BlalockLecture InteractionsEtc 2009 by ib3kJV4

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									          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)
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
                                   Statistical
                                    Analysis
                                               Interaction-Term Usage
                                         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                                     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] macro-structures magnify or mitigated by intermediate-level
                  inst’s… help us…explain the contingent nature of pol-econ
    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
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                                        Result of Outcomes at T0
                                                     (A)
                                            Interest Structure of                                                                On to T+1
      The Cycle of                             the Polity and
                                                                                Elec
                                                                                         tion
                                                                                                s
    Political Economy                            Economy




                                                                                                                 Go orm
                                                                                                                   F
                                                                                                                   ve atio
     Examples of the Elements at




                                                                                                                     rnm n
             Each Stage:




                                                                                                                        en
                                                                                                                           t
        (A) Interests:
            Sectoral Structure of Economy
            Income Distribution
            Age Distribution
            Trade Openness                                                                                        (B)
        Elections:                                         Action at                                            Partisan
            Electoral Law
            Voter Participation                            Time T0                                          Representation in
        Government Formation:                                                                                 Government
            Fractionalization




                                                                                          s
            Polarization




                                                                                            r
                                                                                        cto
                                                                                  lA
        (B) Representation:




                                                                                   ta
                                                                                en
            Partisanship




                                                                                                                           icy
                                                                           rnm
        Policy:

                                                                            e




                                                                                                                         Pol
                                                                         ov
            Fiscal Policy
                                                                     n-G
                                                                    No
            Monetary Policy
            Institutional Adjustment
        Government Termination:
            Replacement Risk
                                                                                                         nt
        (C) Outcomes:                                                                              nme
            Unemployment                            (C)                                     o v e r at i o n )
                                                                                          (G in
            Inflation                                                                      Term
            Growth
                                               Political and
            Sectoral Shift                  Economic Outcomes
            Debt                                                                         Exogenous Factors
            Institutional Change
  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
              Iffully 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:
                    y  h(I) f ( X)  1  h(I) g (Z)
    

   In words…
…
                                         y                  f ( X )
…                                       x    h(I )          x       ;
…i.e., effect of
                                         y                     g ( Z )
  anything
  depends on                             z     h(I )           z        ;
  everything
  else!
                                         y
                                         i      h ( I )
                                                    i        f (X)  g (Z)
        Not Every Argument Is an
          Interactive Argument
   Not Interactive:
       X affects Y through its effect on Z: XZY
            In (political) psychology / behavior, this called
             mediation. Interaction is called moderation in this
             literature.
       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)
            I.e., the outcomes differ across 22 of X and Z.
   Interactive: Effect of X on Y depends on Z (
                          Z on  f  X 
    converse: Effect of Z   YY depends on X):
                  Y
                       f
                           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      DMag

    ¶ {¶ ENPP }
       ¶ SFrag
                      ¶ {¶ ENPP }
                         ¶ DMag    ¶ 2 ENPP      ¶ 2 ENPP
                  º           º              º              ³ 0
    ¶ DMag            ¶ SFrag   ¶ SFrag¶ DMag ¶ DMag¶ SFrag
    Empirical Specification: Lots ways get
          A Typical Linear-Interactive
                 Specification
   Want linear f() w/ these properties; many ways to get
    there: β0  β1 SFrag  β2 DMag  ε
    ENPP
          ENPP
                    β1  f  DMag   α0  α1 DMag
                         ?

          SFrag
          ENPP
                    β2  f SFrag   γ0  γ1 SFrag
                         ?

          DMag
     ENPP  β0  α0  α1 DMag SFrag  γ0  γ1 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):
                                  ¶ ENPP
            Effect (SFrag ) º             = β SF + β SFDM DMag
                                  ¶ SFrag
                              º ΔENPP = β SF ΔSF + β SFDM DMag ×ΔSF
                                  ΔENPP
                              º         = β SF + β SFDM DMag
                                   ΔSF
   Effect of DMag on ENPP (is f of SFrag):
                         ¶ ENPP
      Effect (DMag ) º          = β DM + β SFDM SFrag
                         ¶ DMag
                    º ΔENPP = β DM ΔDM + β SFDM SFrag ×ΔDM

