The Incumbency Advantage in Multimember Districts: Evidence from the U. S. States
Author(s): Gary W. Cox and Scott Morgenstern
Source: Legislative Studies Quarterly, Vol. 20, No. 3 (Aug., 1995), pp. 329-349
Published by: Comparative Legislative Research Center
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GARY W. COX
SCOTT MORGENSTERN
Universityof California-SanDiego
The
Incumbency
Advantage
in Multimember
Districts:
from
Evidence
the
U.S.
States
Studies of the incumbencyadvantagein U.S. state legislative elections have
usually been limitedto the relativelyfew states that use single-memberdistrictsexclusively. In this paperwe providea methodfor computingthe incumbencyadvantagein
multimemberdistricts,based on the Gelman-King(1990) estimatorfor single-member
districts, and use it to estimate the incumbencyadvantagein 40 U.S. states over the
period 1970-86. We find that the incumbencyadvantagehas increasedin states using
multimemberdistricts but at a substantiallylower average rate than in states with
single-memberdistricts.We also find that states in which legislatorshave more opportunity or ability to performcasework services for their constituentsare also those in
which the incumbencyadvantageis larger.
Studiesof the so-called incumbencyadvantage-the additional
electoralsupporta candidategains due to his or her incumbentstatushave recently expandedtheir purviewto include a wide range of state
elections (Breaux 1990; Cox and Morgenstern 1993; Garand 1991;
Holbrook and Tidmarch 1991; Jewell and Breaux 1988, 1991; King
1991a; Niemi, Jackman,and Winsky 1991; Weber,Tucker,and Brace
1991). This expansion in the literatureis importantnot only for our
understandingof state governmentbut also as an opportunityto test
extant theories of the incumbencyadvantageoriginatingin investigations of federalelections.
Thus far, however,in partbecause researchershave sought to
maintain comparabilitywith studies of federal elections and in part
because measurementtechniques have not been developed for true
multimemberdistricts, studies of the incumbency advantagein state
legislative elections have mostly been limited to states with singlememberdistricts or multimemberdistricts with posts (which operate
essentially as if they were single-member districts).' This article
removes this restriction in the literatureby providing a method for
computingthe incumbencyadvantagein multimemberdistrictsthat is
LEGISLATIVESTUDIESQUARTERLY,XX, 3, August 1995
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329
330
Gary W. Cox and Scott Morgenster
directlycomparableto methodsused in single-memberdistricts.In particular,it extendsthe methodin Gelmanand King (1990) so thatit can
be employedin multimemberdistricts.
Ourplan of attackis as follows. First,we brieflyreviewthe literatureon the state-levelincumbencyadvantage.Second, we show how
to adapt Gelman and King's model for use in multimemberdistricts.
Third,we apply the generalizedGelman-Kingmethod to 40 of the 50
states in the period 1968-86. We find that the incumbencyadvantage
increasedin multimemberstates, althoughneitheras much nor as soon
as in single-memberstates. Fourth,we replicateand extend our earlier
analysis of the determinantsof the incumbencyadvantage,continuing
to find supportfor the generalidea that states in which legislatorscan
performmore caseworkservices for theirconstituentsare also states in
which the incumbencyadvantageis larger.Fifth, we discuss some of
the technical difficulties in handling uncontested races. Finally, we
conclude with some commentson our findings and the possibilitiesfor
futureresearch.
Measuring the Incumbency Advantage
The incumbencyadvantagecan be denominatedin eithervotes
or seats. In terms of votes, it can be defined as the differencebetween
the percentageof the vote an incumbentactuallygarneredand the percentage the same candidate would have garneredhad he or she not
been an incumbent.In termsof seats, it can be definedas the difference
between an incumbent's probabilityof victory and the hypothetical
probabilitythat the same candidatewould have won had he or she not
been an incumbent.It is quite possible, as Jacobson(1987) has pointed
out, for the vote advantageof incumbentsto be growing without any
correspondingchange in the seat advantage.In what follows, we shall
focus on the vote advantage and various methods of measuring it,
returningin the conclusionto the issue of the seat-denominatedincumbency advantage.
The vote-denominatedincumbencyadvantagecan be, and has
further
refinedinto a scare-offadvantageand what King (1991b)
been,
has called the "contested-electionincumbencyadvantage."The scareoff advantageis the decreasein the probabilitythatthe election will be
contested when the incumbentcandidateruns (operationally,a race is
contestedwhen the runner-upgarnersat least some minimumshareof
the two-party vote-usually
5% or 10%). The contested-election
advantageis the increased vote share that an incumbentparty enjoys
when the incumbentcandidateruns for reelection in a contested race.
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Incumbency Advantage
331
In this paper,we shall focus on the contested-electionadvantagerather
than the scare-off effect (on which, see Cox and Morgenster 1993;
Jewell andBreaux 1991;Weber,Tucker,and Brace 1991).
Regardless of how one measures it, the (vote-denominated,
contested-election)incumbencyadvantageappearsto have increasedin
the 1970s and early 1980s, at least in states employing single-member
districts or their equivalents. Jewell and Breaux (1988) and Garand
(1991) find that incumbents'averagemarginof victory increasedsignificantly. Holbrook and Tidmarch(1991) find that the sophomore
surge more than doubled. Cox and Morgenster (1993) find that the
incumbencyadvantage,as measuredby Gelman and King's unbiased
regressiontechnique,almostdoubled.
Explanations of the increase in the state-level incumbency
advantagehave focused on the same factors invoked to explain the
increasing advantagein federal elections. In particular,investigators
have looked at the ability of legislators to provide services to constituents (Fiorina 1977). Their findings have been largely consistent.
