Context MattersContext Matters::Empirical Analysis in Social Empirical Analysis in Social

Science,Science,Where the Effect of AnythingWhere the Effect of AnythingDepends on Everything ElseDepends on Everything Else

Interaction Terms,Interaction Terms,

Multilevel Models,Multilevel Models,

& Nonlinear Regression& Nonlinear Regression

Blalock LectureBlalock LectureICPSR Summer School, 6 August 2008ICPSR Summer School, 6 August 2008

(Here for fuller pedagogical slides & materials: (Here for fuller pedagogical slides & materials: http://www-http://www-personal.umich.edu/~franzese/InteractionsLecture.zippersonal.umich.edu/~franzese/InteractionsLecture.zip))

Robert J. Franzese, Jr.Robert J. Franzese, Jr.Associate Professor of Political Science,Associate Professor of Political Science,

The University of Michigan, Ann ArborThe University of Michigan, Ann Arbor

OverviewOverview Interactions in Pol-Sci: Ubiquitous, but more The Linear-Interaction Model

From theory to empirical-model specification: Arg’s that imply interax (& some that don’t), & how write.

Interpretation: Effects=derivatives & differences, not coefficients! Std Errs (etc.): effects vary, so do std errs (etc.)!

Presentation: Tables & Graphs, & Choosing Spec. Elaborations, Complications, & Extensions:

Use & abuse of some common-practice “rules” Sample-Splitting v. (Dummy-) Interacting Variance Implications of Interactions, & Random-effect /

Hierarchical / Multilevel Models Interactions in nonlinear/qualitative (inherently

interactive, latent-variable) models Nonlinear Least-Squares Models

Interactions in Pol-Sci Interactions in Pol-Sci ResearchResearch

Common. ‘96-‘01 AJPS, APSR, JoP: 54% some stat meth (=s.e.’s), of which

24% = interax (so interax ≈ 12.5% total). (N.b., most rest nonlin/qual dep-var &

formal theory, not counted, so this understates tech nature our discipline)

Statistical Analysis

Interaction-Term Usage Journal (1996-2001)

Total Articles

Count % of Tot Count % of Tot % of Stat American Political Science Review 279 274 77% 69 19% 25% American Journal Political Science 355 155 55% 47 17% 30%

Comparative Politics 130 12 9% 1 1% 8% Comparative Political Studies 189 92 49% 23 12% 25%

International Organization 170 43 25% 9 5% 21% International Studies Quarterly 173 70 40% 10 6% 14%

Journal of Politics 284 226 80% 55 19% 24% Legislative Studies Quarterly 157 104 66% 19 12% 18%

World Politics 116 28 24% 6 5% 25% TOTALS 2446 1323 54% 311 13% 24%

Interactions in Pol-Sci Interactions in Pol-Sci TheoryTheory

Our Theories/Substance say should be more; core classes of argument are almost inherently interactive: INSTITUTIONAL: institutions=interactive variables: funnel,

moderate, shape, condition, constrain, refract, magnify, augment, dampen, mitigate… pol processes that translate soc interest struct into effective pol pressures, &/or pressures into pub-pol responses, &/or policies to outcomes.

“…institutionalist model=>policy more than sum of countervailing pressure from soc grps; that pressure mediated by org’l [i.e., inst’l] dynamic… (Hall).”

“[Political struggles] mediated by inst’l set in which they [occur] (Ikenberry)” “institutions… constrain & refract politics… [effects of] macrostructures

magnified or mitigated by intermed-lvl institutions… help us…with explanation of contingent nature of pol-econ dev’p… (Steinmo & Thelen)”

“SIE clearly a move [toward] incorp’ing inst’l features into rat-choice. Structure & procedure combine w/ pref’s to produce outcomes (Shepsle)

STRATEGIC: actors’ choices (outcomes) conditional upon environ., opp. set, & other actors’ choices (see: tomorrow).

CONTEXTUAL: actors’ choices (outcomes) conditional upon environ., opp. set, & other actors’ choices (see: tomorrow).

Across all subfields & areas: Comp Pol… Int’l Rltns… U.S. Pol…; PolEcon… PolBehave… LegStuds… PolDev…

Economy

Society

Polity

Economics Affects Politics and SocietyPolitics Affects Economics and SocietySociety Affects Politics and Economics

(Complex) Context-(Complex) Context-Conditionality: Conditionality: (Hallmark of Modern (Hallmark of Modern

Pol-Sci Theory?)Pol-Sci Theory?) Principal-Agent (Shared Control) Situations, for example:

If fully principal: y1=f(X); If fully agent: y2=g(Z); Institutions=>Monitoring & Enforcement costs principal must

pay to induce agent behave as principal would: 0≤h(I)≤1. RESULT:

( )

( )

( )

( ) ( ) 1 ( ) ( )

( ) ;

( ) ;

( ) ( )

y fx x

y gz z

y hi i

y h f h g

h

h

f g

X

Z

I

I X I Z

I

I

X Z

In words…………i.e., effect

of anything depends on everything else!

Not Every Argument Is Not Every Argument Is an Interactive Argumentan Interactive Argument Not Interactive:

X affects Y through its effect on Z: XZY X and Z affect each other: xz Y depends on X controlling for Z, or Y depends

on X & Z: E(Y|X,Z)=f(Z), E(Y|X)=f(Z), Y=f(X,Z) Interactive:

Effect of X on Y depends on Z (implies converse: Effect of Z on Y depends on X):

Y Yf Z f X

X Z

From Theory/Substance to From Theory/Substance to Empirical-Model Empirical-Model

SpecificationSpecification Classic Comparative-Politics Example:

Societal Fragmentation, SFrag, & Electoral-System Proportionality, DMag, Effective # Parliamentary Parties: ENPP

“Theory”: Hypotheses:

Empirical Specification: Lots ways do it

),( ,DMag,SFragENPP f

0

SFrag

ENPP0

DMag

ENPP

0

2

DMagSFrag

ENPP

SFragDMagDMagENPP

SFragENPP

The Typical Linear-Interactive The Typical Linear-Interactive SpecificationSpecification

Want linear f() w/ these props; many ways get there:

SFragββDMag

ENPP

DMagββSFrag

ENPP

εSFragDMagβDMagβSFragββ

εSFragDMagγαDMagγSFragαβ

εSFragDMagγDMagγDMagSFragαSFragαβ

εDMagSFragγγSFragDMagααβENPP

SFragγγSFragfβDMag

ENPP

DMagααDMagfβSFrag

ENPP

εDMagβSFragββENPP

SFDMDM

SFDMSF

SFDMDMSF0

11000

10100

10100

10

?

2

10

?