                         ΔENPP
                    º          = 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
                 E.g., by “centering” (centering changes nothing, btw…)
            Other-var(s)=0 may not have anything subst’ly main about it
       COEFFICIENTS ARE NOT EFFECTS. EFFECTS ARE
        DERIVATIVES &/OR DIFFERENCES.
            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: NOTES2
   Interactions are logically symmetric:
    

       If argue effect x depends z, must
                                               
        For any function, not just lin-add. y
                                           x    y
                                                  z     2 y
                                                              
                                                                2 y
        also believe effect z depends x. z      x      xz zx
   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…
         Frequent 2nd-Moment Implications
                   Interactions
   DMag permissive ele sys: allows more parties…
     NP   0  1 DM   ; 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
                  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       SF
          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.:
                   y                                     y  ˆ
    Effect  x       β x  β xz z  Est.Eff. x   E    β x  β xz z
                                                                     ˆ
                   x                                     x 
                                                y  
                                                x      
                                                                   ˆ   
             Est.Var. Est.Eff. x   E Var  E     E Var β x  β xz z 
                                                                         ˆ
                                                                                 
                                         
                                         ˆ     ˆ        
                                                          ˆ       ˆ                   
                                                                                     ˆ ˆ       
                                      V β x  β xz z  V β x  V β xz  z 2  2  C β x , β xz z


    In words… More Generally:V (xβ)  x V (β)  x
                                   ˆ        ˆ 
        From Hypotheses to Hypotheses
                   Tests:
                    Does Y Depend on X or Z?
ENPP  β0  βSF SFrag  βDM DMag  βSFDM SFragDMag  ...  ε
                                                                  91
          Hypothesis                Mathematical Expression             Statistical test
          x affects y, or                    y=f(x)                         F- test:
y is a function of (depends on) x     y x   x   xz z  0         H0:  x   xz  0
                                                                       Multiple t-tests:
         x increases y                 y x   x   xz z  0
                                                                       H0:  x   xz z  0
                                                                       Multiple t- tests:
         x decreases y                 y x   x   xz z  0
                                                                         x   xz z  0

          z affects y, or                     y=g(z)                        F- test:
y is a function of (depends on) z      y z   z   xz x  0        H0:  z   xz  0
                                                                        Multiple t-tests:
         z increases y                 y z   z   xz x  0
                                                                       H0:  z   xz x  0
                                                                       Multiple t- tests:
         z decreases y                 y z   z   xz x  0
                                                                       H0:  z   xz x  0
    From Hypotheses to Hypotheses Tests:
    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         y x   x   xz z  g ( z )             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

                                        y x  z   2 y xz   xz  0
The effect of x on y decreases in
                                                                                     t-test: H0:  xz  0
                 z
                                                  y=f(xz,•)
The effect of z on y depends on x          y z   z   xz x  h( x)              t-test: H0:  xz  0
                                        y z  x   2 y zx   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

                                        y z  x   2 y zx   xz  0
The effect of z on y decreases in
                                                                                     t-test: H0:  xz  0
                 x


                           Does Y Depend on X, Z, or XZ?
                                                                          93
           Hypothesis                    Mathematical Expression                      Statistical Test
y is a function of (depends on) z,
                                                   y=f(x,z,xz)                 F-test: H0:  x   z   xz  0
    z, and/or their interaction
        Use & Abuse of Some Common
                  ‘Rules’
   Centering to Redress Colinearity Concerns:
       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
        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), 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
 Presentation: Marginal-Effects / Differences
                   Tables & Graphs
 Plot/Tabulate Effects, dy/dx, over
  Meaningful &/or Illuminating Ranges of z,
    dy / Conf. Int.’s
  with dx  t Var(dy / dx)  ˆ  ˆ z  t V ( ˆ )  V ( ˆ ) z  2C( ˆ , ˆ ) z
     ˆ             ˆ
                df , p                                                           x         xz     df , p         x             xz
                                                                                                                                     2
                                                                                                                                                x     xz

                                                                                Figure 5. Marginal Effect of Runoff, Extending the Range of Groups
                                                                  20



       Explain axes                                              15
                               d (Number of Candidates )/d (Runoff )



       Explain shape                                             10


       Linear-interax:                                                5

           Will cross 0
                                                                       0
        & be insig @ 0.
       Rescaling &                                                    -5


