linear interaction modelslinear interaction modelsfranzese/ps699.linearinteractionmodels.… ·...

Linear Interaction ModelsLinear Interaction Models

PS699, Winter 2010

(Here for further pedagogical slides & materials:http://www.umich.edu/~franzese/SyllabiEtc.html)

Robert J. Franzese, Jr.

Professor of Political Science,

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

Slide 1 of 37

Overview• Interactions in Pol-Sci: ubiquitous, but should be more

F th t i i l d l ifi ti A t• From theory to empirical-model specification: Arguments that imply interactions (& some that don’t), & how to write.

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

• Presentation: Tables & Graphs, & Choosing between equivalent Specificationsq p

• Use & abuse of some common-practice “rules”• Extensions:

– Split-sample v. dummy-interaction– Common 2nd-moment implications of interactions– Interactions with uncertainty = random coefficients ≈ hierarchical.

Slide 2 of 37

Interactions in Pol-Sci Research• Common. ‘96-‘01 AJPS, APSR, JoP:

54% h ( ’ ) f hi h 24% i– 54% some stat meth (=s.e.’s), of which 24% = interax (so interax ≈ 12.5% or 1/8th total).(N b Q lD & f l h d &– (N.b., most rest QualDep & frml thry, not counted, & “thry” in denom) so understate tech nature discipline)

S i i lStatisticalAnalysis 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 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%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% fLegislative Studies Quarterly 157 104 66% 19 12% 18%

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

Slide 3 of 37

Interactions in Pol-Sci Theory1y• Ubiquitous, but our Theories/Substance say

should be even more; core classes of argumentshould be even more; core classes of argument inherently interactive:

INSTITUTIONAL i tit ti i h tl– INSTITUTIONAL: institutions are inherently interactive variables:

I tit ti f l d t h diti t i• Institutions funnel, moderate, shape, condition, constrain, refract, magnify, augment, dampen, mitigate political processes that…p

– …translate societal interest-structures into effective political pressures,

– …&/or pressures into public-policy responses,– …&/or policies to outcomes.

I h ff ff i i• I.e., they affect effects≡interaction.

Slide 4 of 37

Interactions in Pol-Sci Theory2

• Views from across institutionalist perspectives:

y

Views from across institutionalist perspectives:– Hall: “institutionalist model⇒policy more than sum

countervailing pressure from soc grps; that press mediatedg p g p ; pby organizational dynamic.”

– Ikenberry: “[Political struggles] mediated by inst’l setting where [occur]”

– Steinmo & Thelen: “inst’s…constrain & refract politics… [ ][effects of] macro-structures magnify or mitigated by intermediate-level inst’s… help us…explain the contingentnature of pol econ development ”nature of pol-econ development…

– Shepsle: “SIE clearly a move [to] incorporating inst’l features into R-C Structure & procedure combine w/ preferences tointo R C. Structure & procedure combine w/ preferences to produce outcomes.”

Slide 5 of 37


• Ubiquitous but our Theories/Substance say

y

• Ubiquitous, but our Theories/Substance say should be even more; core classes of argument i h tl i t tiinherently interactive:– INSTITUTIONAL: …

– STRATEGIC: actors’ choices (outcomes) conditional upon institutional/structural environ., p / ,opportunity set, & other actors’ choices.

– CONTEXTUAL: actors’ choices (outcomes)CONTEXTUAL: actors choices (outcomes) conditional upon environment, opportunity set, & aggregates of other actors’ choicesaggregates of other actors choices.

Slide 6 of 37


• Across subfields:

y

– Comparative Politics examples:• Electoral system & societal structure ⇒ party system.y p y y

• Divided government & polarization ⇒ legislative productivity.

• Corruption depends institutional & societal structures.

– International Relations examples:p• System polarity & offense-defense balance ⇒ war

propensity.

