1 Advances in Political Economy - Department of Political Science

EDITOR’S PROOF

A Heteroscedastic Spatial Model of the Vote 357

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of different functions, but for the sake of our example we can use a simple parabola

(e.g. a quadratic approximation) estimating the convexity of lenses or the projection

of a ray of light on a parabolic mirror.

As an illustrating example, let us use the case of the Republican Party in the

U.S. In the modelLiRdescribes the reported location of the Republican Party by

respondenti. The self-reported ideological position of the same respondent is given

byxi. The quadratic approximation is thus

LiR=a+bxi+cx^2 i. (1)

We can center the convex lens of the Republican Party at its projected axis; that is,

where there exists an individualxi∗that observes the “true” location of the Republi-

can Party, designatedL∗iR, from a position perpendicular to the principal ideological

axis on which theNrespondents—each with a different image ofR’s position—are

arrayed. This allows us to setL∗iR=x∗i. With this equality, we can use (1)tosolve

forxi∗. The solution is

L∗iR=xi∗=−

1

2

− 1 +b+

√

1 − 2 b+b^2 − 4 ca

c

. (2)

When voting for the party, all respondentsxi=xi∗observe images that are either

closer to or further away fromLiR=L∗iDfor everyxi=xi∗, e.g. magnification.

We can describe thismagnification(M)ofthemirrorthatiattaches toRas:

MiR=

(xi−LiR)^2

(xi−L∗iR)^2

. (3)

Note that magnification is defined as the ratio of two quadratic (Euclidian) distances:

the distance from the voter’s position and her perception of the candidate’s position,

and the distance from the voter’s position to the “true” location of the party. We can

think of the first of these as “reported distance” and the second as “true distance.”

Thus, whenM>1 we have a lens thatstretchesideological, distance and when

M<1 the effect of the lens is tocompressideological distance. Moreover, if we

had information to explain the degree of magnification in reported data, we could

also estimate the “true” rather than the reported distance from the voters to the

candidates.

(

xi−L∗iR

) 2

=

(xi−LiR)^2

MiR

. (4)

While there are many different functional forms that can be used to estimate bi-

ases in the perceived location of parties, the previous example serves two purposes.

First, it provides the intuition for how we might link lessons from physics to models

of voter choice. And second, it provides a point of departure to estimate assimilation

and contrast in proximity models of voting.