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 Robert J. Franzese, Jr., The University of Michigan Jude C. Hays, The University of Illinois
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Spatial Econometric Models of Interdependence Theory & Substance; Empirical Specification, Estimation, Evaluation; Substantive Interpretation & Presentation.
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Spatial Econometric Modelsof Interdependence
Theory & Substance; Empirical Specification, Estimation, Evaluation; Substantive
Interpretation & Presentation
Talk prepared for Blalock Lecture on7 August 2008 at the ICPSR Summer School
based on the joint work ofRobert J. Franzese, Jr., The University of Michigan
Jude C. Hays, The University of Illinois
• 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:
• Specific: a model of inter-jurisdictional tax-competition (P&T ch. 12)– Empirics: “Galton’s Problem”; Estimation, Inference,
Interpretation, & Presentation
Overview
• 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
A Motivating Context:Globalization & Domestic-Policy Autonomy
• Theoretical Contexts (ubiquitous):– ANY Strategic Decision-making: sisj
• 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)
Kayser 2007), candidate qualities, contributions, or strategies (Goldenberg et al. 1986; Mizruchi 1989; Krasno et al. 1994; Cho 2003; Gimpel et al. 2006)
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
*)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.
1( ) 1 (1 )k c ks S U * 1 *( , ) ( )k k f k kf F M * *( , ) ( ) ( , )k k k k kk K S F
* * *( , ) 1 ( ) 1 ( ) ( , ) ( , ) 1 ( ) 1 ( )l k k k k k k k k k k l l lW U S S F M F L V L
* *( ) ( ) ( ) 2 ( , )1 ( ) 1
( ) ( ) ( )
p p p p p p ppk l k k k k
l kp p pk l k
S e L e S F
S L S
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)
• ° 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±).
it ij jt it t it t itj i
y w y
d s dsd β s β d s β
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:
, where and y Qδ ε Q Wy x δ
OLS
cov( , )ˆ ˆplim ; 0var( ) OLS
Wy x
x
S-OLS
cov , var( )1ˆplim where plimcov , cov( , ) n
Wy ε x Q Qδ Ψ
Wy ε Wy xΨ
1
where and
ˆplim plim , which, with ,
cov , var var1ˆplim cov , cov , var
cov , cov , var
n n
y Qδ ε Q Wy My X δ β
Q Q Q εδ δ X = x
Wy ε My x
δ Wy ε Wy My xΨ
Wy ε Wy x My
1
ˆ 0
ˆOVB:
ˆ
WyMy,X
WyMy,X 2
F
β β F
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)
Line: S-ML. Non-spatial OLS results plotted against the larger-scale 2nd y-axis on the right.
Figure 5: Standard-Error Accuracy Plotted Against True Across Representative NxT Sample-Dimensions
Notes: Dashed Line: S-OLS. Dotted line: S-2SLS-IV. Dashed-Dotted Line: S-ML. Standard error accuracy is gauged by the ratio of the average estimated standard error to the true standard deviation of the sampling distribution. Values less than one indicate overconfidence.
Figure 6: Standard-Error S Accuracy Plotted Against True
Across Representative NxT Sample-Dimensions
Notes: Solid Line: Non-spatial OLS. Dashed Line: S-OLS. Dotted line: S-2SLS-IV. Dashed-Dotted Line: S-ML. Standard error accuracy is gauged by the ratio of the average estimated standard error to the sampling-distribution standard deviation. Values less than one indicate overconfidence.
