The m-STAR Model of Dynamic, Endogenous Interdependence and NetworkBehavior Coevolution in Comparative & International Political Economy* Jude C. Hays Assistant Professor of Political Science University of Illinois, Urbana-Champaign
[email protected] Aya Kachi Doctoral Candidate, Department of Political Science University of Illinois, Urbana-Champaign
[email protected] Robert J. Franzese, Jr. Professor of Political Science The University of Michigan, Ann Arbor
[email protected] 5 January 2009 ABSTRACT: Even casual observation reveals obvious spatial patterns in labor-market outcomes and policies across the developed democracies, and within the European Union particularly. Labor-market policies entail significant cross-border spillovers, so strategic interdependence among developed democracies might explain this. However, these countries also faced common or very similar exogenous-external conditions and internal trends, which would also tend to generate spatial patterns in the domestic responses thereto, even without any interdependence. Likewise, membership in the EU itself presents both a series of common external stimuli and a set of strategic interdependencies in common and individual-country labor-market-relevant actions. Furthermore, labor-market policies will themselves shape the patterns of economic interchange by which some of the interdependencies arise. That is, the policies of interest, labor-market policies in this case, may also shape the patterns of connectivity by which others’ policies affect those domestic policies, a complex sort of endogeneity known as selection in the dynamic-networks literature. We have discussed elsewhere the severe empirical-methodological challenges in distinguishing the first two of these possible sources of spatial correlation (a.k.a., Galton’s Problem). This paper extends those analyses, applying the multiparametric spatiotemporal autoregressive (m-STAR) model as a simple approach to modeling the patterns of interdependence simultaneously with its effects, while recognizing their possible endogeneity (i.e., selection). We do so in an empirical application attempting to disentangle the roles of economic interdependence, correlated external and internal stimuli, and EU membership in shaping labor-market policies in recent years.
* This research was supported in part by NSF grant #0318045. We thank Chris Achen, Jim Alt, Kenichi Ariga, Klaus Armingeon, Neal Beck, Jake Bowers, Jim Caporaso, Kerwin Charles, Bryce Corrigan, Tom Cusack, David Darmofal, Jakob de Haan, Scott De Marchi, John Dinardo, Zach Elkins, James Fowler, John Freeman, Fabrizio Gilardi, Kristian Gleditsch, Mark Hallerberg, John Jackson, Jonathan Katz, Mark Kayser, Achim Kemmerling, Gary King, Hasan Kirmanoglu, Herbert Kitschelt, Jim Kuklinski, Tse-Min Lin, Xiaobo Lu, Scott McClurg, Walter Mebane, Covadonga Meseguer, Michael Peress, Thomas Pluemper, David Prosperi, Dennis Quinn, Megan Reif, Frances Rosenbluth, Ken Scheve, Phil Schrodt, Chuck Shipan, Beth Simmons, David Siroky, John D. Stephens, Duane Swank, Wendy Tam-Cho, Vera Troeger, Craig Volden, Michael Ward, Greg Wawro, and Erik Wibbels for useful comments on this and/or other work in our broader project on spatial-econometric models in political science. All remaining errors are ours alone.
