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Spatial Econometric Models of Interdependence Theory & Substance; Empirical Specification, Estimation, Evaluation; Substantive Interpretation & Presentation. Talk prepared for Blalock Lecture on 7 August 2008 at the ICPSR Summer School based on the joint work of

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slide1

Spatial Econometric Modelsof InterdependenceTheory & Substance; Empirical Specification, Estimation, Evaluation; Substantive Interpretation & Presentation

Talk prepared for Blalock Lecture on

7 August 2008 at the ICPSR Summer School

based on the joint work of

Robert J. Franzese, Jr., The University of Michigan

Jude C. Hays, The University of Illinois

overview
Overview
  • Motivation: Integration & Domestic Policy-Autonomy
    • Does economic integration constrain govts from redistributing income, risk, & opportunity through tax & spending policies?
    • In answering this & related questions, scholars have overlooked spatial interdependence of domestic policies as important evidence.
    • Economic integration generates externalities across political jurisdictions, which implies strategic policy interdependence, so policy of one govt will be influenced by policies of its neighbors.
  • Interdependence Substance, Theory, & Empirics: Use contexts econ integration (& related) to explore & explain:
    • Substance:i’s actions depend on j’s. Examples.
    • Theory:
      • General: externalitiesstrategic policy complements/substitutesrace-to-bottom/top/elseearly/late-mover advantagesstrategic delay/rush-for-1st
      • Specific: a model of inter-jurisdictional tax-competition (P&T ch. 12)
    • Empirics: “Galton’s Problem”; Estimation, Inference, Interpretation, & Presentation
a motivating context globalization domestic policy autonomy
A Motivating Context:Globalization & Domestic-Policy Autonomy
  • Standard Argument:
    • ↑ capital mobility & trade integration sharpen capital’s threat vs. domestic govts to flee excessive/inefficient tax & public policy; forces welfare-state retrench & tax shift from more-mob. cap. (esp. finance) to less-mob. lab. (esp. skilled-man.)
  • Recent counter-arguments & findings:
    • Some empirical Q whether constrained or ° constraint from trade/capital integ.
    • Counter-arguments (e.g.):
      • Rodrik (Cameron): Demand (contra supply) SocPol may ↑ w/ integ  indeterminate
      • Garrett ’98/Boix ’98: Left/active govt more/as efficient  capital not flee
      • Hall-Soskice ‘01/Franzese-Mosher ’02: comparative institutional advantage  trade-integ foster divergence; (liquid) cap-integ may foster race to bottom (not nec’ly) zero
      • Swank ’02 (& many others): political & economic barriers &/or advantages  considerable maneuvering room
  • Standard & all counter arguments  spatial interdependence b/c whatever pressures may arise from globalization depend on what neighbors, competitors, partners, substitutes, & complements do
    • Accordingly, appropriate model places others’ policies on right-hand side
    • Basinger-Hallerberg ’04 maybe 1st in C&IPE to notice & incorporate explicitly
  • Interdependence (def):yi=f(yj≠i); note: not merely that yi & yj≠i corr
the broad range of spatial interdependence
The Broad Range of Spatial Interdependence
  • (Simmons et al.’s 06) Mechanisms:
    • Competition
    • Coercion
    • Learning
    • Emulation
    • [Migration/Contagion (F&H Add)]
  • Theoretical Contexts (ubiquitous):
    • ANY Strategic Decision-making: sisj
    • Externalities & Spillovers
    • Learning/Emulation, Demonstration
    • Networks/Epistemic Communities
    • Literal Diffusion, Contagion, Migration
  • Substantive Contexts (ubiquitous):
    • Security Policy (e.g., alliances, wars)
