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\title{\large \bf Connecting Micro and Macro Partisanship}

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\section*{\large \bf  \centering CONNECTING MICRO AND MACRO PARTISANSHIP$^{*}$ \centering \\}

\section*{\large \centering John E.  Jackson and Ken Kollman$^{\dagger}$}

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\noindent ~~~

\noindent * This paper continues a project that began with a  paper on partisanship and path dependence presented at the 2007 Meeting of the Society for Political Methodology, State College, PA, July, 2007.  We express our thanks to Chris Achen, Neal Beck and John Freeman for their comments on that paper and for pushing us to explore this connection.  Jude Hays offered suggestions that improved the model. Lastly, we want to thank the editors and reviewers for their requests and suggestions, which led us to do more that we thought possible.  All errors are, of course, our own.

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\noindent  ${\dagger}$ Dept of Political Science and Center for Political Studies, University of Michigan, Ann Arbor, MI.   \end{titlepage} 

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\begin{abstract}  This paper extends models of micro and macropartisanship in two ways.  It first develops  a model of individual partisanship that accommodates changes in partisan utilities.  Achen's Bayesian partisan updating is a special case of our model.  This more general micro model is then aggregated to create a model of macropartisanship.  This macro model is a more general version of the models of macropartisanship estimated by various authors.  The less restricted version incorporates possible individual and temporal heterogeneity.  We present an example using real data that offers a possible way to estimate the parameters in the full aggregate model.  \end{abstract}

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\markright{\normalsize Micro and Macro Partisanship \hfill \today \hfill}

\section*{\large  \bf  1  Introduction}

This paper proposes a model to improve our understanding of individual and macro partisanship and how they are connected.  Partisanship is a central concept in modern political science, but it is safe to say that a consensus on precisely what partisanship means, on its stability and on whether and how it influences other aspects of political behavior, has been elusive.   We do not wish to enter into nor certainly attempt to settle any of these debates here.   In fact, given particular parameter values, the model is sufficiently general to accommodate many of the prevailing interpretations at both the micro and macro level.  This model, because of its generality, can contribute to this debate.

The individual and the aggregate models contain identical terms measuring individual and temporal heterogeneity in the stability of partisanship and in the response to new information.  The micro and macro models allow individuals to differ in their attachments to a party and in their reactions to events that might alter partisanship.  Both models also allow the relationships between new information and partisanship to vary over time so as to accommodate periods of high stability and periods of realignment.  The consistency of the micro and macro models means that information gained or propositions made at one level can be passed easily to the other level, and researchers can connect the two levels conceptually.  Substantively, consistent micro and macro models facilitate better understanding of the connections between individual partisanship and the balance between the parties at the national level in the United States, for the parties' regional composition, and for presidential, congressional, and subnational  politics.   The methodological strategy can be adopted to study and to connect micro and macro partisanship in other contexts, such as countries with multi-party systems and different electoral rules.

It is valuable to have consistent micro and macro models for empirical analysis as well as for conceptual and  substantive arguments.  Models that incorporate varying amounts of stability are likely to involve time varying parameters.  Accurate estimation of these parameters requires data with temporal variations, which is most readily available with aggregate time series data.   Other parameters, particularly those related to individual heterogeneity, will be better estimated with cross-sectional individual data.  Comparability of results across these two empirical efforts requires consistency between the structure of and parameters in the individual and aggregate models.   

Our model depicts a dynamic process where voters can update their partisanship after observing new information.  In one form this is a Bayesian-style approach but the model incorporates other ways to understand partisanship, partisan change and stability. Individuals may differ in the amount of updating they do in response to this new information.  We also incorporate an expanded version of an error correction model commonly used in macropartisanship studies.   We show how our micro model can be aggregated to specify a model of macropartisanship with precisely the same terms.  We then offer an initial attempt using American partisanship data from the 1950s to 2000 to estimate time-varying parameters and parameters that indicate heterogeneity among individuals.   While our data analysis here is offered as a demonstration of possibilities rather than as an exploration of the substantive debates, our initial attempt points the way toward further progress and reveals the types of data and methods necessary to test for partisanship change at multiple levels of analysis.  

\section*{\large  \bf  2  Individual Partisanship}

\hspace{.25in} Achen (1992) and Gerber and Green (1998) propose a Bayesian type model of individual partisanship in a two party system in which current partisanship is a weighted sum of past partisanship, $P_{i,t-1}$, and the current utility differential associated with the parties, $U_{i,t}$,   \begin{eqnarray} P_{it} & = & \rho_{t} P_{i,t-1} + (1 - \rho_{t})U_{it} + V_{it}. \label{eq:pit1} \end{eqnarray}  The utility term is sufficiently general at this point to accommodate various types of decision processes, such as proximity, directional, or valence voting.  A stochastic term $V_{it}$ is added to make the model more general and because we are interested in empirical applications where the observed values for $P_{it}$ are not given deterministically by $P_{i,t-1}$ and $U_{it}$.$^{1}$ 

In the Bayesian type models the value of $\rho_{t}$ is inversely proportional to the uncertainty of past partisanship and directly proportional to the uncertainty in $U_{it}$ as indicators of true partisanship.  The two models differ in the treatment of $U_{it}$.  Achen assumes it is a constant true utility plus error, $U_{it} = U^{*}_{i} + \omega_{it}$, with $U_{i}^{*}$ varying across individuals but not over time for any given individual.  Achen (1992, p. 199) says his,  ``...model describes a stable period between realignments.''   Gerber and Green also treat realized utility as true utility plus error, $U_{i,t} = U^{*}_{i,t} + \omega_{it}$, but assume individuals' true party utility changes temporally as an autoregressive process, $U^{*}_{it} = \alpha U^{*}_{i,t-1} + \nu_{it}$, where $\alpha$ is the amount of autoregressivity and $\nu_{it}$ is another stochastic term.  Achen's and Gerber and Green's models are identical if $\alpha = 1$ and the variance of $\nu = 0$.  

Neither model accommodates an exogenous shift in conditions that might result in a restructuring of partisan utility and thus in partisanship.  What occurs during periods of party system change and instability is substantively interesting and important.  Burnham (1970, p. 16) argues that abrupt shifts in partisan voting occurred around 1874, 1894, and 1930, which correspond to crises in the U. S. economy.  He further shows (p. 57) these changes occurring on a regional level.  At the individual level both Miller (1992) and Green, et al. (2002) show that among African-Americans and among southern whites, particularly southern white males, there were important partisanship changes among age cohorts beginning with the Goldwater candidacy in 1964.  Even if these systematic changes in partisanship occur infrequently and/or only among subgroups within the electorate they can have significant consequences for the national party structure.$^{2}$  Thus, it is valuable to have a model that accommodates both periods of stability and episodes of systematic change.$^{3}$   

We alter the structure and notation of these models to incorporate explicitly the possibility of systematic shocks to the utility term.  Let $U_{it} = f(S_{t}) + \omega_{it}$, where $S_{t}$ represents the state of the world at time $t$, such as economic conditions, party platforms, foreign policy crises, scandals, or candidate attributes, and $\omega_{it}$ is the same random term used by Achen and by Gerber and Green to indicate the random component in realized utility. Individuals do not know the precise functional form and parameters in $f(S_{t})$.  In any period they only know the state of the world, $S_{t}$, and their utility.      

When people compare the difference in current utility to their previous partisanship and there is no change in $S$ so that $\Delta S_{t} = 0$ this difference is attributable to the draw from the distribution of $\omega_{it}$ and to how previous partisanship differs from their true partisanship.  This is Achen's Bayesian model with $\Delta S_{s} = 0$ for all $s \leq t$, so that $S_{t} = S_{0}$.  Partisanship at $t-1$ is an increasingly accurate predictor of true partisanship.  Over $t-1$ elections individuals acquire $t-1$ realizations of $U_{is} = f(S_{0}) + \omega_{is}$, which correspond to $t-1$ draws from the distribution of $\omega$.  The expectation is that $P_{i,t-1}$ will differ from $f(S_{0})$ by $\bar{\omega}_{i,t-1} =\frac{1}{t-1} \sum_{s=1}^{t-1}\omega_{is}$ and the imprecision, or variance, of $P_{i,t-1}$ as an estimate for $f(S_{0})$ is $\sigma^{2}_{\omega}/(t-1)$.

