Markov Chain Models for Rolling Cross-section Data: How Campaign

Markov Chain Models for Rolling Cross-section Data: How Campaign
Events and Political Awareness Aect Vote Intentions and Partisanship in
the United States and Canada
by
Walter R. Mebane, Jr.y
and
Jonathan Wandz
April 7, 1997
Prepared for delivery at the 1997 Annual Meeting of the Midwest Political Science Association,
April 10{12, Palmer House Hilton, Chicago, IL, Political Methodology Section. Thanks to Jasjeet
Sekhon for helpful discussion of several important points. Thanks to Jonathan Cowden for letting
tron help macht, p2c2e and tempter with the computing. The authors share equal (or at least
equally distributed) responsibility for all errors.
y Associate Professor, Department of Government, Cornell University,
[email protected].
z Doctoral Student, Department of Government, Cornell University,
[email protected].
Abstract
Markov Chain Models for Rolling Cross-section Data: How Campaign Events and Political
Awareness Aect Vote Intentions and Partisanship in the United States and Canada
We use a new approach we have developed for estimating discrete, nite-state Markov chain
models from \macro" data to analyze the dynamics of individual choice probabilities in two
collections of rolling cross-sectional survey data that were designed to support investigations of
what happens to voters' information and preferences during campaigns. Using data from the 1984
American National Election Studies Continuous Monitoring Study, we show that not only did
individual party identication vary substantially during the year, but the dynamics of party
identication changed signicantly in response to the conclusion of the Democratic party's
nomination contest. Party identication appears to have measurement error only when the model
misspecies the dynamics. There are rapid oscillations among some categories of partisanship
that may reect individual stances regarding not only competition between the parties but also
competition among party factions. Using data from the 1993 Canadian Election Study, we show
that the critical events that shaped voting intentions in the election varied tremendously
depending on an individual's level of political awareness, and that the eects of awareness varied
across regions of the country.
Do electoral campaigns matter? Theories of resources and information (\momentum") in American presidential elections suggest some ways that they do (Brams 1978; Aldrich 1980; Gurian 1987;
Bartels 1988), and both spatial theory (Hinich and Munger 1997) and popular wisdom suggest that
in all kinds of elections campaigns have a great potential to make a dierence at least some of
the time. But some have suggested ways that volatility one may observe during campaigns may
be more apparent than real (Green and Palmquist 1990; Gelman and King 1993). We use a new
approach we have developed for estimating discrete, nite-state Markov chain (DFMC) models
from \macro" data to analyze the dynamics of individual choice probabilities in two collections of
rolling cross-sectional survey data that were designed to support investigations of what happens
to voters' information and preferences during campaigns.1 One set of data comes from the United
States and one from Canada. From both countries we nd strong evidence that the answer is, \Yes,
campaigns do matter."
First we estimate the degree of instability and the amount of measurement error in individual
Americans' party identications during 1984, using the 1984 American National Election Studies Continuous Monitoring Study (ANES-CMS) (Miller and the National Election Studies 1985).
Green and Palmquist (1990) argue that individual party identication exhibits a high degree of
stability once measurement error is taken into account. We nd not only that individuals' partisan
propensities varied substantially during the year, but that the dynamics changed signicantly in
response to the conclusion of the Democratic party's nomination contest. During the early part of
the primary season, many Republicans were briey shifting into \Democrat Leaner" status. But
as soon as the contest between Mondale and Hart was eectively decided in favor of the former,
Republican irtations with a Democrat identication ceased, to be replaced by a slow but steady
1
In the usage of the econometrics literature on estimation for Markov chain parameters, \macro" data are data in
which the state-to-state transitions over time for each individual are not observed, but rather one observes at each
of a number of times a distinct cross-section of individuals and their current states.
1
stream of Democrat Leaners becoming Republicans. There is an appearance of substantial measurement error in the survey items that measure party identication only when the model is incorrectly
specied to ignore the qualitative change in the dynamics.
Second we study changes in individual Canadians' voting intentions during the campaign period
of the 1993 Canadian federal election, using the 1993 Canadian Election Study. Several authors have
pointed to a few key dates during the campaign at which events occurred that they say signicantly
eroded support for the incumbent Progressive Conservative party (Frizzel, Pammet and Westall
1994; Johnston, Nevitte and Brady 1994). We verify that the dynamics of support outside of
Quebec for the Progressive Conservative, Liberal and Reform parties did change at the suggested
dates. But the critical events that shaped voting intentions in the election varied tremendously
depending on an individual's level of political awareness, and the eects of awareness varied across
regions of the country.
Previous analyses of Markov chain models of voter transitions have focused on transitions
between two elections, using either panel survey data (Miller 1972; Browne and Payne 1986) or
aggregate vote returns (McCarthy and Ryan 1977). Studies of the former class have an advantage
of observing transitions between states, but both types have suered from small numbers of time
periods. The economics literature on Markov chain models treats estimation problems more akin to
the ones we encounter in trying to model unobserved transitions over numerous time periods (Lee et
al. 1970; McRae 1977; Rosenqvist 1986, Mot 1993). Our new approach avoids the combinatorial
diculties that prevented Rosenqvist and McRae from using full maximum likelihood techniques.
Most directly comparable is the work by Mot (1993) that uses maximum likelihood to estimate
a binary state model for female labor supply. But the assumptions Mot makes about the initial
probability vector constrains his model to the analysis of binary states and would be untenable for
2
most political science research.2
We use maximum likelihood to estimate parameters that represent the initial probability vector
and each row of the transition matrices as functions of individual-level characteristics.3 We use
a multinomial logit functional form to specify the initial probability vector and each row of each
transition matrix. An important feature of our model is that the initial probability vector is
estimated as a function of all of the data, rather than simply the observations from the rst time
period. The initial probability estimates are therefore fully ecient; estimates of the distribution of
voters at the beginning of the Markov chain are not determined solely by the small survey samples
obtained for the rst time period. By allowing transition matrices to change during the time covered
by the data, we can test whether campaign events changed the structure of individuals' decision
making. In particular, we can test whether there were changes in the rate at which individuals
moved between voting preferences or changed partisanship.
Party Identication in the United States
Green and Palmquist (1990) call into question work by many authors4 who in various ways have
argued against the original conception of party identication (PID) as a highly stable individual
orientation that does not respond signicantly to short-term political stimuli (Converse, Campbell,
2
Mot constrains the initial probabilities distribution between the two states to be one and zero. He argues, \as
most outcome variables are dened pi0 will be zero|an individual is unemployed at the beginning of an unemployment
spell, not working prior to entering the labor force, unmarried at the beginning of the lifetime, and so on" (Mot
1993, fn 17).
3
To solve the severely nonconcave optimization problem we use GENOUD (Mebane and Sekhon 1996), a computer
program that combines evolution programming (drawing on both genetic algorithm and homotopy search techniques)
with quasi-Newton methods.
4
Green and Palmquist cite Jackson 1975a, Jackson 1975b, Meier 1975, Page and Jones 1979, Markus and Converse
1979, Fiorina 1981, Erikson 1982, Franklin and Jackson 1983, Franklin 1984, and Brody and Rothenberg 1988.
3
Miller and Stokes 1960). Green and Palmquist argue that impressions of instability and reactivity
in PID are in large part due to failures to take into account measurement error in the PID scales.
They base their argument on a \quasi-Markov simplex" (Joreskog 1970) model for measurement
error introduced by Wiley and Wiley (1970), which they apply to data from the 1980 ANES Major
Panel Study (ANES-MPS) to estimate the variance in the measurement error in the seven-point
PID scale.5 They use the variance estimate to correct for measurement error in lagged PID when
estimating several models that bear on some of the claims about particular \short-term forces" that
aect PID. The typical nding is that eects that appear statistically signicant and substantively
large when measurement error is ignored become insignicant and small when they use the error
variance estimates from their quasi-Markov simplex technique.