Holbrook and Tidmarch,in a study of 32 states, find that the sophomore surgeincreaseswith allowancesfor staff and for tripsback to the
constituency. King (1991a), in a study of 13 states, finds that the
incumbencyadvantageis statisticallyrelatedto the size of legislative
operatingbudgets, measuredon a per legislator basis. Cox and Morgenstern (1993), in a study of 24 states, find that the incumbency
advantageincreaseswith the size of legislativeoperatingbudgets,measured on a per constituent basis (staff allocations and per legislator
budgetsboth havingpositivebut insignificantimpacts).
Evidence on the contested election incumbencyadvantagein
states employing true multimemberdistricts is currentlyunavailable.
There is some evidence pertinentto the scare-off advantagein multimemberdistricts (Jewell and Breaux 1991; Weber,Tuckerand Brace
1991), but none pertinentto the contested-election advantage.In the
next section, we show how this advantagecan be measuredin a way
thatwill yield figurescomparableto those alreadyavailablefor singlememberdistricts.
The Gelman-King Model
Gelmanand King (1990) define the incumbencyadvantageas
the difference between w(l), the "proportionof the vote received by
the incumbentlegislatorin his or her district,if he or she runs against
majorpartyopposition,"and w(0), the "proportionof the vote received
by the incumbentpartyin thatdistrict,if the incumbentlegislatordoes
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Gary W. Cox and Scott Morgenstem
332
not run and all majorpartiescompete for this open seat"(1143). Their
definitionthus correspondsto thatof the vote-denominated,contestedelection advantagegiven above.
Gelmanand King's regression-basedmethodof estimatingthe
incumbencyadvantagehas two main advantagesover its predecessors.
First,it uses all contestedelections, as opposedto the retirementslump
or sophomore surge measures, which respectively look only at races
following the retirementor first legislative term of an incumbent.Second, Gelman and King prove that their estimatorof the incumbency
advantageis unbiased,whereaspreviousestimatorsare not.
Our goal is to generalize Gelman and King's procedure
realm of single-memberdistricts. To this end, we reinterthe
beyond
some
of
Gelman and King's variablesto clarify their application
pret
in multimemberdistricts. As we show in the endnotes, our reinterpreted variablesare precisely equivalentto Gelman and King's original variables, when the analysis deals with the special case of
single-memberdistricts.
Gelmanand King specify theirmodel as follows:
E(v2)= Po+ PIvl + 2Pl+ I2
where E(v2) = the expected two-partyvote percentagefor the Democraticpartyin the current(or second) time period;2
v,
= the two-party vote percentage for the Democratic party
in the previous(or first)election;
= the numberof Democraticvictors in the previous election less the numberof Republicanvictors, divided by
the districtmagnitude;3and
= the numberof Democraticincumbentcandidatesin the
12
currentelection less the numberof Republicanincumbent candidates.4
The coefficient of 12, t, gives the average increment to the
Democraticparty'svote sharedue to a unit increase in I2. Since a unit
increasein I2's value can occur with either the additionof a Democratic incumbentor the subtractionof a Republicanincumbent,41can be
interpretedas the value to a partyof having one more incumbentcandidate of its own (or one less incumbent candidate of the opposite
P,
party).
Previous applications of the Gelman and King model have
been limited to single-memberdistricts, in studies either of federal
elections (Gelmanand King) or of those state elections thatexclusively
use single-member (or equivalent) districts (Cox and Morgenster
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Incumbency Advantage
333
1993; King 1991a). Our reinterpretationof Gelman and King's variables makes clear that their model is applicableto multimemberdistrictsas well.
Thereis, however,an importantdifficultythat is more acute in
the multimemberthanin the single-membercontext.We can explainthis
difficultyby firstreviewinghow Gelmanand King select and handlethe
dataon single-memberdistricts,thennotingthata problemremainseven
afteranalogousproceduresareused in the multimembercase.
The primaryrestrictionon Gelmanand King's analysisof U.S.
House elections is thatthey exclude uncontestedraces.They also focus
attentionon the major partiesby using the Democraticpercentageof
the two-partyvote, ratherthan the Democraticpercentageof the total
vote, as the dependentvariable.We have also excludedcases involving
uncontestedraces and used the Democraticpercentageof the two-party
vote as the dependentvariable.
Gelman and King are left with a fairly homogeneous dataset
afterexcludinguncontestedraces, in the sense thatmost of theirobservationsare from districtsin which one Democratfaced one Republican
in both the current and previous election. There are some cases in
which third-partycandidatescome and go, but such candidatestypically get few votes and, in any event, the analysis uses two-partypercentagesto partiallycompensatefor theirappearance.
For multimemberdistricts,excluding uncontestedraces leaves
a much less homogeneous dataset. In double-memberdistricts, for
example, an election that pits one Democratagainsttwo Republicans,
whereas the previous election pitted two Democrats against two
Republicans,is still in the data-and one would certainly think that
decreasing the number of Democrats running would decrease the
Democratic percentage of the vote. To control for variations in the
numberof candidatesin multimemberstates, we have furtherconfined
attention in those states to contests in which M Democrats faced M
Republicansin both the currentand previouselection, where M is the
numberof seats in the district.
One might interpretthis restrictionas inherentin the notion of
a contested-electionincumbency advantage.After all, some standard
for deciding what constitutesa contestedelection must be established,
and the M-against-Mstandardseems a naturalone. Withthis definition
of a contested election-which coincides with that used by Gelman
and King in the special case of single-memberdistricts-we are simply
excludingall uncontestedraces.
Nonetheless, excluding uncontested races may introduce
sample-selection bias, particularlywhen the category of uncontested
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334
Gary W. Cox and Scott Morgenster
races includes what might more properlybe called partiallycontested
races (e.g., (M-l)-against-M). Thus it may be prudentto seek some
way to include uncontestedraces in the analysis,especially if thereare
lots of them.We returnto this matterin the penultimatesection.
