Context Matters:
Empirical Analysis in Social Science,
Where the Effect of Anything
Depends on Everything Else
Interaction Terms,
Multilevel Models,
& Nonlinear Regression
Blalock Lecture
ICPSR Summer School, 30 July 2009
(Here for fuller pedagogical slides & materials:
http://www.umich.edu/~franzese/SyllabiEtc.html)
Robert J. Franzese, Jr.
Professor of Political Science,
The University of Michigan, Ann Arbor
Overview


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 Specifications
Elaborations, Complications, & Extensions:
 Use & abuse of some common-practice “rules”
 Multilevel/Hierarchical Issues:




Sample-Splitting v. (Dummy-) Interacting
Variances in Interax, & Rndm-effect/Hierarchical/Multilevel, Models
Interax in QualDep (inherent-interax, latent-var) models
Complex Context-Conditionality & Nonlinear Least-Squares
Interactions in Pol-Sci Research

Common. ‘96-‘01 AJPS, APSR, JoP:
54% some stat meth (=s.e.’s), of which 24% = interax
(so interax ≈ 12.5% or 1/8th total).
 (N.b., most rest QualDep & formal theory, not
counted, so understate tech nature of discipline)

Journal (1996-2001)
Total
Articles
American Political Science Review
American Journal Political Science
Comparative Politics
Comparative Political Studies
International Organization
International Studies Quarterly
Journal of Politics
Legislative Studies Quarterly
World Politics
TOTALS
279
355
130
189
170
173
284
157
116
2446
Statistical
Interaction-Term Usage
Analysis
Count % of Tot Count % of Tot % of Stat
274
77%
69
19%
25%
155
55%
47
17%
30%
12
9%
1
1%
8%
92
49%
23
12%
25%
43
25%
9
5%
21%
70
40%
10
6%
14%
226
80%
55
19%
24%
104
66%
19
12%
18%
28
24%
6
5%
25%
1323
54%
311
13%
24%
Interactions in Pol-Sci Theory1

Our Theories/Substance say should be more; core classes of
argument almost inherently interactive:

INSTITUTIONAL: institutions=interactive variables:

funnel, moderate, shape, condition, constrain, refract, magnify, augment,
dampen, mitigate political processes that…




…translate societal interest-structure into effective political pressures,
&/or pressures into public-policy responses,
&/or policies to outcomes. I.e., they affect effectsinteraction.
Views from across institutionalist perspectives:




Hall: “institutionalist model=>policy more than sum countervailing pressure
from soc grps; that press mediated by organizational dynamic.”
Ikenberry: “[Political struggles] mediated by inst’l setting where [occur]”
Steinmo & Thelen: “inst’s…constrain & refract politics… [effects of] macrostructures magnify or mitigated by intermediate-level inst’s… help us…explain
the contingent nature of pol-econ development…”
Shepsle: “SIE clearly a move [to] incorporating inst’l features into R-C.
Structure & procedure combine w/ preferences to produce outcomes.”
Interactions in Pol-Sci Theory2

Thry/Subst: core classes arg inherently interax (cont.):




INSTITUTIONAL: …
STRATEGIC: actors’ choices (outcomes) conditional upon
inst’l/struct’l environ., opp.-set, & other actors’ choices.
CONTEXTUAL: actors’ choices (outcomes) conditional upon
environ., opp. set, & aggregates of other actors’ choices.
Across all subfields & areas:

Comp Pol… Int’l Rltns… U.S. Pol… For examples:




Electoral system & societal structure  party system.
System polarity & offense-defense balance  war propensity.
Divided government & polarization  legislative productivity.
PolEcon… PolBehave… LegStuds… PolDev…




Electoral & partisan cycles depend on inst’l & econ conditions
Gov’t inst’s shape voter behavior: balancing (Kedar, Alesina); economic
voting (Powell & Whitten); etc.
Effects divided gov’t depend presidential v. parliamentary.
Effect inequality on democratization depends cleavage structure.
Interactions in Political Science
Theory & Substance:
Everyone’s Favorite “Model”
Economics Affects Politics and Society
Politics Affects Economics and Society
Society Affects Politics and Economics
Economy
Polity
Society
Interactions in Political Science
Theory & Substance:
An Old (& still) Favorite “Model” of Mine
Result of Outcomes at T-1
The Cycle of
Political Economy
(A)
Interest Structure of
the Polity and
Economy
Result of Outcomes at T0
On to T+1
Elec
tion
t
en
rnm n
ve atio
Go orm
F
Examples of the Elements at
Each Stage:
(A) Interests:
Sectoral Structure of Economy
Income Distribution
Age Distribution
Trade Openness
Elections:
Electoral Law
Voter Participation
s
(B)
Partisan
Representation in
Government
Action at
Time T0
Government Formation:
lA
ct o
rs
Fractionalization
Polarization
nta
(B) Representation:
ov
Pol
ern
i cy
me
Partisanship
Policy:
No
n-G
Fiscal Policy
Monetary Policy
Institutional Adjustment
Government Termination:
Replacement Risk
(C) Outcomes:
Unemployment
Inflation
Growth
Sectoral Shift
Debt
Institutional Change
(C)
Political and
Economic Outcomes
t
men )
v er n
(Go ination
Term
Exogenous Factors
Interactions in Political Science
Theory & Substance:
An Newer Favorite “Model” of Mine

Complex Context-Conditionality:
Effect of (almost) everything depends on
(almost) everything else.
 E.g., Principal-Agent Situations

 If
fully principal, y1=f(X); if fully agent, y2=g(Z);
institutions: 0≤h(I)≤1.
y  h(I ) f ( X)  1  h(I) g (Z)

y
x
 h(I ) f (xX ) ;
y
z
 h(I) g(zZ ) ;
y
i

h ( I )
i
 f ( X)  g ( Z ) 
(Complex) Context-Conditionality:
(Hallmark of Modern 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:
In words…
…
…
…i.e., effect of
anything
depends on
everything
else!
y  h(I ) f ( X)  1  h(I) g ( Z)

y
x
 h (I )
f ( X )
x
y
z
  h(I )
y
i
h ( I )
i

;
g ( Z )
z
;
 f ( X)  g ( Z ) 
Not Every Argument Is an
Interactive Argument

Not Interactive:

X affects Y through its effect on Z: XZY


X and Z affect each other: XZ.


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)


In (political) psychology / behavior, this called mediation.
Interaction is called moderation in this literature.
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):
Y
Y
 f Z  
 f X 
X
Z
From Theory/Substance to
Empirical-Model Specification

Classic Comparative-Politics Example:





Societal Fragmentation, SFrag, &
Electoral-System Proportionality, DMag,
Effective # Parliamentary Parties: ENPP
“Theory”: ENPP  f ( SFrag , DMag ,  ,  )
Hypotheses: ENPP  0 ENPP  0
SFrag
¶ {¶¶ ENPP
SFrag }
¶ DMag

º
DMag
¶ {¶¶ ENPP
DMag }
¶ 2 ENPP
¶ 2 ENPP
º
º
³ 0
¶ SFrag
¶ SFrag¶ DMag ¶ DMag¶ SFrag
Empirical Specification: Lots ways get there...
A Typical Linear-Interactive
Specification

Want linear f() w/ these properties; many ways to get there:
ENPP  β0  β1 SFrag  β2 DMag  ε
?
ENPP
 β1  f  DMag   α0  α1 DMag
SFrag
?
ENPP
 β2  f SFrag   γ0  γ1 SFrag
DMag
 ENPP  β0  α0  α1 DMag SFrag  γ0  γ 1 SFrag DMag  ε
 β0  α0 SFrag  α1 DMagSFrag  γ0 DMag  γ1 SFragDMag  ε
 β0  α0 SFrag  γ0 DMag  α1  γ1 SFragDMag  ε
 β0  βSF SFrag  β DM DMag  βSFDM SFragDMag  ε

ENPP
 βSF  βSFDM DMag
SFrag
ENPP
 β DM  βSFDM SFrag
DMag
Interpretation of Effects:
Derivatives & Differences, Not Coefficients

Standard Linear Interactive Model:
ENPP  β0  βSF SFrag  βDM DMag  βSFDM SFragDMag  ...  ε

Effect of SFrag on ENPP (is a function of DMag):
Effect (SFrag ) º
¶ ENPP
= βSF + βSFDM DMag
¶ SFrag
º ΔENPP = βSF ΔSF + βSFDM DMag ×ΔSF
º

ΔENPP
= βSF + βSFDM DMag
ΔSF
Effect of DMag on ENPP (is f of SFrag):
Effect (DMag ) º
¶ ENPP
= β DM + β SFDM SFrag
¶ DMag
º ΔENPP = β DM ΔDM + β SFDM SFrag ×ΔDM
º
ΔENPP
= β DM + β SFDM SFrag
ΔDM
Interpretation of Effects: NOTES1

“Main Effect”: e.g., SF = “main effect of SFrag”

….but SF is merely the effect of SFrag at other variable(s)
involved in interaction with it=0, so:


Other-var(s)=0 may be beyond range, or sample, or even logically
impossible.
Other-var(s)=0 substantive meaning of 0 altered by rescaling



Other-var(s)=0 may not have anything subst’ly main about it
COEFFICIENTS ARE NOT EFFECTS. EFFECTS ARE
DERIVATIVES &/OR DIFFERENCES.