• Terrorist targeting & counterterrorism responses depend “grievance” & resources

– American Politics …

Slide 7 of 37


• Political Economy:

y

– Electoral & partisan cycles depend on inst’l & econ conditions

• Political Behavior:– Gov’t inst’s shape voter behavior: balancing (Kedar, p g ( ,

Alesina); economic voting (Powell & Whitten); etc.

• Legislative Studies:g– Effects divided gov’t depend presidential v.

parliamentary.parliamentary.

• Political Development:Effect inequality on democratization depends– Effect inequality on democratization depends cleavage structure.

Slide 8 of 37

Theory & Substance:E ’ F it “M d l”

E i Aff P li i d S i

Everyone’s Favorite “Model”

Economics Affects Politics and SocietyPolitics Affects Economics and SocietySociety Affects Politics and Economics(Actually,

thi iEconomy

this is a picture of ubiquitous d

P lit

endogeneity, but probably

many Polityinteractions

within that core set of

Society

endogeneousrelationships!)

Slide 9 of 37

Theory & Substance:( )An Old (& still) Favorite “Model” of Mine

(A)Result of Outcomes at T-1 Result of Outcomes at T0

O t T+1(Actually, this Interest Structure of

the Polity and Economy

Elections

GovF

The Cycle of Political Economy

On to T+1( y,picture doesn’t

explicitly depict vernment

Formation

Examples of the Elements at Each Stage:

(A) Interests:Sectoral Structure of EconomyIncome Distribution

depict interactions

either. Lots of mediation and

(B)Partisan

Representation in Government

Age DistributionTrade Openness

Elections:Electoral LawVoter Participation

Government Formation:

Action at Time T0

mediation and a circular /

sequential sort f bi i Government

Polic

y

FractionalizationPolarization

(B) Representation:Partisanship

Policy:Fiscal Policy Gov

ernmen

tal A

ctors

of ubiquitous endogeneity is shown directly,

P

(Government

ination)(C)

Fiscal PolicyMonetary PolicyInstitutional Adjustment

Government Termination:Replacement Risk

(C) Outcomes:Unemployment

Non-G

but again probably many

interactions (GoTerminatio(C)

Political and Economic Outcomes

UnemploymentInflationGrowthSectoral ShiftDebtInstitutional Change

Exogenous Factors

within that set relationships!)

Slide 10 of 37

Theory & Substance:An Newer Favorite “Model” of Mine

• Complex Context Conditionality:• Complex Context-Conditionality:– Effect of (almost) everything depends on

(almost) everything else.– E.g., Principal-Agent SituationsE.g., Principal Agent Situations

• If fully principal, y1=f(X); if fully agent, y2=g(Z); institutions: 0≤h(I)≤1.y2 g(Z); institutions: 0≤h(I)≤1.

{ }( ) ( ) 1 ( ) ( )= + −y h f h gI X I Z( ) ( )( ) ; ( ) ;∂ ∂ ∂ ∂

∂ ∂ ∂ ∂⇒ = = −y f y gx x z zh hX ZI I

[ ]( ) ( ) ( )∂ ∂∂ ∂= −y hi i f gI X Z

Slide 11 of 37

(Complex) Context-Conditionality: (Hallmark of Modern Pol-Sci Theory?)

P i i l A t (Sh d C t l) Sit ti f l• Principal-Agent (Shared Control) Situations, for example:– If fully principal: y1=f(X);– If fully agent: y2=g(Z);y g y2 g( );– Institutions=>Monitoring & Enforcement costs principal must pay to

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

{ }( ) ( ) 1 ( ) ( )y h f h g= + −I X I Z• In words……

( )( ) ;y fx xh∂ ∂

∂ ∂⇒ = XI………i.e., effect of

( )( ) ;y gz zh∂ ∂

∂ ∂= − ZIanything depends on everything

[ ]( ) ( ) ( )y hi i f g∂ ∂

∂ ∂= −I X Zeverything else!