         “main effect”                              -10
                                                                            -2        -1          0        1         2         3         4      5          6
         “centering”                                                                           Effective Number of Societal Groups (Groups )
             Presentation: Expected-
        Value/Predictions Tables & Graphs
   Predictions, E(y|x,z):
    y  tdf , p Var( y ) 
    ˆ                ˆ
                                                          ˆ           ˆ               ˆ               ˆ
                                                     V (  0 )  V (  x ) x 2  V (  z ) z 2  V (  xz )( xz ) 2
     ˆ     ˆ       ˆ       ˆ
     0   x x   z z   xz xz  tdf , p                    ˆ ˆ                 ˆ ˆ
                                                      2C (  ,  ) x  2C (  ,  ) z  2C (  ,  ) xz  ˆ ˆ
                                                                 0     x                 0     z                 0     xz

                                                             ˆ ˆ                   ˆ ˆ                       ˆ ˆ
                                                      2C (  x ,  z ) xz  2C (  x ,  xz ) x 2 z  2C (  z ,  xz ) xz 2

         Here’s one place a little matrix algebra would help:
                        ˆ     ˆ       ˆ       ˆ                    ˆ ˆ
y  tdf , p Var ( y )   0   x x   z z   xz xz  tdf , p xV (β)x
ˆ                 ˆ
                                                                   ˆ ˆ
                                                               V (0 )          ˆ ˆ ˆ          ˆ ˆ ˆ           ˆ ˆ ˆ
                                                                                C (  0 ,  x ) C (  0 ,  z ) C (  0 ,  xz )   1 
                                                                                                                                
                                                                ˆ ( ,  )
                                                              C 0 xˆ ˆ           V x   ˆ
                                                                                   ˆ ( )       ˆ ( ,  ) C ( ,  )   x 
                                                                                                    ˆ ˆ
                                                                                                C x z           ˆ ˆ ˆ
  ˆ     ˆ       ˆ       ˆ
  0   x x   z z   xz xz  tdf , p   1   x   z   xz  
                                                                ˆ ˆ ˆ            ˆ ˆ ˆ             ˆ ˆ          ˆ ˆ ˆ
                                                                                                                      x     xz      
                                                                                                                                  z 
                                                               C (0 ,  z ) C ( x ,  z )     V ( z )       C (  z ,  xz ) 
                                                                                                                                    
                                                              ˆ ˆ ˆ            ˆ ( ,  ) C ( ,  )
                                                                                    ˆ ˆ         ˆ ˆ ˆ              ˆ (  )   xz 
                                                                                                                        ˆ
                                                              C (  0 ,  xz ) C x xz               z     xz    V xz 

         Use spreadsheet or stat-graph software (…list
                             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:
            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)
       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:
            (full-set,set-1): choose which (& what base if use full-set-
             minus-1) to abet presentation & discussion
                     Examples & Practice:
            Basic Linear-Interactive Model
   Basic Linear-Interactive Model:
        eusup  b0  bedu edu  blftrt lftrt  bedlr edu  lftrt  ...  

   Effect of edueusup  bedu  bedlr lftrt
                 ?
                                     edu
                                                         eusup
                       For the record, the effect of lftrtlftrt  blftrt  bedlr edu
                                                          :

   Std Error of that Effect (of edu on eusupp)?
     ˆ ˆ    ˆ                   
                            ˆ ˆ      ˆ ˆ                  ˆ ˆ ˆ
    V bedu  bedlr lftrt  V bedu  V bedlr lftrt 2  2  C bedu , bedlr lftrt                           
                                      Vˆ
                 Std. Err. effect of edu:blftrt  bedlr edu   V  blftrt   V  bedlr  edu 2  2  C bedu , bedlr  edu
                                       ˆ          ˆ              ˆ ˆ            ˆ ˆ                    ˆ ˆ ˆ


   To do this (e.g. in Stata, using the .dta
    subset)
       First: gen edlr=education*leftright
                             Examples & Practice:1a
               Basic Linear-Interactive Model:
          Education & Leftright Purely Micro-level Model
    . reg eu_support education leftright edlr
          Source |       SS       df       MS                     Number of obs   =    42680
    -------------+------------------------------                  F( 3, 42676)    =   371.33
           Model | 7869.51854      3 2623.17285                   Prob > F        =   0.0000
        Residual | 301474.935 42676 7.06427347                    R-squared       =   0.0254
    -------------+------------------------------                  Adj R-squared   =   0.0254
           Total | 309344.453 42679 7.24816545                    Root MSE        =   2.6579
    ------------------------------------------------------------------------------
      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                            eusup                                 2 eusup
            .5919  .0321 lftrt             .0801  .0321 edu                    bedlr  .0321
     edu                              lftrt                              edulftrt

