Empirical foundation of valence using Aldrich–McKelvey scaling

Abstract

This paper uses data from the 2004 pre-election survey of the American National Election Study to test empirically different ways of incorporating a valence parameter into a Downsian utility function. We call particular attention to the problem of interpersonal incomparability of responses to the liberal-conservative scale, and use Aldrich–McKelvey’s pathbreaking method to obtain accurate distances between respondents and candidates, the key regressors. We find that the utility function the most supported by the empirical evidence, the intensity valence utility function, is the one which permits to make the better predictions for the 2004 presidential election. We also consider counterfactual analyses wherein we test if Bush, the candidate with the highest intensity valence, has dominant strategies which would have insured him to obtain a majority of the popular vote. According to the theory, it is known that the candidate with the highest intensity valence does not have such dominant strategies if the distribution of voters in the policy space is too heterogenous. Nevertheless, we show that the distribution of voters in 2004 is sufficiently homogenous for Bush to have dominant strategies.

Appendices

Appendix A: Estimation results taking issue scale responses at face value

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4. However, the responses to the liberal-conservative scale are taken at face value to obtain the distances, i.e., $d_{i,j}=|{\widetilde{a}}_i-{\widetilde{x}}_{i,j}|$.

Table 7 provides the results. The first result is that the Downsian and the additive valence models are rejected according to the likelihood ratio tests, while the intensity valence model is not. These findings are similar to those obtained in Sect. 4. However, one should clearly have in mind that the fact that they are identical to the one with an Aldrich–McKelvey correction cannot permit to claim or suggest that an Aldrich–McKelvey correction is not necessary. The findings in Table 7 would be credible under the assumption that taking the data at face value is not a problem. This assumption is refutable and has been refuted, i.e., statistically tested and rejected in Sect. 3.1. Indeed, we have shown that the null hypothesis of interpersonal comparability of responses is rejected. Given that there is clear problem of interpersonal incomparability of responses, any econometrics using these data at face value cannot be considered as credible.

The main reason of providing these results is mainly to show that when the data are taken at face value the utility function is linear in distance as it is sometimes mentioned (see, e.g., Adams et al. 2005, p. 17). Indeed, given the estimated coefficient ${\widehat{\gamma }}=1.038$ and its estimated standard error ${\widehat{se}}({\widehat{\gamma }})=0.066$, the 95% confidence interval of $\gamma$ is [0.909, 1.168], so the null hypothesis $\text{ H}_0: \gamma =1$ is not rejected. As the results with the Aldrich–McKelvey correction suggest (Table 4), and as Adams et al. aptly explain, the fact that the utility is linear in distance in Table 7 is due to the fact that “the policy scales from which distance is measured [...] are constrained to specified finite intervals.”

Table 7 Maximum likelihood estimates of the 4 SUR models relaxing the assumption that the utility functions are linear in distance and taking liberal-conservative scale responses at face value

Appendix B: Support for each candidate according to the Downsian and additive valence models

1.1 B.1 Support for each candidate according to the Downsian model

Using the locations of Kerry, Nader and Bush ($x_k=-0.422$, $x_n=-0.394$ and $x_b=0.816$) and the Downsian utility function parameters obtained in Column [2] of Table 4, Fig. 7 describes the estimated Downsian utilities given by the three candidates. As shown in Table 1, note that voters are located between −3.704 and 4.233 on the liberal-conservative space. However, Fig. 7 only considers the interval of the liberal-conservative space ranging from -1 to 1. We do so in order to clearly visualize when the estimated utilities intersect. Indeed, Kerry and Nader are so close ($x_k=-0.422$ and $x_n=-0.394$) that it would have been difficult to visualize these intersections if we had shown all the possible values taken by the respondents in the policy space. The main objective of this Appendix is to fully understand why when the voters vote sincerely, the Downsian model predicts unrealistically that Nader will obtain 31.56% of the votes. Thus, we focus on the situation wherein all the voters vote sincerely.

It is easy to see from Fig. 7, that:

• (i.) The voters whose bliss points are strictly higher than $a_2\simeq 0.211$ obtain the highest level of utility with Bush. If so, these voters will vote for Bush. Then, according to the Downsian model, the relative frequency of voters who will vote for Bush is ${\widehat{S}}_b=\frac{\sharp \{i=1,\ldots , N ;~ a_i>a_2\}}{N}\simeq 0.4468$.