1

210

Interpretation of Interpretation of EffectsEffects::Derivatives & Differences, Derivatives & Differences, NotNot

CoefficientsCoefficients Standard Linear Interactive Model:Standard Linear Interactive Model:

Effect of Effect of SFragSFrag on on ENPPENPP (is (is ff of of DMagDMag):):

Effect of Effect of DMagDMag on on ENPPENPP (is (is ff of of SFragSFrag):):

ε...SFragDMagβDMagβSFragββENPP SFDMDMSF0

ΔSFDMagβΔSFβΔSFΔSFrag

ΔENPP

DMagββSFrag

ENPPSFragEffect

SFDMSF

SFDMSF

ΔDMSFragβΔDMβΔDMΔDMag

ΔENPP

SFragββDMag

ENPPDMagEffect

SFDMSF

SFDMDM

Interpretation of Interpretation of EffectsEffects: : NOTESNOTES

““Main Effect”: e.g., Main Effect”: e.g., ββSFSF = “ = “main effectmain effect of SFrag” of SFrag” …….YEEEE—UCKK!!!.YEEEE—UCKK!!! Just effect at other var(s)=0, so: Just effect at other var(s)=0, so:

May not be in sample or even logically possible.May not be in sample or even logically possible. Other-var(s)=0 substantive meaning moveable by rescalingOther-var(s)=0 substantive meaning moveable by rescaling E.g., “centering” (centering changes nothing, b.t.w.)E.g., “centering” (centering changes nothing, b.t.w.) COEFFICIENTS ARE NOT EFFECTS. EFFECTS ARE DERIVATIVES COEFFICIENTS ARE NOT EFFECTS. EFFECTS ARE DERIVATIVES

&/OR CHANGES. Only = in purely linear-additive model b/c &/OR CHANGES. Only = in purely linear-additive model b/c only there do derivatives = coefficients.only there do derivatives = coefficients.

Not “effect ‘indep. of’…” & def’ly not “effect ‘control. for’…”Not “effect ‘indep. of’…” & def’ly not “effect ‘control. for’…” Cannot substitute linguistic invention for understand Cannot substitute linguistic invention for understand

model & mathmodel & math Interactions are logically symmetric:Interactions are logically symmetric:

Interax often 2Interax often 2ndnd-moment (var./het.) implic’s too:-moment (var./het.) implic’s too: DMagDMag “permissive” elect sys: allows more parties… “permissive” elect sys: allows more parties… Few Few Veto ActorsVeto Actors allows greater policy-change… allows greater policy-change…

xz

y

zx

y

xz

22zy

xy

Interpretation of Interpretation of EffectsEffects::Standard Errors for Standard Errors for EffectsEffects

Std Errs reported with regression output are for coefficients, not for effects. The s.e. (t-stat, p-level) for is std. err. for est’d effect SFrag

at DMag=0 (…which is logically impossible). Effect of x depends on z & v.v. (i.e., which was the

point, remember?), so does the s.e.:

In words… More Generally:

SF̂

ε...SFragDMagβDMagβSFragββENPP SFDMDMSF0

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ2 ,

E

E Var E E Var

V V V C

x xz x xz

x xz

2x xz x xz x xz

y yEffect x β β z Est.Eff. x β β z

x x

yEst.Var. Est.Eff. x β β z

x

β β z β β z β β z

ˆ ˆ( ) ( )V VXβ X β X

From Hypotheses to Hypotheses From Hypotheses to Hypotheses Tests:Tests:

Does Does YY Depend on Depend on XX or or ZZ?? Hypothesis Mathematical Expression91 Statistical test

x affects y, or y is a function of (depends on) x

y=f(x) 0x xzy x z

F- test: H0: 0x xz

x increases y 0x xzy x z Multiple t-tests:

H0: 0x xz z

x decreases y 0x xzy x z Multiple t- tests:

0x xz z

z affects y, or

y is a function of (depends on) z y=g(z)

0z xzy z x F- test:

H0: 0z xz

z increases y 0z xzy z x Multiple t-tests:

H0: 0z xz x

z decreases y 0z xzy z x Multiple t- tests:

H0: 0z xz x

From Hypotheses to Hypotheses From Hypotheses to Hypotheses Tests:Tests:

Is Is YY’s Dependence on ’s Dependence on XX Conditional on Conditional on ZZ & v.v.? & v.v.? How? How? Hypothesis Mathematical Expression92 Statistical test

The effect of x on y depends on z

y=f(xz,•) ( )x xzy x z g z

2 0xzy x z y x z t-test: H0: 0xz

The effect of x on y increases in z 2 0xzy x z y x z t-test: H0: 0xz

The effect of x on y decreases in z 2 0xzy x z y x z t-test: H0: 0xz

The effect of z on y depends on x

y=f(xz,•) ( )z xzy z x h x

2 0xzy z x y z x t-test: H0: 0xz

The effect of z on y increases in x 2 0xzy z x y z x t-test: H0: 0xz

The effect of z on y decreases in x 2 0xzy z x y z x t-test: H0: 0xz

Does Does YY Depend on Depend on XX, , ZZ, or , or XZXZ??Hypothesis Mathematical Expression93 Statistical Test

y is a function of (depends on) z, z, and/or their interaction

y=f(x,z,xz) F-test: H0: 0x z xz

PresentationPresentation: Marginal-Effects / : Marginal-Effects / Differences Tables & GraphsDifferences Tables & Graphs

Plot/Tabulate Plot/Tabulate EffectsEffects, dy/dx, over Meaningful , dy/dx, over Meaningful &/or Illuminating Ranges of z, with Conf. Int.’s&/or Illuminating Ranges of z, with Conf. Int.’s

Explain axesExplain axes Explain shapeExplain shape Linear-interax:Linear-interax:

WillWill cross 0 cross 0 Insig @ d=0Insig @ d=0

Rescaling &Rescaling & ““main effect”main effect” ““centering”centering” Max(Asterisks)Max(Asterisks)

zCzVVtzdxydVartdxyd xzxxzxpdfxzxpdf )ˆ,ˆ(2)ˆ()ˆ(ˆˆ)/ˆ(/ˆ 2,,

Figure 5. Marginal Effect of Runoff, Extending the Range of Groups

-10

-5

0

5

10

15

20

-2 -1 0 1 2 3 4 5 6

Effective Number of Societal Groups (Groups )

d(N

umbe

r of

Can

did

ates

)/d

(Run

off

)

Use & Abuse of Some Use & Abuse of Some Common “Rules”Common “Rules”