    • Environmental (e.g., air-pollution reg)
    • Regulatory (e.g., telecomm stds)
    • Legis reps’ votes depend on others’
    • Elects., cand. qualities or strategies
    • p(∙)&outs coups (Li&Thompson 75), riots (Govea&West 81), revolts (Brinks&Copp 06)
    • Contextual effects in micro-behavior:
      • Braybeck&Huckfeldt 02ab, Cho 03, Huckfeldt et al. 05, Cho&Gimpel 07, Cho&Rudolph 07, Lin et al 06
  • Policy, instit’s, regimes diffusion:
    • Policy: Schneider&Ingram‘88, Rose ‘93, Meseguer ‘04,‘05, Gilardi ‘05
    • Institutional or regime: Implicit/Informal: Dahl’s Polyarchy, Starr’s Democratic Dominoes, Huntington’s 3rd Wave. Explicit/Formal: O’Loughlin et al. ‘98, Brinks & Coppedge ‘06, Gleditsch & Ward ‘06, ’07
  • Int’l diffusion of liberalization:
    • Simmons&Elkins 04, 06a, 06b, Eising 02, Brune et al. 04, Brooks 05…
  • Globalization & interdependence:
    • Genschel 02, Basinger&Hallerberg 04, Knill 05, Jahn 06, Swank 06, F&H 06,07, Kayser 07
  • Tobler’s Law: ‘‘I invoke the first law of geography: everything is related to everything else, but near things are more related than distant things’’ (1970).
    • Plus: “Space More Than Geography” (Beck, Gleditsch, & Beardsley 2006)
substantive theoretical ubiquity centrality 1
Substantive & Theoretical Ubiquity & Centrality (1)
  • US State Policy-innovation diffusion: deep roots & much contemporary interest, & sustained attention between:
    • Crain 1966; Walker 1969, 1973; Gray 1973; Knoke 1982; Caldiera 1985; Lutz 1987; Berry & Berry 1990; Case et al. 1993; Berry 1994; Rogers 1995; Mintrom 1997ab; Brueckner 1998; Mintrom & Vergari 1998; Mossberger 1999; Berry & Berry 1999; Godwin & Schroedel 2000; Balla 2001; Mooney 2001; Wejnert 2002; Coughlin et al. 2003; Bailey & Rom 2004; Boehmke & Witmer 2004; Daley & Garand 2004; Grossback et al. 2004; Mencken 2004; Berry & Baybeck 2005; Garrett et al. 2005; Costa-Font & Ons-Novell 2006; Karch 2006; Rincke 2006; Shipan & Volden 2006; Volden 2006; Werck et al. 2006; Woods 2006; Volden et al. 2007.
  • Similar policy-learning mechanisms underlie some comparative studies of policy diffusion:
    • Schneider & Ingram 1988; Rose 1993; Bennett 1997; Dolowitz & Marsh 2000; True & Mintrom 2001; Tews et al. 2003; Jensen 2004; Meseguer 2004, 2005; Brooks 2005, 2007; Gilardi 2005; Gilardi et al. 2005; Murillo & Schrank 2005; Weyland 2005; Braun & Gilardi 2006; Linos 2006; Parys 2006; Ermini & Santolini 2007; Moscone et al. 2007.
  • Institutional or regime diffusion likewise long-standing & recently much reinvigorated:
    • Dahl’s 1971 Polyarchy (1 of 8 causes dem listed); center-stage Starr’s 1991 “Democratic Dominoes”; Huntington’s 1991 Third Wave; Beissinger 2007; Bunce & Wolchik 2006, 2007; et al. in E. Eur. Transitions; Hagopian & Mainwaring 2005 et al. in LA; O’Loughlin et al. 1998, Brinks & Coppedge 2006, Gleditsch & Ward 2006, 2007 estimated empirically extent, paths, &/or patterns dem diffuse. Kelejian et al. 2007 give institutional diffusion general theoretical & empirical treatment.
  • C&IPE, e.g. globalization≈interdependence:
    • Diffusion of “Liberalization” & Related: Simmons & Elkins 2004, Simmons et al. 2006, Eising 2002; Brune et al. 2004; Brooks 2005, 2007; Jordana & Levi-Faur 2005; Way 2005; Lazer 2006; Prakash & Potoski 2006; Brune & Guisinger 2007; and many others.
    • Glob/Interdep/TaxComp & Dom Policy Auton: Genschel 2002; Guler et al. 2002; Franzese & Hays 2003, 2004b, 2005a, 2007abc, 2008c; Badinger et al. 2004; Basinger & Hallerberg 2004; Heichel et al. 2005; Henisz et al. 2005; Holzinger & Knill 2005; Knill 2005; Polillo & Guillén 2005; Elkins et al. 2006; Jahn 2006; Lee & Strang 2006; Manger 2006; Swank 2006; Baturo & Grey 2007; Cao 2007; Cao et al. 2007; Coughlin et al. 2007; Garretsen & Peeters 2007; Mosley & Uno 2007; Mukherjee & Singer 2007.