E.g., by “centering” (centering changes nothing, btw…)
Only in purely linear-additive-separable model are they equal b/c only there
do derivatives simply = coefficients.
SF is not “effect of SFrag ‘independent of’…” & definitely not its
“effect ‘controling for’…other var(s) in the interax”

Cannot substitute linguistic invention for under-standing the
model’s logic (i.e., its simple math)
Interpretation of Effects:

Interactions are logically symmetric:



For any function, not just lin-add.
If argue effect x depends z, must
also believe effect z depends x.
     2 y
2 y



z
x
xz zx
y
x
y
z
Interactions often have 2nd-moment (variance, i.e.,
heteroskedacity) implications too:



2
NOTES
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…
Level: micro or macro; i, j, k, …
Frequent 2nd-Moment Implications Interactions

DMag permissive ele sys: allows more parties…
NP  0  1DM   ; V ( )  f ( DM ) , e.g.,  0  1DM

Few Veto Actors allows greater policy-change…
y  0  1VP   ; V ( )  f (VP) , e.g.,  0  1VP

I.e., these are Rndm-Coeff &/or Het-sked Props…
Interpretation of Effects:
Standard Errors for Effects
ENPP  β0  βSF SFrag  βDM DMag  βSFDM SFragDMag  ...  ε

Std Errs reported with regression output are for coefficients, not for
effects.
ˆ


 SFstd. err. for est’d effect SFrag at DMag=0
The s.e. (t-stat, p-level) for
is
(…which is logically impossible).
Effect of x depends on z & v.v. (i.e., which was the point, remember?),
so does the s.e.:
y
 y  ˆ
Effect  x  
 β x  β xz z  Est.Eff. x   E    β x  βˆ xz z
x
 x 

  y  
Est.Var. Est.Eff. x   E Var  E     E Var βˆ x  βˆ xz z 


  x  



    
In words… More Generally:
V (xβˆ )  x V (βˆ )  x



 V βˆ x  βˆ xz z  V βˆ x  V βˆ xz  z 2  2  C βˆ x , βˆ xz z

From Hypotheses to Hypotheses Tests:
Does Y Depend on X or Z?
ENPP  β0  βSF SFrag  βDM DMag  βSFDM SFragDMag  ...  ε
Hypothesis
x affects y, or
y is a function of (depends on) x
Mathematical Expression
y=f(x)
y x   x   xz z  0
x increases y
y x   x   xz z  0
x decreases y
y x   x   xz z  0
z affects y, or
y is a function of (depends on) z
y=g(z)
y z   z   xz x  0
z increases y
y z   z   xz x  0
z decreases y
y z   z   xz x  0
91
Statistical test
F- test:
H0:  x   xz  0
Multiple t-tests:
H0:  x   xz z  0
Multiple t- tests:
 x   xz z  0
F- test:
H0:  z   xz  0
Multiple t-tests:
H0:  z   xz x  0
Multiple t- tests:
H0:  z   xz x  0
From Hypotheses to Hypotheses Tests:
Is Y’s Dependence on X Conditional on Z & v.v.? How?
Hypothesis
The effect of x on y depends on z
Mathematical Expression92
y=f(xz,•)
y x   x   xz z  g ( z )
Statistical test
t-test: H0:  xz  0
  y x  z   y xz   xz  0
2
The effect of x on y increases in z
  y x  z   2 y xz   xz  0
t-test: H0:  xz  0
The effect of x on y decreases in
z
  y x  z   2 y xz   xz  0
t-test: H0:  xz  0
The effect of z on y depends on x
y=f(xz,•)
y z   z   xz x  h( x)
t-test: H0:  xz  0
The effect of z on y increases in x
  y z  x   2 y zx   xz  0
t-test: H0:  xz  0
The effect of z on y decreases in
x
  y z  x   2 y zx   xz  0
t-test: H0:  xz  0
  y z  x   2 y zx   xz  0
Does Y Depend on X, Z, or XZ?
Hypothesis
y is a function of (depends on) z,
z, and/or their interaction
Mathematical Expression
y=f(x,z,xz)
93
Statistical Test
F-test: H0:  x   z   xz  0
Use & Abuse of Some Common ‘Rules’

Centering to Redress Colinearity Concerns:



Must Include All Components (if x∙z, then x&z):





Adds 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 samplemean, the new 0. Do only if aids presentation.
Application of Occam’s Razor &/or scientific caution (e.g., greater
flexibility to allow linear w/in lin-interax model), but
Not a logical or statistical requirement.
Safer rule than opposite & to check almost always, but
Not override theory & evidence if (strongly) agree to exclude
Pet-Peeve: Lingistic Gymnastics to Dodge the Math


“Main effect, Interactive effect”: the effect in model is dy/dx.
Discussion of [coefficients & s.e.’s] as if [effects & s.e.’s].
Presentation: Marginal-Effects / Differences Tables
& Graphs
 Plot/Tabulate Effects, dy/dx, over Meaningful &/or
Illuminating Ranges of z, with Conf. Int.’s
2
 dyˆ / dx  tdf , p Var(dyˆ / dx )  ˆx  ˆxz z  tdf , p V ( ˆx )  V ( ˆxz ) z  2C ( ˆx , ˆxz ) z
Figure 5. Marginal Effect of Runoff, Extending the Range of Groups


Explain axes
Explain shape
Linear-interax:
Will cross 0
& be insig @ 0.


Rescaling &
“main effect”
 “centering”
 Max(Asterisks)

20
15
d (Number of Candidates )/d (Runoff )

10
5
0
-5
-10
-2
-1
0
1
2
3
4
Effective Number of Societal Groups (Groups )
5
6
Presentation: Expected-Value/Predictions
Tables & Graphs

Predictions, E(y|x,z):
yˆ  tdf , p Var ( yˆ ) 
ˆ0  ˆx x  ˆz z  ˆxz xz  tdf , p
V ( ˆ0 )  V ( ˆx ) x 2  V ( ˆz ) z 2  V ( ˆxz )( xz) 2
 2C ( ˆ , ˆ ) x  2C ( ˆ , ˆ ) z  2C ( ˆ , ˆ ) xz
0
x
0
z
0
xz
 2C ( ˆx , ˆz ) xz  2C ( ˆx , ˆxz ) x 2 z  2C ( ˆz , ˆxz ) xz2

Here’s one place a little matrix algebra would help:
yˆ  tdf , p Var ( yˆ )  ˆ0  ˆx x  ˆz z  ˆxz xz  tdf , p xVˆ (βˆ ) x
 ˆ0  ˆx x  ˆz z  ˆxz xz  tdf , p

1
x
z
 Vˆ ( ˆ0 )
Cˆ ( ˆ0 , ˆx ) Cˆ ( ˆ0 , ˆz ) Cˆ ( ˆ0 , ˆxz )   1 

 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
V ( x )
C (  x ,  z ) C (  x ,  xz )  x
 C (0 ,  x )
 
xz  

ˆ ˆ ˆ
ˆ ˆ ˆ
Vˆ ( ˆz )
Cˆ ( ˆz , ˆxz )   z 
 C (0 ,  z ) C ( x ,  z )
 