If the observed difference between current utility and past partisanship is accompanied by a change in $S$, individuals are uncertain how much of that difference is a systemic change related to the fact that $\Delta S_{t} \neq 0$ and how much is the realized value for $\omega_{it}$.    The larger the systemic change in $S_{t}$ the larger the change in party utility and in true partisanship and the less weight individuals give to previous partisanship as it becomes a less reliable predictor of their true partisan utility.  The expected squared difference between past partisanship and currently realized utility is related to the variance in the observational error, $\omega_{t}$, and to the absolute difference in the state of the world, $|\Delta S_{t}|$.  In regression terms, the former is the random and the latter is the systematic components of this difference.  The larger the systematic component the more weight individuals give to the currently realized utility and the less weight to past partisanship.  This makes $\rho_{t}$ directly related to $\sigma^{2}_{\omega}$ and inversely related to $|\Delta S_{t}|$.       

This discussion in effect provides a partial specification for $\rho_{it}$ in eq. \ref{eq:pit1}, which determines the relative weights given to past partisanship and currently realized utilities in selecting a current partisanship.  These relationships can be expressed as, \begin{equation} \rho_{it} = \frac{[a_{0} + h(i)_{t}]}{g(|\Delta S_{t}|)} = [a_{0} + h(i)_{t}]\gamma_{t}, \label{eq:alpha} \end{equation} where $a_{0}$ incorporates but may not be identically equal to $\sigma_{\omega}$, $h(i)_{t}$ represents the importance of accumulated experience with $\partial h(i)_{t}/\partial t > 0$, and $1/g(|\Delta S_{t}|) = \gamma_{t}$ is the relationship between $\rho_{t}$  and the change in the state of the world.  A reasonable condition is that $\gamma_{t} = 1$ when $\Delta S_{t} = 0$, indicating that with no change in the state of the world $\rho_{t}$ is only related to the variance of $\omega$ and individuals' experience.  Eq. \ref{eq:pit1} can now be written as, \begin{eqnarray} P_{it} & = & \rho_{it} P_{i,t-1} + (1 - \rho_{it})U_{it} + V_{it} \nonumber \\  & = &   [a_{0} + h(i)_{t}]\gamma_{t} P_{i,t-1} + \{1 - [a_{0} + h(i)_{t}]\gamma_{t} \}U_{it} + V_{it}  . \label{eq:pit2} \end{eqnarray}

Eq.~\ref{eq:pit2}  has several implications for the dynamics of individual partisanship, and ultimately for aggregate partisanship.  It encompasses most  common models of individual partisanship and whether and how partisanship changes over time.  In periods of stability, meaning there is no change in the state of the world so that $\Delta S_{t} = 0$ and $\gamma_{t} =1$, and if $a_{0} = 0$ and $h(i) = (t-1)/t$, where $t$ is the number of elections in which $i$ has participated, eq. \ref{eq:pit2} is Achen's Bayesian model of how people update, or learn, their partisanship during stable periods.  It is also quite general in that $h(i)_{t}$ and $\gamma_{t}$ can accommodate specifications for how experience or other individual specific and temporal factors affect the amount of updating when new information is available.  

Eq. \ref{eq:pit2} is consistent with empirical studies of individual partisanship, though these models do not consider temporal heterogeneity with $\rho_{it}$ varying over time.   Franklin (1984), Franklin and Jackson (1983), Fiorina (1981) and Stewart and Clarke (1998) find that $\rho_{i} < 1$ and that variations in $U_{it}$ are strongly associated with differences in $P_{it}$.  Though the specifications for the composition of $U_{it}$ vary these results imply that voters regularly update their partisanship in response to current utility evaluations and changing conditions.  Green, Palmquist and Schickler (2002) and Schickler and Green (1996), in contrast,  conclude that $\rho_{i}$ is close to 1 and that there is little association between measures of $U_{it}$ and $P_{it}$, meaning individuals' partisanship is temporally stable and not responsive to changes in the state of the world.   Green and Yoon (2002) conclude the stability in individual partisanship is all attributable to stable individual effects and not to a strong autoregressive process.  This result implies that $\rho_{i} = 0$ and $U_{it} = U_{i}$, a constant for each individual that is not responsive to changes in the state of the world or that the state of the world is unchanging.  Wawro (2002) disputes this result and using an estimator he says is more appropriate and reliable estimates $\rho_{i}$ to be about 0.45.  He also finds significant uniform shifts in partisanship at each election period, implying $U_{it} = U_{i} + D_{t}$ where $D_{ t}$ is a time dummy.  

A few of the micro studies include individual heterogeneity in $\rho_{i}$, as proposed in eq. \ref{eq:pit2}.  Franklin and Jackson (1983) model $\rho_{i}$ as a function of age and find that it increases with age, and presumably with political experience, as in Achen's model.  Green and Yoon (2002) estimate their model stratified by age and political interest and find that $\rho_{i}$ does not differ from zero for any group.   Bartels, et al. (2011) estimate a multinomial logit model similar in specification to Green and Yoon's with similar results, except they find that the coefficient on past partisanship is higher among the oldest voters, a result consistent with Achen's Bayesian model.$^{4}$  Clarke and McCutcheon (2009) propose and estimate a model where the electorate is composed of movers and stayers.  The former, who shift their partisanship from one period to the next in a Markovian fashion, are estimated to be thirty to thirty-five percent of the British electorate and forty-five to fifty-five percent of the U. S. and Canadian electorates.  Stayers are those who maintain the same partisanship across periods. The models of Green and Yoon, Bartels, et al. and Clarke and McCutcheon do not include a utility term, so the stability could be interpreted as being the result of stable utilities, as in Achen's model.  These, and any other, individual differences can be incorporated in eq. \ref{eq:pit2} with appropriate specifications for $\alpha_{0}$ and $h(i)_{t}$.  For example, in Clarke's and McCutcheon's model with $\alpha_{0} = 0$, $h(i)_{t} = 0$ for movers and $h(i)_{t} = 1$ for stayers at time $t$.  Other propositions about individual heterogeneity, such as any related to age, political knowledge, or ethnicity, can be accommodated by appropriate specifications for $h(i)_{t}$.  These must be developed for particular political contexts and with appropriate substantive justification.  

If $\gamma_{t} = c$, a constant, meaning the amount of updating does not depend on changes in the state of the world, partisanship updating proceeds in a temporally constant manner as individuals observe new utilities and accumulate experience with their partisanship.  This is the specification estimated (as far as we are aware) in existing models of individual partisanship.   If $\gamma_{t}$ is not a constant but varies with changes in the state of the world then only in the absence of exogenous shocks, $\Delta S_{t}= 0$, will partisan updating take place as in the Achen model designed for periods of stability.  But when there are shocks that alter individual utilities, such that $|\Delta S_{t}| > 0$, people give increased weight to their utilities and less weight to past partisanship in choosing their current partisanship.   

One important implication if $\gamma_{t}$ in eq. \ref{eq:pit2} varies with changes in the state of the world is that models of individual partisanship estimated with data from different elections will estimate different values for the coefficients on lagged partisanship and on any exogenous variables in the model.  These temporal differences will be determined by the magnitude of $\gamma_{t}$,  which is related to the magnitude of $|\Delta S_{t}|$.    

Eq. \ref{eq:pit2} is altered in an important way before developing the model for macro partisanship.  The alteration is the addition of an error correction term.  In the common error correction model the value for $P_{it}$ adjusts to the deviation between $P_{i,t-1}$ and its equilibrium value, given by $U_{i,t-1}$, $(P_{i,t-1} - U_{i,t-1}).$ (See Clarke and Stewart, 1994.)  In our model the adjustment is to the deviation of $P_{i,t-1}$ from its systematic component in eq. \ref{eq:pit2}, $P_{i,t-1} - [\rho_{i,t-1}P_{i,t-2} - (1 - \rho_{i,t-1})U_{i,t-1}]$.  The error correction term relative to $P_{i,t-1}$'s equilibrium value is nested within our specification,  \begin{eqnarray} P_{it} & = &  \rho_{it}P_{i,t-1} + (1 - \rho_{it})U_{it}  -\delta \left\{ P_{i,t-1} - [\rho_{i,t-1}P_{i,t-2} - (1 - \rho_{i,t-1})U_{i,t-1}] \right\} + \epsilon_{i,t} \nonumber \\ & = &  \rho_{it}P_{i,t-1} + (1 -\rho_{it})U_{it}    -\delta(P_{i,t-1} - U_{i,t-1}) + \delta \rho_{i,t-1}(P_{i,t-2} - U_{i,t-1}) + \epsilon_{it}.  \label{eq:pitx}  \end{eqnarray}   The $\delta (P_{i,t-1} - U_{i,t-1})$ term in eq. \ref{eq:pitx} is the error correction term where adjustments are based on how lagged partisanship differs from its equilibrium value that period.  Eq. \ref{eq:pitx} shows that this standard error correction model is nested within the error correction model proposed here.  Another coefficient $d$ added to the last term enables a statistical test to discriminate between the two models, \begin{equation} P_{it} =  \rho_{it}P_{i,t-1} + (1 -\rho_{it})U_{it}   -\delta(P_{i,t-1} - U_{i,t-1})  + d \delta \rho_{i,t-1}(P_{i,t-2} - U_{i,t-1}) + \epsilon_{it}.   \label{eq:pit3} \end{equation}  The error correction based on deviations from expected equilibrium values implies $d = 0$ while the error correction model proposed here implies $d = 1$.  

