Specifying the Model

The method of point-to-block realignment assumes that the observed point-level (person) data and the extrapolated block-level averages (districts) have a joint Gaussian distribution. We start by specifying how the training side of the model works crvey data. Define now $ \mathbf{s} $ as a set of $ n $ observed sites $ \{{\mathbf{s}}_1,{\mathbf{s}}_2,\dots, {\mathbf{s}}_n\} $, where each $ {\mathbf{s}}_i $ represents the location of a survey respondent in space—either in latitude and longitude, or in northings and eastings (as we use in this application).^{Footnote 3} Here, $ \mathbf{Y}(\mathbf{s}) $ is an associated collection of outcomes $ \mathbf{Y}(\mathbf{s})=\{Y({\mathbf{s}}_1),Y({\mathbf{s}}_2),\dots, Y({\mathbf{s}}_n)\} $, the survey response of interest for the survey-taker at each site. $ {\mathbf{X}}^{*}(\mathbf{s})=\{{\mathbf{x}}^{*}({\mathbf{s}}_1),{\mathbf{x}}^{*}({\mathbf{s}}_2),\dots, {\mathbf{x}}^{*}({\mathbf{s}}_n)\} $ is a collection of covariates for each survey respondent observed at his or her respective point in space. We specify a linear model as follows:

(1)$$ \mathbf{Y}(\mathbf{s})=\mu (\mathbf{s})+\omega (\mathbf{s})+\kern0.50em {\mathbf{\epsilon}} \ (\mathbf{s}), $$

where $ \mu (\mathbf{s})=\mathbf{X}(\mathbf{s})\beta $ is the mean structure based on a linear additive component (like a standard regression model), $ \omega (\mathbf{s}) $ are realizations from a mean-zero stationary spatial process that captures spatial association (closer points are more informative than distant points), and ϵ(s) is a regular uncorrelated disturbance term.

An important feature of equation (1) is that the variance is split into two disturbance terms: one that captures spatial association, and the other that is a traditional independent and identically distributed error term with homoscedastic variance. We thereby use the following distributional assumptions for these two terms: $ \omega (\mathbf{s})\sim \mathcal{N}(0,{\sigma}^2\mathbf{H}(\phi )) $ and $ \kern0.50em {\mathbf{\epsilon}} (\mathbf{s})\sim \mathcal{N}(0,{\tau}^2\mathbf{I}) $. Several of the parameters of these variance components have a substantive interpretation. From the idiosyncratic error term, ϵ, we call the variance term $ {\tau}^2 $ the nugget. This is the amount of error variance in the outcome that is independent from spatial separation. We can think of this as the variance in the error when the geographic separation between observations is negligible. Turning to the spatial $ \omega $ error terms, $ {\sigma}^2 $ is called the partial sill. The partial sill reflects the variance that can be driven by geographic distance between two observations, with the assumption being that more distant observations have a higher variance. The partial sill equals the maximum amount of variance among observations due strictly to geographic separation. In fact, the nugget plus the partial sill equals the sill, which is the maximum total variance possible among distant observations. Finally, the other parameter feeding into the spatial $ \omega $ error terms is the range term, $ R=1/\phi . $ ($ \phi $ itself is called the decay term.) The larger the range term is, the farther the distance between observations before the variance among those observations equals the sill. In other words, the range term helps gauge the distance at which error variance is maximized.

The last piece of specifying $ \omega (\mathbf{s}) $ is that we must specify the function $ \mathbf{H}(\phi ) $. This is a parametric spatial correlation function that typically only requires us to estimate the decay parameter $ \phi $. We typically assume an isotropic model, which means that the level of spatial correlation does not depend on direction but only on the distance between the observations $ {d}_{ij}=\parallel {\mathbf{s}}_i-{\mathbf{s}}_j\parallel $. In this case, we must choose a parametric model—the exponential, spherical, wave, and Gaussian are a few common options—that captures the patterns of residual association in our data. Each of these parametric models specifies both a spatial correlation function (stating simply how much observations should correlate given their distance apart) and a semivariogram function (specifying how much observations should vary given their distance apart). The two fit naturally together with a high correlation implying a low variance and vice versa. Once we determine the best-fitting parametric correlation function, the product of the correlation function $ \mathbf{H} $ with the partial sill $ {\sigma}^2 $ builds the spatial covariance structure into the joint distribution of the $ \omega (\mathbf{s}) $ disturbance terms.

