Spatial Econometrics The basics

Transcription

1 Spatial Econometrics The basics Interaction effects spatial econometric models Direct and indirect (spatial spillover) effects Bootlegging effect Basic numerical findings Identification issues Estimation Software Storage/numerical problems The spatial weight matrix W Normalization of W Stationarity conditions Spatial panels Fixed vs. random effects SLX model Parameterizing W Model comparison Model selection Bayesian posterior model probabilities Role economic theory J.Paul Elhorst, University of Groningen, the Netherlands 1

2 - What is a spatial econometric model? - What are spatial interaction effects? - What are spatial spillover effects? - How to estimate a spatial econometric model? - How to interpret the outcomes of a spatial econometric model? - How to select the spatial weights matrix W and the right econometric model? - Economic theory. - Increase level up to level that is required for Spatial Economic Analysis 2

3 Course is based on: - Elhorst J.P. (2014) Spatial Econometrics: From Cross-sectional Data to Spatial Panels. Springer, Heidelberg New York Dordrecht London, Plus - Halleck Vega, S., Elhorst J.P. (2015) The SLX model. Journal of Regional Science 55(3): Halleck Vega S., Elhorst J.P. (2016) A regional unemployment model simultaneously accounting for serial dynamics, spatial dependence and common factors. Regional Science and Urban Economics 60 (2016) Elhorst, J.P. (2016) Spatial Panels. Encyclopedia of GIS (2nd Edition). Springer, Berlin Heidelberg. Forthcoming. - Elhorst J.P., Halleck Vega S.M (2017) The SLX model: Extensions and the sensitivity of spatial spillovers to W, Papeles de Economía Española 152: Plus - Computer lab assignments and exam questions 3

4 The idea that cross-sectional units interact with others has recently received considerable attention, as evidenced in the development of theoretical frameworks to explain social phenomena such as social norms, peer influence, neighborhood effects, network effects, contagion, epidemics, social interactions, interdependent preferences, etc. 4

5 5

6 Spatial econometric model Linear regression model (Y=Xβ+ε) 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 6

7 Linear spatial econometric model for cross-section data in vector notation Y=ρWY+αι N +Xβ+WXθ+u, u=λwu+ε Y denotes an N 1 vector consisting of one observation on the dependent variable for every unit in the sample (i=1,,n), ι N is an N 1 vector of ones associated with the constant term parameter α, X denotes an N K matrix of exogenous explanatory variables, with the associated T parameters β contained in a K 1 vector, and ε = ( ε1,..., εn) is a vector of disturbance terms, where ε i are independently and identically distributed error terms for all i with zero mean and variance σ 2. The total number of interaction effects in this model is K+2. 7

8 8 W is an N N matrix describing the spatial arrangement of the spatial units in the sample. Usually, W is row-normalized Example: Russia China - India Row-normalizing gives W= /2 0 1/ This is an example of a row-normalized binary contiguity matrix for N=3.

9 9 Let, Y = then / 0 2 1/ WY 2 1 = = Similarly, let = X, then / /2 0 1/ WX = = Finally, note that X may not contain a constant, since this constant and the corresponding WX variable = = /2 0 1/ WX would be perfectly multicollinear.

10 Table 1. Spatial econometric models with different combinations of spatial interaction effects and their flexibility regarding spatial spillovers Type of model Spatial # Flexibility spillovers interaction effects OLS, Ordinary least squares model - 0 Zero by construction SAR, Spatial autoregressive model WY 1 Constant ratios SEM, Spatial error model Wu 1 Zero by construction SLX, Spatial lag of X model WX K Fully flexible SAC, Spatial autoregressive WY, Wu 2 Constant ratios combined model SDM, Spatial Durbin model WY, WX K+1 Fully flexible SDEM, Spatial Durbin error model WX, Wu K+1 Fully flexible GNS, General nesting spatial model WY, WX, Wu K+2 Fully flexible 10

