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Spatial Econometric Models of Interdependence Theory & Substance; Empirical Specification, Estimation, Evaluation; S

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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

- 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: externalitiesstrategic policy complements/substitutesrace-to-bottom/top/elseearly/late-mover advantagesstrategic 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

- 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

- (Simmons et al.’s 06) Mechanisms:
- Competition
- Coercion
- Learning
- Emulation
- [Migration/Contagion (F&H Add)]

- Theoretical Contexts (ubiquitous):
- ANY Strategic Decision-making: sisj
- 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)

- 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)

- 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)

- 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

- 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

- 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

- 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, sd (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.
- yiyj => 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±).

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

- 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

- 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

- 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

- Most convenient to work in (Nx1) vector form:
- WN=an NxN of (time-invariant) spatial wts, wij, & WNIT 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

- 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

- 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

- 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 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).

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

- 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)

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

The Estimators: Bayesian Gibbs-MH Sampler (2)

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

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.:

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:

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)

- 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)

- 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

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.

Example Estimated Spatial Effect, with Certainty Estimate, in Binary-Outcome Model

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

- 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

- 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)

- 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

- 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

- 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

- 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

- 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).

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

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