         Only for modifying effect does standard regression output tell us
          directly.
    What are the standard errors of these effects?
         Only for modifying effect does standard regression output tell us
          directly.
     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
         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$3 is absolute reference to cell containing coefficient on
         Ex’s & Practice: Basic Linear-Interact
                        Model:
        Education & Leftright Purely Micro-level
                        to Plot
    Spreadsheet FormulaModel Effect Lines w/ C.I.
       Column Three: standard error (or can skip to bounds)
            1st Cell (A29): =($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
          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:
                  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)
            Finally, you can generate variances & std errors, which you could tabulate:
                  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
                                                                                                     Google
          Procedures
Create the variables which set
                                                  Command syntax
                                               egen xmin = min(x)                     ‘kam franzese michigan
                                               egen xmax = max(x)
the values of the variables x,
z, and xz (& other variables,      gen xh = ((_n-1)/(v-1))*(xmax-xmin) in 1/v                            press’
                                                 egen zh=mean(z)
                      ˆ
 if any) for which y will be
           calculated.
                                                  gen xhzh=xh*zh
                                                    gen col1h=1                              for that data, and
Assemble the variables into a
           matrix, Mh
                                    mkmat xh zh xhzh col1h in 1/v, matrix(Mh)
                                                                                      stata & excel resources
   Create betas, a column
                                               matrix betas=e(b)’
vector with k x 1 dimensions.
    Calculate the predicted                   matrix yhat=Mh*betas
                                                                                              or go directly to
             values.                                                                   http://www.press.umich.edu
   Convert the vector into a
            variable.
                                             svmat yhat, name(yhat)                /KamFranzese/Interactions.html
   Create a matrix from the
 estimated covariance matrix                   matrix V = get(VCE)
 of the coefficient estimates.
Calculate the variance of the
        predicted values.
Extract the diagonal elements
                                               matrix VYH=Mh*V*Mh’

                                            matrix DVYH= vecdiag(VYH)
                                                                                                     Google
       into a row vector.
  Transpose elements into a                   matrix VARYHAT=DVYH’                    ‘Matt Golder interaction’
         column vector.
   Convert the vector into a
            variable.
                                          svmat VARYHAT, name(varyhat)                        or go directly to
    Calculate the estimated                                                             http://homepages.nyu.edu
     standard error of each               gen seyhat1 = sqrt(varyhat1)
     predicted probability.                                                            /~mrg217/interaction.html
 Present a table of predicted
  values with corresponding            tabdisp xh, cellvar(yhat1 seyhat1)
     standard errors of the
        predicted values.
  Generate lower and upper       gen LByhat1=yhat1-invttail(e(df_m),.05)*seyhat1
 confidence interval bounds.
      Graph the predicted
  probabilities and the upper
                                 gen UByhat1=yhat1+invttail(e(df_m),.05)*seyhat1
                                                                                                        Stata
                                    twoway connected yhat1 LByhat1 UByhat1 xh
     and lower confidence
            intervals.
                                                                                                     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.
            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)
       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…)
       Split-sample abets eyeballing, obfuscates statistical analysis, of
                                         s.e.b  by V b   V b  
        the main point: the different effects b   category.2Cb , b   V b   V b 
                                                   1i   1j   1i   1j    1i   1j   1i   1j


            What’s s.e/signif. of b1i-b1j? Need
            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
              Cross-Level Interactions:
         From the (C or G)LRM to the HLM
                     eusupij   j0   lr lftrt ij  ...   ij 
                                          j
                                                                 
                        0   0  1GSPEND j  u 0  or
                          j                                j

                         lr   0   1GSPEND j  u1j 
                           j                                     
    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
    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
    Elaborations, Complications, & Extensions:
    Random-Effect & Hierarchical/Multi-level Models
   R.E. Model: Odd that std. lin-interax model:
        Assumes know y=f(X)+error: β0  β x x  βz z  β xz xz  ε
                                       y
        But dy/dx=f(z) w/o error!:x  β x  β xz xz
                                   y