There is a substantive difference in the two models.  The implication of eq. \ref{eq:pit3} with $d = 1$ is that individuals adjust their partisanship at time $t$ to the difference between their partisanship at $t-1$ and its expected value given their partisanship at $t-2$ and their partisan utility at time $t-1$.   Think of the terms $\rho_{it}P_{i,t-1} + (1 - \rho_{it})U_{it}$ as the sequence or path of values that $P_{it}$ would follow given the exogenously determined values for $U_{it}$.  In the model proposed here corrections are made to deviations from this path rather than to deviations from the equilibrium values, which are $U_{i,t-1}$.  A consequence of this difference is that the model that corrects for deviations from the equilibrium value will adapt to a permanent change in $U_{it}$ faster than the one that corrects for deviations from the expected sequence but both converge to the same equilibrium given by $U_{it}$.  

We have to this point proposed a rich model of the dynamics of individual partisanship.  The model in eq. \ref{eq:pit3} contains both temporal and individual heterogeneity and two possible error correction mechanisms.  By varying the specification of different components, such as the $h(i)_{t}$ and $\gamma_{t}$ functions, and parameters, such as $\alpha_{0}$ and $d$, the equation accommodates the various common models of individual partisanship, including Achen's Bayesian model, the slow random walk proposed by Erikson, MacKuen and Stimson (1998), and the models where some individuals adapt to new information more rapidly than others.$^{5}$  Any macro model derived from this structure should also be general and consistent with these various propositions.  

\section*{\large \bf 3 Aggregate Partisanship}

\hspace{.25in} We now turn to the question of how the model of micro partisanship aggregates to a model of macropartisanship.   Achen and Shively (1995) argue for macroconsistency, meaning that models relating aggregate level variables, such as macropartisanship, should be derivable from theoretically justifiable micro level models and an explicit aggregation process.  ``Without that constraint, macrolevel research too easily slips into studies of the interrelationships of meaningless statistical aggregates.  Only when both macrotheoretical propositions and statistical assumptions are rigorously inferred from the microlevel can we have faith in macrolevel studies.'' (Achen and Shively, 1995, p. 25.)  We agree with Achen and Shively, but this consistency is often elusive.  In their seminal paper on macropartisanship, MacKuen, Erikson and  Stimson state that, ``... [their] findings will neither support nor undermine particular models of individual partisanship.''  (1989, p. 1129)  In later work, Erikson, et al. (1998, 2002) estimate a macro partisanship model and link it descriptively to a micro model, but they do not present the algebra connecting the two models.$^{6}$    

The work on macro partisanship is extensive, methodologically diverse, and often susceptible to Achen's and Shively's criticism about the lack of micro foundations.  Our purpose is not to review or assess these studies and their associated methodologies.  (See Brandt and Freeman, 2009, for a sophisticated critique and results that contradict the proposition that macro partisanship is affected by economic conditions.)  The theme here is to begin with a rich micro foundation, to develop its aggregate properties, and where relevant to relate existing results to the terms in this integrated micro and macro model.

The absence of an explicit link from the micro to the macro model of partisanship has been a missed opportunity, and goes beyond the satisfaction of logical consistency.    Substantive conclusions about empirical changes in partisanship differ depending on whether the focus is the micro-survey based public opinion literature or the macropartisanship literature.  With tight conceptual connections across levels of analysis we might possibly improve some of the debates---what is the typical length of time that a negative opinion of the incumbent president influences individuals' partisanship, for instance?  The desired transportability requires consistent micro and macro models. This objective motivates what follows.  

The model of macropartisanship aggregates eq.~\ref{eq:pit3} for any group of interest, including the electorate as a whole.  Assume this group has $N$ members so that 
\begin{eqnarray} \bar{P}_{t}  & = & N^{-1} \sum P_{it} \nonumber \\ 
& = & N^{-1} \sum \langle [ \alpha_{0} + h(i)_{t}]\gamma_{t}P_{i,t-1} + \{1 - [\alpha_{0} + h(i)_{t}]\gamma_{t}\} U_{it}  \nonumber \\ 
& &   - \delta (P_{i,t-1} - U_{i,t-1}) + d \delta [\alpha_{0} + h(i)_{t-1}]\gamma_{t-1}(P_{i,t-2} - U_{i,t-1}) + \epsilon_{it}  \rangle \nonumber \\ 
& = & N^{-1} \left\{  \alpha_{0} \gamma_{t}  \sum P_{i,t-1} + (1 - \alpha_{0} \gamma_{t}) \sum U_{it} -\delta \left( \sum P_{i,t-1} - \sum U_{i,t-1} \right)    \right. \nonumber \\ 
& & + d \delta a_{0} \gamma_{t-1} \left( \sum P_{i,t-2} - \sum U_{i,t-1} \right)+ \gamma_{t} \left(  \sum h(i)_{t}P_{i,t-1} - \sum h(i)_{t}U_{it} \right) \nonumber \\ 
& &  \left. + d \delta \gamma_{t-1} \left[ \sum h(i)_{t-1}P_{i,t-2} - \sum h(i)_{t-1}U_{i,t-1} \right] + \sum \epsilon_{it} \right\} \nonumber \\ 
& = & \alpha_{0} \gamma_{t} \bar{P}_{t-1} + (1 - \alpha_{0} \gamma_{t}) \bar{U}_{t}  - \delta (\bar{P}_{t-1} - \bar{U}_{t-1}) +d \delta a_{0} \gamma_{t-1}(\bar{P}_{t-2} - \bar{U}_{t-1}) \nonumber \\ 
& & +  \gamma_{t} \left( N^{-1} \sum h(i)_{t}P_{i,t-1} -  N^{-1} \sum h(i)_{t}U_{it} \right) \nonumber \\ 
& & + d \delta \gamma_{t-1} \left[ N^{-1}\sum h(i)_{t-1}P_{i,t-2} - N^{-1}\sum h(i)_{t-1}U_{i,t-1} \right] + \bar{\epsilon}_{t}.   \label{eq:pbar} \end{eqnarray}  
Eq.~\ref{eq:pbar} indicates that current mean partisanship is a function of current and lagged mean utilities, mean partisanship lagged one and two periods, the cross products of the current experience term and past partisanship and current utilities, and the cross products of lagged experience and lagged utilities and the second lag of partisanship plus the mean value of the contemporaneous stochastic element.  The weight given to some of these different components, $\gamma_{t}$ and $\gamma_{t-1}$, varies temporally depending upon the magnitude of the changes in the state of the world, $|\Delta S_{t}|$.  The structure described in eq. \ref{eq:pbar} is very general with important and interesting implications for models of macropartisanship and their estimation. 

\section*{\large 3.1 {\it Individual Homogeneity}}

\hspace{.25in}  First consider the model with individual homogeneity, meaning $h(i)_{t} = 0$, so the cross product terms in eq. \ref{eq:pbar} drop out leaving, \begin{eqnarray}   \bar{P}_{t} & = & \alpha_{0} \gamma_{t} \bar{P}_{t-1} + (1 - \alpha_{0} \gamma_{t})\bar{U}_{t} - \delta (\bar{P}_{t-1} - \bar{U}_{t-1}) + d \delta a_{0} \gamma_{t-1} (\bar{P}_{t-2} - \bar{U}_{t-1}) + \bar{\epsilon}_{t}.   \label{eq:pbar1} \end{eqnarray}     

\section*{\normalsize 3.1.1 {\rm Temporal Homogeneity}}  

\hspace{.25in}  Temporal homogeneity means that $\gamma_{t} = \gamma_{t-1} = 1$.   What remains is the conventional model of macropartisanship that describes aggregate partisanship at time $t$ as a function of lagged aggregate partisanship, aggregate party utilities, and the error correction terms with the parameter $d$ distinguishing between the two forms of the error correction model,  \begin{eqnarray} \bar{P}_{t} & = & \alpha_{0}\bar{P}_{t-1} + (1 - \alpha_{0})\bar{U}_{t} - \delta (\bar{P}_{t-1} - \bar{U}_{t-1}) - d \delta \alpha_{0} ( \bar{P}_{t-2} - \bar{U}_{t-1}) + \bar{\epsilon}_{t}. \label{eq:ecm} \end{eqnarray}  The difference between the common error correction model of macropartisanship and eq. \ref{eq:ecm} is the inclusion of the $d \delta \alpha_{0} (\bar{P}_{t-2} - \bar{U}_{t-1})$ term, as discussed with the individual model.$^{7}$ A subtle but important implicit assumption in all the macro partisanship studies is contained in eq. \ref{eq:ecm}.  The $\bar{P}_{t-1}, \bar{P}_{t-2}$ and $\bar{U}_{t-1}$ terms all refer to the mean lagged values for the respondents in the survey done in period $t$ from which $\bar{P}_{t}$ is determined.  These lagged values are being approximated by the mean values calculated from the surveys done in periods $t-1$ and $t-2$, which constitute different individuals.  