For testing the hypothesis that PID is stable but measured with error, the most important
problem with Green and Palmquist's approach is that it requires the assumption that instability
aects all kinds of partisans equally. This assumption is a consequence of their assumption that PID
is an interval-level variable. The assumption of interval-level measurement emulates the practice
of most (not all!) of the authors that Green and Palmquist target, and so can be defended to
some extent as necessary for a fair comparison between the original results and their new ones
(Miller 1987). Nevertheless the assumption is highly problematic. The problem is that at any
given time some kinds of partisans may be stable in their orientations while others are not. During
a particular election year it may be, for instance, that Democrats are likely to become Republicans
but Republicans are unlikely to return the favor and become Democrats. Treating PID as an
interval-level variable implies that any instability in PID must act on all of the variable's values
in a uniform manner. To see this, imagine that the model for instability in an individual's true,
interval-level PID at time t, yt , is yt = ayt,1 + bxt, for coecients a and b and random variable xt.
5
Green and Palmquist also use data from other ANES panel studies for some of their empirical demonstrations.
4
Such a form covers the models that Green and Palmquist use. There is perfect stability if a = 1
and b = 0; otherwise there is instability. The problem with such a model is that any particular type
of partisanship can be unstable only if all types of partisanship are unstable. It is not possible,
in such a model, for only Democrats to have unstable loyalties. Either every kind of partisan is
subject to instability, or no kind is.6
Instability in PID is best thought of as a propensity for individuals to change from one kind of
partisanship to another. PID should be thought of as a set of discrete categories, perhaps ordered
categories. For PID to be unstable and reactive to short-run political stimuli means that individuals
change categories in response to new political events or new political information.
For such a purely categorical PID to be susceptible to measurement error means that at any
particular time an individual may be observed in a category dierent from the one that actually
characterizes the individual. To distinguish error from truth it is necessary to specify properties
of the true values that the erroneous components do not possess. A basic distinction is to specify
that the errors of observation at dierent times are independent while the true values are not. The
basic Markov assumption for the true values' serial dependence is that the current true value for
an individual depends only on the true value for that individual in the immediately preceding time
period. The quasi-Markov simplex model that Green and Palmquist use has such a structure for
PID presumed measurable at the interval level, with the assumption that the measurement errors
are not only serially independent but identically distributed over time.
A Markov chain model for categorical PID with independent and identically distributed observation errors may be written as follows. Let it be the probability that some individual has PID of
6
The limitation to uniform eects would disappear in a suciently non-linear model of PID's relationships with
other variables. But such a model would require even stronger assumptions about PID's measurement scale. To
specify the values of PID as signicant in non-linear relationships would require PID to be measured on at least a
ratio scale.
5
type i at time t 2 f0; 1; 2; : : : g. To use the conventional terminology for a Markov chain, we say that
it is the probability that the individual is in state i at time t. The vector t = (1t; 2t; : : :; 7t)>
contains the probabilities that the individual is in each of the seven states corresponding to the
conventional seven levels of PID;7 t is the distribution at t of the process that generates the true
values. Let P be a 7 7 matrix in which element pji is the conditional probability that the individual is in state i at time t + 1, given that the individual is in state j at time t. As conditional
probabilities, the values in each row of P sum to one:
i=1 pji = 1; P is
P7
the transition probability
matrix (see the Appendix for more details). The distribution of the true values at time t is given by
t> = 0>Pt , where 0 is the initial distribution, 0> is the transpose of 0 and Pt indicates t postmultiplications of 0> by P. To represent observation error, each element mji of the 7 7 matrix
M is the conditional probability that an individual is observed in state i, given that the individual
P
is truly in state j ; naturally, 7i=1 mji = 1. The observed distribution at t is ~t> = t> M.8 There
is no observation error if M = I, the identity matrix.
The conventionally assumed ordinality of the PID categories9 suggests that it is appropriate to
use a tridiagonal form for P. In such a form, transitions are possible only between immediately
7
In the ANES-CMS data only 40 of the 3496 survey respondents did not select one of the seven conventional levels
of PID. We omit those 40 respondents from our analysis and ignore those types of responses here.
8
M is identiable only up to a row permutation. To identify the parameters of M we use a specication in which
the diagonal value is the largest value in each row.
9
The conventional order is \Strong Democrat," \Democrat," \Democrat Leaner," \Independent," \Republican
Leaner," \Republican," \Strong Republican".
6
adjacent states:
2
p11 p12 0 0
6
6p21 p22 p23
0
6
6
6 0
p32 p33 p34
6
P = 666 0 0 p43 p44
6
0 0 p54
6 0
6
6
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
675
p45
:
p55 p56
0 p65 p66 p
0 0 p76 p77
For instance, to go from state 2 (Democrat) to state 5 (Republican Leaner), an individual must
rst occupy states 3 (Democrat Leaner) and 4 (Independent). For daily observations, the minimum
time to go from being a Democrat to being a Republican Leaner is therefore three days.
Unconstrained maximum likelihood estimates (MLEs) of the three wave-to-wave transition matrices for PID using the 1980 ANES-MPS data provide indirect support for the tridiagonal form.10
In Table 1 one can see that for all three matrices the diagonal value is the largest value in each
row, and the values o the three central diagonals are generally small. Especially in view of the
fact that we estimate P for transitions on a time scale of single days, while the values in Table 1
represent transitions over periods of months, the transitions observed during 1980 give no reason
to doubt the suciency of using the tridiagonal form for the 1984 data.
*** Table 1 about here ***
10
Data from the ANES surveys were made available by the Inter-university Consortium for Political and Social
Research and by the Data Archive of the Cornell Institute for Social and Economic Research. The ANES-MPS and
ANES-CMS data were originally collected by the Center for Political Studies of the Institute for Social Research,
The University of Michigan, for the National Election Studies, under the overall direction of Warren E. Miller. Maria
Elena Sanchez was director of studies in 1980. Santa Traugott was director of studies in 1984. The data were collected
under grants from the National Science Foundation. Neither the original collectors of data nor the Consortium bear
any responsibility for the analyses or interpretations presented here.
7
Dynamics of Party Identication During 1984
The principal political event of 1984 whose eects we examine is the conclusion of the Democratic
nomination contest. The conclusion of the contest is not the same as the end of the nomination process. Counting delegates and superdelegates, Walter Mondale eectively locked up the nomination
by mid-May, even though primaries then remained pending in several large states, many of which
were ultimately won by Gary Hart (Bartels 1988, 247{252). The question is whether that mid-May
conclusion of the contest changed the dynamics of individuals' partisan self-identications. Hart
had run as a \new Democrat," with a \campaign of ideas" rather than a \campaign of personality"
and with clear intentions to reform the Democratic party in ways that would have greatly reduced
the inuence of traditional interest groups (Hart 1983; 1993, 209{219). Mondale stood as much
more the heir to the aura of Roosevelt's New Deal, and especially with his embrace of protectionist
\domestic content" legislation that would have restricted competition in automobile production,
he appeared captive to traditionally Democratic labor unions. Before mid-May, an observer could
have perceived a realistic chance for the more reformist alternative to gain the nomination. It
was therefore uncertain what kind of Democratic party would exist for the general election and
beyond. After mid-May there was no more uncertainty. Anyone who may have been attracted by
the prospect of Hart-style reform in the Democratic party's organization or policy positions should
therefore have lost interest in the party after mid-May.