I. Introduction Labor-market outcomes and policies exhibit obvious spatiotemporal patterns within and across the developed democracies, and among European Union member-states especially. We have shown elsewhere (Franzese & Hays 2006c) that EU member-states’ labor-market policies exhibit significant interdependence along borders, a pattern possibly indicative of appreciable cross-border spillovers in labor-market outcomes inducing strategic interdependence among these political economies in labormarket policies. However, these countries also faced common or very similar exogenous-external conditions and internal trends, which would likewise tend to generate spatial patterns in the domestic policy-responses, even without interdependence. Moreover, EU membership itself likely entails both a series of common external stimuli and a set of strategic interdependencies relevant to labor-market policy. Furthermore, labor-market policies themselves may shape the patterns of economic exchange by which some of the policymaking interdependencies arise. That is, the policies of interest may also shape the patterns of connectivity by which foreign labor-market policies affect domestic ones, an example of a complex sort of endogeneity known as selection in the dynamic-networks literature. Elsewhere (Franzese & Hays 2003, 2004ab, 2006b, 2007abcd, 2008abc), we discussed the severe empirical-methodological challenges (sometimes called Galton’s Problem) in distinguishing the first two of these possible sources of spatial correlation: common exposure and interdependence. This paper extends those analyses, applying the so-called multiparametric spatiotemporal autoregressive (m-STAR) model as a simple approach to estimating simultaneously the patterns and the effects of interdependence, recognizing their possible endogeneity (i.e., selection). We do so in an empirical application aimed to disentangle the roles of political-economic interdependence, correlated external and internal stimuli, EU membership, and selection in shaping labor-market policies in recent years. The paper structures these explorations as follows. Section II reviews a generic theory (due to Brueckner 2003) of strategic policy complementarity and substitutability from positive and negative
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externalities.1 Section III briefly summarizes our previous work regarding specification, estimation, interpretation, and presentation of spatial autoregressive (SAR) and spatiotemporal autoregressive (STAR) models (Franzese & Hays 2003,2004ab,2006abc,2007abcd,2008abc). That work highlighted Galton’s Problem of distinguishing spatial correlation due to interdependence from that arising from common or correlated exogenous internal or external stimuli. Of first-order importance in drawing such distinctions, we showed, was the relative and absolute empirical accuracy and power with which the empirical-model specification reflects the patterns of interdependence on one hand and the exogenous internal and external stimuli on the other. This leads naturally to the extensions offered here, elaborated in Section IV, of parameterizing and estimating the patterns of interdependence. We suggest an application and interpretation of the multiparametric spatiotemporal autoregressive (mSTAR) model as a simple means of doing this, while endogenizing the pattern of interdependence dynamically to the outcome-variable of the model (here: labor-market policy). Section V contains our illustrative empirical analysis, discusses those results, and offers some preliminary conclusions. II. Race-to-the-Bottom Dynamics and Policy Free Riding Consider, first, the actual history of EU member-government ALM policies. Figure 1 shows that average total ALM-expenditures have increased from 1980-2003, while the standardized variance (i.e., the coefficient of variation) of expenditures across countries has decreased.2 One might see a “race-to-the-top” in these trends and be tempted to infer that EU employment-policy coordination has been successful. However, post-Lisbon (1997-2003) spending-growth was slower than the preLisbon period,3 and consensus prevails that, excepting Scandinavia, EU members lag in designing and implementing policies to upgrade the skills of their workers.4 By this view, despite the trends in 1
Franzese & Hays (2006b) review the history of the EES starting with the Luxembourg Jobs Summit. In 1980, average ALM spending barely exceeded $54 (2000, PPP$) per capita. By 2003, it was almost $253, a 370% increase. The coefficient of variation (standard deviation) meanwhile dropped (rose) from .80 (43.5) to .47 (120.1). 3 The relevant amounts are about $30 of the total $150 increase, or only 15% of the total in the last 25% of the period. 4 The 2004 Joint Economic Report asked 6 of the original 15 EU-members to strengthen their ALM policies. 5 later received a C grade (“partial and limited”) and1 a B (“in progress”) for their response. The Council asked every member 2
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Figure 1, spending could and should be higher. If ALM-program spending among EU member-states is, in fact, suboptimal, strategic interdependence in the making of active-labor-market policies could be at fault. Two kinds of strategic interactions in particular, race-to-the-bottom dynamics and policy free-riding, would induce suboptimal expenditures on employment policies.
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Figure 1. Aggregate Active Labor Market Expenditures in the EU, 1980-2003
In theory, race-to-the-bottom (RTB) dynamics occur when policies are strategic complements across jurisdictions—i.e., when policy changes adopted in one create incentives for others to adopt similar changes. RTB arguments have been applied to capital taxation, environmental regulations, and labor standards, inter alia. Cuts, elimination, or reduction in taxes, regulations, or standards in one jurisdiction raise the costs to others of maintaining high taxes, regulations, or standards, causing the effected jurisdictions to follow suit in their own policies. By contrast, free riding occurs when policies are strategic substitutes—i.e., when policy changes by one create incentives for others to move oppositely. E.g., an increase in defense expenditures by one ally lowers the marginal security benefit of defense spending for others, creating incentives for them to free ride (e.g., Redoano 2003). to improve its investment in human capital in one or more ways. The modal response to these recommendations was C, “partial and limited” (European Commission 2005). See Murray & Wanlin (2005) for another disappointing report card. Page 3 of 44
More formally (following Brueckner 2003), consider a two countries (i,j), each with domestic welfare that, due to externalities, are a function of both domestic and foreign policies (pi,pj): W i ≡ W i ( pi , p j ) ; W j ≡ W j ( p j , pi )
(1).