When determining the exact parametric specification of $ \mathbf{H} $, we normally focus on the related semivariogram to determine the right parametric structure. We choose the right model through an empirically driven process, wherein the empirical semivariogram is calculated from the residuals of an initial model. The formula for the empirical semivariogram is (Cressie Reference Cressie1993, 69):

(2)$$ \widehat{\gamma}(d)=\frac{1}{2|N(d)|}\sum_{(i,j)\in N(d)}|z({\mathbf{s}}_i)-z({\mathbf{s}}_j){|}^2, $$

where $ z({\mathbf{s}}_i) $ is the residual term for the respondent located at site $ {\mathbf{s}}_i $ from an initial linear model, $ d $ is an approximate distance of interest (possible distance values are usually coarsened into bins), $ N(d) $ is the set of all pairs of observations such that $ |z({\mathbf{s}}_i)-z({\mathbf{s}}_j)|\approx d $, and $ |N(d)| $ is the number of pairs in the set that are separated by distance $ d $. The semivariogram equals two quantities: the variance of all observations separated by distance $ d $ when pooled together, as well as half the variance of the differences $ (z({\mathbf{s}}_i)-z({\mathbf{s}}_j)) $ between observations separated by distance $ d $. Using the empirical semivariogram, we then determine which parametric model is most appropriate for our data, choosing from the exponential, spherical, wave, Gaussian, or some other parametric semivariogram. Once we have that, we know the related spatial correlation function (Banerjee, Carlin, and Gelfand Reference Banerjee, Carlin and Gelfand2015, 28–29). In our case here, the best-fitting model is the Gaussian semivariogram, so implies that our spatial correlation function should be $ H{(\phi )}_{ij}=\mathrm{exp}(-{\phi}^2{d}_{ij}^2) $.

With all of these elements in place for modeling the responses of survey takers, we now step back and think about where this training model fits relative to our forecasting process of state, congressional district, and state legislative district ideology. As our model assumes the observed point-level data and the extrapolated block-level averages have a joint Gaussian distribution, we get

$$ f((\begin{array}{c}{\mathbf{Y}}_{\mathbf{s}}\\ {}{\mathbf{Y}}_{\mathbf{B}}\end{array})|\beta, {\sigma}^2,\phi )=\mathcal{N}((\begin{array}{c}{\mu}_{\mathbf{s}}(\beta )\\ {}{\mu}_{\mathbf{B}}(\beta )\end{array}),{\sigma}^2(\begin{array}{cc}{\mathbf{H}}_{\mathbf{s}}(\phi )& {\mathbf{H}}_{\mathbf{s},\mathbf{B}}(\phi )\\ {}{\mathbf{H}}_{\mathbf{s},\mathbf{B}}^T(\phi )& {\mathbf{H}}_{\mathbf{B}}(\phi )\end{array})), $$

:

where $ {\mathbf{Y}}_{\mathbf{s}} $ represents the vector of ideology among individual citizens, $ {\mathbf{Y}}_{\mathbf{B}} $ represents the vector of ideology in all block-referenced constituencies of interest, and defines the correlation matrix of observations as before. Note that this presents the simplified case where there is no nugget effect $ ({\tau}^2) $, but the result still holds if the variance–covariance terms do include a nugget. By standard normal theory (e.g., Ravishanker and Dey Reference Ravishanker and Dey2002), the conditional distribution of our extrapolated block averages is:

(3)$$ \begin{array}{l}{\mathbf{Y}}_{\mathbf{B}}|{\mathbf{Y}}_{\mathbf{s}},\beta, {\sigma}^2,\phi \sim \mathcal{N}({\mu}_{\mathbf{B}}(\beta )+{\mathbf{H}}_{\mathbf{s},\mathbf{B}}^T(\phi ){\mathbf{H}}_{\mathbf{s}}^{-1}(\phi )({\mathbf{Y}}_{\mathbf{s}}-{\mu}_{\mathbf{s}}(\beta )),\\ {}\kern0.50em {\sigma}^2[{\mathbf{H}}_{\mathbf{B}}(\phi )-{\mathbf{H}}_{\mathbf{s},\mathbf{B}}^T(\phi ){\mathbf{H}}_{\mathbf{s}}^{-1}(\phi ){\mathbf{H}}_{\mathbf{s},\mathbf{B}}(\phi )]).\end{array} $$