– 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.
t t 1 t t ty y Wy X β ε
0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
0 0 0 0 1 1 0 0 1 0 0 0 0 0 1
0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 1 0 0 1 0 0
0 1 0 0 0 1 0 0 0 0 0 1 0 1 1
1 1 0 0 1 0 0 0 1 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 1 0 0 0 1
AUT BEL DEN FIN FRA DEU GRE IRE NTH NOR PRT ESP SWE CHE GBR
AUT
BEL
DEN
FIN
FRA
DEU
GRE
IRE
NTH
W
0 0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
0 0 1 1 0 0 0 0 0 1 0 0 0 0 0
1 0 0 0 1 1 0 0 0 0 0 0 0 0 0
0 1 0 0 1 0 0 1 1 0 0 0 0 0 0
NOR
PRT
ESP
SWE
CHE
GBR
Swank & Steinmo APSR ’02 Replication
Table 2. Reanalysis of Swank and Steinmo (2002, Appendix Table 2) Effective Tax Rate on Capital Effective Tax Rate on Labor Swank and
Steinmo Reanalysis
(1) Reanalysis
(2) Swank and
Steinmo Reanalysis
(3) Reanalysis
(4) Temporal Lag 0.809** 0.808**
(0.05) 0.864** (0.048)
0.671** 0.66** (0.054)
0.711** (0.054)
Spatial Lag 0.104* (0.054)
0.126** (0.054)
0.017 (0.058)
0.05 (0.055)
Liberalization 1.146 1.235* (0.725)
0.629 (0.702)
-.261** -0.255** (0.102)
-0.168* (0.091)
Trade -0.018 0.009 (0.064)
0.005 (0.061)
-0.009 0.001 (0.023)
-0.001 (0.023)
Structural Unemployment -1.147** -1.218** (0.306)
-1.033** (0.283)
-0.359** -0.38** (0.189)
-0.148 (0.189)
Public Sector Debt 0.089** 0.099** (0.036)
0.046 (0.032)
0.053** 0.056** (0.014)
0.038** (0.013)
Elderly Population 1.264** 1.011 (0.615)
-0.08 (0.481)
-0.018 0.03 (0.23)
0.171 (0.184)
Growth 0.230* 0.242 (0.151)
0.307** (0.147)
-0.008 -0.009 (0.051)
0.009 (0.051)
Percent Change in Profits 0.127** 0.136** (0.055)
0.174** (0.054)
Domestic Investment 0.066 0.045 (0.055)
0.059 (0.049)
Inflation 0.115** 0.115** (0.05)
0.063 (0.043)
Unemployment 0.280** 0.296** (0.084)
0.144* (0.079)
Left Government 0.018** 0.018* (0.01)
0.012 (0.01)
0.008** 0.008** (0.004)
0.007* (0.004)
Christian Dem. Government 0.041** 0.035 (0.028)
0.01 (0.026)
0.001 0.002 (0.011)
0.009 (0.01)
Fixed Effects Country Yes Yes Yes Yes Yes Yes
Year Yes Yes1 No Yes Yes1 No R2 .928 .922 .914 .989 .989 .988
1Biannual Period Effects. Parentheses contain standard errors. **Significant at the 5% Level; *Significant at the 10% Level.
Hays IO ‘03 Replication
Basinger & Hallerberg APSR ’04 ReplicationTable 1. Replication and Reanalysis of Basinger and Hallerberg
APSR Partisanship -2.17
(1.58) -1.83 (2.05)
-1.83 (1.93)
-1.71 (1.91)
Partisanship world -11.2** (3.39)
-11.88* (5.32)
-11.87* (5.01)
-10.23* (5.03)
Capital controls 1.30 (2.70)
1.84 (2.49)
1.84 (2.34)
2.02 (2.32)
Capital controls world 6.35* (3.50)
27.15* (13.39)
27.27* (12.53)
23.18* (12.53)
Ideological distance -3.05** (1.46)
-2.47 (1.77)
-2.47 (1.67)
-2.85* (1.67)
Ideological distance world 9.11* (4.72)
2.65 (3.30)
2.63 (3.10)
2.55 (3.07)
Change in capital taxation in competitor countries
.20* (.09)
.03 (.28)
.02 (.18)
.10* (.05)
Control Variables
Intercept 16.94** (3.05)
18.18** (3.40)
18.16** (3.19)
17.42** (3.17)
Tax Ratet-1 -.29** (.06)
-.29** (.04)
-.29** (.04)
-.29** (.04)
Growtht-1 .27** (.10)
.32** (.12)
.33** (.11)
.30** (.11)
Inflationt-1 -.00 (.01)
-.00 (.02)
-00 (.01)
-.00 (.01)
N 269 269 269 269 R2 / LL .34 .32 -670.83 -668.99
Estimator S-OLS S-OLS S-ML S-ML
Weights Matrix Incorrect
GDP Weights
Correct GDP
Weights
Correct GDP
Weights
Binary Contiguity Weights
Notes: The country dummy variable coefficients are omitted to save space.