substantive theoretical ubiquity centrality 2
Substantive & Theoretical Ubiquity & Centrality (2)
  • Representatives’ votes (Lacombe & Shaughnessy 2005), citizens’ votes (Huckfeldt & Sprague 1991; O’Laughlin et al. 1994; Pattie & Johnston 2000; Beck et al. 2003; Calvo & Escolar 2003; Kim et al. 2003; Schofield et al. 2003; Lacombe & Shaughnessy 2007), election outcomes (Shin & Agnew 2002, 2007; Hiskey & Canache 2005; Wing & Walker 2006; Kayser 2007), candidate qualities, contributions, or strategies (Goldenberg et al. 1986; Mizruchi 1989; Krasno et al. 1994; Cho 2003; Gimpel et al. 2006)
  • Probabilities & outcomes of coups (Li & Thompson 1975), riots (Govea & West 1981), civil wars (Murdoch & Sandler 2004, Buhaug & Rød 2006) &/or revolutions (Brinks & Coppedge 2006)
  • IR: interdep≈definition of subject:
    • States’ entry into wars, alliances, treaties (Murdoch et al. 2003), or organizations.
    • Empirical attention to inherent spat-dep IR greatest in: Shin & Ward 1999; Gleditsch & Ward 2000; Gleditsch 2002; Ward & Gleditsch 2002; Hoff & Ward 2004; Gartzke & Gleditsch 2006; Salehyan & Gleditsch 2006; Gleditsch 2007, and, in different way, Signorino 1999, 2002, 2003; Signorino & Yilmaz 2003; Signorino & Tarar 2006
  • In micro-behavioral work, too, long-standing & surging interest “contextual” or “neighborhood” effects:
    • Huckfeldt & Sprague 1993 review, some of which stress interdep: Straits 1990; O’Loughlin et al. 1994; Knack & Kropf 1998; Liu et al. 1998; Braybeck & Huckfeldt 2002ab; Beck et al. 2002; McClurg 2003; Huckfeldt et al. 2005; Cho & Gimpel 2007; Cho & Rudolph 2007. Sampson et al. 2002 and Dietz 2002 review the parallel large literature on neighborhood effects in sociology
  • At & beyond other disciplinary borders, subjects include:
    • Social-movements: McAdam & Rucht 1993; Conell & Cohn 1995; Giugni 1998; Strang & Soule 1998; Biggs 2003; Browning et al. 2004; Andrews & Biggs 2006; Holmes 2006; Swaroop & Morenoff 2006.
    • Microeconomic preferences: Akerloff 1997; Postlewaite 1998; Glaeser & Scheinkman 2000; Manski 2000; Brock & Durlauf 2001; Durlauf 2001; Glaeser et al. 2003; Yang & Allenby 2003; Sobel 2005; Ioannides 2006; Soetevent 2006
    • Macroeconomic performance: Fingleton 2003; Novo 2003; Kosfeld & Lauridsen 2004; Maza & Villaverde 2004; Kelejian et al. 2006; Mencken et al. 2006
    • Technology, marketing, and other firm strategies: Abramson & Rosenkopf 1993; Geroski 2000; Strang & Macy 2001; Holloway 2002; Bradlow 2005; Autant-Berard 2006; Mizruchi et al. 2006
    • Violence and crime: Grattet et al. 1998; Myers 2000; Baller et al. 2001; Morenoff et al. 2001; Villareal 2002; Baker & Faulkner 2003; Oberwittler 2004; Bhati 2005; Mears & Bhati 2006; Brathwaite & Li 2008
    • Fertility, birthweight, child development, & child poverty: Tolnay 1995 and Montgomery & Casterline 1996; Morenoff 2003; Sampson et al. 1999; Voss et al. 2006
    • Not to mention public health and epidemiology (contagion!).
  • More exotic topics: ordainment of women (Chaves 1996), right-wing extremism (Rydgren 2005), marriage (Yabiku 2006), national identity (Lin et al. 2006), & faculty (Weinstein 2007).
policy interdependence a general theoretical model brueckner 03
Policy Interdependence:A General Theoretical Model (Brueckner ‘03)
  • i’s utility depends pi & pj b/c interdep (& vv):
  • Accordingly, i’s optimal policy, pi*, depends j’s action, pj:
  • So slope best-response function depends on effect of pj on marginal utility of pi:
  • Therefore: Diminishing returns and…
    • …negative externalities =>strategic complements:
      • Positive slope: positive feedback/same-signed reactions