These quantities can be estimated with Monte Carlo integration:

$$ {({\widehat{\mu}}_{\mathbf{B}}(\beta ))}_k={L}_k^{-1}\sum_{\mathrm{\ell}}\mu ({\mathbf{s}}_{k\mathrm{\ell}};\beta ) $$

$$ {({\widehat{H}}_{\mathbf{B}}(\phi ))}_{k{k}^{\prime }}={L}_k^{-1}{L}_{k^{\prime}}^{-1}\sum_{\mathrm{\ell}}\sum_{{\mathrm{\ell}}^{\prime }}\rho ({\mathbf{s}}_{k\mathrm{\ell}}-{\mathbf{s}}_{k^{\prime }{\mathrm{\ell}}^{\prime }};\phi ) $$

$$ {({\widehat{H}}_{\mathbf{s},\mathbf{B}}(\phi ))}_{ik}={L}_k^{-1}\sum_{\mathrm{\ell}}\rho ({\mathbf{s}}_i-{\mathbf{s}}_{k\mathrm{\ell}};\phi ) $$

We conduct this Monte Carlo integration using the technique of Bootstrapped Random Spatial Sampling (BRSS) developed by Monogan and Gill (Reference Monogan and Gill2016). Doing so allows us to forecast the average ideology with:

(4)$$ {\widehat{Y}}_{\mathbf{B}}={\widehat{\mu}}_{\mathbf{B}}(\beta )+{\widehat{H}}_{\mathbf{s},\mathbf{B}}^T(\phi ){\widehat{H}}_{\mathbf{s}}^{-1}(\phi )({\mathbf{Y}}_{\mathbf{s}}-{\widehat{\mu}}_{\mathbf{s}}(\beta )). $$

We account for the spatial element by forecasting $ \widehat{Y}({\mathbf{s}}_{k\mathrm{\ell}};\beta, {\sigma}^2,{\tau}^2,\phi ) $ and using this quantity in our Monte Carlo integration. With the nugget effect, from Y(s) = µ(s) + ω(s) + ϵ(s), we also get $ {\mathbf{Y}}_{\mathbf{s}}\sim \mathcal{N}(\mu, \Sigma ) $, where we still require $ \Sigma ={\sigma}^2\mathbf{H}(\phi )+{\tau}^2\mathbf{I} $, $ H{(\phi )}_{ij}=\rho (\phi, {d}_{ij}) $, and $ {d}_{ij}=||{\mathbf{s}}_i-{\mathbf{s}}_j|| $. Again, for this application, the Gaussian semivariogram function was the best fitting, meaning that our correlation function is $ H{(\phi )}_{ij}=\mathrm{exp}(-{\phi}^2{d}_{ij}^2) $.

Why Proximity Matters for Public Opinion

Tobler’s First Law of Geography states, “Everything is related to everything else, but near things are more related than distant things” (Tobler Reference Tobler1970, 236). Here, we assume that this law holds for individuals’ opinions and ideology also, with more physically proximate Americans holding a more similar political outlook. While there are physical barriers, such as highways and rivers, that separate populations and therefore can change ideology dramatically, our kriging approach connects these smoothly with no sudden shift.

Proximal influence in politics is supported by Gimpel and Schuknecht (Reference Gimpel and Schuknecht2003, 2–4)who describe two different approaches to understanding regionalism in this way. First, Gimpel and Schuknecht give a compositional approach asserting that political behavior is similar within a region due to economic interests, racial origin, ethnic ancestry, religion, social structure, and other related factors (Fischer Reference Fischer1989; Garreau Reference Garreau1981; Gastil Reference Gastil1975; Lieske Reference Lieske1993). Therefore, if all of these factors could be included in an empirical model, then the variability with geographic units in the model would be small. Clearly, though, it is often impossible to measure every relevant demographic and socioeconomic variable, or even to identify every critical variable for inclusion. As some relevant inputs may be overlooked or unmeasured, we assume that neighboring individuals will have a relatively similar political outlook, even holding included covariates constant. Second, Gimpel and Schuknecht give a contextual approach that offers the idea that citizens’ political attitudes and behaviors are influenced by political socialization and by interactions with other citizens in their social network, which is supported by a large literature in political science (DeLeon and Naff Reference DeLeon and Naff2004; Djupe and Sokhey Reference Djupe and Sokhey2011; Huckfeldt and Sprague Reference Huckfeldt and Sprague1995; Putnam Reference Putnam1966; Reference Putnam1993). For instance, “the first place to look for political networks is within the immediate physical proximity of each individual” (Sinclair Reference Sinclair2012, 26). This means that we expect under the contextual approach as well that geographically proximate individuals will have relatively similar opinions, even in a general setting.