\section*{\normalsize 3.1.2 {\rm Temporal Heterogeneity} } 

\hspace{.25in}   Temporal homogeneity may be reasonable in stable periods where there are no large, long-term changes in the state of the world.  In such periods individual partisan utilities, ``are drawn independently over time from a common distribution.'' (Achen, 1992, p. 199.)  Thus, it is likely that successive elections are homogenous with respect to the amount and type of individual partisan change.   

This homogeneity condition is more questionable in periods where there are large changes in the state of the world that might trigger significant systematic shifts in individuals' true party utilities, leading them to change the weights given to past partisanship and to current utility assessments in updating their partisanship.  These different weights are indicated by values for $\gamma_{t}$ that vary over time.  If the factors causing people to change their true party utilities persist over several elections then we expect variations in the persistence of past partisanship over this period.$^{8}$   

Models of macropartisanship with temporal heterogeneity are not prominent in the literature.  Lebo and Box-Steffensmeier (2008) use a dynamic conditional correlation (DCC) technique to show that the monthly correlation between the Index of Consumer Sentiment and Presidential Approval varies between 1978 and 2004, often in association with observable events such as foreign policy crises. Though they do not include macro partisanship in their analysis their findings could easily extend to the relationship between these or other variables and aggregate partisanship.$^{9}$

The Carmines' and Stimson's study of issue evolution (1989) is a natural candidate for the estimation of a macro partisanship model with temporal heterogeneity.  They document both an abrupt shift and a longer term evolution in both public opinion and party positions on civil rights beginning in 1963-64.  They do not, however, estimate an explicit model for macropartisanship that incorporates these two distinct but related shifts.    Their discussion though clearly cites the evolution in civil rights attitudes and party positions as a basis for shifts in partisanship among both the mass public and political elites.   They conclude that the history of opinions and partisanship constitutes a process of punctuated equilibria consisting of short periods of structural change followed by longer periods of stability.  Eq.~\ref{eq:pbar1} captures the possibility of this punctuated equilibria.  For appropriate specifications for $a_{0}$, $\gamma_{t}$, and $f(S_{t})$ there will be periods of substantial partisan change followed by stable periods.  

The presence of temporal heterogeneity has an important implication for the conventional micro level models of individual partisanship estimated with cross-sectional data.  These data, even when using panel data, refer to a given election where the shocks that might lead people to alter their partisan utilities and the weight given to these current evaluations in updating their partisanship are common to all or at least to large groups of respondents.  Consequently the cross-section studies can estimate these coefficients for each election but cannot assess how they change with altered external circumstances, such as a shift in party positions as Carmines and Stimson document with their civil rights study.   Time series data are required to estimate this model as that is the only way to observe the variations in $S_{t}$ that are hypothesized to produce the variations in $\rho_{t}$.   (For a seminal discussion of estimating models with time-varying parameters see Beck, 1983.) The payoff to such an analysis, however, could be high if one's goal is to understand partisan evolution and punctuated equilibria.  

\section*{\large \bf 3.2 {\it Individual Heterogeneity}}

\hspace{.25in} An important part of the models of individual partisanship is the individual heterogeneity in the weights given to past partisanship and current utilities, meaning that $h(i)_{t} \neq 0$.  With this addition eq. \ref{eq:pbar} includes the cross product terms that explicitly model individual differences in the updating of partisanship, as denoted in eq. \ref{eq:pit2}.  In Achen's model this heterogeneity is created by individual differences in the length of experience with their current partisanship.   This heterogeneity, and the variables included in $h(i)_{t}$, can be generalized to any set of factors that create differences in how individuals respond to new partisan evaluations, exogenous events, and to their past partisanship.  Frequently mentioned variables in addition to experience are political interest and knowledge.  More generally, such variables could be social and socialization factors that create attachments to a party independent of utilitarian evaluations.  The $h(i)_{t}$ could also denote purely random variations in individuals' updating processes, but this possibility is not pursued here.  The only restriction on $h(i)_{t}$ is that it is assumed to be linear in its parameters to permit the aggregation in eq. \ref{eq:pbar}.

Box-Steffensmeier and Smith (1996) and Box-Steffensmeier and DeBoef (2001) estimate fractionally integrated models of macropartisanship.  The Box-Steffensmeier and Smith model does not include any exogenous variables that could proxy $\bar{U}^{*}_{t}$ so they are only estimating the autoregressive coefficient, $a_{0}$.  Following proofs from Granger they conclude that fractional integration rules out individual homogeneity in the degree of persistence, meaning $\alpha_{i0} \neq a_{0}$ for all $i$.  They do not  model the individual heterogeneity directly, denoted here as $ h(i)_{t}$, but assume that $h(i)_{t}$ is distributed as a beta distribution on the range zero to one.  Their specification and assumption limit what can be concluded about the sources of individual heterogeneity and about partisan adjustment but their results argue strongly that heterogeneity exists.  

Box-Steffensmeier and DeBoef include a measure of macroideology in their study and stratify respondents  by education.$^{10}$  In the macropartisanship equations they observe fractional integration, meaning heterogeneity, but do not observe significant differences in the degree of fractional integration among their four education levels.  They do, however, observe different Granger causal patterns between macroideology and macropartisanship at different education levels.  These results indicate there is heterogeneity related to years of schooling in the link between macroideology and macropartisanship.  This implies heterogeneity related to education in the relationships in eq. \ref{eq:pbar}.      

The univariate time series specification in the two papers by Box-Steffensmeier and co-authors differs from the structure of the macropartisanship model in eq.~\ref{eq:pbar}.   The analyses in these two papers omit any exogenous variables that might lead to systematic partisan shifts, which are a central part of the micro and macro models in eqs. \ref{eq:pit2} and \ref{eq:pbar}.   Less obviously, the fractionally integrated models consider individual variation in $a_{0}$ but do not associate it with any explicit variables.  In the macropartisanship model in eq.~\ref{eq:pbar} it is the correlations of these variations, the $h(i)_{t}$, with past partisanship and current party utilities each time period that are important.  

Computing the cross product between $h(i)_{t}$ and $U_{it}$ is relatively straightforward, though we are not aware of any macropartisanship model that does so.  Both terms refer to the current election so that appropriate variables should be available in the surveys used to measure aggregate partisanship and attitudes.  For example, and to anticipate our empirical example, if the individual heterogeneity is related to age, $Age_{it}$, and current utilities are a function of attitudes on civil rights, $Att_{it}$, the cross product between age and attitudes for each time period can be calculated from available survey data, $C_{AgeAtt_{t}} = N_{t}^{-1}\sum Age_{it}Att_{it}$.   

The big difficulty is computing the cross product between $h(i)_{t}$ and $P_{i,t-1}, P_{i,t-2},$ and $U_{i,t-1}$ because this involves information from two time periods, which is not usually available in a given survey.  To continue with the age example, the required correlation between age and past partisanship is, $C_{AgeP_{t-1}} = N_{t}^{-1} \sum Age_{it}P_{i,t-1}$.  Unless one has a panel study it is not possible to associate the age of person $i$ at time $t$ with her or his partisanship at $t-1$.  As we suggest in our example it might be possible, by adopting certain assumptions, to estimate these cross products, but only for some variables.

The presence of both temporal and individual heterogeneity, meaning $\gamma_{t} \neq 1$ and $h(i)_{t} \neq 0$ in eq.~\ref{eq:pbar}, adds considerable complications to the model and to any estimation effort.  They also place considerable demands on the type of data required for any estimation.  There are the usual requirements, such as consistently and accurately measured variables at appropriate time periods for partisanship and for the variables that constitute the expression for party utilities.  In addition, the data must permit the calculation, or at least adequate estimation, of the cross product terms in these equations.  These are daunting tasks, and a full exploration is beyond the scope of this paper.  We try, however, to make headway, and use some readily available data to illustrate how such an estimation might proceed and to present the results of this estimation.