Centering to Redress Colinearity ConcernsCentering to Redress Colinearity Concerns:: Adds no info, so changes Adds no info, so changes nothingnothing; no help w\ colinearity or ; no help w\ colinearity or

anything else; only moves substantive content of x=0,z=0.anything else; only moves substantive content of x=0,z=0. Specifically, makes coeff. on Specifically, makes coeff. on xx ( (zz), effect when ), effect when zz ( (xx) at sample-) at sample-

mean, the new 0. Do only if aids presentation.mean, the new 0. Do only if aids presentation. Must Include All Components Must Include All Components (if (if xx∙z∙z, then , then xx&&zz):):

Application of Occam’s Razor &/or scientific caution (e.g., Application of Occam’s Razor &/or scientific caution (e.g., greater flexibility to reflect nonlinear in lin-interax model), butgreater flexibility to reflect nonlinear in lin-interax model), but

Not Not a logical or statistical requirement…a logical or statistical requirement… Safer rule than opposite & to check almost always, butSafer rule than opposite & to check almost always, but NotNot over-ride over-ride theory & evidence if (strongly) agree on theory & evidence if (strongly) agree on

exclusionexclusion.. Pet-Peeve: Lingistic Gymnastics to Dodge the MathPet-Peeve: Lingistic Gymnastics to Dodge the Math

““Main effect, Interactive effect”: Main effect, Interactive effect”: thethe effect in your model is effect in your model is dy/dxdy/dx

Discussion of [coefficients & s.e.’s] as if [effects & s.e.’s].Discussion of [coefficients & s.e.’s] as if [effects & s.e.’s].

Elaborations, Complications, & Elaborations, Complications, & Extensions:Extensions:

Sample-Splitting v. (Dummy-)InteractingSample-Splitting v. (Dummy-)Interacting Split-sample Split-sample ≈ Full-Dummy Interaction≈ Full-Dummy Interaction::

Subsampling by binary (or multinomial) category & est sep’ly Subsampling by binary (or multinomial) category & est sep’ly ≈ ≈ Dummy for each category & interact each dummy w/ each Dummy for each category & interact each dummy w/ each xx..

Coeff’s same (or equal substantive content if Coeff’s same (or equal substantive content if set-1set-1 dummies). dummies). S.E.’s same except S.E.’s same except ss22 part of OLS’s part of OLS’s ss22(X’X)(X’X)-1-1 is is ssii

22 for splitting for splitting Can make exact by allow Can make exact by allow ssii

22 (FWLS) (FWLS) Subsample by hi/lo values some non-nominal var equivalent to Subsample by hi/lo values some non-nominal var equivalent to

nominalizingnominalizing the extra-nominal info & dummy-interact; the extra-nominal info & dummy-interact; I.e., I.e., =crazy=crazy on most grnds (=wasting info; w/ little we have?!) on most grnds (=wasting info; w/ little we have?!) Non-parametric, or Non-parametric, or extremeextreme-measure-err arg’s might justify-measure-err arg’s might justify

Biggest Difference:Biggest Difference: Split-sample abets eyeballing, obfuscates statistical analysis, of the Split-sample abets eyeballing, obfuscates statistical analysis, of the

main point: the different effects by category.main point: the different effects by category. What’s s.e/signif. of bWhat’s s.e/signif. of b1i1i-b-b1j1j? Need ? Need Luckily, cov=0, but, still, squaring 2 terms, sum, & square-root in head?Luckily, cov=0, but, still, squaring 2 terms, sum, & square-root in head?

Can choose Can choose full dummy setfull dummy set to mirror the split-sample estimates to mirror the split-sample estimates directly (& report that way, if wish) or the directly (& report that way, if wish) or the set-less-oneset-less-one to get to get significance of differences b/w samples directly (std rptd significance of differences b/w samples directly (std rptd tt-test)-test)

jijijiji bVbVbbCbVbVbbes 11111111 ,2..

Elaborations, Complications, & Elaborations, Complications, & Extensions:Extensions:

Random-Effect & Hierarchical/Multi-lvl Random-Effect & Hierarchical/Multi-lvl ModelsModels R.E. Model: Odd that std. lin-interax model:R.E. Model: Odd that std. lin-interax model:

Assumes know Assumes know y=f(X)+errory=f(X)+error:: But But dy/dx=f(z)dy/dx=f(z) w/o error!: w/o error!: So, try:So, try:

=> std. lin-interact again!...Except compound error-term…=> std. lin-interact again!...Except compound error-term…

HLM: Same model, except HLM: Same model, except xxijij & & zzjj,&,& So it just std. lin-interact too, although with slightly So it just std. lin-interact too, although with slightly

different compound error stochastic properties.different compound error stochastic properties.

εxzβzβxββy xzzx0

xzββ xzxxy

zεxεεxzzxβ

εzεxxεzβy

εxβ

εzβ

εzβxββy

2100

0210

22z

y

11x

y

0210

1100

1010

10

10

jijijjij zx 210*

Elaborations, Complications, & Elaborations, Complications, & Extensions:Extensions:

Frequent 2Frequent 2ndnd-Moment Implications -Moment Implications InteractionsInteractions DMagDMag permissive ele sys: permissive ele sys: allowsallows more parties… more parties…

Few Few Veto ActorsVeto Actors allowsallows greater policy- greater policy-change…change…

I.e., these are Rndm-Coeff Propositions, as before…I.e., these are Rndm-Coeff Propositions, as before…

0 1 0 1 ; ( ) ( ) , e.g., y VP V f VP VP

0 1 0 1 ; ( ) ( ) , e.g., NP DM V f DM DM

Elaborations, Complications, & Elaborations, Complications, & Extensions:Extensions:

Random-Effect & Hierarchical/Multi-lvl Random-Effect & Hierarchical/Multi-lvl ModelsModels Properties of OLS Estimates Lin-Interact if truly RE/HLM:Properties of OLS Estimates Lin-Interact if truly RE/HLM:

So, OLS coeff. est’s differ from truth by So, OLS coeff. est’s differ from truth by AAεε**, as usual:, as usual:

So, OLS coeff. est’s unbiased & consistent, as usual:So, OLS coeff. est’s unbiased & consistent, as usual:

Note: only works for models w/ additively separable stochastic Note: only works for models w/ additively separable stochastic component; not nec’ly hold for others, like logit/probit, e.g.component; not nec’ly hold for others, like logit/probit, e.g.