Furthermore, Erikson, Wright, and McIver (Reference Erikson, Wright and McIver1993, 48) propose that “the unique political cultures of individual states exert an important influence on political attitudes.” This idea goes as far back as Elazar (Reference Elazar1966), who proposes that U.S. states can be categorized based on an individualist, moralist, or traditionalist view of government’s role. Erikson, Wright, and McIver (Reference Erikson, Wright and McIver1993, 56–68) also demonstrate that a higher proportion of variance in ideology and partisanship is predicted by state-level dummies than by demographic information, although state-level residuals will pick up some of the effects of unmeasured individual-level variables. There also is evidence that this holds in urban areas, where political culture shapes the impact of identity on public opinion and political participation, even in cities with heterogeneous neighborhoods (DeLeon and Naff Reference DeLeon and Naff2004, 703). Our method of kriging increases the ability to accurately model the effects of political culture, omitted predictors, and social context by including weighted neighbors’ residuals in forecasts of public opinion and ideology. For example, Western Kentucky and Southeast Illinois are similar places that are likely to be populated with similar people, both culturally and in demographic terms.

Data: 2008 CCES and 2010 Census

In this study, we use the 2008 CCES as training data to estimate our model of individual ideology as a function of demographics. The 2008 CCES offers 21,849 observations spread across the American states and congressional districts. This survey asks respondents to place themselves ideologically on a scale from 0 to 100, with 0 representing the most liberal and 100 the most conservative. Our training model predicts responses as a function of age, education, race, sex, income, religion, urban–rural status, homeownership, employment status, and a geographic trend term. CCES respondents were located geographically by ZIP code. Our procedure for locating these respondents is described in greater detail in the next section.

After estimating the training model over the CCES, we used 2010 Census data to forecast the ideology of 724,814 simulated citizens throughout the continental United States (Minnesota Population Center, 2011). We simulated by census block, the most precise geospatial unit the Census Bureau keeps track of: the first 701,050 simulated citizens were drawn purely in proportion to the population of the block, and the last 23,764 were drawn with one observation per census tract (proportionally by block within tract). We used Census Bureau maps of census blocks to place simulated citizens in latitude and longitude (or more exactly in eastings and northings), and we drew predictor values based on the variables’ local distribution for that block.

The 11 million census blocks perfectly tessellate all higher level geospatial indicators of which the Census Bureau keeps track, so there are no gaps and no overlaps of areal units. Hierarchically, blocks are nested within block groups, block groups are nested within tracts, and tracts are nested within counties. When simulating covariate values for a kriged point, if a predictor was not reported at the block level, we drew from the most precise level for a given location. More specifically, we simulate age, race, sex, and homeownership based on block-level data. We simulate education and income based on block group-level data. We simulate employment status based on tract-level data. We simulate religion and urban–rural status with county-level data. By using the 2010 census block data, all of these predictors are simulated with greater geographic precision and with more up-to-date data than in Monogan and Gill (Reference Monogan and Gill2016).

Moving Noncontiguous States

Any method of measuring the ideology of public constituencies ought to be comprehensive in covering all 50 states as well as all state legislative and congressional districts falling within each. A major challenge of using spatial data analysis to measure public opinion in the United States is that it is difficult to measure opinion in the noncontiguous states of Alaska and Hawaii. In fact, measuring public opinion in these two states is often difficult anyway on account of having few, if any, survey respondents in many national polls. Kriging, like many methods of spatial analysis, requires observations to have neighbors. If we attempted to train a kriging model using Alaska’s and Hawaii’s data as they are located on a map, then we would have extreme geographic outliers that could distort the estimation of our spatial error process model. This, in turn, would diminish our ability to make accurate predictions of opinion as we turned to forecasts because the partial sill would treat ocean distance as regular geographic space and create a smoothed spatial surface over broad swaths of the Pacific Ocean and Canada. When the goal is to understand and predict the opinions of American citizens, this is not a sound substantive approach.