ˆ ˆ ˆ
  xz 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
V (  xz ) 
C (  0 ,  xz ) C (  x ,  xz ) C (  z ,  xz )
Use spreadsheet or stat-graph software (…list coming...)
Presentation:
Choose Illuminating Graphics & Base Cases

Interpretation same regardless of “type” of interax: effect
always ≡ dy/dx, but present appropriately…

All combos Dummy, Discrete, or Continuous:






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 may
see as x interacting w/ dummy for that condition)
Always plot over substantively revealing ranges.
Esp. w/ dums, have several (identical) specification options:



Dum-Dum=>4 (or 2#interax vars) points estimated, so box&whisk or histograms
Dum-Contin or DiscFew-Contin=>2 (or # categories) slopes, so E(y|x,z) as line or
dy/dx as box&whisker or histograms
Contin(DiscMany)-Contin(DiscMany)=>Effect-lines best or (slices from) contour
plot (i.e., slices from 3D)
(full-set,set-1): choose which (& what base if use full-set-minus-1) to abet
presentation & discussion
(overlap, disjoint): choose to facilitate presentation & discussion.
Scale Effectively: e.g., center only if & to extent that aids
presentation & discussion (b/c centering does nothing else)
Examples & Practice:
Basic Linear-Interactive Model

Basic Linear-Interactive Model:

Effect of edu?
eusup  b0  bedu edu  blftrt lftrt  bedlr edu  lftrt  ...  


eusup
 bedu  bedlr lftrt
edu
For the record, the effect of lftrt:
eusup
 blftrt  bedlr edu
lftrt
Std Error of that Effect (of edu on eusupp)?

    


Vˆ bˆedu  bˆedlr lftrt  Vˆ bˆedu  Vˆ bˆedlr lftrt 2  2  Cˆ bˆedu , bˆedlr lftrt


Std. Err. effect of edu:

    
To do this (e.g. in Stata, using the .dta subset)




Vˆ bˆlftrt  bˆedlr edu  Vˆ bˆlftrt  Vˆ bˆedlr edu 2  2  Cˆ bˆedu , bˆedlr edu
First: gen edlr=education*leftright
Then: reg eu_supp ...education leftright edlr...
Examples & Practice:1a
Basic Linear-Interactive Model:
Education & Leftright Purely Micro-level Model







. reg eu_support education leftright edlr
Source |
SS
df
MS
-------------+-----------------------------Model | 7869.51854
3 2623.17285
Residual | 301474.935 42676 7.06427347
-------------+-----------------------------Total | 309344.453 42679 7.24816545








-----------------------------------------------------------------------------eu_support |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------education |
.5918654
.0302344
19.58
0.000
.5326053
.6511255
leftright |
.080127
.0147551
5.43
0.000
.0512068
.1090472
edlr | -.0320928
.0054447
-5.89
0.000
-.0427646
-.021421
_cons |
5.093097
.0831665
61.24
0.000
4.930089
5.256105
------------------------------------------------------------------------------

What’s effect of edu? Of lftrt? How moderate each other?
eusup
 .5919  .0321 lftrt 
edu


eusup
 .0801  .0321 edu 
lftrt
=
=
=
=
=
=
42680
371.33
0.0000
0.0254
0.0254
2.6579
 2eusup
 bedlr  .0321
edulftrt
Only for modifying effect does standard regression output tell us directly.
What are the standard errors of these effects?


Number of obs
F( 3, 42676)
Prob > F
R-squared
Adj R-squared
Root MSE
Only for modifying effect does standard regression output tell us directly.
Recall: Main Effects refer to beyond-range values; they not direct
evidence on whether effect (generally) positive
Examples & Practice: Basic Linear-Interact Model1b
Education & Leftright Purely Micro-level Model

What are the standard errors of these effects? Need this:


.vce
Covariance matrix of coefficients of regress model
e(V) | education
leftright
edlr
_cons
-------------+-----------------------------------------------education | .00091412
leftright | .00037584
.00021771
edlr | -.00014825 -.00007456
.00002965
_cons | -.00234221 -.00110293
.00037572
.00691666

I prefer spreadsheet at this point:




















Copy (“as Table”) regression-estimation results; Paste into spreadsheet.
Ditto for estimated v-cov of estimated coefficients (as text or table)
Finally, you’ll want summary stats for vars in your model (as text best):
. tabstat dgovpw psupgpw npgovpw PD NPPD NPGS NPPDGS, statistics( mean max min
median iqr skewness sd ) columns(variables)
stats |
dgovpw
psupgpw
npgovpw
PD
NPPD
NPGS
NPPDGS
---------+---------------------------------------------------------------------mean | 25.24783 57.38261 1.965217 .6521739 1.369565 116.3004 765.9826
max |
45.1
80.4
4.3
1
3.8
305.52
2181.2
min |
11
41.1
1
0
0
49
0
p50 |
26.6
57.2
1.8
1
1.6
96
916.2
iqr |
13.7
10.8
1.7
1
2.2
83.47
1108
skewness | .1587892
.821165
.907747 -.6390097 .2729543 1.218438 .1633625
sd | 9.655468 9.103879 .9599284 .4869848 1.213723 69.04308 644.2565
--------------------------------------------------------------------------------
Ex’s & Practice: Basic Linear-Interact Model1c
Education & Leftright Purely Micro-level Model

Spreadsht Formula to Plot Effect Lines w/ C.I.

Col 1: Conditioning Var (lftrt in deusup/dedu)
1st Cell (A29): Enter min of range to consider (smpl min)
 2nd Cell: =A29+1




Or sub “+(max-min)/(#steps)” for +1 (choose big# to smooth)
Copy down until reach max value you want plot to cover.
Column Two: Effect (dGovDur/dNP here)

1st Cell (B29): Enter =$B$3+$B$5*$A29, where:



$B$3 is absolute reference to cell containing coefficient on edu
$B$5 is absolute reference to cell containing coefficient on edlr
$A29 is reference to cell containing value of lftrt for that row


$ optional, but helps if want copy whole block later for other effects
Copy Down
Ex’s & Practice: Basic Linear-Interact Model:

Education & Leftright Purely Micro-level Model
Spreadsheet Formula to Plot Effect Lines w/ C.I.

Column Three: standard error (or can skip to bounds)

1st Cell (A29): =($B$11+$D$13*$A29^2+2*$B$13*$A29)^0.5






$B$11 is absolute reference to cell containing variance of edu coefficient
$D$13*$A29^2 is absolute reference to cell containing variance of edlr
coefficient, times the square of the value of lftrt for that row.
2*$B$13*$A29 is absolute reference to cell containing 2 times the covariance
of edu and edlr coefficients times value of lftrt for that row.
^0.5 turns that estimated variance into a standard error.
Copy down.
Column Four: Lower bound of 95% C.I.

1st Cell (B29): Enter =+$B29-1.96*$C29, where:

$B29 is (column-absolute opt.) reference to cell containing effect est
 $C29 is (column-absolute opt.) reference to cell containing std err est
 1.96 is critical value for 95% C.I. on large-N t-dist or std-norm dist;
 can sub crit.val. for diff. C.I. % or smpl-size or use sprdsht formula
Copy Down


Column Five: Upper bound (analogous, but +1.96*$C29)
Ex’s & Practice: Basic Linear-Interact Model:
Education & Leftright Purely Micro-level Model

In Stata, plot dY/dX w/ c.i. from smpl min-max:




egen zmin = min(z) ; egen zmax = max(z) finds those sample min & max
for variable z. (z=psupgpw in our case, i.e., GS)
gen z0 = (_n-1)/(v-1)*(zmax-zmin)+zmin in 1/v creates var counting v
equal-size steps from sample min to max.
gen dyhatdx=_b[x]+_b[xz]*z0 creates var of dY/dX ests (x=npgovpw)
Stata code tedious to get to s.e.’s & c.i. plots (bit better in matrix form)

First have to work in matrices for bit, then back to vars:







Finally, you can generate variances & std errors, which you could tabulate:




matrix V = get(VCE)
(makes matrix of v-cov mat)
matrix C= V[3,1]
(grabs 3,1 element as covar)
gen column1 = 1 in 1/v
(makes a variable equal to all ones)
mkmat column1, matrix(col1) (makes vector called col1 of that var)
matrix cov_x_xz = C*col1
(makes a constant vector of covar)
svmat cov_x_xz, name(cov_x_xz)
(makes that vector a variable)
gen vardyhatdx=(_se[x])^2+(z0*z0)*(_se[xz]^2)+2*z0*cov_x_xz
gen sedyhatdx=sqrt(vardyhatdx)
(makes variable equal to s.e. of effect)
tabdisp z0, cellvar(dyhatdx sedyhatdx)
(makes table effects & s.e.’s)
Or you can generate the confidence interval bounds & plot:



gen LBdyhatdx=dyhatdx-invttail(e(df_r),.05)*sedyhatdx
gen UBdyhatdx=dyhatdx+invttail(e(df_r),.05)*sedyhatdx
twoway connected dyhatdx LBdyhatdx UBdyhatdx z0
Ex’s & Practice: Basic Linear-Interact Model
Calculating Predicted-Values & Standard Errors
Procedures
Create the variables which set
the values of the variables x,
z, and xz (& other variables,
if any) for which ŷ will be
calculated.
Assemble the variables into a
matrix, Mh
Create betas, a column
vector with k x 1 dimensions.
Calculate the predicted
values.
Convert the vector into a
variable.
Create a matrix from the
estimated covariance matrix
of the coefficient estimates.
Calculate the variance of the
predicted values.
Extract the diagonal elements
into a row vector.
Transpose elements into a
column vector.
Convert the vector into a
variable.
Calculate the estimated
standard error of each
predicted probability.
Present a table of predicted
values with corresponding
standard errors of the
predicted values.
Generate lower and upper
confidence interval bounds.
Graph the predicted
probabilities and the upper
and lower confidence
intervals.
Command syntax
egen xmin = min(x)
egen xmax = max(x)
gen xh = ((_n-1)/(v-1))*(xmax-xmin) in 1/v
egen zh=mean(z)
gen xhzh=xh*zh
gen col1h=1
mkmat xh zh xhzh col1h in 1/v, matrix(Mh)
matrix betas=e(b)’
matrix yhat=Mh*betas
Google
‘kam franzese michigan press’
for that data, and
stata & excel resources
or go directly to
http://www.press.umich.edu
/KamFranzese/Interactions.html
svmat yhat, name(yhat)
matrix V = get(VCE)
matrix VYH=Mh*V*Mh’
matrix DVYH= vecdiag(VYH)
matrix VARYHAT=DVYH’
svmat VARYHAT, name(varyhat)
Google
‘Matt Golder interaction’
or go directly to
http://homepages.nyu.edu
/~mrg217/interaction.html
gen seyhat1 = sqrt(varyhat1)
tabdisp xh, cellvar(yhat1 seyhat1)
gen LByhat1=yhat1-invttail(e(df_m),.05)*seyhat1
gen UByhat1=yhat1+invttail(e(df_m),.05)*seyhat1
twoway connected yhat1 LByhat1 UByhat1 xh
Stata
help mfx
Elaborations, Complications, & Extensions:
Sample-Splitting v. (Dummy-)Interacting

Split-sample (e.g., unit-by-unit) ≈ Full-Dummy Interax:

Subsample by binary (or multinomial, e.g., CTRY in TSCS) category to est
sep’ly ≈ Dummy @ category & interact @ dummy w/ @ x.




Subsample by hi/lo values some non-nominal var equivalent to nominalizing the
extra-nominal info & dummy-interact;


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 si2 for splitting
Can make exact by allow si2 (FWLS)
I.e., wasting information, when usually have too little (non-parametric or extrememeasurement-error arguments might justify…)
Split-sample abets eyeballing, obfuscates statistical analysis, of the main point:
the different effects by category.
s.e.b  b   V b   V b   2C b , b   V b   V b 
 What’s s.e/signif. of b1i-b1j? Need
1i



1j
1i
1j
1i
1j
1i
1j
Luckily, cov=0, but, still, squaring 2 terms, sum, & square-root in head?
Can choose full dummy set to mirror the split-sample estimates directly (&
report that way, if wish) or the set-less-one to get significance of differences b/w
samples directly (in the standard reported t-test)
One big advantage of hierarchical modeling is how it affords, naturally, various
positions b/w these extremes
Cross-Level Interactions:
From the (C or G)LRM to the HLM
eusupij   j0   lrj lftrt ij  ...   ij 

 0j   0  1GSPEND j  u 0j  or
 lrj   0   1GSPEND j  u1j 
eusupij  0  1lftrt ij  2GSPEND j  3lftrt ij  GSPEND j  ...   ij

If (C or G)LRM assumptions apply, then (O or G)LS
unbiased, consistent, and efficient.


I.e., not much changes; +/- as before re: effects, v(effects),
symmetry, neither micro-/macro-level coeff’s=effects, etc.
Two main issues of concern:

Parameter heterogeneity: (see pictures)



systematic &/or stochastic (fixed v. rndm intrcpt/coeff)
can cause bias if pattern unmodeled hetero relates to X,
Non-spherical error var-covar matrix (v-cov() will not be spherical):
efficiency & proper s.e.’s issue, not a bias/consistency issue.

But “mere inefficiency” can be serious, & accurate s.e.’s very important.
From the (C or G)LRM to HLM

Examples of parameter hetereogeneity that covaries
w/ X values, and so LS is biased:

Note re: FE v. RE -- both theoretically could cause bias if
cov w/ X, but RE i.d.’d off orthogonality.
Elaborations, Complications, & Extensions:
Random-Effect & Hierarchical/Multi-level Models

R.E. Model: Odd that std. lin-interax model:



Assumes know y=f(X)+error:
But dy/dx=f(z) w/o error!:
So, try:
y  β0  βx x  βz z  βxz xz  ε
y
x
 β x  β xz xz
y  β0  β1 x  β2 z  ε 0
y
x
 β1   0  1z  ε 1
y
z
 β2   0   1 x  ε 2
 y  β0   0  1z  ε 1 x   0   1 x  ε 2 z  ε 0
 β0   0 x   0 z  1   1 xz  ε 0  ε 1 x  ε 2 z 


=> std. lin-interact again!...Except compound error-term…
HLM: Same model, except xij & zj,&

 *   ij0   1j xij   ij2 z j
So it just std. lin-interact too, but w/ different compound-error
stochastic properties…
Typical 2nd-Moment Implications of Interactions

DMag permissive ele sys: allow more parties…
NP  0  1DM   ; V ( )  f ( DM ) , e.g.,  0  1DM

Few Veto Actors allow greater policy-change…
y  0  1VP   ; V ( )  f (VP) , e.g.,  0  1VP

I.e., these are Rndm-Coeff &/or Het-sked Props…
NP  0  1DM  2 SF  3 DM  SF   ; V ( )  f ( DM ) , e.g.,  0   1DM
From CLRM to Hierarchical Model:
An Example

Std. HLM: Same model, except xij & zj,& different
compound-error stochastic properties.
eusupij   j0   lrj lftrt ij  ...   ij
0 GSPEND
1
2  u0
 0j 


 * 0  ij 1  j xij   ij zj j j
 lrj   0   1GSPEND j  u1j
 eusupij   0  1GSPEND j  u 0j   0 lftrt ij
 1lftrt ij  GSPEND j  lftrt ij u1j  ...   ij
gathering terms :
eusupij   0  ...  1GSPEND j   0 lftrt ij
 1lftrt ij  GSPEND j   u 0j  lftrt ij u1j   ij 

eusup
eusup
 blftrt  blrGS GS &
 bGS  blrGS edu
lftrt
GS
Properties of OLS under HLM Conditions

Properties of OLS Estimates of Lin-Interact Model if truly
RE/HLM:
y  β0  β x x  βz z  β xz xz  ε 0  ε 1 x  ε 2 z   Xβ  ε 0  ε 1 x  ε 2 z 
So, OLS coeff. est’s sill differ from truth by Aε*:
1
1
βˆ LS  X X  X y  XX  X Xβ  ε 0  ε 1 x  ε 2 z 
1
1
1
 X X  XXβ  X X  Xε 0  ε 1 x  ε 2 z   β  XX  Xε*
 So, OLS coeff. est’s unbiased & consistent (iff…):
1
1
E βˆ LS  E β  XX  Xε*   E β  XX  Xε 0  ε 1 x  ε 2 z 
1
1
 β  XX  XE ε 0  ε 1 x  ε 2 z   β  XX  XE ε 0   E ε 1 x  E ε 2 z 
1
1
 β  XX  X0  E ε 1 x  E ε 2 z   β  XX  X0  0  0  β. Q.E.D.