When i chooses its policy, pi, to maximize its social welfare, this affects the optimal policy-choice in j, and vice versa. We can express such strategic interdependence between i and j with a pair of bestresponse functions, giving i’s optimal policies, pi*, as a function of j’s chosen policy, and vice versa:5 pi* =Argmax pi W i ( pi , p j ) ≡ R ( p j ) ; p*j =Argmax p jW j ( p j , pi ) ≡ R ( pi )
(2).
The signs of the slopes of these best-response functions determine whether RTB or free-riding dynamics will occur and depend on these ratios of second cross-partial derivatives: ∂p*j ∂pi* = −W i pi p j / W i pi pi ; = −W j p j pi / W j p j p j ∂p j ∂pi
(3).
If governments are maximizing, the second derivatives in the denominators of (3) are negative, so the slopes depend directly on the signs of the second cross-partial derivatives (i.e. the numerators). If W pii,pjj > 0 , policies are strategic complements, and policy reaction-functions slope upward. If W pii,pjj < 0 , policies are strategic substitutes, so reaction functions slope downward. If W pi ,pj = 0 , the i j
best-response functions are flat as strategic interdependence does not materialize. Interestingly, given diminishing marginal returns in the welfare function, negative externalities induce strategic-complement policy-interdependence, and positive externalities induce strategicsubstitute policy-interdependence. In the national-defense case raised above, for instance, spending in one ally induces free riding by others due to the positive security externalities (and diminishing returns of military expenditures). If ALM expenditures also create positive employment externalities and diminishing returns, the same problem could arise in this context. Diminishing returns—i.e., 5
Explicitly, i’s optimum policy is obtained by maximizing Wi(pi,pj) with respect to pi, taking pj as given; i.e., setting the partial of Wi with respect to pi equal to 0 and solving for pi* as a function of pj (and verifying negative second partials). Page 4 of 44
reducing unemployment requires increasing amounts of spending as the rate declines—seems likely.6 So, if ALM spending in one country, i, helps lower unemployment in another, j, then an increase in expenditures in country i will reduce the marginal benefit to j of its (marginal increment of) spending, inducing lower equilibrium spending in j. Figure 2 illustrates this situation graphically. Notice that such a strategic context also creates first-mover disadvantages—those spending earlier will bear larger portions of the costs of reducing unemployment—and thus the potential for war-ofattrition dynamics that would delay and push equilibrium ALM spending of both i and j even lower. Figure 2. Best Response Functions: Strategic Substitutes
p 2 = R( p1 )
p1 = R( p 2 )
Do cross-border positive employment externalities of ALM policies exist among EU countries; and, if so, are they sufficiently strong to induce fiscal free-riding in ALM policy? On balance, the evidence suggests that ALM policies may have increased employment in Europe and other OECD countries (e.g., Martin 2000; Martin & Grubb 2001; European Commission 2005). A consensus in micro-level research finds ALM-program participants enjoy increased employment probability.7 However, these findings reveal nothing about the effects of ALM programs on non-participants and 6
For instance, if labor-market-training programs increase employment by raising workers’ marginal productivity, then, in any given macroeconomic conditions, some workers will have marginal productivity just below a threshold where firms find hiring them profitable while others will have productivity far below that. A little training might get the first group hired, but much more per worker would be required to make the less-productive group profitably employable. 7 I.e., the average treatment effect for the treated is an increase in the probability of employment (Heckman et al. 1999). Page 5 of 44
so say little about net employment consequences. Aggregate data speaks more clearly to net-effects, but, in macro-level studies, more disagreement about the net employment effects of ALM programs prevails. Some find sizeable displacement rates, particularly for subsidized employment programs (e.g., Forslund & Krueger 1997, Calmfors et al., 2001, Dahlberg & Forslund 2005), whereas others find larger, positive net-employment effects (Kraft 1998, Estevao 2003). Perhaps the best evidence for beneficial ALM-policy effects, though, shows in their mediation of adverse macroeconomic shocks. In a seminal paper on the interaction of shocks and institutions, Blanchard & Wolfers (2000) estimate that an adverse shock that reduces employment by 1% at the sample-mean level of ALMprogram expenditures but reduces employment by just 0.2% at sample-maximum ALM spending. Granting that ALM programs can be effective, should they involve cross-border externalities? A large literature examines the regional patterns of unemployment in Europe (e.g., Elhorst 2003a; Puga 2002; Overman & Puga 2002). This work shows that, generally, employment differences between bordering regions are much smaller, even if the regions lie in different countries, than differences between more distant regions within countries. I.e., geographic proximity is more important than nationality in shaping spatial patterns of unemployment in Europe.8 Consider the implications of French ALM programs for Belgium, for example. Effective French ALM policies enhance Belgian workers’ abilities to obtain training in France and return, more employable, to work in Belgium; they enhance Belgian workers’ abilities to find work in France; and they enhance the pools of workers (quantity, quality, and diversity) available along the Franco-Belgian border, luring employers to both sides. Finally, effective French ALM spending stimulates the French economy, which, through trade, has positive effects on Belgium’s economy. These and other agglomeration effects all yield positive externalities of effective French ALM policies to Belgian workers (and Belgians generally). Notice, 8