    • …positive externalities => strategic substitutes:
      • Negative slope: neg. feedback/opposite-signed reactions
policy interdependence general theory substantive implications
Policy Interdependence:General Theory & Substantive Implications
  • Dimin returns & neg externalities Strategic Complements
    •  Race-to-Bottom (RTB) (or -Top). Examples:
      • Tax Competition
      • Labor Regulation
      • Trade Barriers (politically)
    •  Early-mover advantage  “race to go first”
      • E.g.: Exch.Deprec., tech.stds. (& other focal pts. in coord. or battle sexes)
  • Dimin returns & pos externalities Strategic Substitutes
    • Free-Riding Incentives
      • E.g., Alliance Security:
      • E.g., ALMP:
    •  Late-mover advantage  strategic delay & Wars of Attrition
  • DimRet & both +&– externs:
    • Environmental Reg’s (& CHIPs?):
an international tax competition model as a specific substantive example of interdependence
An International Tax-Competition Model as a Specific Substantive Example of Interdependence
  • Stylized Theoretical Model Cap-Tax Comp. (P&T ‘00, ch.12)
    • 2 jurisdicts, dom & for cap-tax, τk & τk* to fund fixed spend. For-invest mobility costs, M.
    • Inds’ lab-cap endow, ei, & choose lab-leisure, l & x, & save-invest, s=k+f to max ω=U(c1)+c2+V(x), over l, c1, & c2, s.t. time-c., 1+ei=l+x, & b.c.’s, 1–ei=c1+k+f+≡c1+s & c2=(1–τk)k+(1–τk*)f–M(f)+(1–τl)l.
    •  equilibrium economic choices of citizens:
    •  indirect utility, W, defined over policy variables, τl, τk, & τk*:
    • Besley-Coate (‘97) citizen-candidate(s) face(s) electorate w/ these prefs.
      • Stages: 1) elects, 2) cit-cand winners set taxes, 3) all private econ decisions made.
      • Ebm win cand has endow eP such that desires implement this Modified Ramsey Rule:
  • Best-Response Functions: τk=T(eP,τk*) & τk*=T*(eP*,τk) for dom & for pm.
    • Slopes: ∂T/∂τk* & ∂T*/∂τk, pos or neg b/c ↑τk* cap-inflow; can use ↑tax-base to ↓τk or to ↑τk (seizing upon ↓elasticity base).
    • Background of this slide plots case both positively sloped; illustrated comparative static is of leftward shift of domestic government.
empirical models of interdependence galton s problem in c ipe
Empirical Models of Interdependence:Galton’s Problem in C&IPE
  • Interdependence  yi=f(yj)
  • Generic (linear) dynamic spatial-lag model of C&IPE:
  • Galton’s Problem: Extremely difficult disting why C(yi,yj) b/c...
    • 1. Correlated domestic/unit-level conditions, d (CPE)
    • 2. Common/corr’d exposure exog-external shocks/conditions, s (open-CPE)
    • 3. Responses to these 2 may be context-conditional, sd (CC-CPE)
      • 40. Correlated stochastic component (Beck-Katz), nuisance C&IPE
    • 4. Interdependence/diffusion/contagion: yi=f(yj,CC-CPE), substance C&IPE
  • Upshot Empirically (Franzese & Hays ‘03,‘04,‘06ab,‘07abcd,‘08ab):
    • ° omit or mis-specify CPE, tend over-estimate IPE (interdependence) & v.v.
    • yiyj => textbook endogeneity/simultaneity problem w/ spatial lag; analogous:
      • ° fail redress endog sufficiently  mis-est (usu. over-est.) ρ (under)mis-est. β
    • Most Important Conclusion:Model It!TM Insofar as omit or rel’ly mis-spec spatial interdep, tend over-est impact domestic & exog-ext factors & v.v.  most crucial, regardless of CPE/IPE emphasis: well-spec model & measure both.
      • S-OLS may suffice. OVB >> simultaneity bias in any of practical examples we’ve considered, & S-OLS did OK our MCs provided interdep remained modest (|ρ|<.3±).
slide12
The Terms of Galton’s Problem:Omitted-Variable vs. Simultaneity Biases inSpatial- and Spatio-Temporal-Lag Models
  • OVB (rel. mis-spec.) v. simultaneity:
    • OVB (OLS):
    • SimB (S-OLS):
    • In S-T, little more complicated, but:

With all positive S-T dep, ρ space-dep over-est’d & time-dep & β under-est’d

estimating spatial spatio temporal lag models
Estimating Spatial/Spatio-Temporal-Lag Models
  • Inconsistent Estimators:
    • Omit spatial-dep (e.g., OLS): bad idea if ρ non-negligible
    • Ignore simultaneity (e.g., S-OLS): could be OK (in MSE) if ρ not too large & sample-dims benevolent
  • Simplest Option, if Available:
    • Time-Lagged Spatial-Lag OLS easy & unbiased iff…
      • No contemporaneous (i.e., w/in obs period) interdependence.
      • Model of temporal dynamics sufficiently accurate (see Achen)
      • 1st obs pre-determined; if not, spatial-Hurwicz bias (order 1/T)
  • Consistent Estimators
    • S-IV/2SLS/GMM: Use WX to instrument Wy, etc.
    • S-ML: Specify system for Max-Likelihood estimation
estimating spatial spatio temporal dynamic models by s ml
Estimating Spatial- & Spatio-Temporal-Dynamic Models by S-ML
  • S-ML for Spatial-Lag Model:
    • Std, but ε → y by |A| not 1  computational issues, plus
  • Conditional S-ML S-T (ie., given 1stobs, Nx1 form); (unconditional is messy but exists; won’t show):
  • Stationarity (if row-stdzd, & ρ,>0): ρ+<1
  • Spatial Probit: complicated but doable (show if time)
  • m-STAR & Est’d W; endog W: doable (show if…)
applications
APPLICATIONS
  • Globalization, Tax Competition, & Domestic Policy
    • Replicates: Swank&Steinmo 02, Hays 03, Basinger&Hallerberg 04
  • ALMP: Active-Labor-Market Policy in EU (F&H ‘06)
    • DepVar: LMT spend per unemployed worker
    • Hypoth: Positive spillovers (@ borders) effective member-state ALMP  free-riding & underinvest. Appreciable?
    • IndVars: rGDPpc, UE, UDen, Deindustrialization (Iversen-Soskice), Trade, FDI, Pop65, LCab, CDemCab, LLibVote, GCons
  • MIDs & Trade: Beck, Gleditsch, & Beardsley ’06
    • DepVar: Directed trade data;
    • Hypoth: MIDs affect trade in & beyond dyad
    • IndVars: GDPab, POPab, Distance, tau-b, MutualDem, MID, Bi/MultiPoleSys
  • AFDC & CHIPs in U.S. States (Volden ’06)
    • AFDC: Hypoth—“states as laboratories”≈diffusion by learning
      • DepVar: max monthly AFDC benefit
      • Ind Vars: state’s poverty rate, avg monthly wage in retail, govt ideology (0-100, R-L), º interparty competition (.5-1.0, comp-non), tax effort (rev as % tax capacity), & % AFDC bens paid by fed govt.
    • CHIPs: Hypoth—“states as laboratories”≈diffusion by learning
      • DepVar: 1 if state’s CHIP includes monthly premium; IndVars same.
practical model specification estimation
Practical Model Specification & Estimation
  • Most convenient to work in (Nx1) vector form:
  • WN=an NxN of (time-invariant) spatial wts, wij, & WNIT gives W.
  • E.g., 15x15 binary-contiguity from ALM paper:
    • N.b., row-stdz typ., convenient, but not nec’ly subst’ly neutral
    • Ideally, substance, which not nec’ly  geography, in W.
  • Beware of extant software: critical bug in LeSage’s MatLab code; likelihood in some third-party Stata SAR code seems flat wrong.
interpreting spatial spatio temporal effects
Interpreting Spatial/Spatio-Temporal Effects
  • The Model:
    • Model may look linear, but is not; as in all beyond purely linear-additive, coefficients & effects very different things!
    • Convenient, for interpretation, to write model this way too:
    • Coefficients, βx are just pre-spatial, pre-temporal—and wholly unobservable!—impulse from some x to y.
  • Spatio-Temporal Effects:
    • Post-spatial, pre-temporal “instantaneous effect” of x:
    • Spatio-Temporal Response Paths:
    • LR Multiplier/LR-SS:
presenting spatial spatio temporal effects
Presenting Spatial/Spatio-Temporal Effects
  • Standard Errors (Confidence Intervals & Hypothesis Tests) of Effects:
    • Delta Method:
    • …or Simulate!
  • Upshot: Cannot see substance clearly from only the estimated coefficients & their standard errors
  • Effective Presentational Options:
    • SR/LR-Response Grids
    • Spatio-Temporal Response-Paths
    • Maps
conclusion
Conclusion
  • Spatial & Spatio-Temporal Interdependence
    • Important & Appreciable Substance (e.g., globalization & int’l cap-tax compete seems quite real & does constrain), not Nuisance.
      • Therefore: Model them. Interpret them.
  • How specify & estimate models?
    • If space-lag is time-lagged, not problem; but if thry & substance says immediate (w/in an observational period), can handle that too:
    • S-OLS not a bad strategy even then, if ρ not too big & smpl-dims right; S-ML, & in some regards IV-based strategies, seem effective
  • Spatio-Temporal Effects not directly read from coefficients: use graphs & maps & grids
  • Information-demands of Galton’s Problem severe 
    • Standard errors of effects tend big. Suspect delta-method lin-approx. maybe part problem; plan explore performance bootstrap.
    • Max effort & care theory, measure, specification, to both C&IPE
spatial qualdep the econometric problem 1
Spatial QualDep: The Econometric Problem (1)
  • Spatial Qualitative/Categorical/Lmtd-Dep-Var Models in the Lit:
    • Spatial probit: McMillen 1992,1995; Bolduc et. al. 1997; Pinkse & Slade 1998; LeSage 1999, 2000; Beron et al. 2003; Beron & Vijverberg 2004
      • Spatial logit: Dubin 1997; Lin 2003; Autant-Bernard 2006
      • Spatial sample-selection (i.e., s-Tobit/Heckit): McMillen 1995, Smith & LeSage 2004, Flores-Lagunes & Schnier 2006
      • Spatial multinomial-probit: McMillen 1995, Bolduc et al. 1997
      • Spatial discrete-duration: Phaneuf & Palmquist 2003
      • Survival w/ spatial frailty: Banerjee et al. 2004, Darmofal 2007
      • Spatial count: Bhati 2005, including ZIP: Rathbun & Fei 2006
  • The Challenge:
    • Not n indep., unidimensional CDF std normals, so (log-)likelihood=product (sum) thereof, but 1 n-dimensional CDF of non-std (heterosked.) normals
  • Spatial Latent-Variable Models: Estimation Strategies
    • McMillen 1992: EM algorithm, rendered spatial probit estimable, but no std-errs & arb. parameterization of induced heteroscedasticity.
    • McMillen 1995, Bolduc et. al. 1997: simulated-likelihood strategies to estimate spatial-MNP
    • Beron et al. ‘03, Beron & Vijverberg ‘04: recursive-importance-sampling (RIS) estimator
    • LeSage 1999, 2000: Bayesian strategy of Markov-Chain-Monte-Carlo (MCMC) by Metropolis-Hastings-within-Gibbs sampling.
    • Fleming 2004: simpler, if approximate, strategies allowing interdep. in (non)linear probability models, estimable by NLS, GLM, or GLMM
    • Pinkse & Slade’s 1998: two-step GMM estimator (for spatial-error probit).
slide48

