1 Advances in Political Economy - Department of Political Science

EDITOR’S PROOF

374 A. Rozenas

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standard deviation. These constraints turn out to be insufficient to identify the model

in (1)–(2). The following restrictions are imposed instead:

μj∈[c 1 −δ,cM− 1 +δ] forj= 1 ,...,J, (3)

∑N

i= 1

τi=0 and

∑N

i= 1

ψi^2 =1fori= 1 ,...,N. (4)

Here,δis a hyper-parameter estimated in the model. Finally, we assume that the cut-

off points are fixed at equal intervals between−1 and 1 (any other interval would

do as well). Since policy space is defined only up to affine transformation, these

constraints do not result in loss of information.

3.1 Model for Missing Data

The model can be extended to exploit the patterns in the missing data (NA re-

sponses) as an additional source of information about the ideological ambiguity.

In particular, I assume that if a party is perceived to be very ambiguous and/or if a

respondent is not knowledgeable, one is more likely to observe an NA answer. Thus,

in the terminology of Little and Rubin (1987), we assume that the missing data are

non-ignorable. For convenience, letzij=ψiz∗j+τi.Alsoletrij=1 if data entryyij

is missing andrij=0 otherwise. The model for the observed data can be written as

zij∼N

(

ψiμj+τi,ψi^2 σj^2

)

, (5)

yij=

{

m ifcm<zij≤cm+ 1 andrij= 0

NA ifrij= 1 ,

(6)

Pr(rij= 1 )=

(

α 0 +α 1 σjψi

)

. (7)

Notice, first, that if a respondent is not highly knowledgeable (highψi)ora

party is ambiguous (highσj), or both, the answers will exhibit high variation. Sec-

ond,zij’s that are drawn from distributions with low standard deviation (lowψiσj)

are less likely to be reported as NA’s, as implied by the missingness model in (7).

Here,is a standard normal distribution function, resulting in a probit model. By

making missingness dependent both onσjandψiwe allow for data distributions

where some parties and/or some respondents tend to have more missing values than

others. Parameterα 1 measures how much missingness in the data depends on the

respondent-level scaleψiand party-level ambiguityσj.

3.2 Prior Distributions

The model is completed by specifying prior distributions. If a cross-national sur-

vey is used, one can specify hierarchical priors where some party-level parameters