There is no ideal solution for how to deal with the locational and data sparsity issues of Alaska and Hawaii. Every alternative has shortcomings. Omitting them is perhaps the worst option: 4 of the 100 U.S. senators and 3 of the 435 members of the House of Representatives come from these states. To omit them would leave out a substantial portion of America’s representative area in combined population (slightly over 2.1 million people total). Meanwhile, leaving their locations as is poses problems of extreme leverage (in a training model) or out-of-area forecasting (when predicting ideology of districts). We argue that it is substantively important to include all 50 states and that we can do so in a theoretically informed way by searching for the ideological neighbors of these states, in the absence of geographical neighbors. Certainly, the discontinuity of the United States presents a unique boundary value problem in cases like this. In terms of local cultural and economic nuances, there are likely to be similarities between Alaska and western Canada as well as between Hawaii and other Pacific islands.^{Footnote 4} However, our outcome variable of ideology is defined here in the American context as American politicians and journalists would discuss them, so the concepts would not easily traverse international borders. Given that the outcome variable is defined within the U.S. context, the ideological neighbor approach allows us to construct predictions based on similar domestic ideological contexts.

To address this, we proceed in two ways. First, when estimating the model itself, we omit Alaska and Hawaii from the training data. Their extreme outlier values could unduly affect the spatial variance components, so only the continental 48 states were included in the training data. Second, when forecasting ideology in these two states, we relocate Alaska and Hawaii to sit next to the west coast of the United States. Doing so greatly narrows the out-of-sample space that falls within the convex hull of our forecasting space. That is, the areas that are part of Canada, Mexico, or the Pacific Ocean that are included within the space of our smoothed kriging surface are shrunken dramatically relative to a model that uses these two states at their actual geographic location.

For the sake of forecasts, we locate Alaska and Hawaii near their ideological neighbors, or locales on the west coast as similar as possible ideologically. To find these states’ ideological neighbors, we estimated a training model on the continental 48 states and then chose the locations off the west coast that minimized predictive error for the two discontiguous states when using that model to predict Alaska’s and Hawaii’s observations in survey data. (Our full process is described in more depth in Online Appendix A.) Figure 2 shows the result of our procedure, illustrating how we relocated Alaska and Hawaii for the sake of the forecasting data. Specifically, that map shows a dot at the location of each census block’s centroid (a census block serving as our primary unit for sampling forecasting observations in kriging). The census blocks for the continental 48 states are in black at their original locations in eastings and northings. The census blocks for Hawaii (in blue) and for Alaska (in red) are at their new ideological neighbor locations in eastings and northings.

Figure 2. Map of census block centroids when Alaska and Hawaii are placed near their ideological neighbors from kriging forecasts.

Substantively in Figure 2, Hawaii has been relocated so that Honolulu is an ideological neighbor with San Francisco. Alaska has been repositioned so that Juneau is just south of San Diego. These positions, again, are the positions that minimize forecasting errors in the two discontiguous states, as detailed in Online Appendix A. This allows each state to have a west coast neighbor that is ideologically similar without producing any overlap between either state and the continental states. To preserve area and point-to-point distances, the original locations of census blocks (with Alaska and Hawaii at their actual locations) were reprojected from longitude and latitude into eastings and northings first. After this reprojection, Alaska and Hawaii were relocated to the positions shown on the map. This solution of finding ideological neighbors is far superior to the common solution to simply drop these two states from measurement models: according to 2010 Census numbers, dropping these states would mean ignoring over 2 million U.S. citizens (737,438 + 1,420,491 = 2,157,929) as of 2018. It is possible to model these two states separately, but that imposes the assumption that there is no influence back and forth between these two states and the contiguous 48 states. Our solution is a compromise between these two extremes that allows inclusion without deteriorating the quality of the total model.