The Econometric Problem (2)

  • Structural Model:
  • Reduced Form:
  • Measurement Equation:
  • Probability:
    • Or:
  • For Spatial-Error-Probit:
slide49

The Econometric Problem (3)

  • Comments:
    • Notice that, when we come to interpret & , we face the same MVN integration
      • We haven’t seen such substantive interpretation yet attempted fully in the literature, but we suggest an easier way to do it.
    • If can order dependence pattern & ensure only antecedent y* appear on RHS, then std probit ML w/ a spatial-lag works
      • We think usu. indefensible subst’ly/thry’ly, but cf. Swank on capital-tax competition, e.g., where argues US exclusively leads & omits US.
    • Having y, not y*, on RHS may seem subst’ly or thry’ly desirable in some cases, but gen’ly not logically possible:
      • Problem would be that outcome, yi, would indirectly (via spatial feedback) determine yi*, but then yi* would directly determine yi. The stochastic difference b/w them will thus  a logical inconsistency.
    • Notice similar MVN issue w/ time lags; suggests similar strategies (but simpler b/c ordered) may allow model temp dynamics directly rather than nuisance (e.g., BKT splines)
slide50

The Estimators: Bayesian Gibbs-MH Sampler (1)

  • Basic Idea (See Gill’s intro Bayesian textbook, e.g.):
    • Monte Carlo (MC): Given likelihood/posterior, can sample to estimate any quantity of interest, including density, e.g.
    • Markov Chain (MC)MC:
      • Each draw depends on previous, so need only conditional like./post.
      • Some theorems indicate, under fairly gen’l conditions, distribution parameter draws converges to distribution under true like./post.
    • Gibbs Sampler: simplest of MCMC family:
      • Express each parameter like./post. conditional on others.
      • Cycle to draw each conditional on others’ starts or previous draw
      • After some sufficient “burn-in”, all subsequent param-vector draws follow true multivariate likelihood/posterior.
    • Metropolis-Hastings: useful when cond’l param-dist non-std
      • Draws from a seed or jump distribution are accepted or rejected as the next sampled parameters, depending on how they compare to a suitably transformed expression of the target distribution
slide51

The Estimators: Bayesian Gibbs-MH Sampler (2)

  • Bayesian Gibbs-MH (MCMC) Sampler for Spatial Probit:
    • Likelihood:
    • Diffuse Priors => Joint Posterior:
    • Conditional Priors:
slide52

The Estimators: Freq. Recursive Import. Smplr (RIS) (1)

  • Basic Idea:
    • To approx. n-dim. cumulative std-norm.,
    • Re-express as a mean by mult & divide by std dist. truncated to support of desired integral, (=the Importance dist.):
    • This gives probability, p, sought as:
  • We want:
    • So, Imp. dist. is n-dim. MVN truncated at v. (uh-oh! but…)
    • V-Cov u being pos-def => Cholesky decomposition exists s.t.:
slide53

The Estimators: Freq. Recursive Import. Smplr (RIS) (2)

  • So we want to calc. this set of indep. cum. std.norms:
  • Can do so recursively, beginning w/ last obs.
    • First, calculate upper bound for truncated-normal dist. of nth
    • Draw from this dist & use it to calc upper bound for (n-1)th…
    • Since indep., probability of sample observed (0,1) is product of n univariate cumulative std. norms at these bounds, (!)
    • Repeat R times & avg => RIS est. of the log-likelihood to max:
slide54