Overman & Puga (2002) attribute the growing importance of geographic proximity to changes in the demand for labor. They identify, test, and find empirical support for three sources of demand change over the period 1986-96: the regional concentration of skilled and unskilled labor, the spatial clustering of industries, and what they term agglomeration effects. The examples given next in the text illustrate all of these sources. Page 6 of 44
too, that only the first two effects require any cross-national labor mobility, which is notoriously low in Europe, even across common-language borders. Notably, Overman & Puga (2002) find the latter two effects predominate as sources of the spatial correlation in employment patterns they observed. Spillover benefits to Belgians would yield little domestic political returns for French policymakers of course, so the latter will mostly ignore such spillover benefits in setting French ALM policies. In sum, given what we know about spatial patterns and spillovers of/in employment and about the employment effects of ALM policies, fiscal free-riding seems quite plausible. ALM spending by national policymakers should exhibit negative interdependence. In our empirical application, we examine the comparative-historical record to gauge the evidence of its existence and magnitude. III. Spatiotemporal Models of Interdependence: Specification, Estimation, & Interpretation Analyses that recognize interdependence across units of outcomes—here: of ALM policies— must specify empirical models in which outcomes in units i and j affect each other. Elsewhere, we (Franzese & Hays 2004a, 2006a, 2007bc, 2008ab) suggested the following generic model of modern, open-economy, context-conditional political-economy with such features:
yit = ρ ∑ wij y jt + φ yi ,t −1 +β′d dit + β′s st + β′sd ( dit ⊗ st ) + ε it
(4).
j ≠i
yjt is the outcome in another (j≠i) unit, which in some manner (given by ρwij) directly affects the outcome in unit i. wij reflects the relative degree of connection from j to i, and ρ reflects the overall
strength of dependence of the outcome in i on the outcomes in the other (j≠i) units, as weighted by wij. Substantively for ALM-policy interdependence, e.g., the wij, could gauge the sizes, geographic contiguity, EU comembership, and/or trade of i’s and j’s political economies. The other right-handside elements reflect non-spatial components: unit-level/domestic factors dit (e.g., election-year indicators, government partisanship), exogenous-external/contextual factors sit (e.g., technology, oil prices), and context-conditional factors dit ⊗ st (i.e., interactions of the former with the latter).
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Distinguishing these spatial (or network) interdependence and non-interdependence sources of spatial correlation is the essence of Galton’s Problem.9 A third potential source of spatial correlation, to be introduced later, is that the relative connectivity from j to i, that is, the wij, may depend on the outcome(s) in i (and/or j).10 As we summarize below (from Franzese & Hays 2003, 2004ab, 2006b, 2007abcd, 2008abc), obtaining good (unbiased, consistent, and efficient) parameter and certainty estimates in such models—more generally and substantively, distinguishing open-economy, contextconditional comparative political economy (CPE) from interdependence (C&IPE: comparative and international political economy) empirically—by any methodological means, including qualitative methods—is not straightforward.11 The first and prime consideration in weighing these alternatives and estimating the role of the corresponding components in (4) are the theoretical and empirical precision and explanatory power, relatively and absolutely, of the spatial and non-spatial parts of the model. To elaborate: the relative and absolute accuracy and power (i) with which the spatial-lag weights, wij, reflect and can gain leverage upon the interdependence mechanisms actually operating and (ii) with which the exogenous domestic, external, and/or context-conditional parts of the model reflect and can gain leverage upon the alternatives are critical to the empirical attempt to distinguish and evaluate their relative strength. The two mechanisms produce similar effects, so inadequacies or omissions in specifying the one tend, intuitively, to induce overestimates of the other’s importance. Secondarily,12 even with the interdependence and the alternative common-shock mechanisms modeled perfectly, the spatial-lag regressor(s) will be endogenous (i.e., covary with the residuals), so estimates of ρ (or, equally, qualitative attempts to distinguish interdependence from common shocks) 9
