthe squaredtermis significantlyhigherunder
both of these specifications,but the relative results are unchanged.
tematically, but nonlinearly, with the size of
SQFT. When the regression residuals are ordered randomly,the BDS statistic falls to 0.003
with a standard error of 0.06. It is therefore
because of orderingby SQFT that we uncover a
pattern in the residualsthroughthe BDS test.
Possible causes of a significantBDS statistic
include omitted unobservedvariables,higher order terms, or interactionterms. Clearly,it is not
CHAOS THEORY AND MICROECONOMICS
213
TABLE2.-MODIFIED BDS TESTRESULTS
possible to include unobserved variables. We
WITHVARIOUSINTERACTION
TERMS
therefore first search higher order terms of the
ca
included variables.While squared terms have a
(Standard
BDS
theoreticalbasis, to capturevariationin marginal
Error)
statisticb
effects for example, further higher order terms
are not very appealing theoretically. Including Interaction Variable
.18
2.66
terms up to SQFT to the sixth power, however, SQFT
(.07)
reduce the BDS to 3.12 which is still significantly Lotsize
.24
4.16
(.06)
different from zero. We conclude that simple
Bedrooms
.06
0.92
nonlinearities are not the explanation for the
(.07)
observedpatternof the regressionresiduals.
Fire
.23
3.30
(.07)
To refine the model further,we desire a parsiBath
.15
2.18
monious set of interaction terms that will still
(.07)
allow a relatively straightforwardcalculation of
Cond
.29
4.08
(.07)
the marginalprice of SQFT. There is no clear
Age
.30
4.11
guidance as to which interaction terms, or how
(.07)
manyinteractionvariables,might be appropriate. Air
.24
3.60
(.07)
Table 2 presents BDS results when each of the
.30
Range
4.54
house-specific characteristics, and two region(.07)
specific variables,is used as an interactionvariToxic
.21
2.77
(.08)
able with all of the other variablesin the regresCBD
.41
5.16
sion model. In all cases, we ordered the data by
(.08)
SQFT to calculatethe BDS statistic.We find that Other Specifications
.44
5.97
an interaction with the number of bedrooms No Sqft Squared
(.07)
(BED) passes the BDS test.1"The BDS statistic With Higher Order
.23
3.12
of 0.92 shows that the hypothesisof iid residuals Sqft Terms
(.07)
cannot be rejected at the 5% level. This specifi- a Calculated from the regression residuals sorted by SQFT.
cation has the furtherattractionthat the marginal bSignificance test from zero.
price of SQFT calculationonly involves SQFT,
the squared term, and the interaction of these
TABLE 3.-MARGINAL
PRICE OF A SQUARE FOOT
OF LIVING AREAa
two variableswith BED.
Choosing a specificationwith BED as the in- Basic Model
$52.49
teractionvariableis attractiveon a priorigrounds.
(2.34)
The interactionallowsthe marginalpriceof SQFT Bedroom Interaction
-Two Bedrooms
33.42
to vary by the number of bedrooms. Table 3
(4.06)
presents the marginal price of SQFT and its
-Three Bedrooms
50.93
(2.32)
standarderror under the two alternativespecifi-Four Bedrooms
68.44
cations. In the basic model, a marginalprice of a
(3.64)
square foot of living area is estimated to cost
-Five Bedrooms
85.95
(6.33)
$52.49 irrespectiveof other house characteristics.
errors
are
in
standard
aAsymptotic
parentheses.
the
Under
interactive and correctly specified
model, however, the marginalprice varies from
$33.42for a two bedroomhouse to $85.95for five
bedrooms.The modified BDS test allows a cer- has captured any systematic nonlinearitieswith
tain confidencethat the interactionspecification respect to squarefeet. In addition,the advantage
of the parsimoniousspecification is shown be11The BED interactionspecificationis not the only possible cause the marginal prices of SQFT calculated
specificationthat passes the modifiedBDS test, for example, under the interaction model are still estimated
additionalinteractionvariablesalso pass. Brockand Dechert relativelyprecisely.