Note: only works for models with additively separable stochastic
component; not nec’ly others (e.g., logit/probit)
Properties of OLS under HLM Conditions
But, OLS s.e.’s will be wrong; not s2(X'X)-1, but:
1
ˆ

V β LS  V β   XX  Xε* 



 
1
1
*
*






 V β   V  X X  X ε  2C β,  X X  X ε




1

 0  V  XX  Xε*   0


  XX  XV  ε  X  XX 
1
  XX 
1
*
1
1
0
1
2


X V  ε  ε x  ε z   X  XX 
1
0
1
2


  XX  X V  ε   V  ε x   V  ε z   X  XX 
(note : the covariance terms are assumed zero)
1
Sandwich Estimators
V βˆ  XX  XV ε   V ε x   V ε z XXX 
1
0
1
1
2
LS


Not σ2I (even if each ε* is), so whole thing doesn’t reduce to
σ2(X'X)-1, so OLS s.e.’s wrong.
Be OK on avg (unbiased) & in limit (consistent) if that term
varied in way “ orthogonal to xx' ”


But def’ly not b/c [∙] includes x & z, which part of X!
=brilliant insight of ‘robust’ (i.e., consistent) s.e. est’s:


Only need s.e. formula that accounts relation V(ε*) to “X'X”, i.e., regressors,
squares, & cross-prod’s involved in X'[∙]X”
 “, robust” & “, cluster” can work (for RE & HLM, resp’ly)
2
ˆ ˆ

 zztrack
 V βRE   I  xx so
e2 rel xx' & zz'  heterosked.:


Vˆ βˆ HM   02I  12 xx sim+grpng
 22 zz
1 n 2
[  ]   ei X i X i'
n i 1
cluster:
nj
 nj

[  ]   ( eij X ij ) ( eij X ij ) 
i 1
j  1  i 1

J
From the CLRM to HLM
y  β0   0 x   0 z  1   1  xz   ε 0  ε 1 x  ε 2 z 
 


1
1
V βˆ LS  XX  X V ε0   V ε 1 x   V ε 2 z  XXX 

 appropriate “, robust” & “, cluster” can work

I.e., asymptotically s.e.’s right…BUT need large nj & N-k


I.e., coefficients still inefficient.



Specifically, from small-sample corrections, seems need small:
én (n - 1)ùé(N - 1) (N - k )ù
j
êë j
úê
ú
ûë
û
Want/need efficiency, or nj or N-k low? HLM/RE or FGLS/FWLS.
Note: similarity RE and HLM, RE & FWLS. As that sim suggests, RE only
helps efficiency and only rightly does so if that’s all it does. (I.e., if the RE’s
orthogonal to X.)
I.e., “work” thusly only or models with additively-separable
stochastic components

As w/ all such “sandwich” estimators, a sort of logical disconnect in
applying them to models w/o such separability…
Elaborations, Complications, & Extensions:
Interax in QualDep (Inherently Interactive) Models

Probit/Logit Models w/ Interactions


Probit: p(y = 1) = F (x ¢–)Logit:
p(y = 1) =
- 1
exp( x ¢)
= éêë1 + exp(- x ¢)ù
ú
û
1 + exp( x ¢)
Marginal Effects: (nonlinear, so must specify @ what x)

Start w/ x' pure lin-add, model inherently inter. b/c S-shaped:

Probit: ¶ p
¶ xk

=
¶ xk
¶ x ¢
= f (x ¢)×
= f (x ¢)×k
¶ xk
Logit: ¶ p
¢
¢
¶ {e x  [1+ e x  ]- 1 }
¢
ex 
=
=
¢ ×k ¶x
1+ e x 
¶x k
¢
¢
¢
e x  (1+ e x  )
(e x  )2 ù
é
= ê (1+ e x¢ )2 - (1+ e x¢ )2 ú×k =
ë
û
=

¶ F (x ¢)
ex 
¢
1+ e x 
¢
ex 
¢
(1+ e x  )2
×e x ¢ ×x
ex 
¢
(1+ e x  )2
×k
¢
¢
×1+ 1e x¢ ×k = p ×(1 - p )×k
If x' =…+xx+zz+xzxz… same except dx'/dx=xx+xzz;
underlying propensity, i.e., movement along S-shape also interact
explicitly x & z. [Discuss meaning inherent v. explicit interax…]
¶p
 Probit: ¶ p = f x ¢ ×  +• Logit:

z
= p ×(1 - p )×(x + xz z )
(
)
(
)
x
xz
¶x
¶x
Elaborations, Complications, & Extensions:
Interax in Nonlin/Qual (Inherently Interax) Models

Standard Errors?
 Delta

Method:
  β

Probit Marginal-Effect s.e.:
¢
é
ù
ˆ
ê¶ {f (x ¢) }ú
ê
ú
¶ x ¢ˆ
¶ x1
ê
¶ ˆ1
ê
ê
M
ê
ê¶ f x ¢ˆ ¶ x¢ˆ
( ) ¶ xk
ê
ê
êë
¶ k
{



 

ˆ
f  β   V  βˆ   f  βˆ  



AsymVar
. . f (βˆ )
ú éê Vˆ (ˆ1 )
úê
úê M
úê
ú êCˆ (ˆ , ˆ )
1
k
úë
ú
ú
û
}
β
é
¶ x ¢ˆ
ê¶ f (x ¢ˆ ) ¶ x1
ê
L Cˆ (ˆ1, ˆ k )ù
¶ ˆ1
úêê
ú
O
M úê
M
ê
ú
L
Vˆ (ˆ k ) úêê¶ f (x ¢ˆ ) ¶¶xx¢ˆ
ûê
k
êë
¶ k
{
{
ˆ
¶ x ¢
ˆ ) ¶ x¢ˆ
¢
Logit: same, except pˆ (1 - pˆ )replaces
f
(
x

¶x
¶x
For first-difference effects, similar, but need specify
from what x to what x, and not just at what x.
Or you could CLARIFY… or mfx…
}ùúú
ú
ú
ú
ú
ú
ú
ú
ú
û
}
Complex Context-Conditionality and
Nonlinear Least-Squares

Complex Context Conditionality: The effect of anything
depends on most everything else. E.g.:

Policymaking:





Socioeconomic-structure of interests
Party-system and internal party-structures
Electoral system & Governmental system
Socio-economic realities linking policies to outcomes
Voting:



Voter preferences & informational environment
Party/candidate locations & informational environment
Electoral & governmental system

Institutions: Sets of institutions; effect each depends configuration
others present (e.g., that core of VoC claim).

Strategic Interdependence: each actors’ action depends on everyone
else’s; complex feedback (see Franzese & Hays….)
Complex Context-Conditionality

EmpiricallyMulticolinear Nightmare: Options?

Ignore context conditionality (stay linear-additive):


Isolate one or some very few interactions for close study; ignore
rest (stay linear-interactive):


Same, to degree lessened by amount interax allow, but demands on data rise
rapidly w/ that amount.
“Structured Case Analysis”:


Inefficient at best, biased more usually, and, anyway, context-conditionality
is our interest!
May help ‘theory generation’, but, for empirical evaluation, doesn’t help;
worsens problem! (See Franzese OxfHndbk CP 2007).
EMTITM: Lean harder on thry/subst to specify more precisely the
nature interax: functional form, precise measures, etc.