Another question that arises when multimemberdistricts are
admittedto the analysis concernshow to handlethe data from a stateyear in which districts vary in numberof seats. The 1970 election in
Montana,for example,involveddistrictsof varyingmagnitude.Should
a single incumbency advantagebe estimated for all districts in this
state-year?Arguing against such a procedureis the observationthat a
5% incumbencyadvantagemeans something very different in a oneseat as opposedto a nine-seatdistrict.A candidateis guaranteedto win
a seat in a nine-seatdistrictif he or she gets 10%of the vote, so a 5%
advantagewould be enormous. In contrast, the same 5% advantage
would be much less significantin a one-seat district,albeit still nice to
have. The difference in political impact of a given percentage-point
advantage suggests that the incumbency advantage will almost certainlybe smallerin largermagnitudedistricts.
Unfortunately,there are not enough observationsto allow one
to estimate a separateincumbencycoefficient for each districtmagnitude in each state. We have adopteda second best approach;in those
state-yearsin which districtsvary in magnitude,we have pooled them
all, with a single slope coefficient for the value of incumbency.This
approachprovides something like a weighted average of the slopes
thatone would get from separateregressionsfor each magnitudeclass,
where the weights are the number of districts contributing to the
regressionfrom each district-magnitudeclass. It indicatesthe average
experience of incumbentsfrom the given state-year,without controlling for the possibility that incumbentscome from districtsof varying
magnitude.
Given this estimationprocedureat the state-yearlevel and the
previous comments aboutthe varyingpolitical significance of a given
incumbency advantage, some caution in comparing incumbency
advantagesacross state-yearsis indicated.If a given state has virtually
all double-memberdistricts in 1974, but virtually all single-member
districtsin 1984, then one expects the averageincumbencyadvantage
to grow even if there is no growth within either single-memberdistrictsor double-memberdistricts.Similarly,if one state in a given year
uses single-memberdistrictsand has an incumbencyadvantageof 6%
while anotherstate in the same year uses double-memberdistrictsand
has an incumbencyadvantageof 3%, it is misleadingto conclude that
the incumbencyadvantagein the formerstate is twice as large as that
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Incumbency Advantage
335
in the latter state. To make the incumbency advantage in one stateyear comparableto that in another,we adopt a simple normalization,
expressing advantages as percentages of the average Droop quota
ratherthanas percentagesof the total votes cast. Thus, in the example
given above, the normalized incumbency advantage in the singlemember state would be 12% (6% divided by a Droop quota of
1/(M+1) = 1/2), while the normalized incumbency advantage in the
double-memberstate would be 9% (3% divided by 1/3). In the analysis that follows we shall use the normalized incumbency advantage
throughout.
Results
With these mattersof data selection and manipulationunderstood, we turn now to estimate the (normalized, vote-denominated,
contested-election)incumbencyadvantagein 40 of the 50 U.S. statesin
the period 1970-86.5All 40 statesfall into one of two categories:those
with and those without true multimemberdistricts. We have already
estimatedthe incumbencyadvantagefor most of the states in the latter
categoryand simply repeatthe relevantfigures(in normalizedform) in
Table 1. In addition,Table 1 presentsnew figures for 18 states,4 without and 14 with truemultimemberdistricts.6
Figure 1 graphs some summarystatistics from Table 1-the
first,second,andthirdquartilesof the distributionof normalizedincumbency advantages in states possessing multimember districts. The
medianadvantagein such statesheads steadilydownwardfrom 1972 to
1978, but thereaftermoves steadily upward,with a small relapse in
1986. The firstand thirdquartilesalso increaseafter 1978, showingthat
the entire distributionof normalizedincumbency advantagesshifted
upwardin multimemberstatesin the 1980s.
The visual impressionof an upwardtrend that one gets from
Figure 1 is corroboratedby a regressionof the estimatedincumbency
advantagein statej, electiont, on t (measuredin yearsafter 1969) and a
set of dummyvariablesidentifyingthe states. This fixed effects model
(see, e.g., Judgeet al. 1982) allows each state a separatey-interceptbut
pools all multimemberstates when estimatingthe coefficient on t. We
follow previouspracticein the literature(Cox and Morgenstern1993;
King 199la) and weight each observationof the incumbencyadvantage
by its standarderror (see Appendix B for the standarderrors used).