(1988) conclude their paper by speculatingon whether the
Because of the success of the BED interaction
BDS test is a uniquetest for specificationerror.Ourempirical
resultsindicatesthat it is not.
model, we examine other housing characteristics
214
THE REVIEW OF ECONOMICS AND STATISTICS
TABLE4.-MODIFIED BDS TESTRESULTS
WITHINTERACTION
TERMS
Basic Model
Bedroom Interaction
ca
Sorted Variable
SQFT
Lotsize
Bedrooms
Fire
Bath
Cond
Age
Air
Range
Toxic
CBD
Ca
(Standard
Error)
BDS
statisticb
(Standard
Error)
BDS
statisticb
.29
(.07)
.32
(.07)
.51
(.07)
.34
(.07)
.28
(.07)
.29
(.07)
.53
(.07)
.37
(.07)
.44
(.07)
.71
(.07)
.82
(.07)
4.06
.06
(.07)
.28
(.07)
.25
(.07)
.14
(.07)
.11
(.07)
.11
(.07)
.32
(.07)
.11
(.07)
.20
(.07)
.55
(.07)
.49
(.07)
0.92
4.44
7.09
4.69
3.98
4.08
7.44
5.18
6.15
9.87
11.42
4.25
3.78
2.12
1.64
1.65
4.76
1.66
3.01
8.26
7.40
a Calculatedfrom the regressionresidualssorted by indicated variable.
bSignificance test from zero.
and their performanceon the BDS test. Table 4
presents the results. The first two columns show
BDS test results for the basic model when the
residuals are ordered by each of the housing
specific characteristics,plus the Keymapvariable
TOXICas well as the CensustractvariableCBD.12
None of the variables that we use to sort the
residualsshow an insignificantBDS statistic.The
second two columns show the residuals of the
BED interactionmodel when orderedby each of
the variables.In addition to SQFT, three other
variablesnow show BDS statistics insignificantly
differentfrom zero at the 5% level: the number
of baths, condition of the house, and the presence of air conditioning.In fact, with the exception of lotsize, all of the variablesin the model
show a significantimprovementin the C statistics. Thus the numberof bedroomsappearsto be
an importantsourceof nonlinearitieswith respect
to many of the housingcharacteristics.
One of the most valuableaspects of the modified BDS test is that unlike many other tests, the
BDS test provides a stopping rule in the search
for an appropriatespecification.When the residuals are found to be iid, informationexcluded
from the regression is not correlated with the
ordering variable. For example, the BDS test
shows that the residuals from the basic model
augmentedby bedroominteractionsare iid when
ordered by SQFT. In contrast,a standardF-test
about the inclusion of more variables, such as
CBD or TOXICinteractions,would lead the researcherto add unnecessarycomplexity.
We have shown that the modified BDS test is
an importantdiagnostictool when examiningthe
specificationof a regressionequationwith respect
to a single variable.There are, however, important limitationsto using the modified BDS test.
As discussedearlier,when variablemeasurement
exhibits geographic aggregation bias, the BDS
test may show significantnonlinearities for all
12
We omitted from table 4 two house-specificcharacteris- specifications.Further,we found several specifitics, dishwashersand numberof parkingspaces, because the
coefficientestimates in the basic model were quite low and cations where the BDS statistic is larger using
three interactivevariablesthan one.13 This may
imprecise(t-statisticsless than 0.1).
CHAOS THEORY AND MICROECONOMICS
be because some interaction variables induce
nonlinearitiesinto the data rather than control
for them. Even more important,perhaps, is that
in additionto incorrectfunctionalform, another
source of significantBDS statistics can be omitted variables.The finding that the BDS statistic
when the regression residuals are ordered by
lotsize do not fall with the BED interactionis not
very reassuring.The BDS statisticusing the basic
log-linearmodel is 4.44. With the BED interaction, it fell slightlyto 4.25. With three interaction
variables, it is still 4.46. In fact, we failed to
discovera specificationfor lotsize that providesa
significantimprovementin, much less passes, the
modified BDS test. As with all diagnostictests,
the test is able to indicate where more work is
needed without being very specific about how to
correctthe problem.