\section*{\large \bf 4 An Empirical Example}

\hspace{.25in} In what follows we build on the important work of Carmines and Stimson (1989) on civil rights attitudes to illustrate how one might estimate a full model of macropartisanship.  We use data collected by the ANES on partisanship, attitudes towards civil rights policy, and the perceived positions of the two parties on this issue for different subgroups of the electorate between the mid-1950s and 2000.   Extending the Carmines' and Stimson's analysis with the ANES data  provides a natural opportunity to show how one might estimate a model of macropartisanship that is linked to a model of micropartisanship with time-varying parameters and individual heterogeneity.   Given the limitations of these data, discussed below, we stress that this is presented as an illustration of how one can, or might, estimate the model of macropartisanship presented above with time-series data.  Better and more carefully constructed data are required in order to make anything beyond suggestive substantive claims.  

Our basic variable definitions are:  \begin{eqnarray*} 
\bar{P}_{jt} & = & N^{-1}\sum P_{ijt} = {\rm ~mean~ partisanship~ of~ members~ of~ group~} j {\rm ~at~time~} t, \\ 
\bar{A}_{jt} & = & N^{-1}\sum A_{ijt} ={\rm ~mean~ civil~rights~attitudes~ of~ members~ of~ group~} j \\ & &  {\rm ~at~time~} t, \\ 
\bar{D}_{jt} & = & N^{-1}\sum Dem_{ijt} ={\rm ~group~j's~location~of~Democratic~elites~at~time~}t,  \\ 
\bar{R}_{jt} & = & N^{-1}\sum Rep_{ijt} ={\rm ~group~j's~location~of~Republican~elites~at~time~}t, \\
\Delta D_{jt} & = & |\bar{D}_{jt} - \bar{D}_{j,t-1}| \\
\Delta R_{jt} & = & |\bar{R}_{jt} - \bar{R}_{j,t-1}| \\ 
\Delta G^{2}_{jt} & = & \Delta D^{2}_{jt} + \Delta R^{2}_{jt}  \\
\bar{L}_{jt} & = & {\rm ~mean~civil~rights~loss~} =  N^{-1}\sum( |A_{ijt} - R_{ijt}| - |A_{ijt} - D_{ijt}|), \\
 \Delta GDP_{t} & = & {\rm~percent~change~in~national~real~GDP~per~capita~in~the~year} \\ & & {\rm~preceding~the~election,} \\
  I_{t} & = & {\rm ~incumbent~administration~in~year~} t, {\rm~Dem~=~1, ~Rep.~=~-1} \\ 
  E_{t} & = & {\rm ~Retrospective~economic ~evaluation~} = \Delta GDP_{t} \ast I_{t}.
  \end{eqnarray*}  The groups referenced by $j$ are northern Whites, southern Whites, northern Blacks and southern Blacks.  We expect the sequences of partisanship, attitudes, and perception of party positions to be different between these groups.  This would make it unwise to aggregate these into a single time series.  Higher values of $\bar{D}_{jt}$ or $\bar{R}_{jt}$ indicate that members of group $j$ associate the Democrats, or Republicans, with more pro-civil rights policies.  The variable $\bar{L}_{jt}$ is the mean difference in the absolute distance between individual civil rights preferences and the two party's positions for members of group $j$ at election $t$.$^{11}$  Berinsky and Lewis (2007) find that the absolute spatial differences are a better fit to associations between utilities and various outcomes than the more theoretically tractable quadratic utility function, hence our use of it for empirical purposes. The expression for party utility in eq. \ref{eq:pbar} is, \begin{equation} \bar{U}_{jt} = \beta_{0} + \beta_{1}\bar{L}_{jt} + \beta_{2}E_{t}. \label{eq:util1} \end{equation}
  
Temporal heterogeneity is modeled as, \begin{eqnarray} 
\gamma_{jt} & = &  e^{-\gamma( \Delta G^{2}_{jt} + \Delta G^{2}_{j,t-1} + \Delta GDP_{t}^{2} + \Delta GDP_{t-1}^{2})} . \label{eq:A} \end{eqnarray} The rate of adjustment in partisanship is related to the amount of current and lagged change in party positions and to the amount of change in real per capital income in the two years before the election.  The implication, as discussed above, is that the larger the changes in the state of the world, in this case party positions and the economy, the less weight given to previous partisanship and the more weight to current utilities.  The squared values follow the regression analogy given earlier and mean that small changes in party positions or the economy receive disproportionally less weight in calculating how $\gamma_{jt}$ differs from one.

Individual heterogeneity has two components.  The first is taken from Achen's Bayesian model with updating.  Data on how many times individuals participated in elections, and hence might have acquired new information with which to update their partisanship, is not available in ANES surveys.  This information variable is proxied by the number of elections, $n_{ijt}$, people could have participated in, given their age.  The proxy for election information is $I_{ijt} = (n_{ijt} - 1)/n_{ijt}$.  The second source of heterogeneity is individuals' education.  Zaller (1992) proposes a rising then falling relationship between political awareness and the stability of partisanship and the impact of new information.  Awareness is proxied by years of schooling beyond eighth grade, $S_{ijt}$, and the non-linear proposition by a spline function, $S^{*}_{ijt} = (S_{ijt} - S^{*})d_{ijt}$, where $S^{*}$ is the knot in the spline function and $d_{ijt}$ is a dummy variable that equals one if $S_{ijt} \geq S^{*}$.  The expression for individual heterogeneity is, \begin{equation} h(i)_{t} = \alpha_{0} + \alpha_{1} I_{it} + \alpha_{2} S_{it} + \alpha_{3}S^{*}_{it}. \label{eq:ht} \end{equation}  

Denote by $C_{XZ}$ the cross product between the variables $X$ and $Z$ during a given election and by $C_{XZ_{t-1}}$ the cross-product between $X$ and lagged $Z$.  For example, the cross product between information and spatial loss in the current election would be $C_{IL} = N^{-1} \sum I_{ijt}L_{ijt}$ and between schooling and lagged partisanship would be $C_{SP_{t-1}} = N^{-1} \sum S_{ijt}P_{ij,t-1}$, and so on.  Combining terms in eqs. \ref{eq:pbar} and \ref{eq:ht} gives the following macropartisanship equation for group $j$,  \begin{eqnarray} \bar{P}_{jt} 
& = &  \alpha_{0}\gamma_{jt} \bar{P}_{j,t-1} + (1 - \alpha_{0}\gamma_{jt}) \bar{U}_{jt} - \delta(\bar{P}_{j,t-1} - \bar{U}_{j,t-1})  \nonumber \\ 
& & + \gamma_{jt} \left[ a_{1} \left( C_{IP_{t-1}} - C_{IU} \right) + a_{2} \left( C_{SP_{t-1}} - C_{SU} \right) + a_{3} \left( C_{S^{*}P_{t-1}} - C_{S^{*}U} \right) \right] \nonumber \\ 
& & + d \delta \gamma_{j,t-1} \left[ a_{0} \left( \bar{P}_{j,t-2} - \bar{U}_{j,t-1}\right) + a_{1} \left( C_{I_{t-1}P_{t-2}} - C_{I_{t-1}U_{t-1}} \right)  \right. \nonumber \\ & & + a_{2} \left( C_{S_{t-1}P_{t-2}} - C_{S_{t-1}U_{t-1}} \right)\left.  + a_{3} \left( C_{S^{*}_{t-1}P_{t-2}} - C_{S^{*}_{t-1}U_{t-1}} \right)   \right] + \bar{\epsilon}_{jt}.  \label{eq:pbart} \end{eqnarray}
 The estimation substitutes the expression for party utilities in eq. \ref{eq:util1} for the $\bar{U}_{jt}$ and $U_{ijt}$ terms and the expression for temporal heterogeneity in eq. \ref{eq:A} for the $\gamma_{jt}$ terms.  The cross products between information and schooling and past partisanship and the variables that constitute current utilities, $\bar{L}_{ijt}$ and $E_{t}$, must be calculated from survey data.   This is not straightforward  because the terms involving $P_{ij,t-1}, P_{ij,t-2}$ and $U_{ij,t-1}$ require the value of a variable in previous periods.  Part of the reason we refer to the subsequent effort as an illustration is that strong assumptions are required to produce these cross product terms.  