The web appendix to Franzese & Hays (2008c) contains, inter alia, brief intellectual-historical background to the label. One could also allow spatial error-correlation to remain and address it by FGLS and/or PCSE, but optimal strategies will be to model the interdependence and correlation in the first moment insofar as possible. 11 Some might suggest starting with nonspatial models and adding spatial components as the data demand, but tests that can distinguish spatial interdependence from other potential sources of spatial correlation in residuals from non-spatial models are lacking and/or weak (Anselin 2006; Franzese & Hays 2008b; Hendry 2006; cf. Florax et al. 2003, 2006). 12 Simulations (Franzese & Hays 2004a, 2006b, 2007cd) show the omitted-variable/relative-misspecification biases of neglecting interdependence typically far exceed the simultaneity biases of failing to redress adequately the endogeneity of spatial lags, although the latter grow appreciable as interdependence strengthens (e.g., !á.3 for row-standardized W). 10
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will suffer simultaneity biases. Furthermore, as with the primary concern of relative omitted-variable or misspecification bias, these simultaneity biases in estimated strength of interdependence (usually overestimation) generally induce biases in the opposite direction (underestimation) regarding the role of common shocks. Thus, researchers who emphasize unit-level/domestic, exogenous-external, or context-conditional processes to the exclusion or relative neglect of interdependence (comparativists and micro-level scholars?) will tend to get empirical results biased toward the former and against the latter sorts of explanations. Conversely, those who stress interdependence to the relative neglect of domestic/unit-level and exogenous-contextual considerations or who fail to account sufficiently the endogeneity of spatial lags (macro-level and international-relations scholars?) will tend to suffer the opposite biases: underestimating the role of common shocks and overestimating interdependence. Most empirical studies in C&IPE where interdependence may arise, notably the policy diffusion and the globalization, tax-competition, and policy-autonomy literatures, analyze panel or time-seriescross-section (TSCS) data (i.e., observations on units over time). To estimate effects and draw sound causal inferences in such contexts, analysts should specify both temporal and spatial interdependence in their models.13 The easiest and most straightforward way to incorporate this interdependence is with a spatiotemporal-lag model, which we can write as an extension of (4) in matrix notation as:
y = ρ Wy + φ My + Xβ + ε
(5).
The dependent variable, y, is an NT%1 vector of cross sections stacked by periods (i.e., the N firstperiod observations, the next N, up through N in period T).14 ρ is the previously described spatialautoregressive coefficient, and WNT is an NT%NT block-diagonal spatial-weighting matrix.15 Each of the T N%N weights matrices, WN, on the block-diagonal have elements wij(t) reflecting the relative 13
Methodologically, Anselin (2002, 2006) usefully distinguishes spatial-statistical and spatial-econometric approaches, the former being more data-driven and tending toward treating spatial correlation as nuisance and the latter wedded more to theoretically structured models of interdependence. Franzese & Hays (2008c: web appendix) offers fuller discussion of the practical import of this subtle but key distinction. In these terms, ours is a spatial-statistical approach. 14 Nonrectangular and/or missing data are manageable, but we assume full-rectangularity for expository simplicity. 15 WNT is block-diagonal assuming no cross-temporal spatial interdependence. Non-zero off-diagonals are possible and manageable, but perhaps unlikely controlling for contemporaneous spatial-lags and time-lags. Page 9 of 44
connectivity from unit j to i that period.16 Thus, for each observation, yit, the spatial lag, Wy, gives a weighted sum of the yjt, with weights wij(t): direct and straightforward reflection of the dependence of each unit i’s outcome on others’. M is an NT%NT matrix with ones on the minor diagonal, i.e., at coordinates (N+1,1), (N+2,2), …, (NT,NT-1), and zeros elsewhere. My is thus the standard (firstorder) temporal-lag;17 & is its coefficient. X contains NT observations on k independent variables; is its k%1 vector of coefficients. Finally, is an NT%1 vector of i.i.d. stochastic components.18 Franzese & Hays (2004a, 2006b, 2007cd, 2008b) explored analytically and by simulation the properties of four estimators for such models: non-spatial least-squares (i.e., regression omitting the spatial component as is common in most extant research: OLS), spatial OLS (i.e., OLS estimation of models like (5), common in diffusion studies and becoming so in globalization/tax-competition ones: S-OLS), instrumental variables (e.g., spatial 2SLS or S-2SLS), and spatial maximum-likelihood (SML). Both OLS and spatial OLS produce biased and inconsistent estimates, OLS due to the omittedvariable bias and spatial OLS because the spatial lag is endogenous and so induces simultaneity bias. We can view these biases as reflecting the terms of Galton’s Problem. On one hand, by omitting the spatial lag that would reflect the true interdependence of the data, OLS coefficient-estimates will suffer omitted-variable biases, the familiar formula for which is Fβ, where F is the matrix of coefficients obtained by regressing the omitted on the included variables and β is the vector of (true) coefficients on the omitted variables.19 In this case, the omitted-variable bias (OVB) is: −1 ′ OVB ⎡⎣βˆ OLS φˆOLS ⎤⎦ = ρ × ( Q1′Q1 ) Q1′ Wy , where Q1 ≡ [ X My ]′
(6).