Slide 12 of 37

Not Every Argument Is an I t ti A tInteractive Argument

• Not Interactive:• Not Interactive:– X affects Y through its effect on Z: X⇒Z⇒Y

• In (political) psychology / behavior, this called mediation. (p ) p y gy / ,Interaction is called moderation in this literature.

– X and Z affect each other: X⇔Z.I X d Z d t h th N t i l t t• I.e., X and Z endogenous to each other. Note: irrelevant to Gauss-Markov (OLS is BLUE); merely implies care to what partials (coefficients) mean.

– 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)• I e the outcomes differ across 2×2 of X and Z• I.e., the outcomes differ across 2×2 of X and Z.

• Interactive: Effect of X on Y depends on Z (⇒converse: Effect of Z on Y depends on X):converse: Effect of Z on Y depends on X):

( ) ( )Y Yf Z f XX Z

∂ ∂= ⇔ =∂ ∂

Slide 13 of 37

From Theory/Substance to E i i l M d l S ifi tiEmpirical-Model Specification

Cl i C ti P liti E l• Classic Comparative-Politics Example:– Societal Fragmentation, SFrag, &

– Electoral-System Proportionality, DMag,

– ⇒Effective # Parliamentary Parties: ENPP⇒Effective # Parliamentary Parties: ENPP

• “Theory”: ),( ε⋅= ,DMag,SFragENPP f• Hypotheses:

0≥∂∂

SFragENPP 0≥

∂∂

DMagENPP

{ } { } 2 2

0∂ ∂∂ ∂∂ ∂ ∂ ∂

≡ ≡ ≡ ≥∂ ∂ ∂ ∂ ∂ ∂

ENPP ENPPSFrag DMag ENPP ENPP

DMag SFrag SFrag DMag DMag SFrag• Empirical Specification: Lots ways get there...∂ ∂ ∂ ∂ ∂ ∂DMag SFrag SFrag DMag DMag SFrag

Slide 14 of 37

A Typical Linear-Interactive Specification• Want linear f(⋅) w/ these properties; many ways to get there:

εDMagβSFragββENPP 210 +++=

( ) DMagααDMagfβSFragENPP

10

?

1 +=→=∂∂

( )

( ) ( )

SFragγγSFragfβDMagENPP

10

?

2 +=→=∂∂

( ) ( )

( )εSFragDMagγDMagγDMagSFragαSFragαβ

εDMagSFragγγSFragDMagααβENPP

10100

10100

+++++=+++++=⇒

( )εSFragDMagβDMagβSFragββ

εSFragDMagγαDMagγSFragαβ

SFDMDMSF0

11000

++++=+++++=

DMagββSFragENPP

SFDMSF

∂

+=∂∂

⇒

SFragββDMagENPP

SFDMDM +=∂∂

Slide 15 of 37

Interpretation of Effects:Derivatives & Differences Not CoefficientsDerivatives & Differences, Not Coefficients

• Standard Linear Interactive Model:

= + + + × + +EN β β SF β DM β SF DM ε

• Effect of SFrag on ENPP (is a function of DMag):

= + + + × + +0 SF DM SFDMEN β β SF β DM β SF DM ... ε

g ( f g)

( ) SF SFDM

ENEffect SF β β DMSF

∂≡ = +

∂

SF SFDMΔEN β ΔSF β DM ΔSF

ΔEN β β DM

= + ⋅

≡ = +

• Effect of DMag on ENPP (is f of SFrag):SF SFDMβ β DM

ΔSF≡ = +

( ) ∂ENPP( ) ∂≡ = +

∂

≡ = + ⋅

DM SFDM

DM SFDM

ENPPEffect DMag β β SFragDMag

ΔENPP β ΔDM β SFrag ΔDM

≡ = +

DM SFDM

DM SFDM

ΔENPP β β SFragΔDM

Slide 16 of 37

Interpretation of Effects: NOTES1

• “Main Effect” & “Interactive Effect”:– For example βSF = “main effect of SFrag”For example, βSF main effect of SFrag– ….but βSF is merely the effect of SFrag at other

variable(s) involved in interaction with it=0, so:variable(s) involved in interaction with it 0, so:• Other-var(s)=0 may be extreme in the sample, or beyond

sample range, or even logically impossible.• Other-var(s)=0 substantive meaning of 0 altered by

rescalingE b “ t i ” ( t i h thi bt )– E.g., by “centering” (centering changes nothing, btw…)