For individuals to react in such a way to the conclusion of the 1984 Democratic nomination
contest would appear to require that they were basing their partisan self-identications at least in
part on well-informed and forward-looking judgments. Finding evidence that individuals did react
in such a way would support arguments that an individual's PID summarizes a history of relatively
sophisticated judgments about issues, performance and candidates (e.g. Jackson 1975a; Page and
Jones 1979; Fiorina 1981; Franklin and Jackson 1983).
8
We use four Markov chain models estimated using the daily cross-sections obtained in the 1984
ANES-CMS to test whether PID varied over time during 1984, and if so whether the dynamics
changed in mid-May to reect the foreclosed possibility that the Democratic party would undergo
substantial \reform." The rst interview in the ANES-CMS data occurred on January 11, 1984,
the last on December 7, 1984. In the Markov chain models, January 11 is day t = 1. In two of
the models there is a single transition probability matrix for the entire span of 331 days. In the
other two models the transition matrix changes at day 120 (May 10), using the method described
in the Appendix.11 Of each of these pairs of models, one allows for measurement error while in
the other all states are assumed to be perfectly observed. As described in the Appendix, we use
a multinomial logit functional form to specify the initial probability vector and each row of each
transition matrix in each model. The measurement error matrices M are parameterized so that
the conditional probabilities decrease in each row as one moves away from the main diagonal.12
Likelihood ratio (LR) tests support the hypothesized mid-May change in the dynamics and
suggest that once the dynamics are specied to include that change there is no longer any indication
that PID is subject to measurement error. From Table 2 one can see that with only one transition
matrix for the entire time period, there appears to be signicant measurement error: the test
statistic is 16.64, which by a 21 distribution is signicant at any reasonable test level.13 The
11
May 10 is two days after a set of primaries on May 8 (IN, MD, NC, OH). Mondale won MD and NC and lost
very narrowly to Hart in IN (40.9 to 41.8) and OH (40.3 to 42.1). The results put Mondale over the top in delegates
and superdelegates (Bartels 1988, 251{252).
12
For each state i there is one parameter, i 2 f,15; 15g. Dene ai = 1=(1 + exp(i )) and then mii = 1 and
mij = ajii,jj if i 6= j . The conditional probabilities in each row of M are mij = mij = P7j=1 mij .
13
There is a complication because several parameters are xed rather than freely estimated in the nal specication
for each model. This occurs whenever a value in the initial distribution or a conditional probability value in a transition
matrix or measurement error matrix is zero or one. In this case the Markov chain has parameters on the boundary
of the parameter space. The number of free parameters is smaller than in the case of an interior solution. The rank
of the hessian matrix evaluated at the MLE is an estimate of the number of free parameters. If the estimates of the
9
measurement error matrix produced by the MLEs for the one-matrix model is striking:
2
:6215
6
6 :2009
6
6
6 :0380
6
c=6
M
6 :1429
6
6
6 :0000
6
6
4 :0000
:0164
:2357
:3329
:1520
:1429
:0000
:0000
:0282
:0894
:2009
:6083
:1429
:0000
:0000
:0486
:0339
:1213
:1520
:1429
:0000
:0000
:0838
:0129
:0732
:0380
:1429
1:0000
:0000
:1445
:0049
:0442
:0095
:1429
:0000
1:0000
:2491
:0019 3
:0267 777
:0024 777
:1429 777 :
:0000 777
:0000 75
:4294
The matrix indicates that the Independent state is subject to the maximum possible amount of
measurement error. Someone whose true state is Independent is equally likely to be observed in any
state. Four of the other states also exhibit large amounts of measurement error. A Democrat has
only a 33 percent chance of being correctly observed in that state. A Strong Republican has only a
43 percent chance of being correctly observed. The pattern of the errors is dicult to understand.
Why are Democrats observed so imperfectly while Republicans are observed perfectly? Why are
Strong Republicans so much more dicult to observe in their true states than are Republicans or
Republican Leaners? It is not reasonable to attribute such an asymmetric pattern to imperfections
in the survey instrumentation. Whatever the origin of such errors, it is dicult to believe that it
is apt to describe them as \measurement" errors.
*** Table 2 about here ***
The LR tests suggest that there appears to be such a stupendous amount of measurement
error only because the model with one transition matrix is grossly misspecifying the dynamics.
Comparing the one-matrix model to the two-matrix model under the assumption of no measurement
error gives a test statistic value of 23.32, which by the 23 distribution is statistically signicant at
o-boundary parameters are asymptotically normal, then the LR test statistics are asymptotically chi-squared with
degrees of freedom equal to the dierence in the number of free parameters in the respective nested models. We have
used this theory to conduct our hypothesis tests.
10
any reasonable test level. Comparing the two models that allow measurement error to one another
gives a test statistic value of 7.17, which is not signicant by the 24 distribution. The test for
the presence of measurement error given two transition matrices is .49, which is not even remotely
c
signicant by the 22 distribution. If the amount of measurement error were truly as large as the M
estimates suggest, we would not expect the estimated eect of the error to vanish so completely just
because the transition matrix is allowed to change at the conclusion of the Democratic nomination
contest. In combination with the substantive implausibility of the \measurement" error pattern
c , the LR tests convince us that both of the models that allow for measurement error
shown in M
should be rejected in favor of the model that has two transition matrices and no measurement error.
The transition matrices for the two-matrix, no-measurement-errors model do exhibit the kind
of pattern to be expected if the Democratic party appeared attractive to non-Democrats during
the period when there were reasonable prospects that it might be reformed, but repulsive to them
once the reform prospects were terminated by the conclusion of the nomination contest. The MLEs
in Table 3 produce the initial distribution and transition probability matrices in Table 4. The fact
that all the values in the lower diagonal of the transition matrix for days 1{119 are positive while in
that matrix the value for the probability of a transition from Democrat to Democrat Leaner (^p1:23)
is zero means that no Strong Democrat or Democrat abandoned the party during that period, while
non-Democrats moved both toward a Democrat PID and away from it. During this period, every
non-Democrat had a small but positive probability of converting to a Democrat or Strong Democrat
identication. As of day 120 the situation reverses. During that period all the values in the upper
diagonal of the transition matrix are positive while the value for the probability of a transition from
Independent to Democrat Leaner (^p2:43) is zero. That means that neither Independents nor any
kind of Republican took on any kind of Democrat identication after the Democratic nomination
contest concluded. Strong Democrats, Democrats and Democrat Leaners, on the other hand, all
11
had small but positive probabilities of converting to some kind of Republican identication.
*** Tables 3 and 4 about here ***
One way to get a sense of the practical signicance of the mid-May change in the dynamics is to
contrast the distribution the MLEs predict for PID on election day using both transition matrices
to a simulation of what the distribution would have been had the rst transition matrix continued
unchanged until then. The former distribution is
> = :14 :17 :11 :09 :17 :17 :15
^300
while the latter we may write as
> = :20 :24 :15 :09 :11 :14 :08 :
~300
Given the mid-May change, we expect Strong Democrats, Democrats and Democrat Leaners to
comprise 42 percent of the population. In the simulated case in which movement toward Democratic
identication continued until election day, 59 percent of the population has one of those three types
of identication. Had the simulated case occurred with the typical conditional probabilities of
voting Democratic given individual PID continuing to apply, the result would have been a landslide
Democratic victory.