Thus, observationsaboutwhich we have more precise informationare
weightedmoreheavily.7
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336
Gary W. Cox and Scott Morgenster
TABLE1
IncumbencyAdvantagein Electionsto LowerHouses
of U.S. StateLegislatures
(unstandardized
regressioncoefficients)
State
Arizonaa
California
Colorado
Connecticut
Delaware
Florida
Georgiab
Idahoa
1970
1972
1974
1976
1978
1980
0.1
3.8
8.1
1.3
c
c
9.1
3.7
0.3
6.8
-2.1
4.9
-1.7
4.6
8.6
8.6
4.6
-1.4
8.4
0.7
-4.6
0.0
1.7
4.6
1.7
5.0
8.6
-8.6
8.9
5.0
9.3
9.9
11.2
2.1
d
d
d
d
1.1
5.1
c
1.0
4.4
2.9
10.8
11.1
8.3
10.6
2.3
1.6
8.7
c
Illinoisd
Indiana"
-0.1
Iowa
Kansas
0.8
4.3
Kentucky
Mainea
0.9
Massachusettsa 2.4
3.0
Michigan
Mississippib
Missouri
-0.6
Montanaa
1.0
d
Nevadaa
New Hampshire"1.7
New Jerseya
New Mexico
4.2
New York
3.8
NorthCarolinaa d
NorthDakotaa 0.7
Ohio
3.7
Oklahoma
3.4
7.6
Oregon
Pennsylvania -1.6
RhodeIsland 0.6
SouthCarolinab d
SouthDakota" 0.2
Tennessee
1.6
Texas
-3.6
Utah
1.3
d
1.1
4.5
c
0.1
7.8
e
2.1
c
c
5.2
2.6
c
c
1.0
1.1
4.0
1.0
1.7
d
7.1
2.5
0.8
c
1.5
4.5
4.1
-0.3
5.6
-1.9
8.0
2.4
0.6
9.4
" c
1.0
-0.6
3.2
e
-0.3
7.5
8.7
-2.0
c
c
5.1
-0.9
6.3
14.6
1982
1984 1986
c
c
c
d
d
c
8.0
-1.4
3.7
2.9
3.0
6.0
0.6
9.1
8.9
C
4.3
'
-0.5
d
d
c
11.4
c
8.9
13.0
-1.1
5.9
1.8
e
8.8
14.3
8.9
c
0.3
5.3
11.5
3.8
c
e
-7.8
-2.2
2.5
0.9
5.3
-0.6
0.6
0.0
1.8
9.8
0.0
10.5
3.8
9.7
1.3
6.8
3.5
5.6
-0.8
c
2.0
1.8
c
4.8
6.2
0.5
0.8
1.6
11.5
-1.7
6.5
7.5
7.0
12.2
0.2
4.1
10.7
1.5
5.7
5.9
-5.4
8.3
7.2
5.9
1.8
0.4
8.7
c
9.7
9.5
c
c
c
3.0
2.7
d
c
0.0
c
3.5
5.5
0.8
12.1
2.4
d
0.8
7.3
c 18.6
1.5
c
c
9.2
c
2.6
c
1.4
c
6.0
c
9.2
5.4
4.7
1.9
8.3
12.2
9.0
2.5
c
c
2.5
c
c
c
c
C
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4.4
4.4
44.5
93.9
d
0.7
7.4
2.0
10.1
4.5
5.8
6.3
8.6
8.4
-2.0
337
Incumbency Advantage
TABLE 1
(continued)
State
1970
1972
1974
1976
1978
1980
1982 1984 1986
c
c
10.9
c
c
d
d
d
5.1
"
5.5
4.2
5.3
6.5
c
4.2
6.0
WestVirginiaa 0.0
1.4
c
0.0
-0.3
-0.5
c
-0.1
0.9
Wisconsin
c
c
4.1
3.8
5.0
-0.1
8.0
-0.1
4.2
c
c
c
4.8
Virginiab
Washington
Wyominga
0.6
d
-0.4
d
-1.1
d
Note: See AppendixB for standarderrorsand numbersof observationsin this analysis.
See AppendixA for a descriptionof the dataset.
aStatehas truemultimemberdistricts.
bStatehas single-memberdistrictsonly but was excluded from Cox and Morgenstem
1993 for not meetingrequirementof 20 observationsin five years(see AppendixA).
'Redistrictingyear.
dlnsufficient data (fewer than 20 single-member districts or fewer than 10 singlememberand multimemberdistrictscombined).
eNotan election year.(Mississippiuses four-yearterms.)
On average, the normalized incumbency advantage in the multimember states increased by .34 per year or about a two-thirds of a percentage point per election cycle. This average rate of increase over
the 1970-86 period, although it clearly hides some differences in the rate
of increase at different points in time, is nonetheless statistically discernible from zero at conventional levels of significance.
The picture for single-member states (Figure 2) shows almost
monotonic growth over the entire time period (1970-86). On average,
the normalized incumbency advantage in such states increased by .56
per year or a bit over one percentage point per election cycle. This
average increase is about 60% greater than that in multimember states.
Comparison with Figure 1 also reveals that incumbency advantages in
multimember states began to increase almost a decade after those in
single-member states did, but then increased at a more rapid rate,
almost catching up by the mid-1980s (the median advantages in singlemember and multimember states are almost identical in 1984, with the
third quartile still lagging).
Explaining the Increasing Incumbency Advantage
What explains the increasing value of incumbency in states
with multimember districts? What explains the differences between
single-member and multimember states? In our previous work, we
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338
Gary W. Cox and Scott Morgenster
FIGURE1
The Valueof Incumbencyin 14 States
EmployingMultimemberDistricts
Valueof Incumbency,in percentagepoints
20
10 1
0
0 Thirdquartileof distributionof incumbencyadvantage.
a Medianof distributionof incumbencyadvantage.
O Firstquartileof distributionof incumbencyadvantage.
-10
_a
1970
1975
1980
1985
1990
Year
found that much of the upward trend in incumbency advantages in
single-member states could be explained by the size of legislative
operating budgets per constituent, a crude measure of the resources
of which each legislator disposed in processing demands from constituents.Does the same hold true for multimemberstates?
In orderto answerthis question,we regressedthe incumbency
advantagein state j, election t on (1) t, measuredin years after 1969;
(2) BUDPER, the real legislative operatingbudget per constituent in
the bienniumbefore election t; (3) BURSIZE,the numberof state and
local governmentemployees per 10,000 inhabitantsin the year of the
election; and (4) a set of state dummyvariables.8The operatingbudget
per constituentcrudelyindicatesthe resourceswith which legislatorsin
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339
Incumbency Advantage
FIGURE2
The Valueof Incumbencyin 26 States
EmployingOnly Single-MemberDistricts
Valueof Incumbency,in percentagepoints
20 -
15
10 _
X
5
/
0
A
1970
a
I
1975
?
e of distribution of incumbency advantage.
t
Medianof distribution of incumbency advantage.