• Other-var(s)=0 may not have anything substantively main about itmain about it

– Is no Main Effect or separately & Interactive Effect; is just the effect, which conditional, varies:j , ,

( ) ( ) ; ∂ ∂

≡ = + ≡ = +∂ ∂SF SFDM DM SFDM

EN ENEffect SF β β DM Effect DM β β SFSF DM

Slide 17 of 37


• COEFFICIENTS ARE NOT EFFECTS. EFFECTS ARE DERIVATIVES &/OR DIFFERENCES.R C S– Only in purely linear-additive-separable model are

they equal because only there do derivatives simplythey equal because only there do derivatives simply = coefficients.

β i t “ ff t f SF ‘i d d t f’ ” &– βSF is not “effect of SFrag ‘independent of’…” & definitely not its “effect ‘controlling for’…other

i bl ( ) i th i t ti ”variable(s) in the interaction”

• Cannot substitute linguistic invention for under-standing model’s logic (its simple math)

Slide 18 of 37


• Interactions are logically symmetric:– For any function, not just lin-add. { } { } yy 22yy ∂∂∂∂ ∂∂– If argue effect x depends z, must

also believe effect z depends x.

I t ti ft h 2nd t ( i

{ } { }xzy

zxy

xzzx

∂∂∂≡

∂∂∂≡

∂∂

≡∂

∂ ∂∂

• Interactions often have 2nd-moment (variance, i.e., heteroskedacity) implications too:– Larger district magnitudes, DMag, are

“permissive” elect sys: allow more parties…– Fewer Veto Actors allow greater policy-change…

(both need additional assumpts)

• All of this holds for any type of variable:– Measurement: binary, continuous…y,– Level: micro or macro; i, j, k, …

Slide 19 of 37

Frequent 2nd-Moment Implications Interactions

• DMag permissive ele sys: allows more parties…0 1 0 1 ; ( ) ( ) , e.g., NP DM V f DM DMβ β ε ε σ σ= + + = +

– Note: unmodeled interactions look like heteroskedasticity; that’s general, actually. Anything unmodeled gets into e2…

F V t A t ll t li h

0 1 0 1; ( ) ( ) , g ,fβ β

• Few Veto Actors allows greater policy-change…

0 1 0 1 ; ( ) ( ) , e.g., y VP V f VP VPβ β ε ε σ σ= + + = +

• I.e., these are Rndm-Coeff &/or Het-sked Props…Slide 20 of 37

Interpretation of Effects:Standard Errors for EffectsStandard Errors for Effects

ε...SFragDMagβDMagβSFragββENPP SFDMDMSF0 +++++=

• Std Errs reported with regression output are for coefficients, not for effects.

β̂– 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,

SFβ

p ( , p ,remember?), so does the s.e.:

( ) ( ) ˆ ˆE∂ ∂⎛ ⎞≡ = + ⇒ ≡ = +⎜ ⎟⎝ ⎠

x xz x xzy yEffect x β β z Est.Eff. x β β z( ) ( )