• (ii.) The voters whose bliss points are strictly between $a_1\simeq -0.408$ and $a_2\simeq 0.211$ obtain the highest level of utility with Nader. If so, these voters will vote for Nader. The relative frequency of voters who will vote for Nader is then ${\widehat{S}}_n=\frac{\sharp \{i=1,\ldots , N ;~ a_1<a_i<a_2\}}{N}\simeq 0.3156$.

• (iii.) The voters whose bliss points are strictly less than $a_1\simeq -0.408$ obtain the highest level of utility with Kerry. The relative frequency of voters who will vote for Kerry is then ${\widehat{S}}_k=\frac{\sharp \{i=1,\ldots , N ;~ a_i<a_1\}}{N}\simeq 0.2375$.

Fig. 7

Estimated Downsian utility functions. Note: This figure depicts the three estimated Downsian utilities in function of a obtained in Table 4. The (red) solid curve depicts the estimated utility if Bush is elected. The (blue) dashed curve depicts the estimated utility if Kerry is elected. The (green) dotdash curve depicts the estimated utility if Nader is elected. The locations of Kerry, Nader and Bush are $x_k=-0.422$, $x_n=-0.394$ and $x_b=0.816$, respectively. Finally, $a_1\simeq -0.408$ and $a_2\simeq 0.211$

Thus, the Downsian model predicts unrealistically that Nader will obtain 31.56% of the vote (under the assumption of sincere voting) because he is located on the right of Kerry ($x_n=-0.394$ and $x_k=-0.422$) and attracts all the voters located between $a_1\simeq -0.408$ and $a_2\simeq 0.211$. This is an interval wherein a massive heap of voters are located as shown in Fig. 3.

1.2 B.2 Support for each candidate according to the additive valence model

Using the locations of Kerry, Nader and Bush ($x_k=-0.422$, $x_n=-0.394$ and $x_b=0.816$) and the estimated additive valence utility function parameters obtained in Column [3] of Table 4, Fig. 8 describes the estimated additive valence utilities given by the three candidates. The main objective is to understand why, even under the assumption of sincere voting, nobody will vote for Nader.

It is easy to see from Fig. 8 that:

• (i.) The voters whose bliss points are strictly higher than $a_1\simeq 0.022$ obtain the highest level of utility with Bush. If so, these voters will vote for Bush. Then, according to the Downsian model, the relative frequency of voters who will vote for Bush is ${\widehat{S}}_b=\frac{\sharp \{i=1,\ldots , N ;~ a_i>a_1\}}{N}\simeq 0.5249$.

• (ii.) The voters whose bliss points are strictly less than $a_1\simeq 0.022$ obtain the highest level of utility with Kerry. The relative frequency of voters who will vote for Kerry is then ${\widehat{S}}_k=\frac{\sharp \{i=1,\ldots , N ;~ a_i<a_1\}}{N}\simeq 0.4751$.

The results are similar under the assumptions of sincere voting and strategic voting because the locations of Kerry ($x_k=-0.422$) and Nader ($x_n=-0.394$) are very close, and Kerry has an additive-valence advantage over Nader (${\widehat{\delta }}_k>{\widehat{\delta }}_n$), as shown in Column [3] of Table 4. Consequently, the higher additive valence of Kerry implies a higher level of utility with Kerry than with Nader for all voters. So even under the assumption of sincere voting, nobody will vote for Nader according to the additive valence model.

Fig. 8

Estimated additive utility functions. Note: This figure depicts the three estimated additive utilities in function of a obtained in Table 4. The (red) solid curve depicts the estimated utility if Bush is elected. The (blue) dashed curve depicts the estimated utility if Kerry is elected. The (green) dotdash curve depicts the estimated utility if Nader is elected. The locations of Kerry, Nader and Bush are $x_k=-0.422$, $x_n=-0.394$ and $x_b=0.816$, respectively. Finally, $a_1\simeq 0.022$

Appendix C: Estimation results considering the mean placement of each candidate to compute each distance

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4, as well as the ability of the different models to predict the 2004 presidential election, as in Table 5. However, the individual positions are taken at face value and we use the mean of the perceived locations of the candidates to obtain the distances, i.e., $d_{i,j}=|{\widetilde{a}}_i-\text{ Mean }({\widetilde{x}}_{i,j})|$, where Mean(${\widetilde{x}}_{i,j}$) is the mean of the perceived locations of candidate j, i.e., Mean(${\widetilde{x}}_{i,k}$)=2.692, Mean(${\widetilde{x}}_{i,n}$) = 2.745 and Mean(${\widetilde{x}}_{i,b}$ ) =5.591; see Table 1.

