Spatial panel models

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1 Spatial panel models J Paul Elhorst University of Groningen, Department of Economics, Econometrics and Finance PO Box 800, 9700 AV Groningen, the Netherlands Phone: , Fax: , jpelhorst@rugnl March 2012 Abstract This chapter provides a survey of the existing literature on spatial panel data models Both static and dynamic models will be considered The paper also demonstrates that spatial econometric models that include lags of the dependent variable and of the independent variables in both space and time provide a useful tool to quantify the magnitude of direct and indirect effects, both in the short term and long term Direct effects can be used to test the hypothesis as to whether a particular variable has a significant effect on the dependent variable in its own economy, and indirect effects to test the hypothesis whether spatial spillovers exist To illustrate these models and their effects estimates, a demand model for cigarettes is estimated based on panel data from 46 US states over the period 1963 to 1992 Keywords Spatial panels, dynamic effects, spatial spillover effects, identification, estimation methods JEL Classification C21, C23, C51 1

2 Lecture + assignment: wwwregroningennl/elhorst click on spatial econometrics at the right Open rar file Finish assignment tomorrow 2

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4 Spatial econometric model Linear regression model extended to include Endogenous interaction effect (1): ρwy - Dependent variable y of unit A Dependent variable y of unit B - Y denotes an N 1 vector consisting of one observation on the dependent variable for every unit in the sample (i=1,,n) - W is an N N nonnegative matrix describing the arrangement of the units in the sample Exogenous interaction effects (K): WXθ - Independent variable x of unit A Dependent variable y of unit B - X denotes an N K matrix of exogenous explanatory variables Interaction effect among error terms (1): λwu - Error term u of unit A Error term u of unit B 4

5 5 W is an N N matrix describing the spatial arrangement of the spatial units in the sample Usually, W is row-normalized Row-normalizing gives W= /2 0 1/ This is an example of a row-normalized binary contiguity matrix for N=3

6 Spatial econometric models: from cross-section to panel data Y=ρWY+αι N +Xβ+WXθ+u, u=λwu+ε Cross-section data Y t =ρwy t +αι N +X t β+wx t θ+u t, u t =λwu t +ε t Space-time data Y t =ρwy t +X t β+wx t θ+µ+α t ι N +u t Spatial panel data µ: vector of spatial fixed or random effects α t : time period fixed or random effects (t=1,,t) Y t =τy t-1 +ρwy t +ηwy t-1 +X t β+wx t θ+µ+α t ι N +u t Dynamic spatial panel data 6

7 Fixed effects versus random effects specification Experience shows that spatial econometricians tend to work with space-time data of adjacent spatial units located in unbroken study areas, otherwise the spatial weights matrix cannot be defined Consequently, the study area often takes a form similar to all counties of a state or all regions in a country Under these circumstances the fixed effects model is more appropriate than the random effects model The idea that a limited set of regions is sampled from a larger population must be rejected and therefore the random effects models 7

8 8 Direct, indirect, and spatial spillover effects spatial panel data model Y t =ρwy t +X t β+wx t θ+µ+α t ι N +u t β θ θ θ β θ θ θ β ρ = = k k N2 k N1 k 2N k k 21 k 1N k 12 k 1 t Nk N 1k N Nk 1 1k 1 t Nk 1k w w w w w w W) (I x y x y x y x y x Y x Y Direct effect: Mean diagonal element Indirect effect: Mean row sum of non-diagonal elements Problem: Calculation of t-values of indirect effects (bootstrapping)

9 Dynamic spatial panel data model Y t =τy t-1 +ρwy t +ηwy t-1 +X t β+wx t θ+µ+α t ι N +u t Short-term (ignore τ and η) Y x 1k L Y x Nk t = (I ρw) 1 [ β k I N + θ k W] Long-term (set Y t-1 =Y t =Y* and WY t-1 =WY t =WY*) Y x 1k L Y x Nk = [(1 τ)i ( ρ + η)w] 1 [ β k I N + θ k W] 9

10 Empirical illustration: Cigarette Demand in the US Baltagi and Li (2004) estimate a demand model for cigarettes based on a panel from 46 US states log( C it ) = α +β 1 log(p it ) +β 2 log(y it ) +µ i (optional) + λ t (optional) + ε where C it is real per capita sales of cigarettes by persons of smoking age (14 years and older) This is measured in packs of cigarettes per capita P it is the average retail price of a pack of cigarettes measured in real terms Y it is real per capita disposable income Whereas Baltagi and Li (2004) use the first 25 years for estimation to reserve data for out of sample forecasts, we use the full data set covering the period Details on data sources are given in Baltagi and Levin (1986, 1992) and Baltagi et al (2000) They also give reasons to assume the state-specific effects ( µ i) and time-specific effects ( λ t) fixed, in which case one includes state dummy variables and time dummies for each year it, 10