        So, try:
                        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…
                                    *  ij   1 xij  ij z j
    HLM: Same model, except xij & zj,&
                                            0
                                                  j
                                                            2


        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  1 DM   ; 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
     0  1DM
NP Props…   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.     lftrt  ...  
                             0      lr
            eusup    ij      j      j     ij          ij


                      j   0    j xij   ij z
                    0  *  ij1GSPEND jj  u 0
                               0    1          2
                                                   j

                    lr   0   1GSPEND j  u1j
                     j

            eusupij   0  1GSPEND j  u 0   0 lftrt ij
                                            j

                           1lftrt ij  GSPEND j  lftrt ij u1j  ...   ij
       gathering terms :
              eusupij   0  ...  1GSPEND j   0 lftrt ij
                           1lftrt ij  GSPEND j   u 0  lftrt ij u1j   ij 
                                                        j

            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*:
   βLS  XX  Xy  XX  XXβ  ε 0  ε 1 x  ε 2 z 
   ˆ             1              1


         XX  XXβ  XX  Xε 0  ε 1 x  ε 2 z   β  XX  Xε*
                 1                1                                      1


 So, OLS coeff. est’s unbiased & consistent (iff…):
E βLS  E β  XX  Xε*   E β  XX  Xε 0  ε 1 x  ε 2 z 
   ˆ                   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.
                    1                                          1



        Note: only works for models with additively separable
         stochastic component; not nec’ly others (e.g.,
Properties of OLS under HLM Conditions
But, OLS s.e.’s will be wrong; not s2(X'X)-1, but:
     
  ˆ  V β   XX 1 Xε* 
V β LS
                           
           V β   V  XX 1 Xε*   2C β,  XX 1 Xε* 
                                                              
           0 V    XX 1 Xε*   0
                                  
            XX  XV  ε  X  XX 
                      1           *             1



            XX 
                      1
                              V  ε 0  ε 1 x  ε 2 z   X  XX 1
                           X                           
            XX  X 
                      1
                        V  ε 0   V  ε 1 x   V  ε 2 z   X  XX 1
                                                               
          (note : the covariance terms are assumed zero)
               Sandwich Estimators
          V β  XX X ε   V ε x  V ε z XXX
             ˆ                  1          0   1       2                   1
                LS        V

   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,
       V βRE
    resp’ly)  2 I  xx  zz
        ˆˆ
                                  so track [  2 rel xx'i X i' zz' 
                                           e ] 1  ei2 X &
                                                       n
     
       heterosked.2:                               n i 1
          
         ˆˆ
        V β HM   0 I  1 xx   2 zz
                2               2
                                                                    J
                                                                          nj          nj
                                                                                                    
                                                            [  ]   ( eij X ij ) ( eij X ij ) 
                                                                    j 1  i 1       i 1          
              From the CLRM to HLM
                y  β0  0 x   0 z  1   1  xz   ε 0  ε 1 x  ε 2 z 

                                                                
              V βLS  XX X V ε0   V ε 1 x  V ε 2 z  XXX
                ˆ          1                                         1




    appropriate “, robust” & “, cluster” can work
       I.e., asymptotically s.e.’s right…BUT need large nj &
        N-k
                                             é (n - 1)ùé N - 1) ( small:
             Specifically, from small-sample corrections, seems needN - k )ù
                                               n            (
                                                       ê
                                                       ë   j   j       úê
                                                                       ûë          ú
                                                                                   û
       I.e., coefficients still inefficient.
            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
    Elaborations, Complications, & Extensions:
    Interax in QualDep (Inherently Interactive) Models
   Probit/Logit Models w/ Interactions
                                                                                             
                                                                                      exp( x¢ )
                                
                                                                                                                 - 1

        Probit: p(y = 1) = F (x¢ )                   – Logit:     p(y = 1) =
                                                                                    1 + exp( x¢ ) ê
                                                                                               
                                                                                                 = é + exp(- x¢ )ù
                                                                                                   1
                                                                                                   ë          ú û

   Marginal Effects: (nonlinear, so must specify @ what
    x)
        Start w/ x¶ purexlin-add, ׶ x ¢ = f inherently inter. b/c S-
                   ' p = ¶ F ( ¢) = f x ¢ model x ¢ ×
                                       ( ) ¶x     ( ) k
         shaped: ¶ x k      ¶ xk              k