O
First quartile
I
1980
Year
of distribution of incumbency advantage.
1985
1990
each state can respond to constituentdemands, while the numberof
governmentemployees suggests the level of demand-as constituents
do battlewith a largerstate and local bureaucracy,they seek more help
from theirelected representatives.
In Table 2, Model 1, both time and size of bureaucracyhave
positivebut statisticallyinsignificantcoefficients,while operatingbudget per constituenthas a negativeand insignificantcoefficient. Despite
the fact that all regressorshave insignificantcoefficients, however,the
R2of the regressionis reasonablylarge, suggesting that there may be
multicollinearityamong the independentvariables.And, indeed, if one
excludes any two of the threevariablesof interestfrom the regression,
the remaining variable's coefficient is both positive and statistically
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340
Gary W. Cox and Scott Morgenstern
TABLE2
in the Advantageof
Variations
Explaining
in
40
U.S. States
Incumbency
Model 1:
MultimemberStates
IndependentVariable
Coefficient
Estimate
t-value
Model 2:
Single-MemberStates
Coefficient
Estimate
t-value
-8.55
.44
.56
.06
.25
1.09
.21
1.31
Size of LegislativeOperatingBudget
-1.02
per DistrictInhabitant
.12
18.06
2.40
Size of Local and StateBureaucracy
by DistrictPopulation
.50
.02
1.00
Intercept
Time Trend
Numberof Observations
AdjustedR2
.02
76
159
.25
.35
Note: Model 1 includes only states for which our datasetcontains some observations
from true multimemberdistricts. Model 2 includes only states for which our dataset
contains observationsexclusively from single-memberdistricts or multimemberpost
districts.
significant(at the .01 level for time and size of bureaucracy,at the .10
level for operatingbudget per constituent).Although there is insufficient informationin the currentsample to disentangle the respective
effects of time, operatingbudget per constituent, and the size of the
state and local bureaucracy,our results lend some supportto the idea
that an increased ability to do constituency service or an increased
demand for constituency services lies behind the increasing incumbency advantagein multimemberstates.
The resultsfor 26 single-memberstates (the 24 studiedin Cox
and Morgenstern 1993, minus Indiana and Montana, plus Georgia,
Mississippi, SouthCarolina,andVirginia)9are qualitativelysimilarbut
allow a clear decision as to which of the nondummyregressorsis the
most important.Time alone or operatingbudgetper constituentalone
or size of bureaucracyalone (when coupled with the statedummyvariables) has a positive coefficient statistically discernible from zero.
When all three regressors are included in the same model, however
(Table 2, Model 2), a clear victor emerges, as in our previous work.
Neithertime nor the size of the state and local bureaucracyhas a statistically significantimpact,but operatingbudgetper constituentdoes.10
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Incumbency Advantage
341
The patternof resultsin the single-memberobservationsis sufficiently differentfrom that in the multimemberobservationsthat the
two sets of datado not pool. In particular,the coefficienton the operating budgetper constituentvariableis significantlylargerin the singlememberthanin the multimemberstates.
As regardssingle-memberstates, we conclude (as before) that
the primaryengine pushing up incumbency advantagesin the 1970s
and 1980s was an increase in the state-providedwherewithalto perform casework and other services for constituents. In multimember
states,the storyis less clearbut seems also to be relatedto the abilityto
supplyconstituencyservice or the volume of demandfor such service.
Uncontested Races and the Incumbency Advantage
We returnnow to the issue of how uncontestedraces ought to
be handledin estimatingthe contested-electionincumbencyadvantage.
Recall thatthe definitionof a contestedrace underlyingthe analysis in
the thirdand fourthsectionsrequiresthattherebe M Republicancandidates and M Democraticcandidates,whereM is the numberof seats to
be filled in the district. There are two fairly evident problems with
excludinguncontestedraces so defined.
The first problem, which occurs whether one is analyzing
single-memberor multimemberdistricts,has to do with the possibility
of rationalentrydecisionsby potentialchallengers.Districtsare uncontested because potentialchallengersdecided not to run.Assuming that
potentialchallengersbase theirdecisions largely on theirjudgmentsof
how much chance they have of winning a seat and assumingthat their
judgmentsare well informed,one underestimatesthe contested-election
incumbencyadvantageby looking only at contestedelections. For,had
contests occurred in the uncontesteddistricts, the incumbents there
might have done betteron averagethandid the incumbentsin actually
contesteddistricts.
A second (andrelated)problemis specific to multimemberdistricts.As defined, uncontestedraces include (M - 1)-against-Mraces,
(M - 2)-against-(M- 1) races (for M > 2), and so forth.Yet these races
may be competitivein thattheremay be more candidatesthan seats to
be had in the district.Forexample,if one Republicanfaces two Democratic incumbentsin a two-seat district,not all can win. Such partially
contested races are excluded at present, yet they may occur precisely
when the incumbencyadvantageis large (again on the assumptionthat
potentialchallengersshy away frombad odds).
Should partially contested or even uncontested races be
included in the analysis and, if so, how? One approachmight be to
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342
Gary W. Cox and Scott Morgenstem
include them all and imputevalues to missing variablesin some fashion (King 1991b).An opposite approach-taken in the earliersections
of this article-excludes them all. Which approachis better depends
on two considerations:first, how well one can estimatethe counterfactual quantitiesnecessaryto implementthe first approach(in particular,
the vote an unopposed-or partiallyopposed-incumbent would have
gotten had he or she faced a contest); second, how much the actual
entry decisions depended on (sound) judgments of the relevantelectoral probabilities.If potentialcandidatesin state legislative elections
have very poor ability to forecast their chances, so that their entry
decisions vary mostly with opportunitycost and other considerations,
then there will not be much bias in the second approach;the risks
involved in imputing data entailed in the first approachwill not be
worththe potentialgain. If, on the otherhand,potentialcandidatesare
pretty good at figuring their chances and make their entry decisions
mostly on this basis, then the risks of imputingdatamay well be worth
the potentialgain.