( ){ } { }ˆ ˆE Var E E Var

⎜ ⎟∂ ∂⎝ ⎠⎡ ⎤⎧ ∂ ⎫⎛ ⎞ ⎡ ⎤≡ = +⎨ ⎬⎢ ⎥⎜ ⎟ ⎣ ⎦∂⎝ ⎠⎩ ⎭⎣ ⎦

x xz x xz

x xz

ff β β z ff β β zx x

yEst.Var. Est.Eff. x β β z( ){ } { }

{ } { } { } ( )ˆ ˆ ˆ ˆ ˆ ˆ2 ,V V V C

⎢ ⎥⎜ ⎟ ⎣ ⎦∂⎝ ⎠⎩ ⎭⎣ ⎦

= + = + ⋅ + ⋅

z

2x xz x xz x xz

x

β β z β β z β β z

• In words… More Generally: ˆ ˆ( ) ( )V V⎡ ⎤′ ′= ⎣ ⎦x β x β xSlide 21 of 37

From Hypotheses to Hypotheses Tests:Does Y Depend on X or Z?

Hypothesis Mathematical Expression91 Statistical test

ε...SFragDMagβDMagβSFragββENPP SFDMDMSF0 +++++=

Hypothesis Mathematical Expression Statistical testx affects y, or

y is a function of (depends on) x y=f(x)

0x xzy x zβ β∂ ∂ = + ≠ F- test:

H0: 0x xzβ β= = Multiple t-tests:

x increases y 0x xzy x zβ β∂ ∂ = + > Multiple t-tests:

H0: 0x xz zβ β+ ≤

x decreases y 0x xzy x zβ β∂ ∂ = + < Multiple t- tests:

0zβ β+ ≥ 0x xz zβ β+ ≥

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

y=g(z) 0z xzy z xβ β∂ ∂ = + ≠

F- test: H0: 0z xzβ β= =y is a function of (depends on) z xzy β β 0 z xzβ β

z increases y 0z xzy z xβ β∂ ∂ = + > Multiple t-tests:

H0: 0z xz xβ β+ ≤

0β β∂ ∂Multiple t- tests:

z decreases y 0z xzy z xβ β∂ ∂ = + < p

H0: 0z xz xβ β+ ≥

Slide 22 of 37

From Hypotheses to Hypotheses Tests:Is Y’s Dependence on X Conditional on Z & v.v.? How?Is Y s Dependence on X Conditional on Z & v.v.? How?

Hypothesis Mathematical Expression92 Statistical test y=f(xz,•)

The effect of x on y depends on z ( )x xzy x z g zβ β∂ ∂ = + =

( ) 2 0xzy x z y x z β∂ ∂ ∂ ∂ = ∂ ∂ ∂ = =t-test: H0: 0xzβ =

The effect of x on y increases in z ( ) 2 0y x z y x z β∂ ∂ ∂ ∂ = ∂ ∂ ∂ = > t-test: H0: 0β ≤The effect of x on y increases in z ( ) 0xzy x z y x z β∂ ∂ ∂ ∂ = ∂ ∂ ∂ = > t test: H0: 0xzβ ≤The effect of x on y decreases in

z ( ) 2 0xzy x z y x z β∂ ∂ ∂ ∂ = ∂ ∂ ∂ = < t-test: H0: 0xzβ ≥

y=f(xz,•)The effect of z on y depends on x

y f(x , )( )z xzy z x h xβ β∂ ∂ = + =

( ) 2 0xzy z x y z x β∂ ∂ ∂ ∂ = ∂ ∂ ∂ = =t-test: H0: 0xzβ =

Th ff f i i ( ) 2 0β∂ ∂ ∂ ∂ ∂ ∂ ∂ t t t H 0β ≤The effect of z on y increases in x ( ) 2 0xzy z x y z x β∂ ∂ ∂ ∂ = ∂ ∂ ∂ = > t-test: H0: 0xzβ ≤The effect of z on y decreases in

x ( ) 2 0xzy z x y z x β∂ ∂ ∂ ∂ = ∂ ∂ ∂ = < t-test: H0: 0xzβ ≥

Does Y Depend on X, Z, or XZ?Hypothesis Mathematical Expression93 Statistical Testyp p

y is a function of (depends on) z, z, and/or their interaction y=f(x,z,xz) F-test: H0: 0x z xzβ β β= = =

Slide 23 of 37

Use & Abuse of Some Common ‘Rules’• Centering to Redress Colinearity Concerns:

– Adds no info so changes nothing; no help with colinearityAdds no info, so changes nothing; no help with colinearity or anything else; only moves substantive content of x=0,z=0.