11 Figure 1. Comparison of different spatial econometric model specifications GNS Y = ρwy +αι u = λwu +ε N + Xβ+ WXθ+ u λ=0 θ=0 ρ=0 λ=0 SAC Y = ρwy +αι u = λwu +ε N + Xβ+ u θ=0 SDM Y N = ρwy + αι + Xβ+ WXθ+ ε SAR Y N = ρwy + αι + Xβ+ ε ρ=0 SLX ρ=0 θ=0 Y N = αι + Xβ + WXθ + ε OLS Y N = αι + Xβ+ε θ=-ρβ ρ=0 SDEM Y = αι N u = λwu +ε + Xβ+ WXθ+ u λ=0 θ=0 SEM Y = αι N + Xβ+ u u = λwu + ε (if θ=-ρβ then λ=ρ) λ=0 Note: GNS = general nesting spatial model, SAC = spatial autoregressive combined model (SARAR), SDM = spatial Durbin model, SDEM = spatial Durbin error model, SAR = spatial autoregressive model (spatial lag model), SLX= spatial lag of X model, SEM = spatial error model, OLS = ordinary least squares model 11

12 Direct, indirect=spatial spillover effects Cross-section or non-dynamic spatial panel data model Y t =ρwy t +X t β+wx t θ+μ+α t ι N +u t Reduced form: Y t =(I-ρW) -1 [X t β+wx t θ+μ+α t ι N +u t ] E(Y) x1k. E(Y) x Nk t E(y1) x1k =. E(y N ) x1k = (I ρw) 1... ( β k I E(y x. E(y x N + θ 1 Nk N Nk K ) ) t W) = (I ρw) 1 βk w 21θ. w N1θ k k w w 12 β. k N2 θ θ k k.... w w 1N 2N. β k θ θ k k = Direct effect: Mean diagonal element (or of different groups) Indirect effect: Mean row sum of off-diagonal elements Problem: t-values of direct and indirect effects are bootstrapped Note: Error terms (μ+α t ι N +u t ) drop out due to taking expectations 12

13 Table Direct and spillover effects corresponding to different model specifications Model Direct effect Spillover effect OLS / SEM (Wu) β k 0 SAR (WY)/ SAC (WY, Wu) * SLX / SDEM WX / Wu Average diagonal element of (I-ρW) -1 β k β k Average row sum of offdiagonal elements of (I-ρW) -1 β k θ k SDM / GNS WY+WX/Wu Average diagonal element of (I-ρW) -1 [β k +Wθ k ] Average row sum of offdiagonal elements of (I-ρW) -1 [β k +Wθ k ] * Ratio between the spillover effect and the direct effect in the SAR/SAC model is the same for every explanatory variable. 13

14 Two further properties: global and local spillover effects Indirect effects that occur if ρ=0 (of WY) are known as local spillover effects E(y1) x1k. E(y N ) x1k... E(y x. E(y x 1 Nk N Nk ) ) βk w 21θ =. w N1θ k k w w 12 β. k N2 θ θ k k.... w1nθ w 2Nθ. β k k k = β k I N + θ k W. This is local because the indirect effects only fall on spatial units for which the elements of W are non-zero. Local spillovers go together with dense(r) W matrix. 14

15 15 Indirect effects that occur if ρ 0 (of WY) are known as global spillover effects k k Nk N k N Nk k W W W I x y E x y E x y E x y E β ρ ρ ρ β ρ...) W I ( ) ( ) (. ) (... ) (. ) ( = = This is global because the indirect effects fall on all units; even if W contains many zero elements, (I-ρW) -1 will not. Global spillovers tend to go together with sparse(r) W matrix; due to the higher-order terms ρ g W g (g>1) locations farther away are reached anyway even if they are not directly connected.