We believe-unfortunatelyon almostpurelya priorigroundsthatthe abilityof state legislativecandidatesto forecasttheirchancesis
limited. Accordingly,we do not think that much informationis containedin the fact that one incumbentfaces a full contest while another
does not. We concede that our estimates are plausibly downwardly
biased, but we do not think the probablesize of the bias justifies the
risks of imputingdata.
A weak test of our belief can be conductedby taking a third
approachto estimatingthe contested-electionincumbencyadvantage,
one in which partially contested (but not uncontested) races are
allowed into the analysis. This approachof course changes nothing in
single-member states, since there are no partially contested races
in such states. But the numberof observationsavailablefor analysis in
multimemberstates increasesby about 18%.In estimatingthe incumbency advantagein this largerdatasetwe have included an additional
variableto controlfor net changes in the numberof majorpartycandidates. Letting NDt and NRt stand for the numberof Democratic and
Republicancandidates,respectively,runningin election t, define NETINCR = (ND2 - NR2) - (NDi - NRI). NETINCR measures the net
increase in the preponderanceof Democratic over Republicancandidates. The logic behind including this variable is simple: the
Democratic party's percentage of the vote in the focal election (v2)
ought to be greaterthanwhat one would predicton the basis of its vote
percentagein the previous election (vl) if the numberof Democratic
candidatesincreasesrelativeto the numberof Republicancandidates.
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Incumbency Advantage
343
It mightbe objectedthatchangesin the preponderance
of Democraticover Republicancandidatesreflect a scare-offeffect that should
be creditedto the incumbencyvariable(by excluding the net increase
variable).Ourresponseto this criticismcan best be explainedin the context of an example. Suppose that two Democratswin in an Arizonan
double-memberdistrict(districtA, say) at election 1, beatingtwo Republicans. Both then run as incumbentsat election 2, but face only one
Republicanchallenger.Presumably,these incumbentswill gamerhigher
vote percentagesagainst their lone challenger than they would have
against two. How should this incrementin their vote percentagesbe
countedwhen measuringthe incumbencyadvantage?Partof the advantage of incumbencyis that incumbentsare more likely to face partial
this is the scare-offeffect andshould
challengesthanarenonincumbents;
not be counted when calculating the contested-election advantage.
Anotherpart of the advantageof incumbencyis that incumbentsare
more likely to fare betterwhen they do face a diminishedfield of outpartychallengersthan are their nonincumbentcolleagues. To measure
this advantage,one shouldcomparethe race in districtA sketchedabove
to anotherin which two Democratsbeat two Republicansat election 1
and only one or zero of them decide to seek reelection while only one
Republicanchallengeremerges at election 2. How much betterare the
two incumbentsof districtA at exploitingthe absenceof a full Republican challengethanarethe one incumbentandone nonincumbent(or two
nonincumbents)of the otherdistrict?
We have estimatedthe incumbencyadvantagein our sampleof
multimemberstatesusing the net increasevariable(andobservationson
partiallycontested elections) and comparedthe resultingestimates to
those obtainedearlierand tabulatedin Table 1. There are two things to
note. First, the correlationbetween the new and old estimates is .87.
Second, the new estimates are slightly largeron averagethan the old
estimates(by a bit over one-tenthof a percentagepoint), but the difference is not statisticallydiscerniblefromzero.Thatthe new estimatesare
largeron averagelends supportto the suspicionthatthe previousresults
are downwardbiased because strategic entry decisions by potential
challengers determine the sample of races investigated. That the
increasein the estimatedsize of the incumbencyadvantageis not statistically significantis consistentwith our contentionthatpotentialcandidates in state legislative elections do not have very good information
about their chances, so that not much informationabout incumbency
advantageis conveyedby the patternof partiallycontestedraces."I
One might consider extending the analysis to include uncontested as well as partiallycontestedraces, again using the net increase
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344
Gary W. Cox and Scott Morgenstem
variableas a control.The problemwith this idea is purelypractical.In
single-member districts, to take the easiest example, the technique
would work only if there were a reasonable number of uncontested
open seats. In practice,however,in many states there are not. Thus we
do not pursuethe matterhere.
Conclusion
This article has shown that the Gelman-Kingmethod of meathe
incumbencyadvantagein contested elections, first devised
suring
for use in single-memberdistricts,can be adaptedfor use in multimember districts as well. Using our generalization of the Gelman-King
technique, we have measuredthe incumbency advantagein 14 U.S.
states which hold free-for-all multimemberelections to fill at least
some of the seats in theirlower houses. Adding these to directlyanalogous measurementsfor 26 predominantlysingle-member states, we
have observationson the incumbencyadvantagein 40 of the 50 states
between 1970 and 1986.
We find that the incumbencyadvantagehas increasedin states
with multimemberdistricts but at a substantiallylower average rate
thanin stateswith single-member(or equivalent)districts.We also find
thatthe size of the legislativeoperatingbudget,measuredon a per constituentbasis in constantdollars, is positively and statisticallysignificantly related to the size of the incumbency advantage in both
multimemberand single-memberstates but that the relationshipholds
up undermultivariateanalysisonly in the single-memberdataset.
Some topics for futureresearchare clear.First,we have looked
at
the contested-election incumbency advantage, saying little
only
about the scare-off effect. Although there is some evidence on this
score in the literature,theredoes not appearto be a complete analysis.