To get a better sense of the rapidity of movement that the estimated transition matrices suggest
was occurring, consider the quartiles of the distribution of hitting times shown in Table 5. The
hitting time ji is the time it takes for an individual who starts in state j to enter state i for the
rst time. Each nite hitting time has a geometric distribution that can be computed from the
transition probabilities (Feller 1968, 443; Resnick 1992, 98{110). Table 5 shows the rst quartiles
and the medians of those distributions, based on the transition probabilities produced by the
MLEs. The hitting times in many instances show quite rapid transitions. During the period when
12
the Democratic nomination contest was alive, for instance, about a quarter of Republicans took
just seven days to pass through the Republican Leaner and Independent states and arrive in the
Democrat Leaner state, while about half of Republicans took just 18 days to do so. But having
arrived at a Democratic Leaner identication, individuals did not take long to abandon it. About a
quarter of Democrat Leaners took just ve days to arrive in the Republican state; about half took
14 days. After the Democratic contest concluded, no Independent, Republican Leaner, Republican
or Strong Republican ever became a Democrat Leaner, but circulation among the Independent,
Republican Leaner and Republican states became if anything more rapid.
*** Table 5 about here ***
On the Democratic side the hitting times show moderately rapid oscillations during the early
period between the Strong Democrat and Democrat states that accelerated to become almost as
rapid as possible at the conclusion of the nomination contest. The median times that show daily
oscillations between the two states in the later period could mean that there was an extreme
degree of indecision on the part of Democrats and Strong Democrats about the strength of their
commitment to their party, or it could mean simply that the two states had ceased to have distinctive
meanings to individuals once the only presidential choice of any consequence had become the one
between Walter Mondale and Ronald Reagan. But the most interesting hitting time pertaining to
Democratic identiers is probably the time it took to leave the Democratic fold entirely by going
from the Democratic Leaner state to the Independent state. The median value of 204 days indicates
that by election day just less than half of those who were Democrat Leaners at the conclusion of
the Democratic nomination contest had become some kind of Republican.
The daily oscillations between the Strong Democrat and Democrat states after the conclusion
of the Democratic nomination contest, and the nearly as rapid wandering among the Independent,
Republican Leaner and Republican states, are remarkable. The rapid movements may indicate
13
that after all PID was aected by a kind of observational error, albeit not serially independent and
temporally identically distributed measurement error such as M represents. To say that the Strong
Democrat and Democrat states had ceased to be distinctly meaningful is to say that to individuals
there was only one state|\General Election Democrat" (GEDem), say|even though the survey
instrumentation continued to record responses in two dierent categories. Similarly, on the other
side there were not in fact the three states of Independent, Republican Leaner and Republican but
only one, say \Non-Movement-Conservative Republican" (NMCRep).
The most important fact about this error|if that's what it is|is that it is endogenous to
the political process. The error is not purely or merely an artifact of the survey interviewing
instrumentation, but rather reects the developing political process and the strategies individuals
are using within that process. The key detail here is that the Strong Democrat and Democrat
categories do not appear to collapse into one another until the Democratic nomination contest
concludes and therefore the issue of drawing internal distinctions|distinctions among Democratic
party factions|had become as a practical matter moot. It is reasonable to speculate that the
distinctions that remained separated subjective party members, the GEDems, from the Democrat
Leaners who did not consider themselves party members but did feel likely to vote for the party's
presidential candidates. On the Republican side, the separation between the NMCReps and Strong
Republicans persists throughout the span of the data, from January through the general election in
November. It seems reasonable to guess that this pattern reects the then-continuing war within
the Republican party between Movement Conservatives and Republican moderates.14 The rapid
oscillations may reect individual stances regarding not only competition between the parties but
also competition among party factions.
The change in the dynamics of PID at the conclusion of the Democratic nomination contest is
14
This interpretation extends the argument of Keith, Magleby, Nelson, Orr, Westlye and Wolnger (1992) that
independent \Leaners" are in eect partisans.
14
characteristic not only of the MLE but of virtually all the parameter vectors contained in the 95
percent condence ellipsoid (95% CE) of the joint sampling distribution of the parameter estimates.
The plots in Figures 1, 2 and 3 illustrate this. To generate the gures we sampled 50 parameter
vectors from the boundary of the 95% CE.15 The gures show trajectories of the PID distribution
over the 331 days covered by the 1984 ANES-CMS data, for the MLE and six of the 50 sampled
vectors. Each curve in each plot shows the trajectory given the initial distribution and the transition
probability matrices produced by one parameter vector; lines in dierent plots that have the same
dash style represent dierent components of the distribution for the same parameter vector. The
solid black lines show the components of the distribution for the vector of parameter MLEs. The six
other vectors were chosen to illustrate the maximal range of variation in the distribution trajectories
produced by parameter vectors on the boundary of the 95% CE. From the set of 50 sampled
parameter vectors, we chose the vectors that produced either the largest or the smallest average
probability over the 331 days for at least one of the categories of PID.
*** Figures 1, 2 and 3 about here ***
All the curves show a change in the slope of the trajectory at the conclusion of the Democratic
nomination contest. In every case, the proportion of \Democrats" (Strong, not-Strong or Leaner)
is increasing up to day 119 and decreasing thereafter, while the proportion of \Republicans" is
decreasing up to day 119 but thereafter increasing. The variations in the heights of the curves
represent only dierent allocations of probability among the three kinds of Democrat or among the
15
Using the robust estimate ^ =
@ 2 log L ,1 , @ log L , @ log L > @ 2 log L ,1
>
>
@ !^
@ !^
@ !^ @ !^
@ !^ @ !^
of the asymptotic covariance matrix
of the free parameters ! (White 1994, 92), we randomly sampled vectors from the multivariate normal distribution
with mean !^ (the MLE) and covariance matrix ^ . For each sampled vector !~ we computed the statistic ~2 =
(!~ , !^ )> ^ ,1 (!~ , !^ ). Sampling was continued until we obtained 50 vectors such that j~2 , 2(1,) j < :02, where
2(1,) denotes the critical value for an = :05 level test using the 2 distribution.
15
three kinds of Republican.
1993 Canadian Federal Election
Of the political changes that occurred during the 1993 Canadian federal election, the near electoral
annihilation of the Progressive Conservative (PC) party is one of the most remarkable. The PC
party won parliamentary majorities in the two preceding elections, with the latter setting a historic
record 211 of 295 seats. On October 25, 1993|election day|the Progressive Conservative party was
reduced to 2 parliamentary seats. A number of explanations have been oered for the precipitous
decline in PC support, including the loss of Campbell's credibility, higher levels of media scrutiny,
increasing regional polarization, and campaign strategy disasters. Even if strategic weaknesses of
the PC party existed prior to the beginning of the campaign, the early public opinion polls did not
foretell their inevitable demise. The events of the campaign mattered. The focus here is two fold:
to test the impact of selected campaign events on vote intentions, and to contrast the behavior of
dierent awareness groups.
In the context of vote analysis, DFMC models have a number of appealing characteristics.
DFMC models enable tests of hypotheses about the eects of specic events. More generally, DFMC
models allow estimation of properties of the process that generates the changes in voter support
observed during the campaign. While other statistical methods such as vector autoregression and
loess can model only aggregate daily proportions, DFMC models provide estimates of the individuallevel movements between states by directly estimating the transition matrix or matrices. Attempts
to explain the relative importance of campaign events in the 1993 Canadian federal election require
an analysis that can test whether potential voters acted dierently after each particular event. Only
if the probability of transition between parties signicantly changed can one claim that an event
had a signicant impact on vote intentions.
16
Timing
There is a general consensus about what were the major events during the 1993 campaign. Most
accounts rely on anecdotal evidence to assess the consequences of the events, however, so the
extent to which individual campaign events contributed to the decline in support for the PC party
have remained untested. In the most comprehensive analysis, Johnston, Nevitte and Brady (1994)
identify four key points of change in the campaign, using graphical analysis of the 1993 Canadian
Election Survey (CES) data. In order of ascending importance, the dates are: September 11{15
(day 1{5 of the CES), October 3{4 (Party Leader Debates, day 25{26 of the CES),16 October 13{
14 (PC Chretien advertisement debacle, day 35 of the CES), and September 20{22 (Conservative
Party \secret plan to cut social programmes," day 10{12 of the CES).17 They argue that the drop
September 20{22 \may not have been enough to deny the Conservatives control of the opposition,
perhaps not even enough to guarantee the Liberals a majority, but was a cardinal event in that
other events hinged on it." (3) By employing a DFMC model we are able to estimate transition
matrices for each of the periods and test the signicance of these events with respect to their impact
on the distribution of daily vote intentions.