Beck, Gleditsch, Beardsley ISQ ‘06 Replication Beck et al. Beck et al. Reanalysis (1) Reanalysis (2) Reanalysis (3)
LN GDP A 0.03** (.01)
0.02** (.01)
-0.001 (.012)
0.029** (.015)
0.028* (.016)
LN GDP B 0.04** (.01)
0.03** (.01)
-0.001 (.012)
-0.015 (.015)
-0.003 (.016)
LN POP A 0.02 (.02)
0.04** (.02)
0.064** (.023)
0.012 (.06)
0.056 (.065)
LN POP B 0.02 (.02)
0.03 (.02)
0.056** (.023)
0.031 (.055)
0.059 (.059)
LN Distance -0.03**
(.01) -0.04**
(.01) -0.043**
(.009) 0.014 (.073)
-0.004 (.072)
LN Tau-b 0.13** (.06)
0.11 (.06)
0.05 (.058)
-0.053 (.06)
-0.063 (.06)
LN Democracy 0.13** (.03)
0.14** (.03)
0.155** (.031)
0.143** (.034)
0.089** (.037)
LN MID -0.20**
(.04) -0.20**
(.04) -0.19** (.037)
-0.186** (.039)
-0.157** (.039)
LN Multipolar -0.30**
(.05) -0.28**
(.05) -0.229**
(.053) -0.157**
(.053) -0.055 (.056)
LN Bipolar -0.06 (.05)
-0.04 (.05)
-0.011 (.048)
0.054 (.053)
0.032 (.055)
Temporal Lag 0.92** (.01)
0.91** (.01)
0.901** (.007)
0.795** (.01)
0.825** (.01)
Spatial Lag 0.02** (.01)
0.07** (.012)
0.18** (.014)
.097** (.039)
Fixed Effects: No No No Dyad Dyad, Yr Contemporaneous
Spatial Lag No No Yes Yes Yes
Estimator OLS OLS ML ML ML Observations 2565 2565 2565 2565 2565
Log-Likelihood — — 31.57 140.71 282.68 LR Statistic 218.28** 283.94**
Notes: Parentheses contain standard errors. **, * = significant at 5%, 10% levels, respectively. The Likelihood Ratio (LR) Statistics evaluate the null hypotheses that the coefficients on the dyad dummies (41) and year
dummies (67) are jointly zero with 5% critical values of 56.94 ( 2. . 41d f ) and 87.11 ( 2
Obs. 48 48 48 48 48 48 Notes: The spatial lags are generated with a binary contiguity weighting matrix. All the spatial weights matrices are row-standardized.
***Significant at the 1% Level; **Significant at the 5% Level; *Significant at the 10% Level.
Notes: In the first two columns, the models are estimated assuming the spatial lags are exogenous. The model in the first column is estimated using standard ML techniques. The parentheses in this column contain estimated standard errors and the hypothesis tests assume that the asymptotic t-statistics are normally distributed. The models in columns two through four are estimated using MCMC methods with diffuse zero-mean priors. The reported coefficient estimate is the mean of the posterior density based on 10,000 observations after a 1000 observation burn-in period. The number in parentheses is the standard deviation of the posterior density. The p-values are also calculated using the posterior density. The last two models are estimated with true spatial estimators described in the text. In third column, 30 of the 10,000 spatial AR coefficients sampled from the posterior distribution were negative. In the fourth column, none of the 10,000 sampled spatial AR coefficients were negative. ***p-value <.01, **p-value<.05, *p-value <.10.