11 Table 1 Estimation results of cigarette demand using panel data models without spatial interaction effects Determinants (1) (2) (3) (4) Log(P) (-2516) Log(Y) 0268 Pooled OLS Spatial fixed effects (1085) Intercept 3485 (3075) (-3888) (-066) Time-period fixed effects (-2266) 0565 (1866) Spatial and time-period fixed effects (-2563) 0529 (1167) R LogL LM spatial lag LM spatial error robust LM spatial lag robust LM spatial error LR test spatial fixed effects: (23157, with 46 degrees of freedom [df], p < 001) LR test time-period fixed effects: (4731, 30 df, p < 001) (robust) LM test (critical value 384): error model 11

12 Table 2 Estimation results of cigarette demand: specification with spatial and time-period specific effects Determinants (1) (2) (3) Spatial and timeperiod fixed effects Spatial and time-period fixed effects bias-corrected Random spatial effects, Fixed time-period effects W*Log(C) 0219 (667) 0264 (825) 0224 (682) Log(P) (-2502) (-2436) (-2491) Log(Y) 0601 (1051) 0603 (1027) 0593 (1071) W*Log(P) 0045 (055) 0093 (113) 0066 (081) W*Log(Y) (-373) (-393) (-355) Phi 0087 (681) σ R Corrected R LogL Wald test spatial lag 1483 (p=0001) 1796 (p=0000) 1390 (p=0001) LR test spatial lag 1575 (p=0000) 1580 (p=0000) 1448 (p=0000) Wald test spatial error 898 (p=0011) 818 (p=0017) 738 (p=0025) LR test spatial error 823 (p=0016) 828 (p=0016) 727 (p=0026) LR/Wald tests: Hausman/Phi tests: Fixed effects model 12

13 To test the hypothesis whether the can be simplified to the spatial error model, H 0 : θ+δβ=0, one may perform a Wald or LR test The results reported in the second column using the Wald test (898, with 2 degrees of freedom [df], p=0011) or using the LR test (823, 2 df, p=0016) indicate that this hypothesis must be rejected Similarly, the hypothesis that the can be simplified to the spatial lag model, H 0 : θ=0, must be rejected (Wald test: 1483, 2 df, p=0006; LR test: 1575, 2 df, p=0004) This implies that both the spatial error model and the spatial lag model must be rejected in favor of the Conclusion: Spatial Durbin model 13

14 Hausman's specification test can be used to test the random effects model against the fixed effects model The results (3061, 5 df, p<001) indicate that the random effects model must be rejected Another way to test the random effects model against the fixed effects model is to estimate the parameter "phi" ( φ 2 in Baltagi, 2005), which measures the weight attached to the cross-sectional component of the data and which can take values on the interval [0,1] If this parameter equals 0, the random effects model converges to its fixed effects counterpart; if it goes to 1, it converges to a model without any controls for spatial specific effects We find phi=0087, with t-value of 681, which just as Hausman's specification test indicates that the fixed and random effects models are significantly different from each other Conclusion: Fixed effects model 14

15 Table 1 Estimation results of cigarette demand using different model specifications Determinants (1) (2) (3) no fixed effects Intercept 2631 (1582) with fixed effects Dynamic with lag WY t-1 Log(C) (6504) W*Log(C) 0337 (1109) 0264 (825) 0076 (200) W*Log(C) (-029) Log(P) (-2180) (-2436) (-1319) Log(Y) 0554 (1496) 0603 (1027) 0100 (416) W*Log(P) 0780 (1115) 0093 (113) 0170 (366) W*Log(Y) (1109) (-393) (-087) R LogL Notes: t-values in parentheses Dynamic outperforms its non-dynamic counterpart 15

16 Table 2 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) (3) Short-term direct Short-term indirect Short-term direct effect Log(Y) Short-term indirect effect Log(Y) Long-term direct Long-term indirect Long-term direct effect Log(Y) Long-term indirect effect Log(Y) Notes: t-values in parentheses no fixed effects with fixed effects Dynamic with lag WY t (-1148) 0160 (349) 0099 (336) (-045) (-2339) (-2473) (-959) 0508 (727) (-226) 0610 (098) 0530 (1548) 0594 (1045) 0770 (355) (-747) (-215) 0345 (048) 16

17 Results non-dynamic A non-dynamic cannot be used to calculate short-term effect estimates of the explanatory variables 17

18 Table 2 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) (3) Long-term direct Long-term indirect Long-term direct effect Log(Y) Long-term indirect effect Log(Y) no fixed effects with fixed effects Dynamic with lag WY t (-2339) (-2473) (-959) 0508 (727) (-226) 0610 (098) 0530 (1548) 0594 (1045) 0770 (355) (-747) (-215) 0345 (048) The direct effects estimates of the two explanatory variables are significantly different from zero and have the expected signs Higher prices restrain people from smoking, while higher income levels have a positive effect on cigarette demand The price elasticity amounts to -101 and the income elasticity to