– Specifically, makes coeff. on x (z), effect when z (x) at p y, ( ), ( )sample-mean, the new 0. Do only if aids presentation.

• Must Include All Components (if x·z, then x&z):– Application of Occam’s Razor &/or scientific caution (e.g.,

greater flexibility to allow linear w/in lin-interax model), butN t l i l t ti ti l i t– Not a logical or statistical requirement.

– Safer rule than opposite & to check almost always, butNot override theory & evidence if (strongly) agree to exclude– Not override theory & evidence if (strongly) agree to exclude

• Pet-Peeve: Lingistic Gymnastics to Dodge the Math“Main effect Interacti e effect” the effect in model is dy/dx– “Main effect, Interactive effect”: the effect in model is dy/dx.

– Discussion of [coefficients & s.e.’s] as if [effects & s.e.’s].Slide 24 of 37

Presentation: Marginal-Effects / Differences Tables & Graphsp

• Plot/Tabulate Effects, dy/dx, over Meaningful &/or Illuminating Ranges of z with Conf Int ’s&/or Illuminating Ranges of z, with Conf. Int. s– zCzVVtzdxydVartdxyd xzxxzxpdfxzxpdf )ˆ,ˆ(2)ˆ()ˆ(ˆˆ)/ˆ(/ˆ 2

,, ββββββ ++±+=±

– Explain axesE l i h

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

20

– Explain shape– Linear-interax: 10

15s)/d(R

unof

f)

• Will cross 0& be insig @ 0.

R li & 0

5

r of C

andi

date

s

– Rescaling &• “main effect”

“ t i ”-5

0

d(N

umbe

r

• “centering”• Max(Asterisks)

-10-2 -1 0 1 2 3 4 5 6

Effective Number of Societal Groups (Groups )

Slide 25 of 37

Presentation: Expected-Value/Predictions Tables & GraphsTables & Graphs

• Predictions, E(y|x,z):

2220

,

))(ˆ()ˆ()ˆ()ˆ(

)ˆ(ˆ

xzVzVxVV

yVarty

xzzx

pdf

ββββ +++

=±

22

000

0

,0

)ˆˆ(2)ˆˆ(2)ˆˆ(2

)ˆ,ˆ(2)ˆ,ˆ(2)ˆ,ˆ(2

))(()()()(ˆˆˆˆ

xzCzxCxzC

xzCzCxCtxzzx xzzx

xzzx

pdfxzzx

ββββββ

ββββββ

ββββ

ββββ

+++

+++±+++

– Here’s one place a little matrix algebra would help:

),(2),(2),(2 xzCzxCxzC xzzxzxzx ββββββ +++

, 0 ,

0 0 0 0

ˆ ˆ ˆ ˆ ˆˆˆ ˆ( ) ( )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ( ) ( , ) ( , ) ( , )ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ

df p x z xz df p

x z xz

y t Var y x z xz t V

V C C C

β β β β

β β β β β β β

′± = + + + ± x β x

1⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥[ ] 0

0 ,

0

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ( , ) ( ) ( , ) ( , )ˆ ˆ ˆ ˆ 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ( , ) ( , ) ( ) (

x x x z x xzx z xz df p

z x z z z

C V C Cx z xz t x z xz

C C V C

β β β β β β ββ β β β

β β β β β β= + + + ±

ˆ, )ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ( ) ( ) ( ) ( )

xz

xzxzC C C V

ββ β β β β β β

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

– Use spreadsheet or stat-graph software (…list coming...)