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:
y Wy My Xβ ε
1t N t t t t y W y y X β ε
1
1t N N t t t y I W y X β ε
1 1 for some (set of) ; i.e., c i
N N t t i N N kx i I W X β ε I W x β
1
N N
t t t t t t t t
t t
y Wy y X β ε W I y X β ε
I W I X β ε
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
Swank & Steinmo APSR ’02 ReplicationFigure 2. Spatio-Temporal Effects on the German Capital Tax Rate from a Positive One-Unit
Counterfactual Shock to Structural Unemployment in Germany (with a 90% C.I.)
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Cumulative 15-Period Effect: -6.523
Swank & Steinmo APSR ’02 ReplicationFigure 3. Spatio-Temporal Effects on the French Capital Tax Rate from a Positive One-Unit
Counterfactual Shock to Structural Unemployment in Germany (with a 90% C.I.)
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Cumulative 15-Period Effect: -.943
Basinger & Hallerberg APSR ’04 Replication
Table 2. Conditional Coefficients for the Effects of a Change in Tax Rates on Capital in Competitor Countries Given Domestic Ideological Distance (Mendoza et al. Tax Rates, GDP Weights) Ideological Distance APSR Regression Coefficients Coefficient, change in competitor countries
.24* (.12)
.38 (.28)
-.13 (.29)
.15* (.08)
Coefficient, change in competitor countries*distance, party, cap controls
Notes: See Basinger and Hallerberg (2004), Table 3.
Basinger & Hallerberg APSR ’04 Replication Table 3. Conditional Coefficients for the Effects of a Change in Tax Rates on Capital in Competitor Countries Given Domestic Partisanship Level (Mendoza et al. Tax Rates, GDP Weights) Partisanship APSR Regression Coefficients Coefficient, change in competitor countries
-.24 (.40)
1.39* (.71)
-.83 (.89)
-.11 (.21)
Coefficient, change in competitor countries*distance, party, cap controls
.67 (.63)
-2.13* (1.21)
1.47 (1.52)
.40 (.37)
0 partisanship (no country)
-.24 (.40)
1.39* (.71)
-.83 (.89)
-.11 (.21)
0.2 partisanship (Norway, 1989)
-.10 (.27)
.96* (.48)
-.54 (.60)
-.03 (.14)
0.4 partisanship (Netherlands, 1982-88)
.03 (.16)
.53* (.28)
-.24 (.32)
.05 (.07)
0.6 partisanship (Austria, 1987-97)
.16* (.07)
.10 (.21)
.05 (.19)
.13* (.06)
0.8 partisanship (Ireland, 1990-92)
.30* (.13)
-.32 (.36)
.35 (.39)
.21* (.11)
Estimator S-OLS S-OLS S-ML S-ML Weights Matrix
Incorrect GDP
Weights
Correct GDP
Weights
Correct GDP
Weights
Binary Contiguity Weights
Notes: See Basinger and Hallerberg (2004), Table 4.
Table 4. Conditional Coefficients for the Effects of a Change in Tax Rates on Capital in Competitor Countries Given Domestic Use of Capital Controls (Mendoza et al. Tax Rates, GDP Weights) Capital Controls APSR Regression Coefficients Coefficient, change in competitor countries
.26* (.10)
.11 (.28)
.16 (.21)
.08 (.06)
Coefficient, change in competitor countries*distance, party, cap controls
-.96 (.76)
.60 (1.86)
-1.88 (1.51)
.20 (.33)
0 capital controls (United States, 1980-97)
.26* (.10)
.11 (.28)
.16 (.21)
.08 (.06)
0.25 capital controls (France, 1980-89)
.02 (.14)
.26 (.35)
-.31 (.33)
.13* (.07)
0.5 capital controls (Portugal, 1980-85)
-.22 (.32)
.41 (.77)
-.78 (.68)
.18 (.14)
0.75 capital controls (Greece, 1981)
-.46 (.50)
.56 (1.23)
-1.25 (1.05)
.23 (.22)
Estimator S-OLS S-OLS S-ML S-ML Weights Matrix
Incorrect GDP
Weights
Correct GDP
Weights
Correct GDP
Weights
Binary Contiguity Weights
Notes: See Basinger and Hallerberg (2004), Table 5.