But, OLS s.e.’s will be wrong; not But, OLS s.e.’s will be wrong; not ss22((X'XX'X))-1-1, but:, but:

zεxεεzεxεεxzβzβxββy 210210xzzx0 Xβ

*210

210

εzεxεε

zεxεε

XXXβXXXXβXXX

XβXXXyXXXβ111

11LS

ˆ

....0000

ˆ

DEQEE

EEEE

EEE

βXXXβXXXβ

XXXβXXXβ

XXXβXXXββ

11

11

11LS

zεxε

zεxεεzεxεε

zεxεεε

21

210210

210*

ˆ 2 ,

0 0

( 2 cov( ) terms, but all assumed 0)

V V V V C

V V

V

V V V

1 1 1

LS

1 1 1

1 1

1 1

β β X X X β X X X β X X X

X X X X X X X X X

X X X X X X

X X X X X X

* * *

* *

0 1 2

0 1 2

ε ε ε

ε ε

ε ε x ε z

ε ε x ε z

Elaborations, Complications, & Elaborations, Complications, & Extensions:Extensions:

Random-Effect & Hierarchical/Multi-lvl Random-Effect & Hierarchical/Multi-lvl ModelsModels

Not Not σσ22II (even if each (even if each εε** is), so whole thing is), so whole thing doesn’t reduce to doesn’t reduce to σσ22((X'XX'X))-1-1, so OLS s.e.’s wrong., so OLS s.e.’s wrong.

Be OK on avg (unbiased) & in limit (consistent) Be OK on avg (unbiased) & in limit (consistent) if that term varied in way “orthogonal to if that term varied in way “orthogonal to X'XX'X”” But def’ly not b/c [∙] includes But def’ly not b/c [∙] includes xx & & zz, which part of , which part of XX!! =brilliant insight of ‘robust’ (i.e., consistent) s.e. =brilliant insight of ‘robust’ (i.e., consistent) s.e.

est’s:est’s: Only need s.e. formula that accounts relation V(Only need s.e. formula that accounts relation V(εε**) to “) to “X'XX'X”, ”,

i.e., regressors, squares, & cross-prod’s involved in i.e., regressors, squares, & cross-prod’s involved in X'[X'[∙∙]X]X”” “ “, robust”, robust” & & “, cluster”“, cluster” work! (for RE & HLM, resp’ly) work! (for RE & HLM, resp’ly)

so track eso track e22 rel rel xx'xx' & & zzzz'' het.het.: :

sim but grpngsim but grpng het-clusterhet-cluster::

11LS XXXXXXβ zεxεε 210 VVVV ˆ

zzxx Iβ 2ˆˆ REV

n

1i

2ie

n

1][ 'XX ii

zzxx 22

21

20

ˆˆ Iβ HMV( ) ( )

j jn nJ

ij iji 1 i 1j 1

[ ] e e

ij ijX X

Elaborations, Complications, & Elaborations, Complications, & Extensions:Extensions:

Random-Effect & Hierarchical/Multi-lvl Random-Effect & Hierarchical/Multi-lvl ModelsModels

appropriate appropriate “, robust”“, robust” & & “, cluster”“, cluster” work! work! I.e., asymptotically std errs right…I.e., need large I.e., asymptotically std errs right…I.e., need large nnjj

I.e., coefficients still inefficient.I.e., coefficients still inefficient. Want the efficiency? HLM/RE or FGLS/FWLS.Want the efficiency? HLM/RE or FGLS/FWLS. Note: similarity RE and HLM, RE & FWLS. As suggests, RE Note: similarity RE and HLM, RE & FWLS. As suggests, RE

only helps efficiency and only rightly does so if that’s all it only helps efficiency and only rightly does so if that’s all it does. (I.e., if the RE’s orthogonal to X.)does. (I.e., if the RE’s orthogonal to X.)

I.e., “work” thusly for models with additively-I.e., “work” thusly for models with additively-separable stochastic componentsseparable stochastic components As w/ all such “sandwich” estimators, logical disconnect in As w/ all such “sandwich” estimators, logical disconnect in

applying them to models w/o such separability.applying them to models w/o such separability.

11LS XXXXXXβ zεxεε 210 VVVV ˆ

0 0 1 1 0 1 20y β x z xz ε ε x ε z

Elaborations, Complications, & Elaborations, Complications, & Extensions:Extensions:

Interax in Nonlin/Qual (Inherently Interax) Interax in Nonlin/Qual (Inherently Interax) ModelsModels

Probit/Logit Models w/ Interactions Probit: – Logit:

Marginal Effects: (nonlin., so must spec. @ what X) Start w/ XB pure lin-add, model still inherently inter. b/c S-

shaped: Probit: Logit:

If XB=…+βxx+ βzz+ βxzxz… same except dXB/dx=βx+ βxzz; underlying propensity, i.e., movement along S-shape also interact specifically of x&z.

Probit: • Logit:

XB )1(Yp 1exp1

exp1

exp)1(

XB

XB

XBYp

xxxx

p

XBXB

XBXB

xxeee

xe

exe

e

e

ee

xe

exe

ex

ee

pp

ex

p

11

11

111

1

11

1

22

2

2

2

1

XBXB

XB

XB

XB

XB

XB

XB

XBXB

XB

XB

XB

XBXBXB

XB

zxzxxp XB zpp xzxx

p 1

Standard Errors?Standard Errors? Delta Method: Delta Method:

Probit Marginal-Effect s.e.:Probit Marginal-Effect s.e.:

Logit: same, except replacesLogit: same, except replaces For first-difference effects, similar, but need For first-difference effects, similar, but need

specify from what X to what X.specify from what X to what X. Or you could Or you could CLARIFYCLARIFY

ˆ. . ( )

ˆ ˆ ˆ

AsymVar f

f V f

β β

β

β β β

k

x

x

kk

k

k

x

x

kk VC

CV

βX

βX

βX

βX

βX

βX

βX

βX

ˆ

1

ˆ

1

11

ˆ

1

ˆ

ˆ

ˆ

ˆˆˆ,ˆˆ

ˆ,ˆˆˆˆ

ˆ

ˆ11

xpp βX ˆ

ˆ1ˆ x βXβX

ˆˆ

Elaborations, Complications, & Elaborations, Complications, & Extensions:Extensions:

Interax in Nonlin/Qual (Inherently Interax) Interax in Nonlin/Qual (Inherently Interax) ModelsModels

Complex Context-Conditionality & Complex Context-Conditionality & Nonlinear Least-SquaresNonlinear Least-Squares

Complex Context ConditionalityComplex Context Conditionality: The effect of : The effect of anything depends on most everything else. E.g.:anything depends on most everything else. E.g.: PolicymakingPolicymaking::

Socioeconomic-structure of interestsSocioeconomic-structure of interests Party-system and internal party-structuresParty-system and internal party-structures Electoral system & Governmental systemElectoral system & Governmental system Socio-economic realities linking policies to outcomesSocio-economic realities linking policies to outcomes

Voting:Voting: Voter preferences & informational environmentVoter preferences & informational environment Party/candidate locations & informational environmentParty/candidate locations & informational environment Electoral & governmental systemElectoral & governmental system

InstitutionsInstitutions: Sets of institutions; effect each depends : Sets of institutions; effect each depends configuration others present (e.g., that core of configuration others present (e.g., that core of VoCVoC claim). claim).