Second, if the basic Gelman-King model were estimated with twoparty seat percentages,ratherthan two-partyvote percentages,as the
dependentvariable(and this would requiregreaterattentionto the limited nature of the dependent variable, especially in single-member
states), one could see whether incumbents have a significant seatdenominatedadvantage.Thatis, one could see whetherthe presenceof
an incumbent in a contested election boosts not just the incumbent
party's vote percentage (as we show here) but also its probabilityof
retainingthe seat held by the incumbent.Previous work by Jacobson
(1987) on federal elections has shown that increasingvote advantages
do not necessarily translateinto increasingseat advantages;the question remains(empirically)open in the statecontext.
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IncumbencyAdvantage
345
Gary W.Cox is Professor of Political Science and Scott Morgenstern is a Ph.D. candidate in Political Science, Universityof Califoria-San Diego, La Jolla, California92093-0521.
APPENDIX A
The datain this paperare publishedas "StateLegislativeElectionReturnsin
the United States, 1968-1986" by the Inter-UniversityConsortiumof Political and
Social Research(ICPSR).This datasetcontainsalmost 123,000 recordsof state house
and state senate elections; it details, among other variables,candidates'parties, vote
totals, district type (post, multimember,or single-member),and incumbency status.
Fromthis datasetwe filteredout all lowerhouse contestsin single-memberdistricts,districtswith post positions,or free-for-allmultimemberdistricts.We excludedcontestsfollowing state redistricting;we also excluded contests in which a third-partycandidate
won more votes thaneitherthe Democrator the Republicanand those in which a major
partycandidatewon morethan90%of the two-partyvote. Because we neededinformation on the currentand previouselection in our model, we also eliminatedobservations
for which we did not have informationfromthe previoustime period.Finally,we examined only races thatwere fully contested;thatis, in the previousand currenttime periods, we soughtracesin which the numberof availablepositionswas equalto the number
of candidatesfromeach of the two mainparties.If these criterialeft at least 20 observations in five years (20 state-years)in a state thatused single-memberdistrictsor two or
moreyearsof 10 observationsin statesthatutilizedat least some multimemberdistricts,
we calculatedthe incumbencyadvantagein the remainingstate-years.We also included
the four states noted in Table 1 (Georgia,Mississippi, South Carolina,and Virginia),
which fell between these criteria,having 10 to 20 observationsof single-memberdistrictsin two or more stateyears. OurearlierpapermistakenlyincludedOklahoma1980
and Tennessee 1986, both of which fail to meet the 20 observationcriterion.Exclusion
of these two datapointsdoes not changeour substantiveresults,and we have thus kept
themin ourdataset.
The result of our whittling left a databasewith about 11,000 observations.
About90%of the observationsarefrom single-memberdistrictsandpost elections, primarily because many multimemberdistricts,which are concentratedin the South, are
not fully contested.Arkansasis a case in point;of the 74 districtsin 1980 thatmet other
criteria,only 8 had a Republicanchallenger.
The various data requirementslisted above-as well as some to be mentioned presently-leave us with observations on 40 of the 50 states. Arkansas is
excluded because of a high numberof uncontestedraces. In Alabama,Louisiana,and
Maryland,the use of four-yearelection cycles limited the data available,forcing us to
exclude these three states from the analysis also. Additionally,we excluded Illinois
because it used cumulativevoting between 1972 and 1980, Nebraskabecause of its
nonpartisanelections, and Minnesota,which did not put partydesignationson the ballot until 1974, becauseit did not have sufficientraces in which RepublicansandDemocratswere the primarycombatants.Alaskaand Hawaiiwere excludedbecauseof a lack
of state budgetarydata. Vermontis not in our dataset because ICPSR has collected
informationonly on the 1986 Vermontelections.
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APPENDIX B
Standard Errors Associated with Results in Table 1
State
Arizona
California
Colorado
Connecticut
Delaware
Florida
Georgia
Idaho
Indiana
Iowa
Kansas
Kentucky
Maine
Massachusetts
Michigan
Mississippi
Missouri
Montana
Nevada
New Hampshire
New Jersey
New Mexico
New York
NorthCarolina
1970
1972
2.9
2.6
0.8
2.5
1.9
5.0
2.5
0.3
1.8
1974
1976
2.2
1.7
1.9
1.0
1.8
1.1
1.2
3.5
2.6
1.0
1.5
2.0
2.1
1.3
3.7
1.8
6.1
0.9
1.4
2.0
3.2
1.6
2.0
1.6
2.5
3.0
2.1
1978
1980
1.5
2.5
3.1
1.4
5.0
2.0
5.2
4.3
1.0
1.6
2.2
3.6
2.7
2.5
2.5
1.4
2.9
1.9
3.0
3.1
0.9
2.5
1.9
1982
1986
3.2
1.3
5.1
2.6
4.4
1.9
3.1
1.6
6.7
2.6
2.4
2.7
3.0
4.9
2.1
3.4
2.1
1984
2.4
3.8
1.0
1.9
2.5
3.3
2.4
11.7
1.5
0.5
1.5
1.8
1.9
5.8
1.1
2.0
1.6
1.1
3.7
1.4
1.1
0.9
3.1
1.9
1.6
6.2
2.5
1.2
1.5
3.5
2.0
1.2
2.1
2.1
8.8
1.0
1.1
4.2
2.4
1.1
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3.0
2.3
2.7
0.7
2.1
0.9
1.2
2.1
2.8
4.0
2.6
6.1
2.0
4.3
3.0
3.9
3.8
1.6
1.3
2.1
2.5
APPENDIX B (continued)
Standard Errors Associated with Results in Table 1
State
1970
1972
1974
North Dakota
Ohio
Oklahoma
Oregon
Pennsylvania
Rhode Island
South Carolina
South Dakota
Tennessee
Texas
Utah
Virginia
Washington
West Virginia
Wisconsin
Wyoming
0.8
1.7
3.9
5.0
1.2
1.6
0.9
0.6
1.9
4.3
2.9
1.2
4.1
1.1
1.3
3.2
1.9
1.5
2.4
1.0
1.9
0.7
2.6
2.8
2.2
1.3
3.0
1.2
1976
1978
1980
1.9
2.5
2.0
5.1
3.1
1.4
3.1
3.8
1.1
4.6
4.3
2.8
1.1
2.1
4.9
2.9
1.5
2.4
4.2
0.6
3.6
2.9
1.9
2.1
0.4
3.0
1.1
2.8
1.0
2.8
2.6
2.0
3.4
3.3
1.3
1.5
4.2
2.9
1.1
2.1
4.1
0.7
3.5
1982
1984
1986
1.4
2.0
3.3
2.5
1.9
2.6
4.5
0.9
1.5
2.4
3.2
1.9
3.0
5.1
1.8
5.5
4.3
2.2
4.7
1.8
1.1
2.4
4.4
2.0
8.9
1.6
0.8
Note: In our earlier paper, Iowa 1970 and New Mexico 1984 were included in our analysis. They are excl
redistricting.Forsimilarreasons,we now includeRhodeIsland 1974.