\subsection*{\large \bf 4.1 {\it Data}}

\hspace{.25in} We estimate the model using data available in the ANES Time Series Cumulative Data File (ANES, 2010).$^{12}$  The election studies asked two questions about attitudes on aid to African-Americans.   We splice together these two attitude questions to approximate a continuous series. The first, from 1956 to 1968, asks if the respondent supports or opposes federal government actions to insure that Blacks have equal access to jobs and housing.$^{13}$  The post-1968 question asks about more generic support for federal government aid for minorities without reference to a particular venue or policy.  The post-1968 question was also restructured as a seven point scale with two opposing policy options located at each end and respondents were asked to locate themselves and the two parties on this scale, relative to the two policy options.  Responses to all questions are recoded on a minus three to three scale so the overall metrics are comparable.

Assessing party positions also changes in 1970.  For the 1956 - 1968 period, which is critical for any model involving attitudes on civil rights issues, the ANES does not ask respondents to place each party on a civil right dimension, but simply asks which party is more likely to follow a specific policy.$^{14}$  The post-1968 question asks people to place the parties on the same opinion scale measuring respondents' preferences.  For the pre-1970 data, $Dem_{jt} = 2$ and $Rep_{jt} = -2$ if a respondent thinks the Democrats are more likely to pursue a pro-civil rights policy, both equal zero if there is no difference, and $Rep_{jt} = 2$ and $Dem_{jt} = -2$ if the Republicans are more pro-civil rights.  For the post-1970 questions, $Rep_{jt}$ and $Dem_{jt}$ are the respondents' placements of the Republicans and Democrats on the seven point scale, rescaled to the minus three to three format.$^{15}$  The only variable measured consistently over this time period is partisanship, which is the standard ANES seven point scale, with minus three indicating a strong Republican, zero an Independent and three a strong Democrat.     

The terms in eq. \ref{eq:pbart} that include current utilities are constructed from the cross-products from the individual data in each election survey.  For example, one term is the cross product of information and spatial loss, $C_{IL} =(1/N_{t})\sum I_{ijt}L_{ijt}$.  This can be calculated with the responses in each election's ANES survey.  Similarly for the cross products with $S, S^{*}, E$, etc.  The difficulty arises with the cross products that include $P_{ij,t-1}$ as this variable refers to information from the previous election, which is not in most ANES surveys.$^{16}$   Our methodology requires two assumptions.  The first is that the value for the variable measured at $t-1$, such as $I_{ij,t-1}$ or $S_{ij,t-1}$ can be used to calculate the value for that variable at time $t$.  For example, since $I_{ijt}= (n_{ijt} - 1)/n_{ijt}$, in the election at $t-1$ $I_{ijt} = n_{ij,t-1}/(n_{ij,t-1} + 1)$.  From this we estimate $\hat{C}_{IP_{jt}} = (1/N_{t-1}) \sum \frac{ n_{ij,t-1}}{(n_{ij,t-1} + 1)}P_{ij,t-1}$.   In the case of years of schooling we make the assumption that $S_{ijt} = S_{ij,t-1}$, which is reasonable for older voters but less so for younger ones, who may still be in school.  Thus, $\hat{C}_{SP_{t}} =(1/N_{t-1}) \sum S_{ij,t-1}P_{ij,t-1}$.  

The second assumption is that these cross products calculated from the data collected during the survey during the election at time $t-1$ would equal the cross products that would exist in the survey data collected during the election at time $t$, if these data were available.  We are also maintaining the previous set of assumptions from above.   In other words, $(1/N_{t})\sum \frac{ (n_{ijt}-1)}{n_{ijt}}P_{ij,t-1} = (1/N_{t-1})\sum \frac{ n_{ij,t-1} }{(n_{ij,t-1} + 1)}P_{ij,t-1}$ and $(1/N_{t})\sum S_{ijt}P_{ij,t-1} = (1/N_{t-1}) \sum S_{ij,t-1}P_{ij,t-1}$.  The weakness of this assumption is that a representative sample of the electorate at election $t-1$ may not match a representative sample during the election at $t$ because of the changing demographics and mobility of the electorate.  Our defense is one of practicality and necessity, as we do not have the necessary panel data for each election. 

\subsection*{\large \bf 4.2 {\it Estimation}}

\hspace{.25in} Substituting the expressions for $\bar{U}_{jt}$ and $\gamma_{jt}$ from eqs. \ref{eq:util1} and \ref{eq:A} into eq. \ref{eq:pbart} and rearranging some terms gives the full model to be estimated, 
\begin{eqnarray} \bar{P}_{jt} & = & \beta_{0} + \beta_{1}\bar{L}_{jt} + \beta_{2}E_{t} - \delta [\bar{P}_{j,t-1} - ( \beta_{0} + \beta_{1}\bar{L}_{j,t-1} + \beta_{2}E_{t-1})] \nonumber \\ & & + e^{-\gamma(G^{2}_{jt} + G^{2}_{j,t-1} + \Delta GDP_{t}^{2} + \Delta GDP_{t-1}^{2})} \left \{ \alpha_{0} \left[ \bar{P}_{j,t-1} - (\beta_{0} + \beta_{1}\bar{L}_{jt} + \beta_{2}E_{t}) \right] \right. \nonumber \\ & & +\left.  \alpha_{1}\left[ C_{I_{j}P_{j,t-1}} - (\beta_{0} \bar{I}_{jt} + \beta_{1}C_{I_{j}L_{j}} + \beta_{2}C_{I_{j}E_{j}})  \right]  \right. \nonumber \\ & & \left. + \alpha_{2} \left[ C_{S_{j}P_{j,t-1}} - (\beta_{0}\bar{S}_{jt} + \beta_{1}C_{S_{j}L_{j}} + \beta_{2}C_{S_{j}E_{j}})  \right] \right. \nonumber \\ & & \left. + \alpha_{3} \left[ C_{S^{*}_{j}P_{j,t-1}} - (\beta_{0}\bar{S}^{*}_{jt} + \beta_{1}C_{S^{*}_{j}L_{j}} + \beta_{2}C_{S^{*}_{j}E_{j}}) \right] \right\}  \nonumber \\ & & + d \delta e^{-\gamma(G^{2}_{j,t-1} + G^{2}_{j,t-2} + \Delta GDP_{t-1}^{2} + \Delta GDP_{t-2}^{2})}  \left\{ a_{0} \left[ \bar{P}_{j,t-2} - \left( \beta_{0} + \beta_{1}\bar{L}_{j,t-1} + \beta_{2}E_{t-1} \right) \right] \right.  \nonumber \\ & & + a_{1} \left[ C_{I_{j,t-1}P_{j,t-2}} - (\beta_{0} \bar{I}_{j,t-1} + \beta_{1}C_{I_{j,t-1}L_{j,t-1}} + \beta_{2}C_{I_{j,t-1}E_{j,t-1}}) \right] \nonumber \\  & &  + \alpha_{2} \left[ C_{S_{j,t-1}P_{j,t-2}} - (\beta_{0}\bar{S}_{j,t-1} + \beta_{1}C_{S_{j,t-1}L_{j,t-1}} + \beta_{2}C_{S_{j,t-1}E_{j,t-1}})  \right] \nonumber \\ & & \left. + \alpha_{3} \left[ C_{S^{*}_{j,t-1}P_{j,t-2}} - (\beta_{0}\bar{S}^{*}_{j,t-1} + \beta_{1}C_{S^{*}_{j,t-1}L_{j,t-1}} + \beta_{2}C_{S^{*}_{j,t-1}E_{j,t-1}}) \right]  \right\}   + \bar{\epsilon}_{jt}. \label{eq:pest} \end{eqnarray} Eq. \ref{eq:pest} contains a lot of terms but only ten parameters: $(\beta_{0}, \beta_{1}, ~\&~ \beta_{2})$ in the utility equation; $(a_{0}, a_{1}, a_{2}, ~\&~ a_{3})$ in the equation for the individual heterogeneity in the autoregressive parameter, $\rho_{it}$; $\gamma$ for the time varying component of $\rho_{it}$; $\delta$, the error correction term; and $d$, which distinguishes between the two forms of error correction.  This is a total of ten parameters.   The model, however, is non-linear in the coefficients, which is accommodated using a non-linear least squares estimator.  (See Davidson and MacKinnon, 1993 and 2004.)  All of the cross product and lagged cross product terms were constructed in the manner and with the assumptions discussed above.  

Following the procedure in Clarke and Stewart (1994) we first checked to see if the key time series variables -- partisanship, civil rights loss and change in GDP per capita -- are stationary.  Because the data closely resemble a pooled cross-section time-series structure with the same time periods observed for the four groups the augmented Dickey-Fuller tests are conducted within this framework.$^{17}$  The evidence rejects the null hypothesis of unit roots.  The Chi-squared values for the three series are 18.48(8), 34.10(8) and 6.96(2), respectively which with appropriate degrees of freedom shown in parenthese have p-values of 0.018. 0.000, and 0.031.  From this we conclude that the series are stationary and eq. \ref{eq:pest} can be estimated as shown.