Strategic InterdependenceStrategic Interdependence: each actors’ action depends on : each actors’ action depends on everyone else’s; complex feedback (see tomoreveryone else’s; complex feedback (see tomorrow)row)

Complex Context-ConditionalityComplex Context-Conditionality EmpiricallyEmpiricallyMulticolinear Nightmare: Options?Multicolinear Nightmare: Options?

Ignore context conditionality (stay linear-add.)Ignore context conditionality (stay linear-add.) Inefficient at best, biased more usually, and, anyway, Inefficient at best, biased more usually, and, anyway,

context-conditionality is our interest!context-conditionality is our interest! Isolate 1or few interactions for close study; ignore Isolate 1or few interactions for close study; ignore

rest (linear-interactive).rest (linear-interactive). Same, to degree lessened by amount interax allow, but Same, to degree lessened by amount interax allow, but

demands on data rise rapidly w/ that amount.demands on data rise rapidly w/ that amount. ““Structured Case Analysis”: don’t see how helpsStructured Case Analysis”: don’t see how helps EMTI: Lean harder on theory/substance to specify EMTI: Lean harder on theory/substance to specify

more precisely nature interactions: functional form, more precisely nature interactions: functional form, precise measures, etc.precise measures, etc. Refines question put to the data (changes default tests also).Refines question put to the data (changes default tests also). GIVENGIVEN theory/subst. specification into empirical model, can theory/subst. specification into empirical model, can

estimate complex interactivity. Side benefits. But must estimate complex interactivity. Side benefits. But must givegive..

Nonlinear Least-SquaresNonlinear Least-Squares Estimate NLS:Estimate NLS:

Interpreting NLS: (already know): Effects = deriv’s Interpreting NLS: (already know): Effects = deriv’s & 1& 1stst-diff’s; Std.Errs. by Delta Meth or simulation-diff’s; Std.Errs. by Delta Meth or simulation

ˆ ˆ

ˆ

ˆ ˆ ˆ

, + with ~

ˆ ˆ, , so = ,

ˆ ˆˆ ˆ , ,

ˆ ˆ ˆ ˆ SSE= , , , ,

ˆ ˆ ˆFOC: SSE=0 2 , 2 , , 0

f g

E f f

Min Min f f

Min f f f f

f f f

Β Β

Β

Β Β Β

y X Β ε ε ε

y X Β y X Β ε

ε ε y X Β y X Β

y y - y X Β X Β y X Β X Β

X Β y X Β X Β

1*

* *

ˆ

1 1 1*

1 1

ˆ ˆ ˆ ˆSo, if, e.g., , , then:

1 ˆ ˆˆ , , (also, as always)

ˆWriting , as simply , we have:

ˆ

LS

LS LSLS

LSLS

f

f fn k

f

Β

X Β XΒ X y X XΒ Β X X X y

V ε y X Β y X Β

X Β

V Β V y V y

Ω 1 1 12 2

ˆ ˆ, which if , meaning

, &, if = , , as always.

f

X Β XΒ X

X X X ΩX X X Ω I X X

Nonlinear Least-Squares & Nonlinear Least-Squares & EMTIEMTI

EITM: EITM: EEmpirical mpirical IImplications of mplications of TTheoretical heoretical MModelsodels Vision:Vision: Theory => more, sharper predictions => better tests, Theory => more, sharper predictions => better tests,

which therefore inform theory morewhich therefore inform theory more TMEI: TMEI: TTheory-specified heory-specified MModels for odels for EEmpirical mpirical IInferencenference

Vision:Vision: Theory structures empirical models & relations b/w Theory structures empirical models & relations b/w obs => specification & (causal) i.d. of empirical modelsobs => specification & (causal) i.d. of empirical models

TIEM: TIEM: TTheoretical heoretical IImplications of mplications of EEmpirical mpirical MMeasureseasures Vision:Vision: Emp. regularity, finds, measures inform thry dev Emp. regularity, finds, measures inform thry dev

EMTI: EMTI: EEmpirical mpirical MModels of odels of TTheoretical heoretical IIntuitionsntuitions Vision:Vision: Intuitions derived from theoretical models specify Intuitions derived from theoretical models specify

empirical modelsempirical models Note: Strongly counter some alternative moves stats Note: Strongly counter some alternative moves stats

& econometrics, & related; there toward non-& econometrics, & related; there toward non-parametric, matching, & experimentation—there, parametric, matching, & experimentation—there, “model-dependence” a 4-letter phrase. Alternative “model-dependence” a 4-letter phrase. Alternative audiences?audiences? Convince skeptic some causal effect exists, vs.Convince skeptic some causal effect exists, vs. For the convinced, give richer, portable model of how work.For the convinced, give richer, portable model of how work.

Nonlinear Least-Squares:Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model“Multiple Hands on the Wheel” Model

(Franzese, (Franzese, PAPA ‘03) ‘03) Monetary Policy in Open & Institutionalized EconMonetary Policy in Open & Institutionalized Econ

Key C&IPE Insts/Struct: CBI, ER-Regime, Mon. OpenKey C&IPE Insts/Struct: CBI, ER-Regime, Mon. Open ºº CBI CBI≡ º≡ º Govt Delegated Mon Pol to CBGovt Delegated Mon Pol to CB ºº Peg Peg≡ º≡ º Domestic (CB&Gov) Delegate to Peg-Curr (CB&Gov)Domestic (CB&Gov) Delegate to Peg-Curr (CB&Gov) ºº FinOp FinOp≡ º≡ º Dom cannot delegate b/c effectively del’d to globeDom cannot delegate b/c effectively del’d to globe

Effect of ev’thing to which for. & dom. mon. pol-mkrs Effect of ev’thing to which for. & dom. mon. pol-mkrs would respond diff’ly depends on combo insts-structs would respond diff’ly depends on combo insts-structs & v.v., &, through intl inst-structs, for. on dom. & v.v.& v.v., &, through intl inst-structs, for. on dom. & v.v.

Multicolinear Nightmare:Multicolinear Nightmare: 2233=8 inst-struct conds, =8 inst-struct conds, ii, times , times kk factors per factors per ππii((XXii) if lin-) if lin-

interactinteract Exponentially more if all polynominials; k!/2(k-2)! if all pairs.Exponentially more if all polynominials; k!/2(k-2)! if all pairs. Good thing can lean on some thry to specify more precisely!Good thing can lean on some thry to specify more precisely!