IV. Summaryand Conclusions
This paper has shown that an application of
chaos theory providesan excellent tool for investigating microeconomicmodel specification.The
modified BDS test proposed here provides an
importantdiagnostictest for the presence of nonlinear structurein microeconomicdata, and provides an important indication about whether a
particular variable in a regression is correctly
specified. The modified BDS test examines
whether the residuals of a regression equation
exhibit a systematicpatternwhen ordered by the
right hand side variable of interest. The test
therefore determineswhether excluded information from the specified regression is correlated
with the variable of interest; a determinationof
no correlationindicatesthat the resultingparameter estimates are unbiased. The modified BDS
test applied to micro data is an importantspecification test with respect to a single variablebecause possible sources of excluded information
consist of alternativefunctionalforms employing
includedvariables.
Specificationtests for a single variable are especiallyimportantin applicationswhere a precise
coefficient on a specific variable is desired. We
have applied the modifiedBDS test to a hedonic
housing price regression,and are able to find a
13
For example,interactionswith BED, TOXIC,and CBD
produce a BDS statistic of 0.93, despite the fact that BED
only as an interactionvariableproducesa BDS of 0.92.
215
relatively parsimoniousspecification where the
residualsare iid when ordered by square feet of
living area. Thus we offer the first attempt to
mediate the trade-offbetween a complexspecification of the overallequationand ease of manipulation for a particularregressioncoefficient.
The test for nonlinearstructure,when used on
macroeconomic data, can give evidence about
whether the residualsare iid, but does not indicate how the model can be transformedto purge
the systematiccomponentfrom the residuals.In
the microeconomiccontext, the modified BDS
test is an excellentsignpostabout potentialerrors
in modellingbecause orderingby the variableof
interestprovidesa methodfor uncoveringsystematic nonlinearitiesin the residuals.
REFERENCES
Baumol,WilliamJ. and Jess Benhabib,"Chaos:Significance,
Mechanism,and EconomicApplications,"TheJournal
of EconomicPerspectives3 (Winter1989),77-106.
Box, G. E. P. and D. R. Cox, "An Analysisof Transformations,"Journalof the RoyalStatisticalSocietySeriesB,
26 (1964),211-252.
Brock,WilliamA., "Introductionto Chaosand OtherAspects
of Nonlinearity,"in W. A. Brockand A. G. Malliaris,
DifferentialEquations,Stabilityand Chaosin Dynamic
Economics(New York:North-Holland,1988).
Brock,WilliamA., and W. Davis Dechert, "A GeneralClass
of SpecificationTests: The Scalar Case," Proceedings
of the Businessand EconomicStatisticsSection,American StatisticalAssociation(1988),70-79.
Brock, William A., W. Davis Dechert, and Jose A.
Scheinkman,"A Test for IndependenceBased on the
CorrelationDimension,"SSRI WorkingPaper #8702,
1987.
Universityof Wisconsin-Madison,
Brock, WilliamA., and Chera L. Sayers, "Is the Business
CycleCharacterizedby DeterministicChaos?"Journal
of MonetaryEconomics22 (1988),71-90.
Cassel,Eric, and RobertMendelsohn,"The Choice of FunctionalFormsfor HedonicPriceEquations:Comment,"
Journalof UrbanEconomics18 (Sept. 1985),135-42.
Dechert, W. Davis, "A Programto CalculateBDS Statistics
for the IBM PC,"Universityof Houston,1987.
Grandmont, Jean-Michel, "On Endogenous Competitive
Business Cycles," Econometrica 53 (Sept. 1985),
995-1045.
Hsieh, David A. and Blake LeBaron,"FiniteSampleProperties of the BDS StatisticI: DistributionUnder the Null
Hypothesis,"Universityof Chicagomimeo, May 1988.
Kmenta,Jan, Econometrics(New York:Macmillan,1986).
Kohlhase,Janet E., "The Impact of Toxic Waste Sites on
HousingValues,"Journalof UrbanEconomics30 July
1991),1-26.
MacKinnon,J. G., H. White, and R. Davidson,"Tests for
Model Specificationin the Presence of Alternative
Hypotheses:Some FurtherResults,"Journalof Econometrics21 (Jan. 1983),53-70.
Murphy,KevinM. and FinisWelch,"EmpiricalAge-Earnings
Profiles,"Journalof LaborEconomics8 (Apr. 1990),
202-229.