0( , ) ( , ) ( , ) ( )xz x xz z xz xzxC C C Vβ β β β β β β ⎣ ⎦⎣ ⎦

Slide 26 of 37

Presentation1:Choose Illuminating Graphics & Base CasesChoose Illuminating Graphics & Base Cases

• Interpretation same regardless of “type” of interax: p g ypeffect always ≡ dy/dx, but present appropriately:– All combos Dummy Discrete or Continuous:All combos Dummy, Discrete, or Continuous:

• Dummy-Dummy=>4 (or 2#interacting variable) points estimated, so box & whisker or histograms effectiveestimated, so box & whisker or histograms effective

• Dummy-Continuous or Discrete(few)-Continuous=>2 (or # categories) slopes, so E(y|x,z) as line or dy/dx as box # g ) p , (y| , ) y/& whisker or histograms effective

• Continuous – Continuous (or DiscMany)=>Effect-lines best or (slices from) contour plot (i.e., slices from 3D)

– Powers (e.g., X & X2=>parabola) viewable as interax w/ self; certain slope shifts too (e.g., dy/dx=a for x<x0

& b for x>x0 is x interact w/ dummy for condition)Slide 27 of 37

Presentation2:Choose Illuminating Graphics & Base CasesChoose Illuminating Graphics & Base Cases

• Interpretation same regardless of “type” of i t ff t l d /d b t tinterax: effect always ≡ dy/dx, but present appropriately…– Always plot over substantively revealing ranges.– Especially with sets of dummies, have several

(identical) specification options:• (full-set or set-less-1): choose which (& what base if use

t l 1) t b t t ti & di iset-less-1) to abet presentation & discussion• (overlapping or disjoint): choose to facilitate presentation

& discussion& discussion.

– Scale Effectively: e.g., center only if & to extent that aids presentation & discussion (b/c centeringthat aids presentation & discussion (b/c centering does nothing else)

Slide 28 of 37

Presentation3: Choose Illuminating Graphics & Base Cases ExamplesGraphics & Base Cases. Examples.

Dummy-Continuous Interaction: could also plot two E(Cands|Groups) lines, with c.i.’s, effectively.

Slide 29 of 37


Dummy-Dummy Interaction: could also plot four E(Supp|gender,party) box-whiskers effectively.

Slide 30 of 37


Quadratic Model of Government Duration as Function of Parliamentary Support of Governing Parties.

Slide 31 of 37


Slide 32 of 37

Elaborations, Complications, & Extensions:S l S litti (D )I t ti 1Sample-Splitting v. (Dummy-)Interacting1

S l l ( b ) F ll D I• Split-sample (e.g., unit-by-unit) ≈ Full-Dum Interax:– Subsample by binary (or multinomial, e.g., CTRY in TSCS)

l I l d d f hcategory to estimate seperately ≈ Include dummy for each category (or set-less-1) & interact each dummy with each x (and include x by itself also if set-less-1)(and include x by itself also if set-less-1)

• Coeff’s same (or equal substantive content if using set-1 dummies).• S.E.’s same except s2 part of OLS’s s2(X'X)-1 is si

2 for splittingp p ( ) i p g

• Can make exact by allow si2 (FWLS)

– Subsample by hi/lo values some non-nominal var equivalent to nominalizing the extra-nominal info & dummy-interact;

• I.e., wasting information, when usually have too little (non-parametric or extreme measurement error arguments might justify)parametric or extreme-measurement-error arguments might justify)

• So usually a bad idea…

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Elaborations, Complications, & Extensions:S l S litti (D )I t tiSample-Splitting v. (Dummy-)Interacting

Split sample abets eyeballing obfuscates statistical analysis– Split-sample abets eyeballing, obfuscates statistical analysis, of the main point: the different effects by category.

• What’s s.e/signif. of b1i-b1j? Need:What s s.e/signif. of b1i b1j? Need:

• Luckily, cov=0, but, still, squaring 2 terms, sum, & root in head?