\subsection*{\large \bf 4.3 {\it Estimated Parameters}}

\hspace{.25in} Five different models are estimated for comparative purposes.  The first is the more conventional macro partisanship model with no heterogeneity as shown in eq. \ref{eq:ecm}.  The next two introduce temporal and individual heterogeneity, eq. \ref{eq:pbar1} and eq. \ref{eq:pbar} with $\gamma_{t} = 1$, respectively.  The fourth model contains both temporal and individual heterogeneity, as shown in eq. \ref{eq:pest}. The last model is the full model but with the specification parameter $d = 1$, which is the stochastic error correction model proposed here. The estimated equations, sum of squared residuals, log likelihood, and $R^{2}$ corrected for degrees of freedom are shown in Table \ref{tbl:results}.   The results support the error correction model proposed here, in contrast to the model with corrections to the equilibrium value, as in all four models $d > 0$ and in fact $d > 1$.   The value of $d = 1.17$ in the fourth model is not statistically significant from one according to a likelihood ratio test computed with from the log likelihood functions.  This is consistent with our expectation and thus we discuss this fourth model subsequently.

The equation with both temporal and individual heterogeneity fit the partisanship series much better than the equation without heterogeneity and better than the equations with only one type.  All the coefficients assessing heterogeneity of both types are statistically significant at well below the 0.001 level.  To the extent the data and method for estimating the cross product terms are credible the results indicate that heterogeneity, both over time and across individuals, is an important part of macropartisanship.  

\begin{center} \center (Table~\ref{tbl:results}  About Here.) \end{center}

The estimates of individual heterogeneity are consistent with expectations.  The knot in the relationship between schooling and the weight given to past partisanship occurs at twelve years of schooling, or a high school education.$^{18}$  The weight given to past partisanship increases by 0.19 for each year of schooling up to twelve years and then decreases by 0.07 for each year above twelve.  

The effect of accumulated information from more elections increases the weight of previous partisanship, as the Bayesian model predicts.  The coefficient is not statistically significant, but this illustrates a difficulty with estimating some relationships with aggregate data. The electorate's age distribution does not vary much, even over forty-five years, which means that the information variable has only a small variance for this sample.  The range, for example, is only 0.77 to 0.91 and a standard deviation of 0.024.  The result is consistent with the Bayesian model and some individual level studies that relate partisan stability to age but we are left with little confidence in this estimate.   

The estimate for $\gamma$ indicates there is considerable temporal heterogeneity, with $\gamma_{jt}$ varying from 0.60 to 0.99.  The latter value implies very stable aggregate partisanship if party positions and the economy are stable.  The values for $\gamma_{jt}$ were greater than 0.95 in 1958, 1990, 1994, and 1996.  The first three are mid-term elections with stable presidential administrations and the fourth was President's Clinton's reelection campaign. By contrast, values of $\gamma_{jt}$ were lowest in 1964, 1972, 1974 and 1984.  The first two featured presidential candidates who departed dramatically from traditional party positions, the third was the first post-Watergate election, and the fourth was after Reagan's first term, where he was trying to lead the Republicans, if not the country, in new directions.  

Macropartisanship is sensitive to the utilities people associate with the civil rights positions of each party and to changes in real income per capita.   Members of groups who support aid to minorities and perceive the Democrats as more likely to provide that aid are stronger Democrats.  Conversely, those who oppose aid and see the Republicans as opposing aid are stronger Republicans.   Income growth increases the partisan advantage of the incumbent party, and conversely for income decreases, which benefit the opposition.  One sample standard deviation difference in utilities and in income change are associated with 0.76 and 0.53 differences in long run partisanship respectively.  The yearly change then depends on the estimated values for the individual and temporal heterogeneity terms. The estimates for $\delta$ provide evidence that people adjust their partisanship if it deviated from its expected value in the previous period.  The estimated values for $\delta$ are consistently about 0.4 and are statistically significant.  An alternative model with no error correction terms, the null hypothesis that $\delta = 0$, was soundly rejected.  

The important conclusion is that a properly specified model of macro partisanship derived from a micro model includes both temporal and personal heterogeneity.  The empirical results are consistent with this specification, indicating that such heterogeneity likely is present in the electorate.  If the data, and therefore the results, are credible the empirical results imply that it is possible with aggregate data to obtain estimates for the evolution of partisanship that can be interpreted at the individual level.  More importantly aggregate data are necessary to estimate the temporal heterogeneity, as that requires substantial information from variables that only vary across many elections.  The results from this estimation are only useful in understanding the evolution of individual behavior if the micro and macro models are consistent with each other.        

\section*{\large  \bf 5  Conclusion}

\hspace{.25in}   This paper specifies a model of aggregate partisanship derived from a model of individual partisanship that contains the individual heterogeneity found in most micro models and that accommodates periods of partisan stability and restructuring.  The individual heterogeneity is maintained in the aggregate model through complex cross product terms.  The ability of the individual model to capture both partisan stability and change requires temporal heterogeneity in that these same relationships are likely to vary over time depending upon the political context.  This feature of the individual model is represented in the aggregate model with time varying parameters.  The advantage of this model is that information about these parameters and how they vary can be passed between the models at each level, which provides conceptual and empirical consistency and can incorporate empirical studies at both levels.  

Estimating the fully specified model poses serious challenges and likely requires analysis of individual and aggregate data.  Estimation of the individual heterogeneity, such as that associated with electoral experience, is better done with the variations available at the micro level.    Estimation of the time varying parameters, however, requires time series data, which is likely to be available only at the aggregate level.  

Creation of adequate data with which to estimate the full macro model is a daunting task, as should be evident from the illustration presented here.  This requires the calculation of cross product terms, some of which must be estimated from data in different surveys because of the lagged variables.  For certain types of variables some approximations should be acceptable, as illustrated here.  

Even defining the right data is a challenge.  The model with partisan change is predicated on there being a change in the state of the world, $S_{t}$, that leads to a restructuring of partisan utilities such that there might be a new equilibrium partisanship.  The civil rights issue from the late 1950's onward fits this description nicely, particularly during the 1960's.  The party's positions on foreign and defense policy before and after the Vietnam War might be another possibility.  The heavy reliance of most work on macro partisanship on short term economic fluctuations as measures of partisan utility is unlikely to capture the type of restructuring envisioned here.  These economic changes are largely cyclical fluctuations around a relatively stable economic equilibrium that is unlikely to stimulate long-term shocks to partisanship the way the civil rights issue did.  Were one to have the data available to examine macropartisanship before, during, and after the Depression of the 1930's one would likely see quite different results compared to what is seen in the macropartisanship literature.  

Developing data that captures this type of permanent change and where the variables and measures are consistent over a long time period for this issue is likely to require herculean efforts, and maybe herculean budgets.  The inability of the available ANES data to meet these needs for the civil rights issue is apparent in the previous discussion.  Compiling the necessary data requires careful decisions about the relevant time periods.  It requires assumptions about the frequency with which people update their partisanship, about what constitutes the state of the world and changes in that state, and about the level of aggregation.  We chose to aggregate by region and race for the example here as that seemed to be the best balancing of within and between group differences with respect to perceptions of the state of the world and partisan change and sample size within the aggregate.  Too high a level of aggregation will miss important partisan realignment if different groups of voters are moving between both parties. 

We wish to emphasize that this is primarily a methods paper designed to develop a model of macropartisanship that is consistent with a micro foundation and to explore the types of data and methods that could be used to estimate that model.  The model's value goes beyond just arguing for better data.  Accepting Achen's and Shively's (1995) caution that credible macro models require strong micro foundations means that it is important to have consistent micro and macro models.  This paper provides that consistency for macropartisanship research in the U. S., which should improve assessments of conclusions based on previous research.  Macro models that lack either or both individual and temporal heterogeneity are subject to potentially serious omitted variable bias.  The model also provides insight into the limitations of much micro level research on partisanship which cannot accommodate temporal heterogeneity and may well miss the importance of factors that contribute to shifts in individual and aggregate partisanship. 

We further wish to emphasize that this paper is not intended as a substantive contribution to the debate about macro partisanship, though we have pointed to comparisons between our model and results with those in this literature. The model, as formulated, is specific to a two-party system like the U. S., with its first-past-the-post system.  The modeling strategy, however, could be followed for other electoral contexts.  Such efforts must begin with a model of individual partisanship specified for the particular electoral context, such as the existence of multiple viable parties, the votes-to-seats rules, and the factors contributing to individual heterogeneity.  The aggregate implications of this model can then be derived following the strategy shown here.   This paper is presented as a challenge to scholars in this field to explore the micro-macro connections with better data than we have used here;  to some extent this paper poses rather than fulfills an agenda.  The structure offered here and its challenges should move the study of partisanship forward. 