Evaluating the Estimators (One Quick MC)

  • DGF: (n.b., same W, diff. coeffs. For x & y)
  • Conditions:
    • Row-stdzd contig. wts U.S. 48; =0.5, =1.0, n={48,144},θ={0.0, 0.5}
  • You can’t see this, but:
    • Rel’ly poor bias perf. BG
    • In fact, std ML w/ Wy
    • seems dominate, but this
    • b/c 2 biases, meas./spec.
    • err & simult. Simult incr
    • in , meas-err decr or flat
    • in n, so over- to under-est.
    • (Checked & it’s true) B&V ‘04
    • do MC like #2 for RIS & find
    • =-18%, =+10%, so better.
calculating presenting effects 1
Calculating & Presenting Effects (1)
  • If confine discussion to y*, then as prev. F&H:
  • And s.e.’s/c.i.’s by delta method as:
calculating presenting effects 2
Calculating & Presenting Effects (2)
  • But we (should) want to discuss:
  • Note: given probit, must know xi; given spatial interdependence, must know X (!).
  • Given interdep, calc these will req. MVN cdf!
  • Or… better idea?
an example application us state chip premia
An Example Application: US State CHIP Premia

Notes: 1. Informative U(0,1) prior on  helps. We’ve qualms.

2. Difference in Bayesian vs. frequentist significance also.

3. Note measurement/specification-error seem to have dom’d here for ML.

in lieu of conclusions
In lieu of conclusions…
  • S-QualDep (latent-y*) models hard, doable
  • We have a lot of work yet to do:
    • Illustrate calculation of effects & s.e.’s;
    • Explore estimator properties systematically;
    • Compare non-spatial probit & spatial-lag ML-probit & approximate specifications
  • Next Crucial Extensions:
    • Extend to other QualDep models…
    • Estimated-W models… (see next for a start)
    • System-of-Equations in Space…
the m star model as an approach to modeled dynamic endogenous interdependence
The m-STAR Model as an Approach to Modeled, Dynamic, Endogenous Interdependence
  • Spatial Econometrics and (Political) Economy & Network Analysis and (Political) Sociology
  • Co-Evolution Models in Network Analysis
    • (Node) Behaviors/Attributes & Network (Edges)
  • Spatial-Statistical Approaches to Est’d-W
  • A Simple Spatial-Econometric Proposal:
    • Estimated W ≈ Multiple W (m-STAR)
    • Endogenize W means W(y)=>S-IV in m-STAR
spatial econometrics