Beck, Gleditsch, Beardsley ISQ ‘06 Replication Figure 1: Temporal Effects with Spatial Feedback (E.g., US Exports to Russia response to US-Russia MID)
counterparts. These numbers are calculated using the long-run spatio-temporal-multiplier matrix 111 WIW)(II . Parentheses contain standard errors
calculated by the delta method.
Some Other Presentations (3)
+1 Shock to Germany
-0.748 - -0.649
-0.648 - -0.549
-0.548 - -0.449
-0.448 - -0.349
-0.348 - -0.249
-0.248 - -0.220
-0.219 - -0.149
-0.148 - -0.049
-0.048 - 0.015
0.016 - 0.049
0.050 - 0.149
+1 Shock to Germany
-0.748 - -0.649
-0.648 - -0.549
-0.548 - -0.449
-0.448 - -0.349
-0.348 - -0.249
-0.248 - -0.220
-0.219 - -0.149
-0.148 - -0.049
-0.048 - 0.015
0.016 - 0.049
0.050 - 0.149
Figure 1. Short-run Spatial Effects of a Positive One-unit Shock to German LMT Expenditures
Figure 2. Steady-state Spatial Effects of a Positive One-unit Shock to German LMT Expenditures
Volden AJPS ‘06 AFDC Replication Table 4. Spatial Effects on AFDC Benefits from a $100 Counterfactual Shock to Monthly Retail Wages in Missouri Neighbor
Immediate Spatial Effect
Long-Run Steady State Effect
Arkansas
.51 [.16,.87]
4.26 [1.01,7.52]
Illinois
.62 [.19,1.04]
5.11 [1.25,8.97]
Iowa
0.52 [.15,.88]
4.37 [.99, 7.75]
Kansas
0.77 [.23,1.31]
6.38 [1.60,11.17]
Kentucky
0.44 [.13,.75]
3.68 [.87,6.50]
Nebraska
0.52 [.15,.89]
4.44 [.99,7.90]
Oklahoma
0.52 [.15,.89]
4.47 [.96,7.98]
Tennessee
0.38 [.12,.65]
3.21 [.75,5.67]
Notes: Effects calculated using estimates from the spatial AR lag model in Table 3. Brackets contain a 95% confidence interval.
Volden AJPS ‘06 AFDC ReplicationFigure 1. Spatio-Temporal Effects on AFDC Benefits in Missouri from a
$100 Counterfactual Shock to Monthly Retail Wages in Missouri (with 95% C.I.)
0
5
10
15
20
25
30
1 2 3 4 5 6 7 8 9 10
Cumulative 10-Period Effect: $55.75
Volden AJPS ‘06 AFDC ReplicationFigure 2. Spatio-Temporal Effects on AFDC Benefits in Nebraska from a
$100 Counterfactual Shock to Monthly Retail Wages in Missouri (with 95% C.I.)
• 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).