( ) ( ) ( ) ( ) ( ) ( )jijijiji bVbVbbCbVbVbbes 11111111 ,2.. +=−+=−• Luckily, cov 0, but, still, squaring 2 terms, sum, & root in head?

– Can choose full dummy set to mirror the split-sample estimates directly (& report that way, if wish) or the set-y ( p y, )less-one to get significance of differences b/w samples directly (in the standard reported t-test)

• Same thing, so choose for to optimize presentational efficacy.

– One advantage of hierarchical modeling is how it affords, t ll i iti b/ th tnaturally, various positions b/w these extremes.

• E.g., can “borrow strength” across units.

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Typical 2nd-Moment Implications of Interactions

DM l ll• DMag permissive ele sys: allow more parties…0 1 0 1 ; ( ) ( ) , e.g., NP DM V f DM DMβ β ε ε σ σ= + + = +

• Few Veto Actors allow greater policy-change…

• I e these are Rndm Coeff &/or Het sked Props0 1 0 1 ; ( ) ( ) , e.g., y VP V f VP VPβ β ε ε σ σ= + + = +

• I.e., these are Rndm-Coeff &/or Het-sked Props…0 1 2 3 0 1 ; ( ) ( ) , e.g., β β β β ε ε σ σNP DM SF DM SF V f DM DM= + + + × + = +

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Sandwich Estimatorsβ β β ε= + + +EN SF DM

0 1 2

1 0 1 1

β β β ε

β α α ω

= + + +

∂ ∂ = = +

EN SF DM

EN SF DM +

( ) ( )2 1 2

β γ ω

β α α ω γ γ ω ε

∂ ∂ = = +

⇒ = + + + + +

0EN DM γ SF +

EN DM + SF SF + DM( ) ( )( ) { }

0 0 1 1 0 1 2

0 0 1 1 0 1 2

β α α ω γ γ ω ε

β α α γ γ ε ω ω

⇒ = + + + + +

= + + + × + + + +

EN DM + SF SF + DM

SF DM SF DM SF DM*

0 1 2 3b b b b ε= + + × + +SF DM SF DM

• Notice the compound error term:– V(ε*) will not be σ2I even if ε is, so V(b) doesn’t reduce to σ2(X'X)-1, so OLS s.e.’s wrong.

– Be OK on average (unbiased) & in limit (consistent) if l t f V( *) “ th l t ' ”element of V(ε*) “orthogonal to xx' ”

– But def’ly not because ε* includes x & z, which part of X!Slide 36 of 37

Sandwich Estimators

1 2( ) ( ) ( ( )

LSV V ω ω− −⎡ ⎤′ ′ ′= + +⎣ ⎦

1 1b X X X S D X X Xε

• Brilliant insight of ‘robust’ (i.e., consistent) “sandwich” estimators:sandwich estimators:

• Only need formula that accounts relation V(ε*) to “X'X”, i e regressors squares & cross-prod’s involved in X'[·]X”i.e., regressors, squares, & cross prod s involved in X [ ]X

• ⇒ “, robust” (or, in HM: “, cluster”) can work:t k 2 l ' & '* 2 2 2 2 2( )V + +– so track e2 rel xx' & zz' ⇒

White’s heteroskedasticity-consistent s.e.’s: 1 2

2 2 2 2 2( )i RE i i

V x zε ω ωε σ σ σ= + +

n1

– similar +grouping⇒ cluster:=

⋅ = ∑n

2i

i 1

1[ ] en i ix x '

* 2 2 2( )V ε σ σ σ′ ′= + +x x z z similar, +grouping⇒ cluster:0 1 2( )

i HM i i i iV ε σ σ σ= + +x x z z

( ) ( )= ==

⎧ ⎫′⋅ = ∑ ∑⎨ ⎬⎩ ⎭

∑j jn nJ

ij ij ij iji 1 i 1j 1

[ ] e ex x

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linear interaction modelslinear interaction modelsfranzese/ps699.linearinteractionmodels.… ·...

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