1 1 2 2

3 3 4 4

5 5 6 6

7 7 8 8

( ) (1 ) ( )

(1 ) ( ) (1 ) (1 ) ( )

(1 ) ( ) (1 ) (1 ) ( )

(1 ) (1 ) ( ) (1 ) (1 ) (1 ) ( )

P E C P E C

P E C P E C

P E C P E C

P E C P E C

X X

X X

X X

X X

Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model

• CB & Govt Interaction (Franzese, AJPS ‘99):

• Full Monetary Expose & Atomistic=>no auton=>

• s.t. that, full e.r. fix=>CB&Govt must match peg=>

c c

Examples & Practice: Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model

• Compact & intuitive, yet gives all the theoretically expected interactions, in the form expected

Examples & Practice: Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model

• Effectively Estimable, yet gives all the theoretically expected interactions, in the form expected

• Just 14 parameters (plus intercepts & dynamics, assuming those constant), just 3 more than lin-add!

• Parameters substantive meaning, too:– Degree to which…constrains certain set of actors.– Yields est. of inflation-target hypothetical fully indep CB

• => general strategy for estimating/measuring unobservables– If know role factor will play & explanators of factor well enough, can estimate

unobservables conditional on both those thrys, if both powerful enough & enough empirical variation.

Examples & Practice: Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model

• Neat, but does it work? (Try it!) Est’d Eq. w/ Std. Errs.:

• Estimated Effects (highly conditional):

Examples & Practice: Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model

Examples & Practice: Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model

Examples & Practice: Nonlinear Least-Squares:“Multiple Hands on the Wheel” Model

Multiple Policymakers:Veto Actors Bargaining in

Common Pools

Overview• Implications for policy outcomes of policymaking-

authority dispersal across multiple, diverse actors:– Veto-Actor Theory (Tsebelis 2002) emphasizes how it:

• Privileges status quo, & so retards policy adjust, reduces change.– Collective-Action / Common-Pool instead highlight:

• Externalities & so over-exploit / underinvest public resources.– Bargaining & Delegation Theories rather tend to stress:

• Bargaining Strengths/Positions & Weighted Compromise.

• This model attempts synthesis:– Distinguish multifarious effects policymaker number &

diversity (i.e., fractionalization, polarization, part’ship).– Offers empirical approach to model these manifold effects

distinctly & effectively.– A (very) initial exploration of this synthetic-theoretical &

structured-empirical approach in evolution fiscal policy (namely, public debt) in dev’d dems, 1950s-90s.

Essential Arguments Veto Actor

• ↑ # &/or ideological/interest polarization policymaking actors whose approval required to ∆SQ, i.e., veto actors, loosely, ↓ probability &/or magnitude policy ∆.– I.e., strictly, as size W(SQ) ↓, which

generally does as # &/or polarization VA ↑, range possible policy ∆(SQ) ↓.

following empirical prediction (Tsebelis 2002, Fig. 1.7):

Possible Distances of New Policy from Status Quo as a Function of the Size of

W(SQ)

• Suggests both mean/expected policy- & variance pol- ↑↓ as W(SQ) ↑↓• No prediction of pol-level or of direction pol-Δ! Only E(|p|), V(p).• Only suggests smaller E(|p|) & V(p): strictly predicts range possible

– Need agenda-setter(s) & amender(s) pref’s & pows & SQ locate to say more;– If assume large-N cases randomize over, or 1 case is random draw from, those

conditions, then get these further E(|p|) & V(p) predicts.– If symmetry in dist conditions, get center line in figure; need something else

(more?) restrictive to get variance prediction, but symmetry not necessary.

Dispersion Policymaking Authority across Multiple Actors: Veto-Actor

Implications • VA Implications: ↑# (Fractlztn) & Pol. VA Privileges SQ

– Retards pol-adjust rates/delays stabilization– ↓range of possible pol-Δ & so, possibly,– ↓ magnitude/variance pol-Δ (1st- & 2nd-order E(Δ)).

• But most not model core theoretical prediction, that fract &/or polar retards policy-adjustment, correctly...– (F ’00, ’02) How model: policy-adjustment-rate effect =

conditional coefficient on lag in dynamic model, not level effect, support.

– (F ’00, ’02) How measure frag/polar in VA thry: raw #, not effective # (size-wtd) actors; max range prefs, not V(prefs) or s.d.(), (size-wtd)

yt=…+(#VA,Range(pref(VA)) yt-1 …; V(Δyt)=f(#VA,Range(pref(VA))

Dispersion Policymaking Authority across Multiple Actors: Common-Pool

Implications • Weingast, Shepsle, Johnsen (1981: WSJ): districting &

distributive/pork-barrel spending (law of 1/n) optimal project-size from district i’s view ↑ in # districts

• Alternative Decision Rules/Processes:– Pure majority rule all pork-barrel projects lose (n-1) to 1

…– Universalism/Log-Roll district-by-district optimal…– Rikerian MWC side-pays sufficient to induce bare-maj: (n-

1)/2 others receive C/n+ε, which also law of 1/n but more marginally so.

– Uncertainty re: MWC Supermajority, i.e. b/w Uni & MWC.– Legislative procedures:

• All coals MWC-Universalism possible;• keys: amend openness, presidents or 2nd chambers, etc.,

anything that add leg step where veto or amend may occur

– Rational Ignorance: makes sidepays/logrolls easier…

Dispersion Policymaking Authority across Multiple Actors: Common-Pool

Implications

• Law of 1/n is general, & stronger as legislative behavior more Uni & less MW, which tendency ↑ as rational ignorance, winning-coalition uncertainty, or legislative-rule closure to amend or veto ↑

• Total pub rev = common pool for n reps, overused to distrib bens; this collective-action prob worsens proportionally by law 1/n; more exactly, by somewhere b/w rates at which (n+1)/2n (MWC) & 1/n (uni) ↓ in n

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 Number of Constituencies

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5

Ratio of B

enefits to C

osts of Projects P

assed

Minimum-Winning-Coalition Decision-Making Universalistic Decision-Making

Manifestation of Common Pools• Manifestation of Common Pool Emphasized Here:

– Common Pool in ELECTIONEERING:• Magnitude incentive to electioneer fades w/ n (Goodhart)• Control over electioneering diminishes w/ n.

– Note: CP not arise in Tsebelis’ VA Theory b/c # & pref’s of VA’s exogenous & predetermined, whereas in CP theory: prefs=f(#).

• CP Effects distinguishable from VA Effects:– Effects on levels, not in dynamics, magnitudes (w/o sign), or

variances.– Proportional to 1/n for equal-sized actors. Std Olsonian

encompassing logic proper n here size-wtd (effect # & sd/var)– Fract (#) & Polar (het.) relate to VA effects; CP relate primarily

to fract (#), although het. can exacerbate some CA probs.