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348
Gary W. Cox and Scott Morgenstem
NOTES
The senior authorgratefullyacknowledgessupportby the National Science
FoundationundergrantnumberSES-9208753.
1. Thereare severaltypes of election held in multimemberdistricts.Free-forall elections, in which each voterhas as manyvotes as thereare seats to be filled, are the
most common. In these elections, voters may cast fewer thanthe total numberof votes
they are allotted,but may not cumulate(cast more thanone vote for a single candidate).
It is this kind of multimemberdistrictthatwe referto as true.Electionsin multimember
districtscan also be conductedby establishinga numberof posts equalto the numberof
seats to be allocatedin the district,requiringeach candidateto file for one and only one
post, and giving each voterone vote in each post. The resultis thatthereare M simultaneous at-largeelections,all held underthe usualsingle-memberrules.
2. The vote percentagefor the Democraticpartyis the sum of the vote percentages of all Democraticcandidates.In a single-memberdistrict,of course, there is
only one such candidate.
3. Gelmanand King define P as a dummy variabledescribingpartyincumbency statusat the secondelection:-1 if the Republicanswon the previouselection,+1 if
the Democratsdid. Ournew definitionof P, in a single-memberdistrictheld by the Democrats,would be 1 (the numberof Democraticvictors)minus0 (the numberof Republican victors),dividedby 1 (the districtmagnitude),yielding +1, the same value as under
the old definition.Similarly,a valueof-1 is obtainedwhenthe Republicanshold the seat.
4. Gelman and King define I as the interactionbetween R2, a dummy variable indicatingthe decision of the incumbentto seek reelection(1 if she seeks reelection and 0 otherwise), and P2, the party which won the previous election (1 if
Democratsand -1 if Republicans).In single-memberdistricts,the value is the difference between the numberof Democraticand Republicanincumbents(divided by the
districtmagnitudeof 1).
5. The states excluded (and the reasonsfor their exclusion) are describedin
AppendixA.
6. Two states-Indiana and Montana-were includedin our previousstudy
of single-memberstates because they have a large numberof single-memberdistricts.
In thatanalysis,only the single-memberdistrictsin Indianaand Montanawere used in
estimating the incumbencyadvantagein those states. In the currentanalysis, all districts from both states were included in the analysis. In Indiana,there are significant
changes in the estimated value of incumbency for all years (with about 50% of the
observationsfrom multimemberdistricts) and in Montanafor three years (with only
about5% of the observationsfrommultimemberdistricts).
7. King (1991a, 127, n. 20) discusses the efficiency of the weighted least
squaresestimator.Our regressionincludes two states-Virginia and Mississippi-for
which only two years of dataon the incumbencyadvantagewere estimable.Excluding
these stateshas no significanteffect.
8. Each observationin the regression was again weighted by the standard
errorof the coefficient estimate of the incumbencyadvantage,as in King (1991a) and
Cox and Morgenstem (1993). The standarderrors used as weights are reportedin
AppendixB.
9. Recall that, even if a state had multimemberdistricts during the time
periodunderinvestigation,it is classified as a single-memberstate if none of its multimemberdistrictscontributedusable data to the regressionscalculatingthe incumbency
advantage.
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Incumbency Advantage
349
10. There are some noteworthy differences between the single-member
resultsin Table2 and those in our previouswork. Removingthe 11 observationsfrom
IndianaandMontanaincreasesthe coefficienton time froman insignificant-.01 (in Cox
and Morgenstem1993) to an insignificant+.11 (in Table2) and simultaneouslycuts the
coefficienton operatingbudgetper constituentalmostin half. These changesreflectthe
removalof the five observationsfromMontana.Estimatesof the incumbencyadvantage
in Montana'ssingle-memberdistricts(only these districtswere examinedin our previous work)put it at the bottomof single-memberstatesin most years.At the same time,
Montanahad one of the smallest legislative operatingbudgets among single-member
statesin all years.Thus the Montanaobservationscontributedstronglyto the size of the
coefficienton operatingbudgetperconstituent,andremovingthemhas a largeimpact.
11. These resultsare for the 54 state-yearsin which relaxingthe M-againstM requirementyielded a gain in the numberof usable datapoints. If the 21 state-years
in which no gain occurredare included,then of course the correlationbetweenthe new
andold estimatesincreasesand the differencebetweenthemdiminishes.
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