1 Advances in Political Economy - Department of Political Science

EDITOR’S PROOF

Inferring Ideological Ambiguity from Survey Data 377

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whereSjk=

∑

ik(zijk−τik)/ψikandDjk=Njk+σ

2

jk/ημ. The respondent lo-

cation parameters are sampled as follows:

τik|· ∼N

(∑

j(zijk−ψikμjk)/σ

2

jk

∑

jkσ

− 2

jk+ψ

2

ik

,

ψ^2 ik

∑

jkσ

− 2

jk+ψ

2

ik

)

.

3. Full conditionals forσandψdo not have a recognizable form, thus Metropolis-
   Hastings algorithm is employed. The log-posterior ofσj^2 is proportional to

lnπ

(

σj^2 |z, μ, τ, ψ, ak,α

)

∝−(Njk/ 2 + 3 )lnσjk^2 −σjk−^2

(

Sjk/ 2 ψik^2 +bk

)

+

∑

i∈Nk

rijklnpijk+( 1 −rijk)ln( 1 −pijk),

wherepijk=(α 0 +α 1 φjkσjk)andSjk=

∑

i∈Njk(zijk−ψikμjk−τik)

(^2) .The
log-posterior forψ^2 has a similar form:
lnπ
(
ψik^2 |z, μ, σ, τ
)
∝−(Jik/ 2 + 1 / 2 )lnψik^2 +S 1 /ψik^2 −S 2 /ψik

• ∑
  i∈Nk
  rijklnpijk+( 1 −rijk)ln( 1 −pijk),
  whereS 1 =. 5
  ∑
  jk(zijk−τik)
  (^2) σ− 2
  jk andS^2 =
  ∑
  jk(zijk−τik)μjkσ
  − 2
  jk. Proposal
  values σjk^2 (t) are sampled from the inverse gamma with shapeλand scale
  (λ− 1 )σjk^2 (t−^1 ). Here,λis the tuning parameter that set to achieve an acceptance
  rate between 30 and 50 %.

4. ln(δ)is sampled from the left-truncated normal distribution with meanci+ 1 −ci,
   unit variance and lower bound equal to ln(max{μjk}−cM− 1 ). The conditional
   distribution for the hyper-parameterbkis gamma with scale 2Jk+and rate
   (a− 1 )

∑

jσ

− 2

j +^1 /.

5. The coefficients in the missing data model,α 0 andα 1 , are sampled using the
   standard data augmentation method by Albert and Chib (1993).
   To implement the identification constraints in (4), after each block of iterations,
   eachψikis divided by the country average ofψik’s; similarly, from eachτikthe
   country average ofτik’s is subtracted. This procedure is similar to hierarchical cen-
   tering by “sweeping” and is known to improve the convergence of MCMC algo-
   rithms in weakly identified models (Robert and Casella 2004 , p. 397). The conver-
   gence can be monitored using the standard tools, as, for example, Geweke ( 1992 )or
   Heidelberger and Welch ( 1983 ) diagnostics.

5 Application: Ideological Ambiguity and Electoral Performance

We apply our model to the expert data by Benoit and Laver ( 2006 ). The survey

was conducted in 48 countries with about 8 parties and 30 experts per country on