* * * ; 1i ij j i i ij i
y w y p y y
Xβ
• Structural Model:
• Reduced Form:
• Measurement Equation:
• Probability:– Or:
• For Spatial-Error-Probit:
The Econometric Problem (2)* * y Wy Xβ ε
1
1
* ( )
where ( )
y I W Xβ u
u I W ε
1 if * 0
0 if * 0i
ii
yy
y
1 1( 1| ) ( ) ( ) 0i i ip y p X I W Xβ I W ε
1
1
( 1| ) ( )
where ~MVN ,[( ) ( )]
i i i iip y p u
x I W Xβ
u 0 I W I W
1 1* ; with ( ) , so ~MVN ,[( ) ( )]
( 1| )i i i i ip y p u
y Xβ u u I W ε u 0 I W I W
x x β
• 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 Econometric Problem (3)
p pD
DX
• 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 (1)
• Bayesian Gibbs-MH (MCMC) Sampler for Spatial Probit:– Likelihood:
– Diffuse Priors => Joint Posterior:
– Conditional Priors:•
•
–
•
The Estimators: Bayesian Gibbs-MH Sampler (2)
122* 2
2( /2)
* *
1, | , , , with
2
in s-lag; in s-err
nn
n n
L e
ε εy W β I W
ε I W y Xβ ε I W y Xβ
122* ( 1), , | , n
np e
ε εβ y W I W
122( 1) 2 2| , , so ~ , which is std, so Gibbsn
np e
ε εβ
2 1
-1 *
1 *
| , ~ , ( )
with and ( ) for s-lag, and, for s-err:
& ( ( ) ( ) ) ( ) ( )
n n
n n n n n
p N
β β X C CX
C I β X X X I W y
C I W β X I W I W X X I W I W y
122( 1)| , ,
w/ for s-lag & s-err as before. Non-std, so Metropolis-Hastings.
nnp e
ε ε
β I W
ε
* 2ˆ| , , ~ ( , ), left- or right-truncated at 0 as 1 or 0i i i if z N y y β
• 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) (1)
0
( ) np f d
x
x x
cng x
0 ( ) ( )
( )cnnc
n
fp g d
g
x
xx x
x1
( ) ( )1ˆ
( ) ( )
Rn n rc c
rn n r
f fp E
g R g
x x
x x
1
1
( ), with ~ , ( ) ( ) ,
, with a diag mat having 2 1i i
p MVN
q y
u v u 0 I W I W
v Q I W Xβ Q
1
1
, with upper-triangular and independent!
So substituting gives: p p
-1
Σ A A A η Au
u A η Bη Bη v η B v
• 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:
The Estimators: Freq. Recursive Import. Smplr (RIS) (2)
1 1 1 111,1 1,2 1,3 1,
12,2
1 1 12,
11,1,
0
, where 0
0 0 0
n
n n
n n
nn n
vb b b b
b
p p b
b
vb
Bη v η B v B v
,1 1
ˆnR
j rr j
l R
υ
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.
1 1* , where and , ~ 0,1n n N y I W x β ε x I W z z ε
ML with Wy ML with Wy* Bayesian Gibbs
Experiment #1: n=48, =0.0 Mean Coefficient Estimate 1.02 0.32 1.13 0.74 1.23 0.30 Actual SD of Estimates 0.33 0.69 0.41 0.36 0.28 0.16 Mean of Reported SE 0.30 0.41 0.35 0.30 0.42 0.21 Experiment #2: n=48, =0.5 Mean Coefficient Estimate 1.22 0.35 1.13 0.69 1.21 0.28 Actual SD of Estimates 0.56 0.76 0.61 0.33 0.24 0.14 Mean of Reported SE 0.36 0.46 0.42 0.29 0.39 0.20 Experiment #3: n=144, =0.0 Mean Coefficient Estimate 0.94 0.42 1.01 0.68 1.14 0.34 Actual SD of Estimates 0.17 0.27 0.19 0.16 0.15 0.10 Mean of Reported SE 0.16 0.22 0.18 0.15 0.22 0.12 Experiment #4: n=144, =0.5 Mean Coefficient Estimate 1.08 0.48 0.97 0.64 1.13 0.32 Actual SD of Estimates 0.19 0.29 0.21 0.16 0.14 0.09 Mean of Reported SE 0.18 0.23 0.20 0.15 0.21 0.12
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:
* -1
1
1,2 1,
2,1
( 1),
,1 ,( 1)
( ) ( )
1
1
1
1
n
n
n n
n n n
w w
w
w
w w
y Wy Xβ ε I W Xβ ε
Xβ ε S Xβ ε
ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ, where and ˆ ˆ ˆ ˆ ˆ
i k i k i k i ki k i
k
V V
s s s ss θ θ s
θ θ θ
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?