• Suggests Proper Model of Policy-Response to public demand should be: …[factor]x(1-f(ln(Eff#))…

Bargaining, Delegation, & Compromise

• Explicit extensive-form delegation & bargaining games: huge theoretical & growing empirical literature.

• Franzese (‘99,‘02,‘03): offer less context-specific empirical strategy (b/c comparativist’s goal is thry that travels, not that requires different model each context) based on Nash Bargaining:– Most ext forms policy bounded by ideal points of actors:

convex set/hull, upper-contour set (=core of coop. game thry), so like Tsebelis, but further, though short of explicit ext-form

– Policy outside that range possible, e.g., if uncertainty resolved unfavorably, but that highly unlikely that E(pol) outside this range

– Thus, E(pol)=some convex-combo (wtd-avg) pol-mkrs’ ideals convex-combinatorial empirical models ≈ compromise

– If Nash Barg, e.g., & n.b. NB is coop. game-thry but equiv. sev. reasonable ext-form non-coop barg. games (Rubinstein 1982), (geometric) wtd-influence pol-mkng; i.e., simple wtd-avg.

Manifestations of Compromise Policymaking

• Spec’ly re: def&debt, Cusack (‘99, ‘01; but, cf., Clark ‘03)– Arg/thry: left more Keynesian actively counter-cyclical, right less,

maybe even pro-cyclical.– Add Nash-Barg Model wtd-avg pol-mkr partisanship conditions °

Keynesian counter-cyc fisc response to macroecon

• Franzese (’99, ’03): Monetary-Policy Delegation– Two Hands on the Wheel (CPE): C×πc(xc)+(1-C)×πg(xg)

– Mult Hnds (CIPE): {[C×πc(xc)+(1-C) ×πg(xg)] ×(1-P)+P×πp} ×(1-E)+E×πw

• Empirical Implementation:– Ideally:

• Describe barg pow prty i as f of chars i & barg environ, j, fi(vij)

• Describe how policy respond to conditions xk if i had exclusive pol-mkng authority, qi(xk), then could embed Nash-Barg sol’n, ∑ifi(vij)qi(xk), in model for estimation rather than…

– Currently:• Assume wtd-avg compromise outcome pre-estimation.• I.e., just assume Policy responds WtdPartCurrGovt × Macroecon.

Empirical Model of the Theoretical Synthesis• Different aspects policy-maker fragmentation, polarization, &

partisanship:– V-A Effects: raw # (frag) and ideological ranges (polar)– C-P Effects: effective # (frag) &, perhaps, ideol. std-dev/var (polar)– D-B Effects: weighted-mean ideologies (partisanship)

• Different ways these three distinct effects manifest in policies:– V-A (primarily) to slow policy-adjustment (delay stabilization);– C-P induces over-draw from common resources (incl. from future as in

debt); under-invest in common properties (less electioneering), log-prop’ly

– D-B induces convex-combinatorial (compromise) policies, incl. greater left-activist/right-conservative Keynesian-countercyclical relative conservative pro-cyclical, to degree left/right controls policymaking

• So:– Absolute # veto actors & their ideol range modify policy-adjustment

rates– Effective # incumbents & std dev ideol (i.e., a wtd measure) affect

intensity of C-P problem in electioneering, & may have level effect on debt via C-P

– Some bargaining process among partisan policymakers (e.g., Nash wtd-influence) determines combination reflected in overall govt’s policy responsiveness to macro conditions (i.e., degree of Keynesian activism)

Model:

( ) ( )( ) ( )( ) ( )

1 , 1 2 , 2 3 , 3

, , ,

1 2 , 1

1

1

1

it i an it ar it i t i t i t

Y i t U i t P i t cg it it

e it e i t en it sd it it it

D NoP ARwiG D D D

Y U P CoG

E E ENoP SDwiG Z

a r r r r r

b b b b

g g g g e

- - -

D D D

-

= + + + × + + +

D + D + D × + + C B

+ × + + + G+

( ) ( )( ) ( )( ) ( )

1 , 1 2 , 2 3 , 3

, , ,

1 2 , 1

1

1

1

it i an it ar it i t i t i t

Y i t U i t P i t cg it it

e it e i t en it sd it it it

D NoP ARwiG D D D

Y U P CoG

E E ENoP SDwiG Z

a r r r r r

b b b b

g g g g e

- - -

D D D

-

= + + + × + + +

D + D + D × + + C B

+ × + + + G+

Empirical Model Specification & Data

• Dit = Debt (%GDP);

• NoP & ARwiG = raw Num of Prtys in Govt & Abs Range w/i Govt:– VA conception, so modify dynamics. Expect ρn & ρar positive.

– By thry & for efficiency: modify all lag dynamics same.

• ∆Y = real GDP growth; ∆U = ∆ unemp rate; ∆P = infl rate. Macroecon• CoG (govt center, left to right, 0-10):

– CoG enters to modify (=‘ly, by thry & for eff’cy) response to econ: βcg<0.

• X = set pol-ec conds (e.g., E(real-interest)-E(real-growth), ToT) response to which not partisan-differentiated or pub-cred/blame gov:

• ENoP & SDwiG = Effective Num of Prtys in govt & Std Dev w/i Govt:– Frag & Polar by weighted-influence concept. CP level-effects modify (at

same rate) electioneering, Et, pre-elect-year, & Et-1, post-elect-yr.: γen & γsd<0.

• ZГ = set of constituent terms in the interactions:– ENoP, SDwiG may have positive coeff’s by CP effect level debt, but issue is

temporal fract more than current govt fract. Thry o/w suggests omit.

