a brief primer on ideal point estimatesjoshua d. clinton princeton university [email protected]...

MotivationTwo Probabilistic Voting Models

ComputationComplications

Conclusion

A Brief Primer on Ideal Point Estimates

Joshua D. ClintonPrinceton University

[email protected]

January 28, 2008

Josh Clinton, Princeton University A Brief Primer on Ideal Point Estimates



Conclusion

Really took off with growth of game theory and EITM.Claim: ideal points = player preferences.

Widely used in American, Comparative and International Relations:Measure legislator/judicial/voter/party/country preferences, gridlockintervals, uncovered sets, polarization, heterogeneity, representation,party pressure.

Outline:

I Statistical Models

I Computational Considerations

I Reasons for Concern




Conclusion

Claim 1:

The maps [of ideal points] are useless unless the userunderstands both the spatial theory that the computer programembodies and the politics of the legislature that produced theroll calls (Poole, 2005).




Conclusion

Claim 2:

Legislative voting is one of the most intensively studied areas ofquantitative political science. Many have assumed that thebasic answers were discovered long ago. Others will interpretthe recent research as filling in ‘everything they always wantedto know about legislative voting’ and (often) didn’t care to ask.But the challenge of the field remains (Weisberg, 1977).




Conclusion

Statistical Model

Uit(θy(t)

)= f (xi , θy(t)) + ζit

Uit(θn(t)

)= f (xi , θn(t)) + νit

Legislator i votes Yea if f (xi , θy(t))− f (xi , θn(t)) > νit − ζit

Pr(yit = 1) = Pr(νit − ζit < f (xi , θy(t))− f (xi , θn(t)))

Ideal point estimators vary in assumptions about deterministic parts(f(xi , θy(t)),f(xi , θn(t))) and stochastic parts (ζit , νit) of legislator i ’s utilityfunction.




Conclusion

NOMINATE & QuadraticQuadratic:

f (xi , θy(t)) = −`xi − θy(t)

´2f (xi , θn(t)) = −

`xi − θn(t)

´2f (xi , θy(t))− f (xi , θn(t)) = −

`xi − θy(t)

´2+`xi − θn(t)

´2= βtxi − αt

Where: αt = θ2n(t) − θ2y(t) and βt = 2

`θy(t) − θn(t)

´.

NOMINATE:

f (xi , θy(t)) = βexp“− 1

2

`xi − θy(t)

´2”f (xi , θn(t)) = βexp

“− 1

2

`xi − θn(t)

´2”f (xi , θy(t))− f (xi , θn(t)) = β

hexp

“− 1

2

`xi − θy(t)

´2”− exp

“− 1

2

`xi − θn(t)

´2”i




Conclusion

NOMINATE & QuadraticQuadratic:

f (xi , θy(t)) = −`xi − θy(t)

´2f (xi , θn(t)) = −

`xi − θn(t)

´2f (xi , θy(t))− f (xi , θn(t)) = −

`xi − θy(t)

´2+`xi − θn(t)

´2= βtxi − αt

Where: αt = θ2n(t) − θ2y(t) and βt = 2

`θy(t) − θn(t)

´.

NOMINATE:

f (xi , θy(t)) = βexp“− 1

2

`xi − θy(t)

´2”f (xi , θn(t)) = βexp

“− 1

2

`xi − θn(t)

´2”f (xi , θy(t))− f (xi , θn(t)) = β

hexp

“− 1

2

`xi − θy(t)

´2”− exp

“− 1

2

`xi − θn(t)

´2”iJosh Clinton, Princeton University A Brief Primer on Ideal Point Estimates



Conclusion

Stochastic Elements

Assume ζit and νit are: uncorrelated and drawn from a symmetric,unimodal distribution. Suppose νit − ζit ∼ N(0, σ2):

f (xi , θy(t))− f (xi , θn(t)) ∼ N(νit − ζit , σ2)

Therefore:

Pr(yit = 1) = Pr(νit − ζit < f (xi , θy(t))− f (xi , θn(t)))

)= Φ

(σ−1(f (xi , θy(t))− f (xi , θn(t))))

)Implying:

PrN(yit = 1) = Φ[β[exp

(− 12ω

(xi − θy(t)

)2) − exp(− 12ω (xi − θn(t))2)]]PrQ(yit = 1) = Φ

[σ−1(βtxi − αt)

]Josh Clinton, Princeton University A Brief Primer on Ideal Point Estimates



Conclusion

Likelihood Functions

Quadratic Model:

L(x,θ|y) = ΠLi=1ΠTt=1PrQ(yit = 1)yit × (1− PrQ(yit = 1))1−yit

= ΠLi=1ΠTt=1Φ (βtxi − αt)yit × (1− Φ (βtxi − αt))1−yit

NOMINATE Model:

L(x,θ|y) = ΠLi=1ΠTt=1PrN(yit = 1)yit × (1− PrN(yit = 1))1−yit

= ΠLi=1ΠTt=1Φ

hβhexp

“− 1

2

`xi − θy(t)

´2” − exp“− 12

`xi − θn(t)

´2”iyit×„

1− Φhβhexp

“− 1

2

`xi − θy(t)

´2” − exp“− 12

`xi − θn(t)

´2”i”1−yit




Conclusion

Parameter Identification:Clearly unidentified without further restrictions:

xβ − α = x∗β∗ − α with x∗ = xs and β∗ = β/sImplies L(β,α, x|y) = L(β∗,α, x∗|y)

Scale and Rotational Invariance:




Conclusion

Parameter Identification:

Clearly unidentified without further restrictions:

xβ − α = x∗β∗ − α with x∗ = xs and β∗ = β/sImplies L(β,α, x|y) = L(β∗,α, x∗|y)

Solution: two linearly independent restrictions on x.I Sen. Helms = 1, Sen. Kennedy = -1I E [x] = 0, V [x] = 1, and median Democrat < 0.