16 Two basic identification problems: (1) How to find the right/best spatial econometric model specification. (2) How to find the right/best specification of the spatial weights matrix. Logical approach (general-to-specific approach) would be take GNS as point of departure. However, the GNS model is often (but not always) overfitted; the parameters of the GNS model can be estimated, but they have the tendency either to blow each other up or to become insignificant, as a result of which this model does not help to choose among simpler models with less spatial interaction effects, especially the SDM and SDEM specifications. Furthermore, there is a slight identification problem in the GNS model; sometimes ρ and λ are interchanged (see GNS2 column in Table below). One basic question is: Do global spillover effects make sense from a theoretical viewpoint, SDM vs. SDEM, and related to this is W dense or sparse? 16

17 Empirical illustration: Cigarette Demand in the US Baltagi and Li (2004) estimate a demand model for cigarettes based on a panel from 46 U.S. states (N=46) log( C it ) = a +β 1 log(p it ) +β 2 log(y it ) +µ i (optional) + l 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 (T=30). 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. We have reasons to believe that spatial interaction effects need to be included in this model! it, 17

18 BOOTLEGGING The main motivation to extend the basic model to include spatial interaction effects is the so-called bootlegging effect; consumers are expected to purchase cigarettes in nearby states, legally or illegally (smuggling), if there is a price advantage. This smuggling behavior is a result of significant price variation in cigarettes across US states and partly due to the disparities in state cigarette tax rates. Baltagi and Levin (1986, 1992) incorporate the minimum real price of cigarettes in any neighboring state as a proxy for the bootlegging effect. A limitation is that this proxy does not account for cross-border shopping that may take place between other states than the minimum-price neighboring state (Baltagi and Levin, 1986). This can be due to smuggling taking place over longer distances by trucks since cigarettes can be stored and are easy to transport (Baltagi and Levin, 1992) or due to geographically large states where cross-border shopping may occur in different neighboring states. To take this into account, other studies have extended the model to explicitly incorporate spatial interaction effects. However, while the specification originally adopted by Baltagi and Levin (1992) resembles the SLX model but then with only one exogenous interaction effect (price), applied spatial econometric studies have either included: (i) endogenous interaction effects, (ii) interaction effects among the error terms or (iii) a combination of endogenous and exogenous interaction effects. 18

19 Basic findings TABLE Model comparison of the estimation results explaining cigarette demand OLS SAR SEM SLX SAC SDM SDEM GNS GNS2 ln(p) (-25.63) (-24.48) (-24.68) (-24.77) (-24.49) (-24.60) (-24.88) (-25.40) ln(i) (11.67) (9.86) (11.07) (10.38) (10.51) (10.33) (10.57) (11.02) W ln(c) (6.79) (-0.22) (6.85) (-7.01) W ln(p) (-2.95) (0.62) (-2.24) (-5.97) W ln(i) (-2.80) (-3.70) (-2.12) (0.85) W u (7.26) (4.73) (6.95) (14.60) R Log-likelihood Notes: t-values are reported in parentheses; state and time-period fixed effects are included in every model, W = pre-specified binary contiguity matrix 19

20 In this case including endogenous interaction effects (SAR model) implies that state cigarette sales directly affect one another, which is difficult to justify. The resulting global spillovers would mean that a change in price (or income) in a particular state potentially impacts consumption in all states, including states that according to W are unconnected.1 Pinkse and Slade (2010, p. 115) argue that an empirical problem like this is insightful precisely because it is difficult to form a reasonable argument to include endogenous interaction effects even though they are easily found statistically. Given the research question of whether consumers purchase cigarettes in nearby states if there is a price advantage, this example points towards a local spillover specification such as the SLX model rather than a global spillover specification. The model is aggregated over individuals since the objective is to explain sales in a particular state, as in Baltagi and Levin (1986, 1992), Baltagi and Li (2004), Debarsy et al. (2012), and Elhorst (2014), among others. If the purpose, on the other hand, is to model individual behavior (e.g., the reduction in the number of smokers or teenage smoking behavior) then this is better studied using micro data. 1This implies that e.g., price changes in California would exert an impact on cigarette consumption even in states as distant as Illinois or Wisconsin. 20

21 TABLE Model comparison of the estimated direct and spillover effects on cigarette demand Direct effects OLS SAR SEM SLX SAC SDM SDEM GNS ln(p) (-25.63) (-25.10) (-24.68) (-24.77) (-24.47) (-24.84) (-24.88) (-25.43) ln(i) (11.67) (10.18) (11.07) (10.38) (10.56) (10.88) (10.57) (10.35) Spillover effects ln(p) , (-5.63) (-2.95) (0.17) (-2.39) (-2.24) (-1.89) ln(i) (5.51) (-2.80) (-0.20) (-2.30) (-2.12) (-2.16) W = pre-specified binary contiguity matrix Basic findings (W pre-specified, fixed!!!) 1. The direct effects produced by the different models are comparable. 2. The spatial spillover effects produced by the different models differ widely. 3. For various reasons the OLS, SAR, SEM, SAC and GNS models need to be rejected. 21