             Probit: p
                     ¶      ¶ { x  [1+ e x  ]- 1 }
                               e
                                   ¢        ¢

                                                      = 1+ e x¢ ×k -
                                                             ex            ex 
                                                                                        × x ¢ ×x
                                                                ¢               ¢
                          =           ¶x                                 (1+ e x  )2
                                                                                ¢        e
                     ¶x k
             Logit:
                            ée x¢ (1+ e x¢ ) - (e x¢ )2 ù× =
                          = ê (1+ e x¢ )2                                  ex 
                                                                               ¢
                                                                                        ×k
                                                     (1+ e x  )2 ú      (1+ e x  )2
                                                            ¢                   ¢
                            ë                                     û k
                                   ex 
                                             ×1+ 1x¢ ×k = p ×(1 - p )×k
                                      ¢
                              =   1+ e x 
                                        ¢
                                                 e




        If x' =…+xx+zz+xzxz… same except
         dx'/dx=xx+xzz)×( +  z )propensity, i.e.,1movement  z )
                  ¶p       ;
                     = f (x¢ underlying      ¶p
                                                = p ×( - p )×(x + xz
                  ¶x            x    xz      ¶x
    Elaborations, Complications, & Extensions:
    Interax in Nonlin/Qual (Inherently Interax) Models
   Standard Errors?
                                                        ˆ
                                            AsymVar. f (β)
                                                .              
                                                                  
                                                            ˆ   V  β   f  β  
                                                         f β
                   Delta Method:             β                    ˆ          ˆ
                                                                              β
                                                                                     
        Probit Marginal-Effect s.e.:
                                        ¢
                          é        ˆ ) }ù                                      é
                          ê¶ { (x ¢
                          ê
                              f        ú
                                        ú
                                         ¶ x¢ˆ
                                             
                                          ¶ x1                                 ê¶ f (x ¢ˆ ) ¶ x1
                                                                                     {   ¶ x ¢
                                                                                                ˆ
                                                                                                    }ù
                                                                                                     ú
                             ê       ˆ
                                   ¶ 1          ú é V (1 )
                                                      ˆ ˆ          ˆ ˆ ˆ ê
                                                                 L C (1, k )ùê       ˆ
                                                                                     ¶ 1
                                                                                                     ú
                                                                                                     ú
                             ê                   úêê                          úê                     ú
                             ê                   úê M                         ú
                                     M                           O     M úê            M             ú
                             ê                   úê                            ê                     ú
                             ê¶ f x ¢ˆ ¶ x ¢    ú êˆ ( ,  )
                                                   C ˆ1 ˆk           ˆ ˆ ú
                                                                    V (k ) úê¶ f (x ¢ˆ ) ¶¶xx¢     ú
                             ê {  ( ) ¶ xkˆ     }
                                                 úë              L
                                                                              ûê     {  
                                                                                                ˆ
                                                                                                    }ú
                             ê                   ú                             ê               k
                                                                                                     ú
                             ê
                             ë     ¶ k          ú
                                                 û                             ê
                                                                               ë     ¶ k            ú
                                                                                                     û

         Logit: same, except(1 - p) ¶¶xx¢ replaces¢ˆ ) ¶¶xx¢
                                                     ˆ                                   ˆ
                           ˆ
                            p    ˆ              f (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…
     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):
            Inefficient at best, biased more usually, and, anyway, context-
             conditionality is our interest!
       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”:
            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
                Nonlinear Least-Squares
                                    y  f ( X, )+  wit h  ~ g( )
   Estimate NLS: E  y   f (X, ), so y= f (X, )  ˆ
                                                 ˆ
                                                                   
                                     Min ˆ ˆ  Min [y  f ( X, )][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

                                                              
                                                                   1
                                                  ˆ
So, if, e.g., f ( X, )  X, t hen: Xy  XX  LS  XX             Xy, and if
                                                             
V( )     2I, t hen V(ˆ )LS 
                          ε
                                    n k
                                        1 
                                                     
                                               y  f X, LS   y  f X, LS  (also, as always).
                                                        ˆ
                                                                      ˆ   
                                                                             
                                            ˆ
T hat is, int uit ively, writ ing  f ( X, LS ) as simply , we hav e:
                                   ˆ
    ˆ
      ( ) 1 y
        LS

       ˆ  
     V LS
            LS
                 V ( ) 1 y   () 1 V( y)() 1,
                                  
                     ˆ        ˆ
     which if f ( X, LS )  XLS meaning   X, &, if =  2I, gives t he famiar
ˆ                       ˆ
LS  ( XX) 1 Xy & V(LS )LS   2( XX) 1, as always.