The null hypothesis is that a single transition matrix adequately represents campaign dynamics
and thus P1r = P2r = : : : PKr for all K . The substantive interpretation of this hypothesis is that
the Progressive Conservatives were destined from the beginning of the campaign to be reduced
to near electoral extinction: none of the campaign events signicantly increased or decreased the
16
LeDuc (1994) notes that \one should not conclude that the debates determined the outcome of the election.
Various polls taken at dierent times during the campaign show that the conservatives had already entered a steep
slide in public support well before the debates took place..." (137)
17
Woolstencroft (1994) describes in detail the controversy over Kim Campbell's statement on September 23 that
\an election is no time to talk about social policy." Johnston, Nevitte and Brady (1994) point out that the drop in
PC vote intentions had largely occurred prior to September 23.
17
magnitude of their defeat. Although there are numerous alternative hypotheses,18 we restrict
ourselves to the four most substantively interesting dates outlined by Johnston, Nevitte and Brady
(1994).
Political Awareness
Political awareness is dened as an individual's cognitive engagement in politics, \as against emotional or aective engagement, or no engagement at all" (Zaller 1992). With Zaller we view an
individual's level of political awareness as aecting both the probability that the individual receives
political information and the individual's ability to process and contextualize new information. We
can test for heterogeneity of behavior associated with awareness in two ways: by estimating separate transition probabilities for each level of awareness; and by allowing the transition matrices
for each group to change at dierent dates. The null hypothesis here is that both the transition
matrices and the dates of change are the same for dierent levels of political awareness.
There are three broad alternative hypotheses with substantively dierent interpretations. First,
the transition matrices may dier while the dates of change are uniform. This could reect that
low awareness individuals are not signicantly delayed in responding to events, but do not have
the cognitive abilities or contextual information to react in the same manner as the highly aware.
Second, the transition matrices may be the same but the dates dier. As opposed to the rst
alternative, low awareness people do not react dierently than the highly aware to events, but their
limited exposure to the media impedes the rate at which they receive the information the event
provides. Third, it may be that neither transition probabilities nor dates of change are shared by
dierent awareness groups. This nal alternative encompasses all remaining possible patterns of
18
There are in fact
P45 ,45
r=1 r
dierent ways to select the number of transition matrices to estimate and the date
of their change. A search over dates is certainly possible for a xed number of transition matrices, however, the
practical implementation is a project for future research.
18
behavior.
The empirical results suggest that patterns akin to the logic of the second hypothesis are
dominant. Events early in the campaign quickly sort out the positions of high awareness people,
while it is not until much later that less aware people begin to make the same choices. In some
cases the disparity in vote distributions between awareness groups becomes small by election day,
but individuals in the two groups have arrived at similar points by very dierent paths.
To measure political awareness we draw on three factual survey questions that probed each
respondent's knowledge of the 1993 rates of Canadian ination and unemployment, as well as of the
size of the federal government's decit. These questions have the appealing property of separating
respondents into awareness groups whose proportions remain stable over the entire period of the
campaign. The more politically-oriented awareness questions all have increasing correct response
rates over the campaign; to use these would likely confound tests of homogeneity with the changing
composition of each group. In addition, there is a growing consensus that \factual awareness is the
best single indicator of [political] sophistication and its related concepts" (Delli Carpini and Keeter
1993). For this analysis, respondents have been divided into two roughly equally sized groups,
designated as high and low awareness.
Data
Data for measuring vote intentions, as well as other respondent characteristics, are drawn from the
Canadian Election Study, 1993.19 The CES was a rolling cross-sectional survey that spanned 45
days of the 1993 Canadian federal election campaign period. The survey interviewed 70-90 new
19
The data were collected by the Institute for Social Research, York University for Richard Johnston, Andre Blais,
Henry Brady, Elisabeth Gidenl, and Neil Nevitte. The project was funded by the Social Sciences and Humanities
Research Council of Canada. The investigators, SSHRCC and the Institute bear no responsibility for the analyses
and interpretations presented here.
19
respondents daily from a random sample of eligible voters across Canada. The campaign-period
survey began 3 days after the ocial election writ was dropped and terminated the day preceding
the election.
Although the PC party was represented nationally, we believe that Quebec and the maritime
provinces have unique electoral environments and therefore exclude them from the present analysis.
The sample is restricted to Western Canada. Heterogeneity in voter preferences between these
provinces is modeled by use of dummy variables for BC and Ontario residency.
Canonical Form
For the case of Canadian multiparty elections, there is no obvious ordering of the parties and
therefore no ordinal structure to the states of the Markov chain. We adopt a transition matrix with
the form
2
p11 0 0 p14
6
6
P = 666 00 p022 p0 pp24
33
34
4
p41 p42 p43 p44
3
7
7
7
7
7
5
(1)
The rst three diagonal elements (p11, p22 and p33) represent the probability of maintaining the
same vote intention between intervals. This form is generalizable to any number of states, but we
focus on the vote proportions for three parties: PC, Liberal, and Reform. We reduce the number of
parameters by constraining all transitions to take place through the last category. The elements in
the last column of the rst three rows represent the probability of renouncing ones' vote intention
and moving to the residual category. The elements in the last row represent the probability of
entering the state corresponding to each column. All other cells are zero. This abstraction requires
defection to another party to occur over the course of two intervals, but avoids imposing an ordinal
structure on the parties. The state corresponding to the last row and column includes primarily
20
those who respond \Don't know" or abstain as their vote intention.20 For estimation, the rst
three rows are parameterized as binary logits and the fourth row is a multinomial logit (see the
Appendix for details).
Results
We estimate a Markov chain for each of the two levels of political awareness with dummy variables
for Ontario and BC. Tests between non-pooled models with dierent specications indicate that
the days at which transition matrix changes occur are not the same for each level of awareness.21
Table 6 provides a summary of solution values. For the present analysis we focus on the estimates
from models with three transition matrices, where high aware individuals have the potential to
change matrices at days 10 and 25 of the CES, and less aware individuals have the potential to
change at days 25 and 35.
*** Table 6 about here ***
Figure 4 and Figure 5 plot the time paths of the estimated vote proportions in BC and Ontario,
respectively, for each awareness level. These case studies provide insight into two key elements that
the DFMC model takes into account. The model not only reveals dierent vote proportions in each
province and awareness group, but distinct patterns of behavior over the course of the campaign.
20
For the present analysis we also include the New Democratic vote in this residual category. This arbitrary
residual category is not of interest itself, but constructed such that we can include respondents in at least one state.
In so doing we attempt to minimize the additional problems inherent in open systems, where individuals could exit
from the states analyzed, by not allowing the pool of potential respondents analyzed by the DFMC model to change
systematically over the course of the campaign. We feel that for this initial analysis, the benets of this strategy
outweigh the potential violations of lumpability assumptions
21
The LR test of the pooled against non-pooled estimation does not reject the hypothesis that they are the same.
As noted in the analysis of PID, we treat the accuracy of these LR tests with circumspection.