22 The OLS model is outperformed by other, more general models. The spillover effects of the SEM model are zero by construction, while the results of more general models show that the spillover effect of the price and income variable are significant. The SAR and SAC models suffer from the problem that the ratio between the spillover effect and the direct effect is the same for every explanatory variable. Consequently, the spillover effect of the income value variable gets a wrong and significant sign. Finally, the GNS model is overparameterized, as a result of which the t-values of the coefficient estimates and the effects estimates have the tendency to go down (not here!, see case below), or its parameters cannot be reproduced using Monte Carlo simulation experiments (see columns GNS2). In sum, only the SLX, SDM and SDEM model produce acceptable results. However, it is not clear which of these three models best describes the data. Even though they produce spillover effects that are comparable to each other, both in terms of magnitude and significance, this is worrying since these models have a different interpretation (global or local). Furthermore, the price spillover effect has a negative sign rather than the expected positive sign; could it be that W=binary contiguity matrix is wrong? ****************************************************** 22

23 Software and methods of estimation Software packages and/or routines to estimate spatial econometric models are Stata, Geoda, R and Matlab. The latter three are all freely downloadable (except for Matlab itself). The assignment Spatial Modeling is meant to learn to work with Matlab and Stata. In addition, there is an assignment on economic growth in Europe. 23

24 Storage and numerical problems: W is an N N matrix with N 2 entries. N N 2 12 provinces municipalities 246, zip codes 16,000,000 1,000,000 points 1,000,000,000,000 However, many elements of W will be zero since many spatial units are not considered to be neighbors of each other. In practice, less than 5% of the observations are nonzero. Estimation of spatial econometric models involves the manipulation of N N matrices, such as matrix multiplication, matrix inversion, the computation of characteristic roots and/or Cholesky decomposition. These manipulations may be computationally intensive and/or may require significant amounts of memory if N is large. Software that does not store zero observations and that uses sparse matrix algorithms is much faster. Software packages to estimate spatial econometric models are Matlab, R, GeoDa, and (partly) Stata; R and GeoDa are freely downloadable. Some of these packages, among which Matlab, also have routines to quickly compute the Jacobian term. 24

25 Method 1: OLS (sometimes called spatial OLS) of the SAR model Create the WY variable (e.g., in excel) and then regress Y on the [WY, X] variables using one of the standard software packages, such as SPSS, Eviews, Stata, etc. Result: Parameter estimates will be (severely) biased, since this estimation method ignores the endogeneity of the WY variable. Recommendation: Not to be recommended! Unless there is no or slow distance decay: common factors or foreign variables: Lee (2002) Consistency and efficiency of least squares estimation for mixed regressive, spatial autoregressive models. Econometric Theory 18, , Pesaran M.H., Schuermann T., Weiner S.M. (2004) Modeling Regional Interdependencies using a Global Error-Correcting Macroeconomic Model. Journal of Business and Economic Statistics 22,

26 Method 2: Maximum likelihood (ML) SAR model The model Y=ρWY+αι N +Xβ+ε can be rewritten as Y = (I ρw) αι + (I ρw) Xβ + (I ρw) ε. N The log-likelihood function of this model can be seen to be N 2 1 LogL = log(2πσ ) + log I ρw 2 2 2σ where e=y-ρwy-αι N -Xβ e'e The term log I-ρW represents the Jacobian term of the transformation from ε to Y taking into account the endogeneity of WY. It is this term which the OLS estimator ignores and why the OLS estimator is biased, even though ρwy is accounted for in the residual e. 26