\clearpage
\section*{\normalsize \bf \centering References}
\begin{description}
\addtolength{\itemsep}{-1em}
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\item Mickey, Robert, 2011. {\it Paths Out of Dixie: The Democratization of Authoritarian Enclaves in America's Deep South, 1944-1972}. Princeton, NJ:  Princeton University Press.

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\end{description}

\clearpage
\section*{\normalsize \bf \centering Endnotes}

\begin{enumerate}

\item If the variance of $V_{it}$ equals zero, eq. \ref{eq:pit1} is the Achen and the Gerber and Green model.

\item See Mickey, 2011, for an innovative study of the national consequences of this southern realignment.

\item The model, and we, are agnostic on whether systematic partisan shifts occur abruptly as Burnham contends or over an extended period as both are possible and consistent with the model.

\item The Green and Yoon model includes an individual specific constant term while the Bartels, et al. model treats this as a random coefficient specific to each respondent.

\item  See Franklin and Jackson (1983), Jackson (1975), Mebane and Wand (1997), and Zaller (1992).

\item We tried to derive the Erikson, et al. (1998) macro model from the equations for individual partisanship but do not believe it aggregates to their macro model without additional assumptions.

\item There are two other minor differences between the most common macropartisanship models and eq. \ref{eq:ecm}.  In eq. \ref{eq:ecm} aggregate partisanship is the mean of individual partisanship, presumably measured on some interval scale.  In studies of macropartisanship this variable is routinely measured as the proportion of Democrats among those who identify with a party, defined formally as $\bar{P}_{t} = \%Dem/(\%Dem + \%Rep)$, where $\%Dem$ and $\%Rep$ are the proportions of Democrats and Republicans in the survey at time $t$.  Though this conventional measure omits Independents and is bounded between zero and one it correlates very highly with measures of mean partisanship constructed by assigning arbitrary values to the NES seven point scale and aggregating that measure for electoral groups, suggesting very little empirical difference.  Also Erikson, et al. (1998, 2002) use a cumulative measure of partisan utility based on the indices of consumer sentiment and presidential approval to define the lagged equilibrium rather than the lagged values themselves. 

\item The model can be expanded easily to incorporate propositions about the length of the instability by including lagged values of $\Delta S$ in the model for the updating weights, $g(\Delta S_{t}, ... , \Delta S_{t-s})$.  In such a model, individuals' updating at time $t$ is related to current and lagged changes in the state of the world, allowing for a more dynamic learning process.

\item The DCC method is of limited use here, however, because it estimates the variations in the correlations between variables rather than in a specified set of parameters in the relationship among these variables.  This will make it difficult if not impossible to talk about the consistency of micro and macro models of partisanship, our main objective, because correlations are sample specific and there is no model that explicitly links these correlations.

\item Macroideology is defined as $\%Liberal/(\%Liberal + \%Conservative)$, where the variables refer to the percent of respondents in each survey who identify themselves as Liberals and Conservatives, respectively.  As with macropartisanship, this measurement approach omits respondents who do not claim to be in either of the two categories, e.g. moderates for ideology, or independents for partisanship.

\item Recent literature shows the importance of proximity measures in electoral decisions; for example, Jessee (2009), Lacy and Paolino (2010) and Tomz and Van Houweling (2008).  Sanders, et al. (2011) say that valence considerations are more important than proximity calculations in vote choices, though both are important.  They then show that proximity assessments are important contributors to valence considerations, making proximity calculations important in a reduced form vote equation. 

\item The party placements for 1956 and 1958 had to be obtained from the separate study files for those two years.  

\item There is a change in the question wording and response categories between 1960 and 1964 as well.  Before 1964 the question asked only about access to jobs and responses were based on a five point Likert agree/disagree scale.  In 1964 and 1968 the reference to housing was added and the question format changed to ask if this was something the federal government should do or not do. 

\item  There are actually two variations.  In 1956 and 1958 respondents were asked with party was most likely to pursue the policy the respondent preferred, which with the reponses to the policy question we can infer which party the person perceived as more pro-civil rights at the federal level.  Beginning in 1960 the ANES asked the respondent which party was most likely to promote equal treatment for Blacks.

\item The ANES did not ask for party placements in 1986, 1996 and 1998.  The placements for these years have been interpolated from the preceding and succeeding surveys.

\item The exception would be the few years where the ANES conducted a panel study, but we want a methodology that is consistent for all elections.

\item The Stata xtunitroot program with the Fisher test and the augmented Dickey-Fuller option is used.  The Fisher test permits gaps in the data, which exist in 1962 and 1966.  For change in GDP per capita the test was conducted for only one group as the variable is identical for all groups. 

\item The fit was best with the knot equal 12.5 years, but the difference  in the log likelihood function was only 0.08.  Because twelve years of schooling has a direct substantive interpretation and schooling is only measured in two year increments it seemed best to set the knot at twelve years.


\end{enumerate}

\clearpage

\begin{table} \centering \begin{tabular}{lccccc} & \multicolumn{4}{c}{Model Heterogeneity} \\ 
                      & None & Temporal & Individual & \multicolumn{2}{c}{Both}      \\
                      & Eq. \ref{eq:ecm} & Eq. \ref{eq:pbar1} & Eq. \ref{eq:pbar},  & \multicolumn{2}{c}{Eq. \ref{eq:pest}} \\ 
                      &                               &                                  & $\gamma_{t} = 1$ \\
 Party Utility \\                      
 ~~$b_{0}$ -- Constant & 0.49   & 0.38          & 0.38           &  0.35  & 0.38   \\
                                         & (0.14) & (0.19)        & (0.20)        & (0.18) & (0.16)  \\
~~$b_{1}$ -- Civil Rights & 0.89    &  0.77         & 0.73          & 0.71  & 0.68      \\
                                             & (0.14)  & (0.14)       & (0.14)       & (0.13) & (0.11)    \\ 
~~$b_{2}$ -- Economic Evaluations & 1.54    & 1.61          & 1.85         & 1.77  & 1.70       \\
                                                                 & (0.57)  & (0.67)       & (0.68)       & (0.69)  & (0.68)  \\
 Autoregressive Term \\
 ~~$a_{0}$ -- Constant & 0.75    & 1.05          & -0.41        & -0.18  & -0.25      \\
                                     & (0.06) & (0.07)       & ( 0.44)      & (0.54)   & (0.58)  \\
 ~~$a_{1}$ -- Experience & -          & -                & 0.79          & 0.76 & 0.89          \\
                                               &            &                   & (0.42)        & (0.51)   & (0.54)         \\
 ~~$a_{2}$ -- Schooling  & -      &          & 0.20          & 0.18     & 0.19          \\
                                             &         &          & (0.05)       & (0.05)   & (0.06)       \\
~~$a_{3}$ -- Schooling $\geq$ 12 & -         & -                & -0.24         & -0.23    & -0.26          \\
                                                              &            &                   & (0.08)        & (0.09)  & (0.10)        \\
~~$\gamma$ -- Temporal Heterogeneity   & -          &  0.19         & -                 & 0.11   & 0.13          \\
                                                               &             & (0.06)       &                    & (0.05)    & (0.05)      \\
Error Correction Term \\       
 ~~$\delta$ & 0.26   & 0.31          & 0.38           & 0.39   & 0.43   \\
                      & (0.11) & (0.10)       & (0.11)         & (0.11) & (0.10)     \\ 
 Model Specification Test \\
 ~~$d^{a}$ & 1.47 & 1.18 & 1.27 & 1.17 & 1 \\ 
                    & (0.33) &(0.19) & (0.16) & (0.14) &  \\~~ \\  
        L$^{*b}$ & 3.14   & 15.75        & 18.66         & 21.93 & 20.62  \\
    SSR$^{c}$ & 4.33 & 3.16        & 2.94         &  2.71 & 2.80   \\ 
    $\tilde{R}^{2^{d}}$ &  0.91 &  0.97 & 0.97 & 0.98 & 0.98  \\ ~~ \\
    \multicolumn{6}{l}{a.  Test of equilibrium error correction ($d$ = 0) v. stochastic error correction ($d$ = 1).} \\
                \multicolumn{6}{l}{b.  Value of the log likelihood function.} \\
              \multicolumn{6}{l}{c.  Sum of squared residuals.}  \\
              \multicolumn{6}{l}{d. $R^{2}$ Corrected for degrees of freedom.}
\end{tabular} \caption{Estimated Macropartisanship Equations w/o and with Heterogeneity.} \label{tbl:results} \end{table}

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