ρˆ OLS ≡ 0 , of course, which is biased by − ρ . Thus, insofar as the spatial lag covaries with the nonspatial regressors—which is (i) highly likely if domestic conditions correlate spatially, (ii) certain for 16
If the pattern of connectivity is time-invariant, then WNT can be expressed as the Kronecker product of a T%T identity matrix and the constant N%N weights-matrix, IT1WN, with the elements wij reflecting relative connectivity from j to i. 17 Higher-order temporal dynamics would simply add further properly configured weights matrices. 18 Alternative (non-i.i.d.) distributions of are possible but add complication without illumination. 19 Likewise, maximum-likelihood estimates of limited- or qualitative-dependent-variable models, like logit or probit, which exclude relevant spatial lags will suffer analogous omitted-variable biases, although Fβ would not describe those. Page 10 of 44
exactly common exogenous-external shocks, and (iii), given non-zero spatial correlation from any source, certain for the time lag also—OLS will overestimate domestic, exogenous-external, or context-conditional effects, including the temporal adjustment-rate, while ignoring interdependence. Note, as did Sir Galton, that this applies equally to qualitative analyses that ignore interdependence. On the other hand, including spatial lags in models for OLS estimation—or tracing putative diffusion processes or otherwise considering cross-unit correlation qualitatively—raises inherent endogeneity biases. Spatial lag, Wy, covaries with the residual, , making S-OLS estimates inconsistent, because it is a weighted average of outcomes in other units and so places some observations’ left-hand sides on the right-hand sides of others: textbook simultaneity. In simplest terms by example: Germany causes France, but France also causes Germany. These asymptotic simultaneity biases (SB) are: −1 ′ SB ⎡⎣ ρˆ ϕˆ βˆ ⎤⎦ = ( Q′Q ) Q′ε, where Q ≡ [ Wy My
X]
(7).
In the case where X contains just one exogenous explanator, x, we can rewrite these biases thus: ⎡ ρˆ ⎤ ⎡ cov ( Wy, ε ) × var ( My ) × var ( x ) ⎤ ⎢ ⎥ 1 ⎢ ⎥ ⎛ Q′Q ⎞ SB ⎢ϕˆ ⎥ = − cov , × cov , × var , where = plim Wy ε Wy My x Ψ ( ) ( ) ( ) ⎜ ⎟ ⎢ ⎥ ⎝ n ⎠ ⎢ βˆ ⎥ Ψ ⎢ − cov ( Wy, ε ) × cov ( Wy, x ) × var ( My ) ⎥ ⎣ ⎦ ⎣ ⎦
(8).