21
*** Figure 4 and Figure 5 about here ***
The estimated distribution of intentions to vote for the PC party in BC was below the historic
mean22 for the early portion of the campaign,23 but stable for both awareness groups. A loss of
some of the 12 PC seats would be expected, but annihilation was not imminent. The debates
did provide a ripple among the low awareness group, while for the high aware the event resulted
in a quick and near complete abandonment of the party. The small percent of individuals who
remained in the party following the debates were loyalists who were not subsequently dissuaded by
the Chretien ad debacle. In contrast, the support for the party among the less aware was cut nearly
in half following day 35. A similar pattern among awareness groups is also evident for Liberal and
Reform support. The Reform party enjoys signicant growth among the less aware following the
party leader's debates, although the gains recede following day 35. For the aware, all changes in
transitions are accomplished by day 25.
PC vote intentions among the less aware in Ontario resemble the behavior seen among their
BC counterparts: the debates gave a considerable boost primarily at the expense of the Liberals.
However, these new converts, along with the other low awareness Tories, quickly ed following the
negative press coverage of the Chretien ad debacle at day 35. The debates had only a slight eect
on high awareness voters, the major drop having already occurred around day 10.
What would have been the outcome on election day had these events not occurred? Among the
innite number of counterfactuals, we propose just one to highlight the importance of these events:
projecting the election day results with and without the nal transition matrix change for each
22
23
See Fiegert 1989, Table 13-1
By estimating the initial probability distribution, the DFMC model is particularly good at tting the early data
points in the survey. Due to concerns over smaller samples and extreme values in the early days we do not place too
much substantive weight on the initial time path (day 1{3), especially since the stationary distribution (Feller 1968,
394) is achieved by day three in most cases.
22
group. Using the same three-transition matrix estimates that created the time paths in Figure 4
and Figure 5, we can calculate \hypothetical" election day results by using only a subset of the
estimated matrices. Instead of using the third transition matrix we continue to use the second, and
thus track the proportions as if each group's nal event never happened. Table 7 shows election
day proportions for both the true and hypothetical estimates by province and awareness group.
The nal transition changes cost the PC party a majority among less aware individuals in both
Ontario and BC.
*** Table 7 about here ***
Figure 6 summarizes the dierences in behavior by awareness group. We subtract the less aware
daily vote proportions, again using the same estimates used in Figure 4 and Figure 5, from those
of the high aware. In BC, there is remarkable convergence by election day among PC and Liberal
supporters. In Ontario, the results for convergence are less convincing, although individuals in each
group are closer than they had been even a week prior to election day. Although low awareness
individuals are usually heading in the same general direction as their aware counterparts, election
day still nds these two groups with substantially dierent vote proportions.
*** Figure 6 about here ***
To answer the question of which were the critical events in the 1993 election requires conditioning
on awareness levels. September 20 was important for only half of the population: those with the
interest, abilities and background necessary to observe and evaluate the early crises in the PC
party. Events in the nal two weeks of the campaign, in contrast, caused no signicant change in
the preferences of the aware group. For the less aware, it was not until media coverage and interest
peaked during the party leader's debates that opinion began to be reshaped in anticipation of the
upcoming election. Extreme volatility characterizes the vote intentions of the less aware, with up
23
to 20 point shifts following the debates and reversals by election day. Events early in the campaign
quickly sort out the positions of high awareness people, while it is not until much later that less
aware people begin to make the same choices.
The question that cannot be resolved by the present analysis is why individuals are making
these changes in response to events. Are the qualitative dierences in the events revealing distinct
decision rules based on either style or substance? Were aware people who abandoned the PC party
early in the campaign responding to the Progressive Conservative policy failures, while less aware
people were merely reacting to a supercial media frenzy near the end of the campaign? Or are
less aware individuals rationally relying on media cues about the prospects and abilities of each
party: a party with a viable chance at governing would not be so brutally treated in the nal days
of the campaign. The condemnations of the anti-Chretien advertisement, both by the media and
from within the PC party itself, could be a clear signal to those who had not been closely attuned
to the campaign that the Progressive Conservatives were not a viable party. The DFMC model
provides a framework for the further investigation of these possibilities. Including a richer array
of covariates from both survey and media data, and allowing for the simultaneous estimation of
endogenous variables, are the next extensions for the analysis of pooled cross-sectional data.
Appendix: Estimating Markov Chains
We use the theory of Finite Markov Chains (Feller 1968; Kemeny and Snell 1960; Bartholomew
1982) to analyze the movement of individuals between discrete states over time. The fundamental
property of a Markov chain is that the probability of event Yit at time t depends solely upon the
event of the directly preceding time, Yjt,1 . Thus
Pr(Yit jYjt,1 ) = Pr(YitjYjt,1 ; Ykt,2) = Pr(Yit jYjt,1; Ykt,2; Ylt,3) = etc : : :
24
The probability of an ordered set of sequences is obtained by the multiplication law of conditional
probability, Pr(Y0 ; Y1 ; :::; YT ) = Pr(Y0 )Pr(Y1 jY0 ) Pr(YT jYT ,1 ).
To calculate in generality for any time (t = 1; 2; :::; T ) the probabilities of being in one of
a nite number of states (i = 1; 2; :::; m), one needs only to know the initial distribution 0> =
10 20 30 : : : m0 and the transition probability matrix,
2
3
p11 p12 : : : p1m
6 p
7
6 21 p22 : : : p2m 7
P = 64
where
..
.
..
.
...
..
.
7
5
pm1 pm2 : : : pmm
Pm
i=1 i0 = 1 and i0 0 for i = 1; :::; m, and i=1 pji = 1 for j = 1; :::; m and pji 0
Pm
for i; j = 1; 2; :::; m. The constraints reect the assumptions that the states are mutually exclusive
and exhaustive, and that probabilities cannot be negative. Repeatedly multiplying the initial
probability vector by t transition matrices gives the probabilities for each state at time t,
t> = 0>Pt
Allowing the transition matrix to change in K successive time periods requires the calculation
of multiple matrixes
P1, P2, ... PK
each covering t1 ; t2; :::tK days respectively. An individual
P interviewed on day t = kk=1,1 tk + w, w tk , would have probabilities t calculated as follows,
t> = 0>
,1
kY
k=1
Ptkk
!
Pwk
(2)
The greatest diculty with using pooled cross-sectional surveys is that the transition of each
individual between states is not observed. We use maximum likelihood to estimate the initial
probability vector and transition matrices that generate the series of vectors of probabilities t>
that presumably describe the distribution of survey respondents on each day. For a random sample
of R independent observations the log-likelihood is
log L =
R X
m
X
r=1 i=1
yritr log ritr
(3)
25
where yritr = 1 if respondent r is in state i on the day of interview tr , and yritr = 0 otherwise.
rtr = (r1tr ; : : :; rmtr )> is the vector of probabilities for respondent r.
We use a multinomial logit (MNL) (Maddala 1983, Theil 1969) form to represent the initial
probabilities, each row of the transition matrices, and the jump vectors. A major advantage of
the MNL parameterization is that the sum of probabilities are constrained to equal one.24 Each
MNL can be estimated as a function of individual respondent attributes. The result is transition
matrices and prior probabilities that are functions of individual level characteristics.
The initial probability that respondent r is in state i is calculated by
ri0 =
1
exp x>r i
and rm0 =
Pm,1
>
1 + s=1 exp xr s
1 + s=1 exp x>r s
Pm,1
(i = 1; 2; :::; m , 1) (4)
where xr is a vector of observations on the variables x for individual r, and i is a vector of
coecients for state i. Similarly each row h of the kth transition matrix is calculated as,
exp log (pk,1;rhi =pk,1;rhm ) + x>r khi
(i = 1; 2; :::; m , 1) and (5)
pkrhi =
Pm,1
1 + s=1 exp flog (pk,1;rhs =pk,1;rhm ) + x>r khs g
pkrhm =
1
1 + s=1 exp flog (pk,1;rhs =pk,1;rhm ) + x>r khs g
Pm,1
Note that log (pk,1;rhi =pk,1;rhm ) is the log odds of a transition by individual r in state h to state i
rather than to state m. We change each transition probability by adding the sum of the products
of the variables in x>r and the coecients khi to the value of the log odds according to the previous
transition matrix. The probability in the cell does not change if khi = 0.