        NLS is BLUE under same conditions OLS, w/  for
         X.
        Interpreting NLS (already know how): Effects =
         Generalized Nonlinear Least-
                y  f (Squares V ( )      I
                       X, )+  wit h                       2       2
   GNLS:
                    ˆ
                 GNLS  ( 1) 1  1y
                   ˆ
               V (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             ˆ        ˆ
                 ( 1)1  1y
                FGNLS
             ˆ             ˆ           ˆ         ˆ       ˆ
         V (FGNLS )  ( 1) 1  1V ( y) 1( 1) 1
                             ˆ
                        ( 1)1
    Nonlinear Least-Squares & EMTI
   EITM: Empirical Implications of Theoretical Models
       Vision: Theory  more, sharper predictions  better tests,
        which therefore inform theory more, which…
   TMEI: Theory-specified Models for Empirical
    Inference
       Vision: Theory structures empirical models & relations b/w
        obs  specification & (causal) i.d. of empirical models
   TIEM: Theoretical Implications of Empirical Measures
       Vision: Emp. regularities, findings, measures inform theory
        dev’p.
   EMTI: Empirical Models of Theoretical Intuitions
       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 &
                   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.
              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 (x2 )   5 (x5 )   6 (x6 )   a
                                      E   a  P  (1  E )  C   3 (x3 )  P  (1  E )  (1  C )   4 (x4 )
                                     
                                    
                                     (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,
            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:
            Privileges S.Q., & so retards policy adjustment, reduces
             change.
       Collective-Action/Common-Pool Theories (WSJ ‘81):
            Externalities & so overexploit/underinvest public goods.
       Bargaining & Delegation Theories rather stress:
            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 &
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…
             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
       (…log-linearly?)
   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
                                                                           Minimum-Winning-Coalition Decision-Making                    Universalistic Decision-Making
                     Ratio of Benefits to Costs of Projects Passed




                                                                     0.9

                                                                     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    65   70   75   80   85   90   95