With positive interdependence and positive covariance of spatial-lag and exogenous regressors, a likely common case, one would overestimate the interdependence-strength, ρˆ , and correspondingly underestimate temporal dependence, φˆ , and domestic/external/contextual effects, βˆ . In sum, Galton’s Problem implies that empirical analyses that ignore substantively appreciable interdependence will also thereby tend to overestimate the importance of non-spatial factors; in fact, the effect of factors that correlate spatially the most will be most over-estimated. On the other hand, simply controlling (or considering qualitatively) spatial-lag processes will introduce simultaneity biases, usually in the opposite direction, exaggerating interdependence effects and understating unitlevel/domestic, exogeneous-external, and context-conditional impacts. Again, those factors that Page 11 of 44
correlate most with the interdependence pattern will have the most severe induced deflation biases. These conclusions hold as a matter of degree as well; insofar as the non-spatial components of the model are inadequately specified and measured relative to the interdependence aspects, the latter will be privileged and the former disadvantaged (and vice versa). Accurate and powerful specification of
W is obviously of crucial empirical, theoretical, and substantive importance to those interested in interdependence, but also to those for whom domestic/unit-level, exogenous-external/contextual, or context-conditional factors are of primary interest. Conversely, optimal specification of the unitlevel/domestic, contextual/exogenous-external, and context-conditional non-spatial components is of equally crucial importance to those interested in gauging the importance of interdependence. Our simulations (Franzese & Hays 2004a, 2006b, 2007cd) showed the omitted-variable biases of OLS are almost always worse and often far, far worse than S-OLS’ simultaneity biases. In fact, SOLS may perform adequately for mild interdependence strengths (ρΣjwijá0.3), although standarderror accuracy can be problematic, and in a manner for which PCSE (Beck & Katz 1995, 1996) will not compensate. S-OLS’ simultaneity biases grow sizable as interdependence strengthens, however, rendering use of a consistent estimator, such as S-2SLS or S-ML, highly advisable. Choosing which consistent estimator seems of decidedly secondary importance in bias, efficiency, and standard-erroraccuracy terms. Since S-ML proved close to weakly dominant,20 we introduce only it here. 21
20
See Franzese & Hays (2007b, 2008b) regarding S-ML estimation; they correct some misleading conclusions from our earlier work on S-ML, stemming from a coding error (see note 21). The instrumental-variables (IV), two-stage-leastsquares (2SLS), generalized-method-of-moments (GMM) family of estimators relies on the spatial structure of the data to instrument for the endogenous spatial lag. Assuming that what we call cross-spatial endogeneity—y’s in some units cause x’s in others—does not exist, WX are ideal instruments by construction. Cross-spatial endogeneity may seem unlikely, until one realizes that vertical ties from yi to yj and horizontal ties from yj to xj (the usual sort of endogeneity) combine to give the offending diagonals from yi to xj. On the other hand, S-GMM should improve upon S-2SLS primary weakness in efficiency, so it may compare more favorably to S-ML. We have not yet explored the possibility. 21 We use J.P. LeSage’s MatLabTM code to estimate our spatial models, having found existing StataTM code for spatial analysis, third-party contributed .ado files, untrustworthy and/or extremely computer-time intensive. We have written StataTM code, which we believe more reliable and efficient, to implement many of our suggestions. Regarding LeSage’s MatLab code, sar.m, note that the line of code calling the standard errors from the parameter-estimate variancecovariance matrix must be corrected to reference the proper element for the ρˆ estimate. We have made other adjustments to the code to enhance its operation. We will make all of our code publicly available once we have tested its reliability more thoroughly and made it more generic and user-friendly. Currently, some of our application-specific Page 12 of 44
The conditional likelihood function for the spatiotemporal-lag model,22 which assumes the first observations non-stochastic, is a straightforward extension of the standard spatial-lag likelihood function, which, in turn, adds only one mathematically and conceptually small complication (albeit a computationally intense one) to the likelihood function for the standard linear-normal model (OLS). To see this, start by rewriting the spatial-lag model with the stochastic component on the left: y = ρ Wy + Xβ + ε ⇒ ε = ( I − ρ W ) y − Xβ ≡ Ay − Xβ
(9),
where X now includes My, the time-lag of y, as its first column, and β includes & as its first row.23 Assuming i.i.d. normality, the likelihood function for ε is then just the typical linear-normal one: NT
⎛ 1 ⎞2 ⎛ ε′ε ⎞ L(ε) = ⎜ 2 ⎟ exp ⎜ − 2 ⎟ ⎝ σ 2π ⎠ ⎝ 2σ ⎠
(10),
which, in this case, will produce a likelihood in terms of y as follows: NT
⎛ 1 ⎞2 ⎛ 1 ⎞ L(y ) =| A | ⎜ 2 ⎟ exp ⎜ − 2 ( Ay − XB ) ' ( Ay − XB ) ⎟ ⎝ σ 2π ⎠ ⎝ 2σ ⎠
(11).
This resembles the typical linear-normal likelihood, except that the transformation from ε to y is not by the usual factor, 1, but by |A|=|I-ρW|.24 Written in (N%1) vector notation, the spatiotemporalmodel conditional-likelihood is mostly conveniently separable into parts, like so:
1 1 Log f yt ,yt−1 ,...,y 2 y1 = − N (T − 1) log 2πσ 2 + (T − 1) log I − ρ W − 2 2 2σ where ε t = y t − ρ WN y t − φ I N y t −1 − Xt β.