1 11 0( ) ( )( 1)i i i
i ii i
p yp u p u
I W X β I W X β
X
p
1 1 1 11 0( ) ( )n i n i
pΦ I W X β Φ I W X β
X
* *1 0 1 0
1 0
ˆ ˆˆ ˆCalculate & at some & , & draws , , .
ˆ ˆApply measurement rule to convert these to and .
Notes: The first two columns’ estimators assume the spatial lags exogenous. The first column gives the standard probit ML estimates. Its parentheses contain estimated standard-errors, and its hypothesis tests assume asymptotic normality of calculated t-statistics. The models in columns two through four apply MCMC methods with diffuse priors, except for a Uniform(0,1) prior on ρ. The reported coefficient estimates are posterior-density means based on 10,000 samples after 1000-sample burn-ins. Parentheses contain sample standard-deviations of these posteriors. The p-values are calculated directly from the posterior density rather than from t-statistics of assumed asymptotic-normality. The last three columns report estimates from true spatial estimators described in the text. In column three, 30 of the 10,000 sampled spatial-lag coefficients were zero; in column four, none of the 10,000 were. Column five reports estimated standard-errors and p-values based on t-statistics assumed asymptotically negative. ***, **, * indicate p-value <.01, <.05, <.10.
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;
– 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
0 0 0 1 1 1 , 1 , 1 ; ...e.g, ij ij ij ij ij i t j td d d y y
1t y Wy y Xβ ε
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.
– 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).
Active Labor Market Program Expenditures in OECD Countries, 1981-2002Dependent variable
NoteAll regressions include fixed country effects. In addition to the country fixed effects, Model (3), (6) and (9) also include fixed year effects.All the spatial weights matrices are row-standardized.The parentheses contain standard errors.*** Significant at the .01 level; ** Significant at the .05 level; * Significant at the .10 level.
Total ALM LMT SEMP
Table 3: First-Period Spatial Effects of Union Density on Logged LMT per Capita (2000 PPP$) AUS AUT BEL CAN DNK FIN FRA GER GRC IRE ITA JPN NTH NWZ NOR PRT ESP SWE CHE GBR USA
Notes: The cells report the first-period spatial effect of a 1% increase in the column country’s union density on its own subsidized employment expenditures ( 100) and the expenditures ( 100) of its OECD counterparts (identified by the rows) based on the model (8) estimates.
Table 4: Estimated ALM-Policy Interdependencies/Network in 1981 AUS CAN FIN FRA NTH NWZ ESP SWE GBR USA
Note: Dependent variable: SEMP. Actual weights multiplied them by 100 (and standard errors adjusted accordingly) to improve table formatting. [XXXX: These are for c-dums, no t-dums model]
1981
Table 5: Estimated ALM-Policy Interdependencies/Network in 1991 AUS BEL CAN DEN FIN FRA DEU IRE ITA JPN NTH NWZ NOR PRT ESP SWE CHE GBR USA
Note: Dependent variable: SEMP. Actual weights multiplied them by 100 (and standard errors adjusted accordingly) to improve table formatting. [XXXX: These are for c-dums, no t-dums model]
1991
Table 6: Estimated ALM-Policy Interdependencies/Network in 2001 AUS AUT BEL CAN DEN FIN FRA DEU GRE IRE ITA JPN NTH NWZ NOR PRT ESP SWE CHE GBR USA
Note: Dependent variable: SEMP. Actual weights multiplied them by 100 (and standard errors adjusted accordingly) to improve table formatting. [XXXX: These are for c-dums, no t-dums model]