Conclusion

Identification Summary

Estimator Comparability ConstraintsInterest Group Scores Not∗ NoneWNOMINATE Not NoneDNOMINATE Across time, not inst. (except pres.) Parametric ChangeDWNOMINATE Across time, not inst. (except pres.) Parametric ChangeCommon Space Across time and instit. Fixed xQuadratic Depends Depends


·Sarbanes ·Reed ·KERRY ·Harkin ·Boxer

·Levin ·Clinton ·Dayton ·Lautenberg ·Mikulski

·Feingold ·Durbin ·Corzine ·EDWARDS ·

Kennedy ·Stabenow ·Johnson·Daschle ·Leahy ·

Wyden ·Murray ·Akaka·Byrd ·GRAHAM, B. ·

Dodd ·Rockefeller ·Hollings ·Schumer ·Cantwell

·Kohl ·Bingaman ·Biden ·Reid ·

Dorgan ·Inouye ·Nelson, B.·Feinstein ·LIEBERMAN ·

Conrad ·Pryor ·Landrieu·Carper ·Jeffords ·

Lincoln ·Baucus ·Bayh ·Breaux ·Nelson, D.

·Chafee ·Snowe

−3 −2 −1 0

−3 −2 −1 0

· McCain· Specter·Collins· Campbell· Murkowski

· Stevens· Sununu·Smith· Shelby·Gregg· Fitzgerald· Graham, L.

· Hutchison· Coleman·Warner· Hagel·Ensign· DeWine· Brownback

· Talent· Allard·Roberts· Dole· Voinovich

· Chambliss· Miller·Lott· Kyl· Inhofe

· Enzi· Bond·Domenici· Crapo· Alexander

· Craig· Nickles·Frist· Bennett· Burns

· Grassley· Bunning·Santorum· Cochran· Allen

· McConnell· Thomas·Sessions· Hatch·

Lugar· Cornyn

0 1 2 3

0 1 2 3

·Sarbanes ·Reed ·KERRY ·Harkin ·Boxer

·Clinton ·Levin ·Dayton ·Lautenberg ·Mikulski

·Feingold ·Durbin ·Corzine ·EDWARDS ·

Kennedy ·Stabenow ·Johnson·Leahy ·Daschle ·

Wyden ·Murray ·Akaka·Byrd ·Dodd ·

GRAHAM, B. ·Rockefeller ·Hollings ·Schumer ·Cantwell

·Kohl ·Bingaman ·Biden ·Reid ·

Dorgan ·Inouye ·Nelson, B.·Feinstein ·LIEBERMAN ·

Conrad ·Pryor ·Landrieu·Carper ·Jeffords ·

Lincoln ·Baucus ·Bayh ·Breaux ·Nelson, D.

·Chafee ·Snowe

0 20 40

0 20 40

· McCain· Specter·Collins· Campbell· Murkowski

· Stevens· Sununu·Smith· Shelby·Gregg· Fitzgerald· Graham, L.

· Hutchison· Coleman·Warner· Hagel·Ensign· DeWine· Brownback

· Talent· Allard·Roberts· Voinovich· Dole

· Chambliss· Miller·Lott· Kyl· Inhofe

· Enzi· Bond·Domenici· Crapo· Alexander

· Craig· Nickles·Frist· Bennett· Burns

· Grassley· Bunning·Santorum· Allen·McConnell

· Cochran· Lugar·Hatch· Thomas· Sessions

· Cornyn

50 70 90

50 70 90



Conclusion

Computation

Much easier now:

I http://voteview.com/dwnl.htm

I WNOM9707.EXE (WNOMINATE executable)

I WNOMINATE for R

I pscl (ideal) in R

I TurboADA for STATA (see Jeff Lewis, UCLA)

I winBUGS




Conclusion

Recommendations

All produce the same answer (in 1d) for “well-behaved” legislatures.Slightly prefer quadratic for computational reasons:

I Easier to customize the estimator.

I Measures of parameter uncertainty are recovered directly.

I Code is well-understood (and replicable in other programs).




Conclusion

Model Evaluation

I Error structure.

I Functional form.

I Goodness-of-fit (degrees of freedom).




Conclusion

Exogeneity

How to interpret ideal point estimates:

I Agenda is endogenous & institutional rules change over time.

I Decision to vote is endogenous & roll calls not always recorded.

I Often related to characteristics being explained.


All Statutes

Congress

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of L

aws/

Vot

es

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50 60 70 80 90 100

040

010

00

1887 1907 1927 1947 1967 1987

Top 3500 Statutes

Congress

Num

ber

of L

aws/

Vot

es

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50 60 70 80 90 100

050

150

1887 1907 1927 1947 1967 1987



Conclusion

Exogeneity








Conclusion




Conclusion

Exogeneity








Conclusion

Conclusion

Political science is heavily dependant on measures of latent concepts(e.g., ideal points, party manifestos, expert surveys).

I Computational issues are largely resolved.

I Theoretical and interpretative issues remain.

I Data generating process important (and not well understood).


MotivationTwo Probabilistic Voting ModelsComputationComplicationsConclusion

a brief primer on ideal point estimatesjoshua d. clinton princeton university [email protected]...

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