(
)
T
∑ ε′ε
t t
t =2
(12).
The unconditional (exact) likelihood function, which retains the first time-period observations as non-predetermined, is more complex (see Elhorst 2005; Franzese & Hays 2007c, 2008b): MatLabTM and StataTM code, and ExcelTM spreadsheets, possibly useful as templates, are available at: http://www-personal.umich.edu/~franzese/FranzeseHays.CPS.InterdependenceCP.TemplateImplementationFiles.zip. 22 Derivation of these likelihoods for spatiotemporal-lag models is due to Elhorst (2001, 2003b, 2005). 23 We do this because, in translating the likelihood in terms of t into terms of yt, Wy enters but My does not. 24 This difference complicates estimation in that the determinant |A| involves ρ , and so requires recalculation at each iteration of the likelihood-maximization routine. Two strategies to simplify are to use an eigenvalue approximation for the determinant (Ord 1975) and to maximize a concentrated likelihood function (Anselin 1988). We discuss both procedures and estimation more generally elsewhere (Franzese & Hays 2004, 2006a, 2007b, 2008b). Page 13 of 44
(
)
N 1 1 N 2 Log f yt ,...,y1 = − NTlog ( 2πσ 2 ) + ∑ log (1 − ρωi ) − φ 2 + (T − 1) ∑ log (1 − ρωi ) 2 2 i =1 i =1 −1
⎛ ⎞ − 2 ∑ ε t′ε t − 2 ε1′ ⎛⎜ ( B − A )′ ⎞⎟ ⎜ B′B − B′AB −1 B′AB −1 ′ ⎟ 2σ t = 2 2σ ⎝ ⎠ ⎝ ⎠
1
T
1
(
−1
) (B − A)
(13). −1
ε1
With small/large T, the first observation contributes greatly/little to the total likelihood, so scholars should/can use the unconditional/conditional likelihood for estimation purposes. One easy way to ease or even erase the simultaneity problem with S-OLS is to lag temporally the spatial lag (Beck et al. 2006; see Swank 2006 for an appropriate application). Insofar as temporal lagging renders the spatial lag pre-determined—i.e., to the extent spatial interdependence does not incur instantaneously, where instantaneous means within an observation period, given the model— S-OLS’ bias disappears asymptotically. That is, if T is large, if the spatial-interdependence process does not operate within an observational period but only with a time lag, and if spatial and temporal dynamics are modeled well enough to prevent that problem arising via measurement/specification error,25 OLS with a time-lagged spatial-lag on the RHS is an effective estimation strategy. However, even in this best case, OLS with time-lagged spatial-lags only provides unbiased estimates if the first observation is non-stochastic (i.e., if initial conditions are fixed across repeated samples).26 Regarding stationarity, the conditions and issues arising in spatiotemporally dynamic models are reminiscent if not identical to those in the more familiar solely time-dynamic models. Let A1h&I, B1hI-ρW, and ω be an eigenvalue of W; then the spatiotemporal process is covariance stationary if: ⎧⎪ φ < 1 − ρωmax , if ρ ≥ 0 A1B1−1 < 1, or, equivalently, if ⎨ ⎪⎩ φ < 1 − ρωmin , if ρ < 0 25
(14).
Testing for remaining temporal and spatial correlation in residuals from the time-lagged spatiotemporal-lag model is possible and highly advisable. Standard Lagrange-multiplier tests remain valid for remaining temporal correlation. Franzese & Hays (2004a, 2008b) describe several tests for/measures of spatial correlation, some of which retain validity when applied to estimated residuals from models containing spatial and temporal lags. 26 Note that the same condition that complicates ML estimation of the spatiotemporal-lag model if the first observations are stochastic, also invalidates the use of OLS to estimate a model with a time-lagged spatial-lag under those conditions. Elhorst (2001:128) derives the likelihood for the spatiotemporal lag model with time-lagged spatial-lag and showed it to retain the offending Jacobian. Hence, this consideration offers no econometric reason to prefer either S-OLS over S-ML estimation of spatiotemporal-lag models or the reverse. Page 14 of 44
For example, with positive temporal and spatial dependence and W row-standardized, the maximum characteristic root is 1, so stationarity familiarly requires &+ρ0), as in the network analyst’s homophily, or less (γ