19 Table 1 Estimation results of cigarette demand using different model specifications Determinants (1) (2) (3) Dynamic no fixed effects with fixed effects with lag WY t-1 Log(P) (-2180) (-2436) (-1319) Log(Y) 0554 (1496) 0603 (1027) 0100 (416) Note that these elasticities (direct effects estimates) of -101 and the income elasticity to 0594 are different from the coefficient estimates of and 0603 due to feedback effects that arise as a result of impacts passing through neighboring states and back to the states themselves 19

20 Table 2 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) (3) Long-term direct Long-term indirect Long-term direct effect Log(Y) Long-term indirect effect Log(Y) no fixed effects with fixed effects Dynamic with lag WY t (-2339) (-2473) (-959) 0508 (727) (-226) 0610 (098) 0530 (1548) 0594 (1045) 0770 (355) (-747) (-215) 0345 (048) The spatial spillover effects (indirect effects estimates) of both variables are negative and significant Own-state price increases will restrain people not only from buying cigarettes in their own state, but to a limited extent also from buying cigarettes in neighboring states (elasticity -022) By contrast, whereas an income increase has a positive effects on cigarette consumption in the own state, it has a negative effect in neighboring states 20

21 The first result is not consistent with Baltagi and Levin (1992), who found that price increases in a particular state due to tax increases meant to reduce cigarette smoking and to limit the exposure of non-smokers to cigarette smoke encourage consumers in that state to search for cheaper cigarettes in neighboring states However, whereas Baltagi and Levin s (1992) model is dynamic, it is not spatial; and whereas our model so far contains spatial interaction effects, it is not (yet) dynamic 21

22 Results: Dynamic spatial panel data model Table 2 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) (3) Short-term indirect no fixed effects with fixed effects Dynamic with lag WY t (349) The short-term spatial spillover effect of a price increase turns out to be positive; the elasticity amounts to 016 and is highly significant (t-value 349) This finding is in line with the original finding of Baltagi and Levin (1992) in that a price increase in one state encourages consumers to search for cheaper cigarettes in neighboring states 22

23 Table 2 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) (3) Short-term direct Short-term direct effect Log(Y) Long-term direct Long-term direct effect Log(Y) no fixed effects with fixed effects Dynamic with lag WY t (-1148) 0099 (336) (-2339) (-2473) (-959) 0530 (1548) 0594 (1045) 0770 (355) Consistent with microeconomic theory, the short-term direct effects appear to substantially smaller than the long-term direct effects; versus for the price variable and 0099 versus 0770 for the income variable 23

24 Table 2 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) (3) Long-term direct Long-term direct effect Log(Y) no fixed effects with fixed effects Dynamic with lag WY t (-2339) (-2473) (-959) 0530 (1548) 0594 (1045) 0770 (355) The long-term direct effects in the dynamic, on their turn, appear to be greater (in absolute value) than their counterparts in the non-dynamic ; versus for the price variable and 0770 versus 0594 for the income variable Apparently, the non-dynamic model underestimates the long-term effects 24

25 Table 2 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) (3) Long-term indirect Long-term indirect effect Log(Y) no fixed effects with fixed effects Dynamic with lag WY t (727) (-226) 0610 (098) (-747) (-215) 0345 (048) Although greater and again positive, we do NOT find empirical evidence that the long-term spatial spillover effect is also significant A similar result is found by Debarsy et al (2011) The spatial spillover effect of an income increase is not significant either A similar result is found by Debarsy et al (2011) 25

26 Table 2 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) (3) Long-term indirect Long-term indirect effect Log(Y) no fixed effects with fixed effects Dynamic with lag WY t (727) (-226) 0610 (098) (-747) (-215) 0345 (048) Interestingly, the spatial spillover effect of the income variable in the non-dynamic spatial panel data model appeared to be negative and significant Apparently, the decision whether to adopt a dynamic or a non-dynamic model represents an important issue 26

27 Determinants (1) (2) (3) Dynamic no fixed effects with fixed effects with lag WY t-1 Log(C) (6504) W*Log(C) 0337 (1109) 0264 (825) 0076 (200) W*Log(C) (-029) Log(P) (-2180) (-2436) (-1319) Log(Y) 0554 (1496) 0603 (1027) 0100 (416) W*Log(P) 0780 (1115) 0093 (113) 0170 (366) W*Log(Y) (1109) (-393) (-087) LogL To investigate whether the extension of the non-dynamic model to the dynamic spatial panel data model increases the explanatory power of the model, one may test whether the coefficients of the variables Y t-1 and WY t-1 are jointly significant using an LR-test The outcome of this test (2 ( )=18638 with 2 df) evidently justifies the extension of the model with dynamic effects 27