( ) ( )( ) ( )( ) ( )

1 , 1 2 , 2 3 , 3

, , ,

1 2 , 1

1

1

1

it i an it ar it i t i t i t

Y i t U i t P i t cg it it

e it e i t en it sd it it it

D NoP ARwiG D D D

Y U P CoG

E E ENoP SDwiG Z

a r r r r r

b b b b

g g g g e

- - -

D D D

-

= + + + × + + +

D + D + D × + + C B

+ × + + + G+

Empirical Model Estimation & Hypoth’s

• Veto-Actor Effects:– Abs Frag & Polar VA’s Retards Pol-Adj Rate:

• Base Pol-Adj Rate:

• Common-Pool Effects:– Eff # (&s.d.prefs) Pol-Mkrs Dampen E:

• Electioneering: • CP Induces Overborrowing:

• Delegation & Bargaining Effects:– Proportionate Conditioning of Keynesian

Activism by Wtd Left-Rightness of Govt:• Keynesian Activism:

, 0an arr r >

, 0enop sdwigg g >

, 0en sdg g <1 2, 0e eg g >

1 2 30 1r r r< + + <

0, 0, 0y U pb b bD D D< > <

0CoGb <

Table 1: Nonlinear-Least-Squares Estimation of Equation (1) Coefficient Standard Error t-Statistic Prob(T>| t| )

Dt-1 1.212 0.060 20.112 0.000 Dt-2 -0.153 0.085 -1.792 0.074 Dt-3 -0.121 0.045 -2.677 0.008

ρn 0.007 0.006 1.089 0.277 ρar -0.000 0.006 -0.013 0.990 ΔY -0.336 0.111 -3.033 0.003 ΔU 0.992 0.308 3.219 0.001 ΔP -0.188 0.063 -2.965 0.003

βcg -0.037 0.037 -0.988 0.323 X1 (open) 15.891 5.279 3.010 0.003 X2 (ToT) 0.388 1.744 0.222 0.824

X3 (openToT) -10.681 5.156 -2.072 0.039 X4 (dxrig) -0.036 0.066 -0.544 0.587 X5 (oy) 2.064 1.094 1.886 0.060

Et 0.687 0.568 1.210 0.227 Et-1 1.490 0.645 2.310 0.021

γen -0.547 0.182 -3.001 0.003 γsd 0.573 0.486 1.179 0.239

Z1 (CoG) 0.051 0.131 0.390 0.697 Z2 (ENoP) 0.281 0.446 0.629 0.530 Z3 (SDwiG) 0.542 0.437 1.242 0.215 Z4 (NoP) 0.181 0.277 0.654 0.514

Z5 (ARwiG) -0.312 0.259 -1.205 0.228 Summary Statistics

N (Deg. Free) 735 (691) Std. Err. Regression 2.525 R2 ( 2R ) 0.991 (0.990) Durbin-Watson 2.101

Table 1: NLS Estimation of Equation (1) with Insignificant Constituent Terms Removed Coefficient Standard Error t-Statistic Prob(T>| t| )

Dt-1 1.207 0.060 20.290 0.000 Dt-2 -0.158 0.085 -1.851 0.065 Dt-3 -0.117 0.045 -2.577 0.010

ρn 0.011 0.005 2.369 0.018 ρar -0.002 0.004 -0.437 0.662 ΔY -0.375 0.087 -4.332 0.000 ΔU 1.095 0.286 3.829 0.000 ΔP -0.207 0.053 -3.889 0.000

βcg -0.051 0.020 -2.484 0.013 X1 (open) 16.128 5.314 3.035 0.002 X2 (ToT) 0.414 1.728 0.239 0.811

X3 (openToT) -10.780 5.194 -2.076 0.038 X4 (dxrig) -0.038 0.066 -0.578 0.563 X5 (oy) 1.898 1.100 1.724 0.085

Et 0.475 0.420 1.133 0.258 Et-1 1.146 0.562 2.040 0.042

γen -0.570 0.209 -2.727 0.007 γsd 0.881 0.586 1.503 0.133

Summary Statistics N (Deg. Free) 735 (696) Std. Err. Regression 2.522

R2 ( 2R ) 0.991 (0.990) Durbin-Watson 2.099

Table 1: Estimated Veto-Actor, Bargaining, and Common-Pool Effects of Multiple Policymakers Veto-Actor Effects: Estimates of Policy-Adjustment Rate

Adjustment Rates NoP=1 NoP=2 NoP=3 NoP=4 NoP=5 NoP=6 Lag Coefficienta 0.943 0.952 0.960 0.969 0.978 0.986

Policy-Adjust/ Yrb 0.057 0.048 0.040 0.031 0.022 0.014 Long-Run Mult.c 17.498 20.639 25.154 32.200 44.727 73.208

½-Lifed 11.778 13.956 17.087 21.971 30.654 50.397 90%-Lifee 39.127 46.362 56.761 72.985 101.832 167.415

Bargaining Effects: Estimates of Keynesian Fiscal Responsiveness

Mean Econ. Performance -2 std. dev.

Mean Econ. Performance -1 std. dev.

Mean Economic

Performance

Mean Econ. Performance +1 std. dev.

Mean Econ. Performance +2 std. dev.

Growth -2.354 0.454 3.261 6.069 8.877 d(UE) 1.915 1.034 0.153 -0.728 -1.608 Infl -3.593 1.230 6.054 10.877 15.701

CoG E(D| Econ)f E(D| Econ) E(D| Econ) E(D| Econ) E(D| Econ) Fiscal-Cycle Magnitudeg

3.0 3.157 0.599 -1.959 -4.516 -7.074 10.231 4.2 2.930 0.556 -1.818 -4.192 -6.566 9.496 5.4 2.703 0.513 -1.677 -3.867 -6.058 8.761 6.6 2.476 0.470 -1.536 -3.543 -5.549 8.026 7.8 2.250 0.427 -1.396 -3.218 -5.041 7.291 9.0 2.023 0.384 -1.255 -2.894 -4.533 6.555

Collective-Action/ Common-Pool Effects: Estimates of Electoral Debt-Cycle Magnitude ENoP=1 ENoP=2 ENoP=3 ENoP=4 ENoP=5

Electoral-Cycle Magnitudeh 1.07410 0.86454 0.65497 0.44541 0.23585

a Calculated as the sum of the coefficients on the dependent-variable lags times (1+ρnNoP+ρarARwiG), where ARwiG=0.97106?NoP-0.67913. b Calculated as one minus the lag coefficient. c Calculated as one over the policy-adjustment per year. d Calculated as ln(.5)/ ln(lag coefficient). e Calculated as ln(.9)/ ln(lag coefficient).

f Predicted deficits given the state of the macroeconomy listed in that column and the partisanship of government listed in that row. g Calculated as the difference between the predicted deficits in the worst macroeconomic scenario minus those in the best. h Calculated as the sum of the coefficients on Et and Et-1 times (1+γenENoP+ γenSDwiG), where SDwiG=0.500551?ENoP-0.23653.

Extension & Refinement

0 0 0 1 2 1

1 1

0 1 2

2

0 1 2

3

0 1 2

ˆ ( )

(1 ln( #))

( )

{ ( ) ( )}

( )

{ ( ) ( )} (1 ln( #)

( )

t t t t

t t

i i i i

t t

i i i i

t t

y NoP AR y

Eff

a a NoP a AR

g f

a a NoP a AR

g f Eff

a a NoP a AR

r r r -= + + +

-+

+ +

S+

+ +

é ùS -ë û++ +

x b

x b

x v

x v