24
Similar uses of the MNL form in Markov chain models occur in MacRae (1977) and Brown and Payne (1986).
26
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31
Table 1: Unconstrained Wave-to-wave Transition Matrix Estimates, 1980 ANES Major Panel Study
January-February to June
SDem
Strong Democrat .7214
Democrat
.1364
Democrat Leaner .0361
Independent
.0179
Republican Leaner .0120
Republican
.0000
Strong Republican .0000
Dem
.2143
.6515
.2892
.0982
.0843
.0217
.0149
DemL
.0500
.1061
.4337
.1875
.0361
.0072
.0149
Ind
.0000
.0202
.0964
.4196
.0723
.0580
.0000
RepL
.0071
.0253
.1325
.1875
.4819
.0942
.0149
Rep
.0071
.0404
.0120
.0714
.2771
.6304
.2239
SRep
.0000
.0202
.0000
.0179
.0361
.1884
.7313
Dem
.1870
.7119
.2222
.1061
.0920
.0221
.0274
DemL
.0407
.0565
.4691
.1515
.1034
.0074
.0000
Ind
.0000
.0395
.1358
.5758
.1149
.0147
.0548
RepL
.0000
.0113
.0864
.1667
.5287
.1250
.0274
Rep
.0000
.0282
.0247
.0000
.1494
.6838
.1370
SRep
.0000
.0056
.0123
.0000
.0115
.1397
.7534
September to November (Post-election)
SDem Dem DemL
Strong Democrat .7886 .1707 .0081
Democrat
.1978 .6484 .0440
Democrat Leaner .0541 .1351 .5541
Independent
.0267 .0667 .0933
Republican Leaner .0000 .0482 .0241
Republican
.0000 .0248 .0248
Strong Republican .0000 .0000 .0139
Ind
.0081
.0549
.1622
.6667
.0964
.0331
.0000
RepL
.0000
.0220
.0811
.1067
.6506
.0826
.0417
Rep
.0244
.0275
.0135
.0400
.1446
.6116
.1111
SRep
.0000
.0055
.0000
.0000
.0361
.2231
.8333
June to September
SDem
Strong Democrat .7724
Democrat
.1469
Democrat Leaner .0494
Independent
.0000
Republican Leaner .0000
Republican
.0074
Strong Republican .0000
Note: Maximum likelihood estimates. Number of cases for each pair of waves: Jan-Feb/Jun, 821;
Jun/Sep, 743; Sep/Nov, 730.
Table 2: Log Likelihoods for Markov Chain Specications for Party Identication During 1984
Specication
No Measurement Error
One Transition Matrix
Two Transition Matrices
With Measurement Error One Transition Matrix
Two Transition Matrices
Note: Number of cases: 3456.
Parameters
,2 log-likelihood Total Free
13267.82
13244.50
13251.18
13244.01
18
30
25
37
13
16
14
18
Table 3: Parameter Estimates for the Two Transition Matrix Model without Measurement Error,
1984 ANES-CMS Data
parameter
1
2
3
4
5
6
111
121
122
132
133
143
144
154
155
165
166
176
211
221
222
232
233
243
244
254
255
265
266
276
MLE
SE
{.043 .289
.569 .211
{4.997
|
.382 .486
{9.992
|
.948 .309
3.938 .140
15.912
|
19.991
|
{4.165 1.066
1.984 .412
{1.394 .403
{11.938
|
.643 .135
{2.674 3.299
14.000
|
14.977
|
{6.338 .909
{13.999
|
{9.946
|
{15.500
|
1.586 3.918
3.442 .491
{10.979
|
{1.000
|
{.541 .234
{7.000
|
{8.102 .490
{9.022 .495
{7.000
|
Note: The second transition matrix begins at day t = 120 (May 10, 1984). SEs are robust. SEs
shown as \|" indicate a xed (boundary-constrained) parameter. Number of cases: 3456.
Table 4: Party Identication Markov Chain Estimates During 1984
SDem
Initial Distribution .1232
Dem DemL Ind RepL Rep SRep
.2271 .0009 .1884 .0000 .3319 .1286
Transition Probability Matrices
Days 1{119
SDem Dem DemL Ind RepL Rep SRep
Strong Democrat
.9809 .0191 .0000 .0000 .0000 .0000 .0000
Democrat
.0166 .9834 .0000 .0000 .0000 .0000 .0000
Democrat Leaner
.0000 .0019 .8774 .1207 .0000 .0000 .0000
Independent
.0000 .0000 .1987 .0000 .8013 .0000 .0000
Republican Leaner .0000 .0000 .0000 .6402 .0232 .3366 .0000
Republican
.0000 .0000 .0000 .0000 .2733 .7267 .0000
Strong Republican .0000 .0000 .0000 .0000 .0000 .0018 .9982
Days 120{331
SDem Dem DemL Ind RepL Rep SRep
Strong Democrat
.0000 1.0000 .0000 .0000 .0000 .0000 .0000
Democrat
.8121 .1858 .0021 .0000 .0000 .0000 .0000
Democrat Leaner
.0000 .0003 .9953 .0044 .0000 .0000 .0000
Independent
.0000 .0000 .0000 .0000 1.0000 .0000 .0000
Republican Leaner .0000 .0000 .0000 .5255 .0000 .4745 .0000
Republican
.0000 .0000 .0000 .0000 .4849 .5138 .0013
Strong Republican .0000 .0000 .0000 .0000 .0000 .0000 1.0000
Note: Probabilities are evaluated at the parameter MLEs. Day 1 is January 11, 1984. Election
day is day 300 (November 6, 1984).
Table 5: Party Identication Hitting Time Distributions During 1984, First Quartiles and Medians
First Quartile Days 1{119
SDem Dem DemL Ind RepL Rep SRep
Strong Democrat
0
15
1 1
1 1
1
Democrat
17
0
1 1
1 1
1
Democrat Leaner
524 507
0
2
3
5
1
Independent
530 512
5
0
1
3
1
Republican Leaner
531 513
6
1
0
2
1
Republican
532 514
7
2
1
0
1
Strong Republican
695 678
171 165 164 163
0
Days 120{331
SDem Dem DemL Ind RepL Rep SRep
Strong Democrat
0
1
251 335 335 336 892
Democrat
1
0
250 335 335 336 892
Democrat Leaner
61
61
0 84
84 85 641
Independent
1 1
1 0
1
2 557
Republican Leaner
1 1
1 1
0
1 556
Republican
1 1
1 2
1
0 555
Strong Republican
1 1
1 1
1 1
0
Median
Days 1{119
SDem Dem DemL Ind RepL Rep SRep
Strong Democrat
0
36
1 1
1 1
1
Democrat
42
0
1 1
1 1
1
Democrat Leaner
1263 1222
0
5
8 14
1
Independent
1277 1235
13 0
2
8
1
Republican Leaner 1279 1238
15 2
0
6
1
Republican
1282 1240
18 5
2
0
1
Strong Republican 1676 1634
412 399 396 394
0
Days 120{331
SDem Dem DemL Ind RepL Rep SRep
Strong Democrat
0
1
604 808 809 811 2150
Democrat
1
0
604 807 808 810 2149
Democrat Leaner
147 147
0 204 204 207 1545
Independent
1 1
1 0
1
3 1342
Republican Leaner
1 1
1 2
0
2 1341
Republican
1 1
1 4
1
0 1338
Strong Republican
1 1
1 1
1 1
0
Note: Hitting time distributions are evaluated at the parameter MLEs. Day 1 is January 11,
1984. Election day is day 300 (November 6, 1984).
Table 6: Log Likelihoods for Markov Chain Specications for Vote Proportions in Canada, 1993
Parameter Counts
Specication
,2 log-likelihood Total
Free
Pooled, Four Transitons Matrices
Changes at Days 10,25,35
5736:57
108
88
More Aware, Four Transitons Matrices
Changes at Days 10,25,35
2734:1
81
60
Less Aware, Four Transitons Matrices
Changes at Days 10,25,35
2970:01
81
61
More Aware, Three Transitons Matrices
Changes at Days 10,25
2743:0
63
49
Less Aware, Three Transitons Matrices
Changes at Days 25,35
2984:63
63
53
Note: Number of cases: 1212 (Unware), 1068 (Aware).