ZIP Codes versus ZCTAs

Prior kriging work placed survey respondents on a map using ZCTAs, as computed by the census, when respondents’ geographic identifier was ZIP code. Mechanically, if a respondent is known to reside in a geographic area, he or she has to be placed at a specific coordinate using eastings and northings. This has been done by starting at the centroid of the areal unit and jittering within the radius of the unit’s area. The problem of doing this with ZCTAs when ZIP code is the true geographic identifier is that the area of ZCTAs does not exactly overlap with the areas covered by ZIP codes themselves (Beyer, Schultz and Rushton Reference Beyer, Schultz, Rushton, Rushton, Armstrong, Gittler, Greene, Pavlik, West Dale L. and Zimmerman2008; Grubesic Reference Grubesic2008; Grubesic and Matisziw Reference Grubesic and Matisziw2006). Hence, respondents could be placed at a position on the map that puts them in the wrong ZIP code, adding unnecessary measurement error to the model.

Figure 3 illustrates the problem. This figure draws the real map of the 30601 ZIP code in Georgia using data obtained by TomTom as well as the 30601 ZCTA using data obtained from census. The solid blue line shows the ZIP code boundary, and the dashed red line shows the ZCTA boundary. As can be seen, if we knew a resident lived in the 30601 ZIP code but placed them at a location in the ZCTA, we could make several mistakes. First, there are observable points in the ZCTA that are outside of the ZIP code. In the east (right) and the north (top) in particular, there are several places in the ZCTA that stray well outside of the ZIP code. If we placed a survey respondent who identified 30601 as his or her ZIP code in one of these portions of the ZCTA, we would have placed him or her in the wrong ZIP code. A second problem that emerges is that the ZCTA does not cover all of the ZIP codes. In the southeast (bottom-right) in particular, there is a large block of land where residents of the 30601 ZIP code could live. If we proceed to locate these individuals using the ZCTA, then we have no chance of putting them at the correct location on the map.

Figure 3. Map illustrating nonoverlap of an example ZIP code compared with corresponding ZCTA.

Note. ZCTA = ZIP Code Tabulation Area.

This problem emerges because of the nature of ZIP codes and how the census has had to deal as best as possible with the issue of investigators’ need of ZIP code-referenced demographic data. ZIP codes themselves are not areal units with defined borders. Rather, ZIP codes are routes defined by the U.S. Postal Service prescribing how to deliver mail efficiently. Hence, there is no official map of where one ZIP code ends and the next begins.^{Footnote 5} To create demographic and geographic data by ZIP code (because it is a common locator recorded for Americans), the Census Bureau created ZCTAs for the 2000 Census—mindful to warn users that ZIP codes crosscut even census blocks, the smallest geographic unit the Bureau records. As a best alternative, the census records the ZIP code that a majority of addresses in a census block use. A ZCTA then is formed as a combination of all census blocks with the same majority ZIP code. This is certainly an important tool that the Census Bureau provides, and in cases in which demographics need to be measured by ZIP code, it is the best alternative available. However, residents who have a ZIP code that is held by a minority of addresses in their census block will be placed in a ZCTA that differs from their ZIP code. Furthermore, Grubesic and Matisziw (Reference Grubesic and Matisziw2006, Table 2) show for the state of New York that ZCTAs differ substantially from ZIP codes in terms of the number of cases as well as the mean and standard deviation of the area they cover. With measurement error like this, it is much safer to use ZIP codes themselves when possible.^{Footnote 6}

In our case, we only need ZIP code locations to locate respondents of the CCES training data. Hence, we turn to a new alternative that deals with the issue of locating the position of ZIP codes themselves in space. Specifically, we use a 2014 dataset that draws from TomTom navigation services. This map defines ZIP code boundaries based on actual addresses, drawing a border around the complete set of addresses with a particular ZIP code. We therefore were able to compute the centroid and radius of actual ZIP codes and then link this information to the CCES to place survey respondents in space. This allowed us to estimate a model over our CCES training data that allowed for spatial correlation among nearby respondents.

Importantly, we only use ZIP codes at the training stage of estimating the model. When forecasting (or kriging) ideology, we use extremely precise census block data from the 2010 Census. At the forecasting stage, we can sample from the population using any geographic unit we wish, as long as we know both the location of the unit and the distribution of demographic predictors within that unit. A census block is the smallest possible geographic unit we can sample from, with 11 million of them defined in 2010. By forecasting using census block data, we can make predictions in places that are often as small as a city block using records of the U.S. Census recorded from that small area to sample demographic predictors. This maximizes predictive accuracy and avoids the ZIP code question altogether at the predictive stage of the model. Between the TomTom ZIP code data for the training stage and the U.S. Census Bureau’s block data for the forecasting stage, we maximize the accuracy in estimation and prediction.