Table 8 Maximum likelihood estimates of the 4 SUR models considering the mean placement of each candidate to compute each distance

Table 9 Popular vote: Reality and point estimates according to the different models (based on the results considering the mean placement of each candidate to compute each distance)

Fig. 9

Estimated additive valence and intensity valence utility functions (based on the results considering the mean placement of each candidate to compute each distance). Notes: Panel (A) depicts the three estimated additive valence utilities in function of ${\widetilde{a}}$ obtained in Table 8 and ${\widetilde{a}}_1\simeq 4.005$. Panel (B) depicts the three estimated intensity valence utilities in function of ${\widetilde{a}}$ obtained in Table 8 and ${\widetilde{a}}_1\simeq 4.018$. In both panels, the (red) solid curve depicts the estimated utility if Bush is elected. The (blue) dashed curve depicts the estimated utility if Kerry is elected. The (green) dotdash curve depicts the estimated utility if Nader is elected. The locations of Kerry, Nader and Bush are Mean(${\widetilde{x}}_{i,k}$)=2.692, Mean(${\widetilde{x}}_{i,n}$) = 2.745 and Mean(${\widetilde{x}}_{i,b}$ ) =5.591, respectively

Table 8 provides the estimates. The first results are that the Downsian and the additive valence models are rejected according to the likelihood ratio tests, while the intensity valence model is not. These results are similar to those obtained in Sect. 4. However, one should clearly have in mind that the fact that these results are identical to the one with an Aldrich–McKelvey correction cannot permit to claim or suggest that an Aldrich–McKelvey correction is not necessary. As already noticed, using the mean placement of each candidate to compute the distances is problematic. As pointed out by Armstrong II et al. (2014, p. 43), it is prone to failure in case of heteroskedasticity. Furthermore, recall that each respondent i provides his placement ${\widetilde{a}}_i$ on a 7-point scale, ranging from 1 to 7. By considering the mean placement of each candidate to compute the distances, we have only 7 possible distances. The lack of variability may be a problem for the estimations.

Table 9 illustrates the inability of all the models to make good predictions when we use the mean of the perceived locations of the candidates to obtain the distances. Recall that in Table 5, when one uses an Aldrich–McKelvey correction, the point estimates of the fractions of vote obtained via the intensity valence model under the assumption of sincere voting (i.e., ${\widehat{S}}_b\simeq 0.5099$, ${\widehat{S}}_k\simeq 0.4652$, ${\widehat{S}}_n\simeq 0.0249$) or under the assumption of strategic voting (i.e., ${\widehat{S}}_b\simeq 0.518$, ${\widehat{S}}_k\simeq 0.482$, ${\widehat{S}}_n=0$) are very close to the percentages of vote obtained by the candidates in reality (i.e., $S_b=0.5073$, $S_k=0.4827$, $S_n=0.0038$). In Table 9, the intensity valence model implies that 55.64% of the voters will vote for Kerry under the assumption of sincere voting and under the one of strategic voting (Lines [5a] and [5b]). The point estimates obtained with the Downsian (Lines [3a] and [3b]) and the additive valence models (Lines [4a] and [4b]) are also unrealistic.

Finally, note that the intensity valence and the additive valence models provide exactly the same predictions. It may appear surprising given that the estimated coefficients of the intensity valence utility functions and the additive valence utility functions are very different, as shown in Table 8. In particular, the estimated slope coefficients are candidate-specific and very different in the case of the intensity valence while they are identical in the case of the additive valence. The reason of these similar predictions is easy to see from Fig. 9. The additive and the intensity valence utility functions are very different as shown in Figure C1, but the predictions are similar mainly because the policy space for the respondents is the 7-point scale. In the case of the additive valence, the voters whose bliss points are higher than ${\widetilde{a}}_1\simeq 4.005$ obtain the highest level of utility with Bush. If so, these voters will vote for Bush. The voters whose bliss points are less than ${\widetilde{a}}_1\simeq 4.005$ obtain the highest level of utility with Kerry and will vote for him. Concerning the intensity valence, the voters whose bliss points are higher than ${\widetilde{a}}_1\simeq 4.018$ obtain the highest level of utility with Bush and will vote for him. The voters whose bliss points are less than ${\widetilde{a}}_1\simeq 4.005$ obtain the highest level of utility with Kerry and will vote for him. Thus, in both cases, the respondents who place themselves at ${\widetilde{a}}\in \{1,2,3,4\}$ vote for Kerry and respondents who place themselves at ${\widetilde{a}}\in \{5,6,7\}$ vote for Bush.