Table 7: Dierence between Expected Vote Proportions on Election Day, With and Without
Last Transition Matrix Change, by Province and Awareness Level in Canada 1993
PC
BC
Liberal Reform Other
PC
:24
:35
,0:11
:49
:29
:20
:02
:32
,0:29
:47
:23
:24
:32
:41
:20
:41
:11 ,0:00
:12
:20
,0:08
:44
:33
:10
Less Aware
Change Day(s)
25 & 35
:06
:21
25 only
:13
:24
Dierence
,0:06 ,0:03
More Aware
Change Day(s)
10 & 25
:05
:22
10 only
:24
:14
Dierence
,0:19
:08
Ontario
Liberal Reform Other
:08
:13
,0:05
:43
:32
:10
:19
:26
:14
:33
:05 ,0:07
Figure 1: Trajectories of the Distribution of PID for the MLEs and Selected Parameter Sets from
the 95% Condence Ellipsoid, Democrat PIDs
0.2
0.0
0.1
Proportion
0.3
Strong Democrat
0
100
200
300
200
300
Day
0.2
0.0
0.1
Proportion
0.3
Democrat
0
100
Day
0.2
0.1
0.0
Proportion
0.3
Democrat Leaner
0
100
200
Day
300
Figure 2: Trajectories of the Distribution of PID for the MLEs and Selected Parameter Sets from
the 95% Condence Ellipsoid, Independent PID
0.2
0.1
0.0
Proportion
0.3
Independent
0
100
200
Day
300
Figure 3: Trajectories of the Distribution of PID for the MLEs and Selected Parameter Sets from
the 95% Condence Ellipsoid, Republican PIDs
0.2
0.0
0.1
Proportion
0.3
Republican Leaner
0
100
200
300
200
300
Day
0.2
0.0
0.1
Proportion
0.3
Republican
0
100
Day
0.2
0.1
0.0
Proportion
0.3
Strong Republican
0
100
200
Day
300
Figure 4: B.C. Daily Vote Proportions by Party and Awareness, DFMC estimates
1.0
1.0
0.8
PC / More Aware
0.8
PC / Less Aware
•
0.2
••
•
• ••
••
•
•
20
•
•• •• ••
• ••••
30
40
0.6
•
••
•• •
0
• • •
•
•
•
10
•
•
•
•
• •
•
•• • •
20
••••••• •• •
30
40
Day
Liberal / Less Aware
Liberal / More Aware
0.8
0.8
•
•
•
•
•
• •
•
•
•
•
10
••• ••
20
•
•
••
•
• ••
30
•
40
•
••
• ••
0
•
•
•
10
•
•
•
• •
20
•
40
Reform / More Aware
1.0
Reform / Less Aware
•
0.8
•
•
0.0
•••
0
•••
10
•
•
• •••••
•
•
• •
20
•
•
•
•
30
Day
• ••
40
••
0.6
•
•
••
•
•
•
•
•
•
•
••
0.2
0.2
•
• •
••
•
•
•
•
•
0.0
0.4
•
•
•
•
0.4
Proportion
0.6
0.8
• • •
30
Day
•
•
•
Day
•
•
•
• •
•
••
•••
•
•
•
•
•
0.0
••
•
•
•
••
1.0
0.0
•
•
••
0
•
•
•
•
•
••
•
•
•
0.2
•
0.2
•
•
•
0.4
Proportion
0.6
•
0.6
•
•
0.4
Proportion
•
•
Day
•
Proportion
•
•
•
•
•
•
•
•
•
•
•
• ••• •
10
1.0
0
•
•
•
1.0
0.0
•
•
•
0.2
•
•
•
0.0
••
•
•
0.4
Proportion
0.6
•
0.4
Proportion
•
••
0
•
• ••
10
•
•
•
• •
•
• •
20
•
•
••
•
• •
30
Day
•
•
••
•
•
•
40
•
•
Figure 5: Ontario Daily Vote Proportions by Party and Awareness, DFMC estimates
1.0
1.0
0.8
PC / More Aware
0.8
PC / Less Aware
•
•
•
•
•
0.0
• •
•
•
•
• •
•
•
• •
•
•
•
10
•
••
20
•
•
•
30
•
•
•
•
•
•
•
•
•
••
•
•
•
•
•
••
10
•
•
•
•
0
•
•
• ••
20
•• ••
30
•
••
40
Day
Liberal / Less Aware
Liberal / More Aware
0.8
•
•
•
0.2
• •
•
•
•
0.0
•
••
•
•
•
•
10
20
30
40
•
•
•••
•
•
•
•
•
••
• •
•
0
•
•
•
•
•
•
•
10
20
30
40
Day
Reform / Less Aware
Reform / More Aware
0.8
0.8
1.0
Day
1.0
•
•
• •
•
•
•
0
•
•
•
•
•
•
•
•
• •
•
•
•
•
•
•
•
••
•
•
•
•
•
•
•
0.2
•
•
•
0.0
0.4
••
••
•
•
0.6
•
• •
•
•
•
•
Proportion
0.6
•
•
•
0.4
0.8
1.0
Day
•
Proportion
0.6
40
•
•
•
• •
• • • •
1.0
0
•
••••
•
•
•
••
•
• •
•
•
0.2
0.2
•
•
•
0.0
•
•
0.4
Proportion
0.6
•
0.4
Proportion
•
•
•
•• • •••
0
10
•
•
••••
••• ••
• •
•• •••
20
•
30
Day
•
•••
• •
• ••
40
• •
•
0.6
0.2
•
•
•
•
• ••••••• •
0
• •
• • ••
•
•
10
•
••
•• • •
•
•
•
•
•
0.0
0.2
•
•
0.4
Proportion
0.6
0.4
•
0.0
Proportion
•
• •
•
•
•
•
•
•
••
20
30
Day
•
•
40
•
•
-0.3
-0.3
0.0
0.1
0
0.2
0.3
10
10
-0.3
-0.3
0.0
0.1
20
30
20
Day
30
40
0.0
0.1
0.2
0.2
0.3
0.3
30
-0.1
Difference in Proportion
-0.1
20
0.1
0.3
0
10
0.0
0.2
Difference in Proportion
0
-0.1
Difference in Proportion
-0.1
Difference in Proportion
-0.3
-0.3
0.0
0.1
-0.1
0.0
0.1
Difference in Proportion
-0.1
Difference in Proportion
0.2
0.2
0.3
0.3
Figure 6: Dierence in Daily Vote Proportions for BC and Ontario, High minus Low Awareness
DFMC estimates
BC PC
Ontario PC
Day
40
0
40
0
0
10
10
10
20
Day
30
BC Liberal
Ontario Liberal
Day
20
Day
30
BC Reform
Ontario Reform
20
Day
30
40
40
40