Refines question put to the data (changes default tests also).
GIVEN thry/subst. specification into empirical model, can estimate complex
interactivity. Side benefits. But must give.
Nonlinear Least-Squares

Estimate NLS:
y  f ( X, )+  wit h  ~ g( )
 
 E y  f ( X, ), so y= f ( X, ˆ )  ˆ
 y  f ( X, )]
 Min ˆˆ  Min [y  f ( X, )][


 Min SSE= yy - yf ( X, )  f ( X, ) y  f ( X, ) f ( X, )

 FOC: SSE= 0  2 f ( X, ) y  2 f ( X, ) f ( X, )  0

So, if, e.g., f ( X, )  X, t hen: Xy  XX   ˆ LS  XX

1
Xy, and if

1 
y  f X, ˆ LS   y  f X, ˆ LS  (also, as always).
 

n k 
T hat is, int uit ively, writ ing ˆ f ( X, ˆ LS ) as simply , we hav e:
ˆ  () 1 y
V( )     2I, t hen V(ˆε)LS 




LS
 
V ˆ LS
 V () 1 y   () 1 V( y)() 1,
LS
which if f ( X, ˆ LS )  Xˆ LS meaning   X, &, if =  2I, gives t he famiar
ˆ LS  ( XX) 1 Xy & V(ˆ LS )LS   2( XX) 1, as always.


NLS is BLUE under same conditions OLS, w/  for X.
Interpreting NLS (already know how): Effects = deriv’s & 1stdiff’s; s.e.’s by Delta Method or simulation…
Generalized Nonlinear Least-Squares

GNLS:
y  f ( X, )+  wit h V ( )   2   2I
 ˆ GNLS  ( 1)1  1y
 V (ˆ GNLS )  ( 1) 1  1V ( y) 1( 1) 1
 ( 1)1  1 1( 1)1
 ( 1)1  1( 1)1  ( 1)1



GNLS is BLUE in same cond’s NLS, but  for I.
…don’t know , so need consistent 1st stage (e.g., NLS)
FGNLS is asymptotically BLUE:
y  f ( X, )+  wit h V ( )   2   2I
ˆ 1) 1 
ˆ 1y
 ˆ
 (
FGNLS
ˆ 1) 1 
ˆ 1V ( y)
ˆ 1(
ˆ 1) 1
 V (ˆ FGNLS )  (
ˆ 1) 1
 (
Nonlinear Least-Squares & EMTI

EITM: Empirical Implications of Theoretical Models


TMEI: Theory-specified Models for Empirical Inference


Vision: Emp. regularities, findings, measures inform theory dev’p.
EMTI: Empirical Models of Theoretical Intuitions


Vision: Theory structures empirical models & relations b/w obs 
specification & (causal) i.d. of empirical models
TIEM: Theoretical Implications of Empirical Measures


Vision: Theory  more, sharper predictions  better tests, which
therefore inform theory more, which…
Vision: Intuitions derived from theoretical models specify empirical
models. I.e., empirical specification to match intuitions, not model.
Note: Strongly counter some alternative moves stats &
econometrics, & related; there toward non-parametric, matching,
& experimentation—there, “model-dependence” a 4-letter word.
Alternative audiences & rhetorical purposes?


Convince skeptic some causal effect exists, vs.
For the convinced, give richer, portable model of how world works.
Nonlinear Least-Squares:
“Multiple Hands on the Wheel” Model (Franzese, PA ‘03)

Monetary Policy in Open & Institutionalized Econ

Key C&IPE Insts/Struct: CBI, ER-Regime, Mon. Open




º CBI ≡ º Govt Delegated Mon Pol to CB
º Peg ≡ º Domestic (CB&Gov) Delegate to Peg-Curr (CB&Gov)
º FinOp ≡ º Dom cannot delegate b/c effectively del’d to globe
Effect of ev’thing to which for. & dom. mon. pol-mkrs would
respond diff’ly depends on combo insts-structs & v.v., &, through
intl inst-structs, for. on dom. & v.v.
 P  E  C   1 ( X1 )  P  E  (1  C )   2 ( X2 )
 P  (1  E )  C   ( X )  P  (1  E )  (1  C )   ( X )

3
3
4
4
 
(1  P)  E  C   5 ( X5 )  (1  P)  E  (1  C )   6 ( X6 )
(1  P)  (1  E )  C   7 ( X7 )  (1  P)  (1  E )  (1  C )   8 ( X8 )

Multicolinear Nightmare:



23=8 inst-struct conds, i, times k factors per πi(Xi) if lin-interact
Exponentially more if all polynominials; k!/2(k-2)! if all pairs.
Good thing can lean on some thry to specify more precisely!
Nonlinear Least-Squares:
“Multiple Hands on the Wheel” Model

CB & Govt Interaction (Franzese, AJPS ‘99):
E ( )  c   c (xc )  (1  c)   g (x g )
 c   c  g (x g )   g (GP,UD, BC , TE , EY , FS , AW ,  a )

Full Monetary Exposure & Atomistic  zero
domestic autonomy 
 1 (x1 )   2 (x 2 )   5 (x5 )   6 (x6 )   a
 E   a  P  (1  E )  C   3 (x3 )  P  (1  E )  (1  C )   4 ( x 4 )

(1  P)  (1  E )  C   c  (1  P)  (1  E )  (1  C )   g ( x8 )

s.t. that, full e.r.fixCB&Gov match peg
 3 ( x3 )   4 ( x 4 )   p
 E   a  P  (1  E )   p

(1  P)  (1  E )  C   c  (1  C )   g (x8 ) 
Nonlinear Least-Squares:
“Multiple Hands on the Wheel” Model

Compact & intuitive, yet gives all theoretically
expected interactions, in the form expected
Nonlinear Least-Squares:
“Multiple Hands on the Wheel” Model

Effectively Estimable, yet gives all 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 theories, if both powerful enough &
enough empirical variation.
Nonlinear Least-Squares:
“Multiple Hands on the Wheel” Model

Neat, but does it work? (Try it! Data online; stata: help
nl). Estimated Equation, w/ Std. Errs.:

Estimated Effects (highly context-conditional):
Nonlinear Least-Squares:
“Multiple Hands on the Wheel” Model
Nonlinear Least-Squares:
“Multiple Hands on the Wheel” Model
Nonlinear Least-Squares:
“Multiple Hands on the Wheel” Model
Multiple Policymakers:
Veto Actors Bargaining in Common Pools

Multiple implications for policy outcomes dispersal of
policymaking-authority across diverse actors:

Veto-Actor Theory (Tsebelis ‘02) emphasizes:


Collective-Action/Common-Pool Theories (WSJ ‘81):


Externalities & so overexploit/underinvest public goods.
Bargaining & Delegation Theories rather stress:


Privileges S.Q., & so retards policy adjustment, reduces change.
Bargaining Strengths/Positions, yielding Weighted Compromise.
This project attempts a synthesis:



Disting. theoretically/conceptually many effects of # (fragment.) &
diversity (polar., partisan) policymakers.
Empirical model of many effects distinctly & effectively.
Preliminary application to evolution fiscal policy (pub debt) in
developed democracies, 1950s-90s.
Veto Actors: Deadlock, Delayed Stabilization,
& Policy-Adjustment Retardation

Tsebelis (‘95b, ‘99, ‘00, ‘02): Essential Argument:





 # &/or ideological/interest polarization of pol-mkng 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):
Suggests both mean/expected policy-∆ & variance pol & pol-∆  as size
of W(SQ)  (aside: why only suggests)
No prediction of pol-level or of direction pol-∆, only of E(|∆p|), V(∆p).
Veto-Actor Implications

 # (=Frag) & Polar of VA Privileges SQ 




Retards policy-adjustment rates/delays stabilization,
 range of possible policy-, & so, possibly,
 magnitude/variance policy-  (1st- & 2nd-order E()).
Results, e.g. in fiscal policy, deficits & debts; originally
mixed, but tighter specify thry into empirical analysis:


(F ’00, ’02) How model: policy-adjustment-rate effect =
conditional coefficient on LDV in dynamic model, not level.
(F ’00, ’02) How measure: frag & polar in VA theory =



raw #, not eff. # (size-wtd) VA;
max range pref’s, not V(pref’s) or sd(pref’s), (size-wtd)
 Model: yt=…+(#VA,Range{pref(VA)})yt-1… &/or
V(yt)=f(#,Range)  empirical support.
Common-Pool Theory (1)