                                                                                                          Number of Constituencies
        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:
       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…)
   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:
       Magnitude incentive electioneer fades w/ n (see, e.g., Goodhart)
       Control over electioneering diminishes w/ n.
    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 size-weighted (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,
               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:
            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-combo emp. models 
        compromise
            If Nash Bargain, e.g., (n.b. NB=coop. game-thry but equiv. sev.
             reasonable ext-form non-coop barg. games: Rubinstein ‘82), 
               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:
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
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  zit 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
                                                            Coeff.          Std. Err.    t-Stat.   Pr(T> | t| )
         Lagged                           D t-1              1.212            0.060      20.112      0.000
       Dependent                          D t-2             -0.153            0.085      -1.792      0.074
       Variables                          D t-3             -0.121            0.045      -2.677      0.008
    ρn (veto-actor effect: fractionalization)                0.007            0.006       1.089      0.277
      ρar (veto-actor effect: polarization)                 -0.000            0.006      -0.013      0.990
                                           ΔY               -0.336            0.111      -3.033      0.003
    Macroeconomic
                                          ΔU                 0.992            0.308       3.219      0.001
      Conditions
                                           ΔP               -0.188            0.063      -2.965      0.003
     βcg (partisan-compromise bargaining)                   -0.037            0.037      -0.988      0.323
                                        x1 (open)           15.891            5.279       3.010      0.003
                                        x2 (T oT )           0.388            1.744       0.222      0.824
        Controls                     x3 (open∙ T oT )      -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
Pre- and Post-Electoral                     Et               0.687            0.568       1.210      0.227
       Indicators                          Et-1              1.490            0.645       2.310      0.021
  γen (common-pool effect: fractionalization)               -0.547            0.182      -3.001      0.003
    γs d (common-pool effect: polarization)                  0.573            0.486       1.179      0.239
                                        z1 (CoG)             0.051            0.131       0.390      0.697
      Constituent                      z2 (ENoP)             0.281            0.446       0.629      0.530
         T erms
                                      z3 (SDwiG)             0.542            0.437       1.242      0.215
        from the
      Interactions                      z4 (NoP)             0.181            0.277       0.654      0.514
                                      z5 (ARwiG)            -0.312            0.259      -1.205      0.228
                                                    Sum m ary Statis tics
             N (Deg. Free)                              735 (691)                          s e2       2.525
                 R2 ( R 2 )                           0.991 (0.990)                     DW-Stat.      2.101
    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.   t-Stat. Pr(T>| t| )
           Lagged                             D t-1             1.207         0.060    20.290    0.000
          Dependent                           D t-2            -0.158         0.085    -1.851    0.065
          Variables                           D t-3            -0.117         0.045    -2.577    0.010
        ρn (veto-actor effect: fractionalization)               0.011         0.005    2.369     0.018
          ρar (veto-actor effect: polarization)                -0.002         0.004    -0.437    0.662
                                               ΔY              -0.375         0.087    -4.332    0.000
        Macroeconomic
                                              ΔU                1.095         0.286     3.829    0.000
          Conditions
                                               ΔP              -0.207         0.053    -3.889    0.000
         βcg (partisan-compromise bargaining)                  -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
           Controls                      x3 (open∙ ToT)       -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
    Pre- and Post-Electoral                     Et              0.475         0.420     1.133    0.258
          Indicators                           Et-1             1.146         0.562     2.040    0.042
      γen (common-pool effect: fractionalization)              -0.570         0.209    -2.727    0.007
        γsd (common-pool effect: polarization)                  0.881         0.586    1.503     0.133
                                                       Sum m ary Statistics
                 N (Deg. Free)                             735 (696)                     s e2    2.522
                    R2 ( R 2 )                           0.991 (0.990)                DW-Stat.   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.           Mean Econ.                Mean                Mean Econ.           Mean Econ.
                                   Performance          Performance             Economic              Performance          Performance
                                    -2 std. dev.         -1 std. dev.          Performance            +1 std. dev.         +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
                                                                                                                                                Fiscal-Cycle
             CoG                 E(D|Econ) f           E(D|Econ)               E(D|Econ)             E(D|Econ)            E(D|Econ)
                                                                                                                                                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
                            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  xt0b 0   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:
Preliminary Results of Fuller Model                   Coeff.        Std. Err.    t-Stat.   Pr(T>| t| )
                                         D(t-1)        1.197          0.059      20.144      0.000
              Temporal
                                         D(t-2)       -0.139          0.085      -1.629      0.104
              Dynamics
                                         D(t-3)       -0.121          0.045      -2.698      0.007
          Veto-Actor Effect
                                          NoP          0.008         0.004       1.883       0.060
 On Outcom e-Adjustm ent Rate
    x 0 : Variables with “Direct”        Open          16.624         3.758       4.423      0.000
          Effect on Outcome            Open*ToT       -11.190         3.135      -3.569      0.000
                                         Ele(t)         0.315         0.363       0.867      0.386
    x 1 : Variables with Effects via
                                        Ele(t-1)        0.873         0.399       2.186      0.029
     Non-Partisan-Differentiated
            Policy Response               OY            2.833         1.295       2.187      0.029
                                       DXRIG3          -0.073         0.072      -1.009      0.314
     Com m on-Pool Effect on
                                       ln(ENoP)       -0.277         0.071       -3.903      0.000
           Policy Response
    x 2 : Variables with Effects via    Growth         -0.238         0.084      -2.815      0.005
         Partisan-Differentiated         d(UE)          0.749         0.228       3.289      0.001
            Policy Response             Inflation      -0.137         0.047      -2.947      0.003
Bargaining-Com prom ise Effects
                                          CoG         -0.049         0.026       -1.893      0.059
 on Partisan Policy-Responses
       Veto-Actor Effect
                                          NoP          0.215         0.121       1.773       0.077
  On Policy-Adjustm ent Rate
   Common-Pool Effect on Debt Level     ln(ENoP)       1.128          0.486       2.320      0.021
                                       Sum m ary Statistics
                   N (Deg. Free)                      735 (697)                    s e2      2.510
                       R2 ( R 2 )                   0.991 (0.990)               DW-Stat.     2.090

								
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