Measures of Ideology and Validity Checks

Figure 5 presents our estimates of ideology in all 50 states. In both panels of the figure, the horizontal axis represents our estimates for the average state ideology, with higher values meaning more conservative. In Figure 5(a), the vertical axis represents the percentage of the two-party vote that Obama won in 2012. Each state is represented by its two-letter postal code, and the line represents the best fit from a regression that models Obama’s vote share as a function of our kriged ideology scores. As the scatterplot and best fit line both show, there is a close relationship between our measures of kriged ideology and presidential vote share, which serves as an external validation of our scores.

Figure 5. Scatterplots of 2012 presidential vote by state and 2011 U.S. Senators’ ideology, each against kriged measure of 2010 state public ideology: (a) Obama vote share 2012 and (b) state ideology 2011.

In addition, Figure 5(b) illustrates how well our measures of public ideology predict the ideology of U.S. senators elected from these respective states. In Figure 5(b), the horizontal axis again is our kriged measure of public ideology. The vertical axis is the first dimension score of DW-NOMINATE, which is frequently used as a measure of member ideology (McCarty, Poole, and Rosenthal Reference McCarty, Poole and Rosenthal1997; Poole and Rosenthal Reference Poole and Rosenthal1997). Higher values of NOMINATE are generally interpreted as more conservative, while lower values are more liberal. Republicans are represented by a red “R” and Democrats by a blue “D.” The line represents a regression predicting each senator’s NOMINATE score with public opinion ideology. As can be seen, more conservative states are more likely to elect conservative members and more likely to elect Republicans. Even within party, the scatterplot shows that within-party variance conforms to expectations: moderate Republicans are elected from more liberal states, and moderate Democrats are elected from more conservative states. Notably, these measures of state ideology are similar to those reported by Monogan and Gill (Reference Monogan and Gill2016): for the continental 48 states, the measures correlate at $ r=.9893 $ $ (SE=0.0211) $. These measures, however, have the advantage of offering scores for Alaska and Hawaii and using more precise input measures.

Figure 6 turns to the 435 districts for the U.S. House of Representatives and displays our measure of public ideology by district. The horizontal axis displays our measure of public opinion ideology by district, and the vertical axis represents each House member’s first dimension DW-NOMINATE score. Again, every Democrat is represented with a blue “D” and every Republican is represented with a red “R.” The line shows the results of a regression of elected members’ ideology as a function of district ideology. Even at this smaller level of geographic precision, we still see that we can use a constituency’s ideology to predict whether those voters will choose a Republican or Democrat and how conservative or liberal the member will be. Again, moderate members of each party tend to be drawn from districts that normally would not elect a member of their respective party. Hence, for both chambers of Congress, we see a relationship between voters’ ideology and the ideology of their members. The fact that this well-established electoral connection continues to be supported by our data further validates our kriging approach.

Figure 6. Scatterplot of 2011 U.S. House of Representatives members’ ideology against kriged measure of 2010 state public ideology.

Finally, we applied our kriging technique to constituencies as precise as state legislative districts. Figure 7 illustrates our measures of constituency ideology in both lower and upper chambers of the state legislatures. In both panels, the horizontal axis captures public ideology with our kriging measure, whereas the vertical axis measures state legislators’ ideology with the common space measure developed by Shor and McCarty (Reference Shor and McCarty2011). In both panels, red dots represent Republican legislators, and blue dots represent Democratic legislators. Districts and legislators for lower chambers are presented in Figure 7(a), whereas upper chambers are presented in Figure 7(b). In each panel, the regression line shows a positive association between electoral conservatism and legislator conservatism. Even at this most precise level where many geographic constituencies are no larger than a neighborhood, our measures of public opinion correspond to the electoral connection that we would expect for state legislators. In addition, for the lower chamber measures of district ideology, our measure correlates with the measures reported by Tausanovitch and Warshaw (Reference Tausanovitch and Warshaw2013) at $ r=.6117 $ $ (SE=0.0119) $. Thus, the measures both appear to be capturing the same concept of constituency ideology. Overall, for many sizes of electoral constituencies, our measures of public ideology pass the external validity checks we consider.

Figure 7. Scatterplots of 2011 state legislators’ ideology in lower and upper chambers, each against kriged measure of 2010 state public ideology: (a) Lower chambers and (b) upper chambers.