Weingast, Shepsle, Johnsen (1981): districting & distributive/pork-barrel
spend (law of 1/n)
 Benefits concentrate district i: Bi=f(C), f’>0 & f”<0
 Costs disperse across n districts: Ci=C/n
 optimal project-size from i’s view  in # districts: f’(C*)=1/n (…loglinearly?)
Alternative Decision Rules/Processes […] 
 […] Law of 1/n is general, & stronger as legislative behavior more

Universalistic & less Minimal-Winning, which tendency  as rational
ignorance, winning-coalition uncertainty, or legislative-rule closure to amend
or veto .
E.g., PubRev = common pool for n reps, overused to distribute bens; this CA
prob worsens “proportionally” by law 1/n, i.e. at rate b/w those at which
(n+1)/2n (MWC) & 1/n (uni)  in n
1
Ratio of Benefits to Costs of Projects Passed
0.9
Minimum-Winning-Coalition Decision-Making
Universalistic Decision-Making
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
5
10
15
20
25
30
35
40
45
50
55
60
Number of Constituencies
65
70
75
80
85
90
95
Manifestations of Common Pools


Velasco (‘98, ‘99, ‘00): inter-temporal totality pub rev is C-P to
today’s policymakers  deficits & debts also law of 1/n
Peterson & co’s, Treisman: federalism  multiple tax authorities 
several common-pool problems:



Again, should be quite general:



Anything that gains set of pol-makers credit  underinvested as n
Anything that gains set of pol-makers blame  overexploited as n
E.g., (thry 2nd-best), ELECTIONEERING:



Inter-jurisdiction competition (w/ high factor mobility)  C-P of investment
resources  over-fishing: taxes too low.
National govt as lender last resort  subnational jurisdictions see fed guarantee
& funds as common pool  excessive borrowing by subnat’l units. (e.g., EU,
EMU & Euro  common pools…)
Magnitude incentive electioneer fades w/ n (see, e.g., Goodhart)
Control over electioneering diminishes w/ n.
Notice: CP not arise in Tsebelis’ VA Theory b/c # & pref’s of VA’s
exog & predetermined, whereas in CP theory: prefs=f(#).
Modeling Common-Pool Effects

CP Effects distinguishable from VA Effects:




C-P Effects on levels, not (as in VA) in dynamics.
Proportional to 1/n for equal-sized actors. Standard
Olsonian encompassingness logic  proper n here is sizeweighted (effective & s.d./var.)
Fractionalization (#) & esp. polarization (het.) relate to VA
effects; CP, conversely, relate primarily to #, although het.
can exacerbate some CA probs.
Suggests Proper Model of Policy-Response to some
public demand for, x1’1, or against, x2’2:


…+(x1’1)(1-f(ln(Eff#))+(x2’2)(1+f(ln(Eff#))+…
Same f(ln(Eff#), b/c overexploit/underinvest same º
Bargaining, Delegation, & Compromise


Explicit extensive-form delegation & bargaining games:
huge theoretical & empirical literature
F (‘99,‘02,‘03): less context-specific empirical strategy…


Because broad comparativist seek thry that travels, not that requires
different model each context.
Offering is roughly equivalent Nash Bargaining.
 Most ext forms  eqbm bounded by actors’ ideal pts:



Policy outside that range possible,



Convex set/hull, upper-contour set (=core of coop. game thry),
So like Tsebelis, but further, though short of explicit ext-form
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-combo emp. models  compromise

If Nash Bargain, e.g., (n.b. NB=coop. game-thry but equiv. sev. reasonable extform non-coop barg. games: Rubinstein ‘82),  (geometric) wtd-influence
pol-mkng; i.e., simple wtd-avg.
Empirical Manifestations & Model
of Compromise Policymaking

Re: def’s & debt, Cusack (‘99, ‘01; cf., Clark ‘03)



Arg: left more Keynes-active counter-cyc; right less, even pro-cyc
Add Nash-Barg Model  wtd-avg pol-mkr partisanship conditions º
Keynesian cntr-cyc fisc-pol response to macroecon.
Empirical Implementation:

Ideally:




Describe barg power party i as f(charact’s i & barg envir, j,  f(vij)
Desc. pol response to conditions xk if i sole pol-mkng control: qi(xk)
Then embed Nash-Barg sol’n, if(vij)qi(xk), in emp. model to est.
Currently:


Assume wtd-avg compromise outcome pre-estimation.
I.e., simply assume by measure & specification that Policy responds to
WtdPartisanshipCurrGovt  MacroeconomicConditions.
Empirical Model of the Theoretical Synthesis (1)

Different aspects of policy-maker fragmentation,
polarization, & partisanship:




V-A Effects: raw # (frag) and ideological ranges (polar)
C-P Effects: eff # (frag) &, maybe, ideol. s.d./var (polar)
D-B Effects: power-wtd mean ideologies (partisanship)
Different ways these distinct effects manifest in pol:



V-A (primarily) to slow pol-adjust (delay stabilization);
C-P induces over-draw from common resources (incl. from
future as in debt); under-invest in common properties (less
electioneering), log-proportionately
D-B induces convex-combinatorial (compromise) policies, incl.
greater left-activist/right-conservative Keynesiancountercyclical/conservative pro-cyclical, in proportion to degree
left/right controls policymaking
Empirical Model of the Theoretical Synthesis (2)

…implies specification where:




Abs # VA & ideol range modify pol-adjust rates
(log) Eff # pol-mkrs & s.d. ideol (wtd measures) gauge
C-P prob in electioneering (+debt-lvl effect?)
Some barg process among partisan pol-mkrs (e.g., Nash
 wtd-influence) determines combo reflected in net
policy responsiveness to macro (º K-activism)


Dit   i  1  n NoPit  ar ARwiGit   1Di ,t 1  2 Di ,t  2  3 Di ,t 3

 
 
 Y Yi ,t  U U i ,t  P Pi ,t  1  cg CoGit
e1



Eit   e 2 Ei ,t 1  1   en ENoPit   sd SDwiGit   xit η  z it ω   it
Empirical Model Specification & Data


Dit   i  1  n NoPit  ar ARwiGit   1Di ,t 1  2 Di ,t  2  3 Di ,t 3  xit η  z it ω   it

 
 

 Y Yi ,t  U U i ,t  P Pi ,t  1  cg CoGit   e1Eit   e 2 Ei ,t 1  1   en ENoPit   sd SDwiGit 






Dit = Debt (%GDP);
NoP & ARwiG = raw Num of Prtys in Govt & Abs Range w/i Govt:
 VA conception, so modify dynamics. Expect n & ar >0.
 By thry & for efficiency: modify all lag dynamics same.
CoG (govt center, left to right, 0-10):
 Modifies response to macroecon (equally, by thry & for eff’cy) : βcg<0.
 Macroec: Y = real GDP growth; U =  unemp rate; P = infl rate.
x’ = controls: set pol-econ cond’s response to which not partisan-differentiated or
gov comm-pool: (e.g., E(real-int)-E(real-grow), ToT)
ENoP & SDwiG = Effective Num of Prtys in govt & Std Dev w/i Govt:
 Frag & Polar by wtd-influence concept. CP lvl-effects modify (at same rate)
electioneering, Et, pre-elect-year, & Et-1, post-elect-yr.: en & sd<0.
z’ω = set of constituent terms in the interactions:
 ENoP, SDwiG may have positive coeff’s by CP effect lvl debt, but issue is
temporal fract more than curr. govt fract. Thry o/w says omit.
Coeff.
D t-1
1.212
Lagged
Dependent
D t-2
-0.153
Variables
D t-3
-0.121
0.007
ρn (veto-actor effect: fractionalization)
-0.000
ρar (veto-actor effect: polarization)
-0.336
ΔY
Macroeconomic
0.992
ΔU
Conditions
-0.188
ΔP
-0.037
βcg (partisan-compromise bargaining)
x1 (open)
15.891
x2 (T oT )
0.388
Controls
x3 (open∙ T oT )
-10.681
x4 (dxrig)
-0.036
x5 (oy)
2.064
Et
0.687
Pre- and Post-Electoral
Indicators
Et-1
1.490
-0.547
γen (common-pool effect: fractionalization)
0.573
γs d (common-pool effect: polarization)
z1 (CoG)
0.051
Constituent
z2 (ENoP)
0.281
T erms
z3 (SDwiG)
0.542
from the
z4 (NoP)
0.181
Interactions
z5 (ARwiG)
-0.312
Sum m ary Statis tics
N (Deg. Free)
R2 ( R 2 )
735 (691)
0.991 (0.990)
Std. Err.
0.060
0.085
0.045
0.006
0.006
0.111
0.308
0.063
0.037
5.279
1.744
5.156
0.066
1.094
0.568
0.645
0.182
0.486
0.131
0.446
0.437
0.277
0.259
t-Stat.
20.112
-1.792
-2.677
1.089
-0.013
-3.033
3.219
-2.965
-0.988
3.010
0.222
-2.072
-0.544
1.886
1.210
2.310
-3.001
1.179
0.390
0.629
1.242
0.654
-1.205
Pr(T> | t| )
0.000
0.074
0.008
0.277
0.990
0.003
0.001
0.003
0.323
0.003
0.824
0.039
0.587
0.060
0.227
0.021
0.003
0.239
0.697
0.530
0.215
0.514
0.228
se2
2.525
2.101
DW-Stat.