Network Analysis

Spatial Econometrics
  • Sociologists & Political Sociologists
  • Core Questions:
    • How do nets form?
    • What expl. net struct.?
    • How ego’s position in net & net struct affect?
  • Core Tools
    • Net stats (measures), graphics, ERGMs, …
  • Economists & Political Economists
  • Core Question:
    • How alters’ actions affect ego’s via network & v.v.?
    • Contagion v. Common Exposure (Galton’s Problem)
  • Core Tools:
    • SAR, STAR, S-QualDep…
    • S-GMM, S-ML
  • INTERDEPENDENCE
    • Definition: yi=f(yj≠i); i’s actions depend on j’s.
    • Seems subset of “Network Effects”, which also:
      • Effects of structure network per se (e.g., # transitive triplets)
      • Effects of position i in network per se. (e.g., betweenness i)
where spatial econometrics needs to go network analysis is or needs to go also
Where Spatial Econometrics Needs to Go (& Network Analysis is or Needs to Go also)
  • Two Things Always Asked Do Next
    • Qualitative & Limited Dependent Variables
      • Bigger estimation challenges because:
        • Cannot place y itself on RHS, can only place y*.
        • N-dimensional integration to get probabilities
      • Considerable progress: S-Probit/Tobit etc., S-MNP, S-…
    • Estimate/Parameterize &, ideally, Endogenize W:
      • This essence of network analysis…
      • However, challenges in many contexts (e.g., C&IPE) differ:
        • W not always (or usually) binary or categorical
        • W not always (or usually) observed.
        • T not always (or usually) very long.
        • Temporal precedence not always (or usually) suffice=>causal prec.
leenders 1997 co evolution model
Leenders’ (1997) Co-evolution Model
  • Selection:
    • Arc forms or not in continuous time Markov process:
  • Contagion:
    • =STAR model
  • => Co-Evolution Model:
    • Identification strategy: time lag
    • Findings of MC’s
      • Coarse obs periodicity => big biases
      • If selection & model contagion => big biases
      • If contagion & model selection => biases, less big
snijders 97 07 co evolution model
Snijders’ (‘97-‘07) Co-evolution Model
  • Steglich et al. (‘07): two threefold empirical challenges
    • contagion, selection, context (1st+3rd=Galton’s Problem; 2nd=coevolution => similar implications)
      • In gen., any omissions or inadequacies in modeling one tends against that & favor others looks most like it
    • coarse periodicity, alternative mech’s & paths, net dependence precludes assume independence.
  • Observed Data:
    • N actors connected by observed, binary, endogenous, & time-variant connectivity matrix
    • Vector of N observed, ordinal behaviors
    • Further exogenous explanators may exist at unit or dyadic level
  • Model Components:
    • Exponential (constant hazard-rate) model of opp to act:
      • One change (or not) by one person @ one time; Can parameterize the rate; Conditionally independent
    • When opp act, multinomial w/ N network options—change tie or none
      • Compares objective with current behaves & net to current behaves & net w/ 1 switch his row: non-strategic
      • Can parameterize, including as function of actions; Conditionally independent
    • When opp act, could instead change behavior/attribute
      • Compares object w/ current net & behaves to his alternative behave, w/ switch of +1,0-1 only: non-strategic
      • Can parameterize, including as function network &/or of others’ actions; Conditionally independent
  • Parameters to Estimate:
    • Coefficients of hazard-rate model and of the two multinomial logits (n.b., IIA)
    • Estimated by simulated method of moments; recently, by simulated likelihood
  • Identification: (IIA and…)
    • Assume temporal precedence implies causal precedence, in particular condition on first obs
    • Each actor’s action or opp to act takes all else as given, i.e., conditional independence
    • In gen., strategy seems: control for (condition on) possible sources dependence; no stochastic dep.
issues from c ipe perspective
Issues from C&IPE Perspective
  • Many behaviors or attributes of interest as dependent variables, & relative connectivity between units less likely binary or ordinal.
  • Strengths of relative connectivity not always observed, or even observable, directly.
    • Under these conditions, for estimation purposes, the left-hand side of the selection component of the model would have no data. Could only estimate them off implications for behavior.
  • Temporal precedence often not suffice assure causal precedence
    • Strategic interdep often operates literally simultaneously or even E(future)
    • In estimation, simultaneous generally means within an observational period & many contexts high frequency behavior relative to obs periodicity.
    • Time lagging suffices only if & insofar as spatiotemporal dynamics fully & properly specified in model (&1st non-stoch, & not w/in period “simult”).
    • Condition on 1st obs needs T large for efficiency & for small-sample bias.
co evolution models in m star format
Co-Evolution Models in m-STAR Format
  • Wr = covariates expected explain network
  • Co-evolution models=models with W=f(y): larger challenges.
    • Our first cut: same poor man’s exogeneity, time-lag the y in W=f(y)…
    • Our plan: two-step estimation-procedure.
      • First, apply spatial-GMM (see, e.g., Anselin 2006, Franzese & Hays 2008b) to obtain by spatial instrumentation consistent estimates of endogenous wij and their estimated variance-covariance.
      • Then draw from that estimated multivariate distribution of instrumented W estimates to insert in the conditional or unconditional m-STAR likelihood.
      • Maximize likelihood under each of q draws from that first-stage S-GMM instrumented estimated distribution of W estimates.
      • Point estimates of parameters then just average of q 2nd stage S-ML estimates
      • Estimated variance-covariance of parameter-estimates is average of estimated variance-covariance matrices from each iteration plus (1+q) times sample variance-covariance in the point estimates across iterations (King et al. 2001).
      • First stage consistent, & asymptotically efficient, so estimator should inherent nice properties of S-ML and S-GMM, but no proof yet.
      • Monte Carlo assessment will follow; so will direct comparison to Snijders et al. approach (near as two models can approx each other).
conclusion1
Conclusion
  • Spatial & Spatiotemporal Interdependence
    • Important & Appreciable Substance (e.g., globalization & int’l cap-tax compete seems quite real & does constrain), not Nuisance.
      • Therefore: Model them. Interpret them.
  • How specify & estimate models?
    • If space-lag is time-lagged, maybe not problem; but if thry & substance says immediate (w/in an observational period), can/should handle that too:
    • S-OLS not a bad strategy even then, if ρ not too big & smpl-dims right; S-ML, &, in some regards, IV-based strategies seem effective
  • Spatiotemporal Effects not directly read from coefficients: use graphs & response-plots & maps & grids
  • Info-demands Galton’s Problem big, + Coevolution  REALLY big
    • Standard errors of effects tend big. Suspect delta-method lin-approx. maybe part problem; plan explore performance sim/boot/jack.
    • Max effort & care theory, measure, specification, to both C&IPE