Appendix D: Estimation results excluding the extreme voters

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4, as well as the ability of the different models to predict the 2004 presidential election, as in Table 5. However, the extreme voters, i.e., those whose bliss points are strictly less than $a_1$ or higher than $a_4$ in Figure 5 are excluded. The sample is now composed of 592 respondents. Note that according to the Aldrich–McKelvey correction, the location of Kerry is $x_k=-0.420$, the one of Nader is $x_n=-0.396$, and the one of Bush is $x_b=0.816$ with this subsample; hence, the locations of the candidates are very similar to those obtained with the Final sample described in Table 1.

Table 10 provides the estimates of the different models. The first result is that, again, the intensity valence is the sole model which is not rejected by the data, considering the simple likelihood ratio tests or the bootstrapped likelihood ratio tests (the procedure is similar to the one presented in Sect. 4.2).

Table 10 Maximum likelihood estimates of the 4 SUR models excluding the extreme voters

Table 11 provides the point estimates of the fractions of vote obtained via the different models under the assumption of sincere voting and under the assumption of strategic voting. The results are very close to the percentages of vote obtained in Table 5.

Table 11 Popular vote: Reality and point estimates according to the different models (extreme voters excluded)

Appendix E: Estimation results excluding the respondents who rate Nader at zero

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4, as well as the ability of the different models to predict the 2004 presidential election, as in Table 5. However, those who rate Nader at zero at the thermometer questions are excluded. The sample is now composed of 536 respondents. Note that according to the Aldrich–McKelvey correction, the location of Kerry is $x_k=-0.425$, the one of Nader is $x_n=-0.391$, and the one of Bush is $x_b=0.816$ with this subsample; hence, the locations of the candidates are very similar to those obtained with the Final sample described in Table 1.

Table 12 provides the estimates of the different models. The first result is that, again, the intensity valence is the sole model which is not rejected by the data, considering the simple likelihood ratio tests or the bootstrapped likelihood ratio tests (the procedure is similar to the one presented in Sect. 4.2).

Table 12 Maximum likelihood estimates of the 4 SUR models excluding the respondents who rate Nader at zero

Table 13 provides the point estimates of the fractions of vote obtained via the different models under the assumption of sincere voting and under the assumption of strategic voting. The results are very close to the percentages of vote obtained in Table 5.

Table 13 Popular vote: Reality and point estimates according to the different models (respondents who rate Nader at zero are excluded)

Appendix F: Estimation results excluding the liberal respondents who rate Nader at zero

This appendix provides maximum likelihood estimates of the four SUR models similar to those in Table 4, as well as the ability of the different models to predict the 2004 presidential election, as in Table 5. However, the liberal respondents who rate Nader at zero at the thermometer questions are excluded. By “liberal respondents” we mean those whose bliss points are strictly less than zero according to the Final sample. The sample is now composed of 571 respondents. Note that according to the Aldrich–McKelvey correction, the location of Kerry is $x_k=-0.418$, the one of Nader is $x_n=-0.398$, and the one of Bush is $x_b=0.816$ with this subsample; hence, the locations of the candidates are very similar to those obtained with the Final sample described in Table 1.

Table 14 provides the estimates of the different models. The first result is that, again, the intensity valence is the sole model which is not rejected by the data, considering the simple likelihood ratio tests or the bootstrapped likelihood ratio tests (the procedure is similar to the one presented in Section 4.2).

Table 14 Maximum likelihood estimates of the 4 SUR models excluding the liberal respondents who rate Nader at zero

Table 15 provides the point estimates of the fractions of vote obtained via the different models under the assumption of sincere voting and under the assumption of strategic voting. The results are very close to the percentages of vote obtained in Table 5.

Table 15 Popular vote: Reality and point estimates according to the different models (liberal respondents who rate Nader at zero are excluded)