Pace Brambor et al. (‘06), but joint-significance of multiple-policymaker
conditioning effects (en, sd, n, ar, cg) overwhelmingly rejects excluding
(p.001), whereas joint-sig coeff’s on constit. terms, z, clearly fails reject (p.602)
exclusion. (Almost) All theory says should be zero, so…
Coeff.
Std. Err.
D t-1
1.207
0.060
Lagged
Dependent
D t-2
-0.158
0.085
Variables
D t-3
-0.117
0.045
ρn (veto-actor effect: fractionalization)
0.011
0.005
ρar (veto-actor effect: polarization)
-0.002
0.004
ΔY
-0.375
0.087
Macroeconomic
ΔU
1.095
0.286
Conditions
ΔP
-0.207
0.053
βcg (partisan-compromise bargaining)
-0.051
0.020
x1 (open)
16.128
5.314
x2 (ToT)
0.414
1.728
Controls
x3 (open∙ ToT)
-10.780
5.194
x4 (dxrig)
-0.038
0.066
x5 (oy)
1.898
1.100
Et
0.475
0.420
Pre- and Post-Electoral
Indicators
Et-1
1.146
0.562
γen (common-pool effect: fractionalization)
-0.570
0.209
γsd (common-pool effect: polarization)
0.881
0.586
Sum m ary Statistics
N (Deg. Free)
R2 ( R 2 )
735 (696)
0.991 (0.990)
t-Stat.
20.290
-1.851
-2.577
2.369
-0.437
-4.332
3.829
-3.889
-2.484
3.035
0.239
-2.076
-0.578
1.724
1.133
2.040
-2.727
1.503
Pr(T>| t| )
0.000
0.065
0.010
0.018
0.662
0.000
0.000
0.000
0.013
0.002
0.811
0.038
0.563
0.085
0.258
0.042
0.007
0.133
se2
2.522
2.099
DW-Stat.
Table 1: Estimated Veto-Actor, Bargaining, and Common-Pool Effects of Multiple Policymakers
Adjustment Rates
Lag Coefficienta
Policy-Adjust/Yrb
Long-Run Mult.c
½-Lifed
90%-Lifee
Veto-Actor Effects: Estimates of Policy-Adjustment Rate
NoP=1
NoP=2
NoP=3
NoP=4
NoP=5
0.943
0.952
0.960
0.969
0.978
0.057
0.048
0.040
0.031
0.022
17.498
20.639
25.154
32.200
44.727
11.778
13.956
17.087
21.971
30.654
39.127
46.362
56.761
72.985
101.832
NoP=6
0.986
0.014
73.208
50.397
167.415
Bargaining Effects: Estimates of Keynesian Fiscal Responsiveness
Growth
d(UE)
Infl
Mean Econ.
Performance
-2 std. dev.
-2.354
1.915
-3.593
Mean Econ.
Performance
-1 std. dev.
0.454
1.034
1.230
Mean
Economic
Performance
3.261
0.153
6.054
Mean Econ.
Performance
+1 std. dev.
6.069
-0.728
10.877
Mean Econ.
Performance
+2 std. dev.
8.877
-1.608
15.701
CoG
E(D|Econ) f
E(D|Econ)
E(D|Econ)
E(D|Econ)
E(D|Econ)
3.0
4.2
5.4
6.6
7.8
9.0
3.157
2.930
2.703
2.476
2.250
2.023
0.599
0.556
0.513
0.470
0.427
0.384
-1.959
-1.818
-1.677
-1.536
-1.396
-1.255
-4.516
-4.192
-3.867
-3.543
-3.218
-2.894
-7.074
-6.566
-6.058
-5.549
-5.041
-4.533
Fiscal-Cycle
Magnitudeg
10.231
9.496
8.761
8.026
7.291
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
1.07410
0.86454
0.65497
0.44541
0.23585
Magnitudeh
a
Calculated as the sum of the coefficients on the dependent-variable lags
f
Predicted deficits given the state of the macroeconomy listed in that column and
Extension & Refinement
E ( yt )   0  xt0b0   0  1 ln( NoPt )  2 ln(1  ARwiGt )  yt 1
 1 1 I

2 
  xt b   p(cit )  qi (xt )

 
i 1

  1  1 ln( NoPt )   2 ln(1  ARwiGt ) 
 1   ln( ENoP )   ln(1  SDwiG ) 
1
t
2
t 
 






x0 = factors that affect policy-outcomes unless pol-mkrs act to
change status quo, i.e., that have effect on pol-out directly.
x1 = factors affecting policy-outcomes if policymakers act to change
status quo, without partisan-differentiated response
x2 = factors affecting policy-outcomes if policymakers act to change
status quo, with partisan-differentiated response
{NoP,ARwiG} = sources of veto-actor effects; as before
{ENoP,SDwiG} = sources of common-pool effects; as before
{p(cit),qj(xt)} = sources of bargaining & delegation effects:


p(cit): Effective policy-influence of party i in context t. (E.g., as now: cabinet
seat-shares, but could become richer model.)
qj(xt): Model of response of party i to pol-econ conditions xt. (E.g., as now:
CoGiMacroecont, but could become richer model.)
Preliminary Results of Fuller Model
Temporal
Dynamics
Veto-Actor Effect
On Outcom e-Adjustm ent Rate
x 0 : Variables with “Direct”
Effect on Outcome
x 1 : Variables with Effects via
Non-Partisan-Differentiated
Policy Response
Com m on-Pool Effect on
Policy Response
x 2 : Variables with Effects via
Partisan-Differentiated
Policy Response
Bargaining-Com prom ise Effects
on Partisan Policy-Responses
Veto-Actor Effect
On Policy-Adjustm ent Rate
Common-Pool Effect on Debt Level
N (Deg. Free)
R2 ( R 2 )
D(t-1)
D(t-2)
D(t-3)
Coeff.
1.197
-0.139
-0.121
Std. Err.
0.059
0.085
0.045
t-Stat.
20.144
-1.629
-2.698
Pr(T>| t| )
0.000
0.104
0.007
NoP
0.008
0.004
1.883
0.060
Open
Open*ToT
Ele(t)
Ele(t-1)
OY
DXRIG3
16.624
-11.190
0.315
0.873
2.833
-0.073
3.758
3.135
0.363
0.399
1.295
0.072
4.423
-3.569
0.867
2.186
2.187
-1.009
0.000
0.000
0.386
0.029
0.029
0.314
ln(ENoP)
-0.277
0.071
-3.903
0.000
Growth
d(UE)
Inflation
-0.238
0.749
-0.137
0.084
0.228
0.047
-2.815
3.289
-2.947
0.005
0.001
0.003
CoG
-0.049
0.026
-1.893
0.059
NoP
0.215
0.121
1.773
0.077
ln(ENoP)
1.128
Sum m ary Statistics
0.486
2.320
0.021
se2
2.510
2.090
735 (697)
0.991 (0.990)
DW-Stat.