27 Method 3: Instrumental variables (IV), Generalized Method of Moments (GMM) SAR model Since WY is endogenous, first regress WY on [X,WX,W 2 X,,W g X] variables, and use this first-stage regression to predict WY, to get W Ŷ. In a model without WX variables g=1 is sufficient. In a model with WX variables, g should at least be 2. Note: this may cause a weak instrument problem! Then regress Y on [ W Ŷ,X] (second-stage regression) to get the coefficient estimates of ρ and β of the SAR model (note: in SDM Y is regressed on on [ WŶ,X,WX]. This method is called 2SLS (two-stage least squares) or spatial 2SLS. Note that WY and W g X for g=1,2,.. can be calculated in advance and then be stored. This was one way to overcome storage problems and explains why this method has been developed. 27

28 The spatial weight matrix W The chosen specification of W is of vital importance: 1. The value of the interaction parameter depends on the specification of W. 2. The significance level of the interaction parameter depends on the specification of W. 3. The specification of W should follow from the theory at hand. In principle, different theories imply different W. However, although one may look to economic theory for guidance, it often has little to say about the specification of W. Therefore, empirical researchers often investigate whether the results are sensitive to the specification of W. 28

29 w W=. w 11 N1... w w 1N. NN. The row elements of a weights matrix display the impact on a particular unit by all other units, while the column elements of a weights matrix display the impact of a particular unit on all other units 29

30 Spatial weights matrices most often used in empirical research in spatial econometrics: 1. p-order binary contiguity matrices (if p=1 only first-order neighbors are included, if p=2 first and second order neighbors are considered, and so on). 2. Inverse distance matrices (with or without a cut-off point) or exponential distance decay matrices. 3. q-nearest neighbor matrices (where q is a positive integer). 4. Block diagonal or group interaction matrices where each block represents a group of spatial units that interact with each other but not with observation in other groups. 5. Leader matrices. Be careful that W is not endogenous, otherwise consult Qu and Lee (2014) on endogenous spatial weight matrix in Journal of Econometrics. 30

31 Row-normalization normalized wij w ij =. N w j= 1 ij Row-normalization is standard in spatial econometrics! 31

32 As an alternative to row-normalization, one may divide the elements of W by its largest characteristic root, ω max, W S =1/ω max W, which might be labeled as matrix normalization (This has the effect that the characteristic roots of W are also divided by ω max, as a result of which ω S max=1, just like the largest characteristic root of a row- or columnnormalized matrix). The advantage of matrix normalization is that the mutual proportions between the elements of W remain unchanged. For example, scaling the rows or columns of an inverse distance matrix so that the weights sum to one would cause this matrix to lose its economic interpretation for this decay. 32

33 Take care of the following conditions with respect to W: (a) the row and column sums of the matrices W, (I-ρW) -1 and (I-λW) -1 before W is row-normalized should be uniformly bounded in absolute value as N goes to infinity, or (b) the row and column sums of W before W is row-normalized should not diverge to infinity at a rate equal to or faster than the rate of the sample size N. Condition (a) is originated by Kelejian and Prucha (1998, 1999), and condition (b) by Lee (2004). Both conditions limit the cross-sectional correlation to a manageable degree, i.e., the correlation between two spatial units should converge to zero as the distance separating them increases to infinity. Under condition (a) or (b), the endogeneity of WY should be accounted for, either by ML or by IV/GMM estimation. 33

34 (c) Strong divergence: the row and column sums of W before W is normalized should diverge to infinity at a rate faster than NN. Condition (c) is originated by Lee (2002). Under this condition, the correlation between two spatial units converges so slowly to infinity that all units may say to affect each other. If this is the case estimation by OLS is consistent (coefficient estimates are not biased if sample size is large enough; if sample size is small one may still have a small sample bias). Instead of OLS, one may still estimate the coefficients ML or by IV/GMM to avoid small sample problems (See ECB document, eq.(10), in the assignment, the bias of the OLS estimator is inversely related to N, so if N goes to infinity, the bias disappears). When the spatial weights matrix is a binary contiguity matrix, (a) is satisfied. Normally, no spatial unit is assumed to be a neighbour to more than a given number, say q, of other units. 34

35 By contrast, when the spatial weights matrix is an inverse distance matrix, (a) may not be satisfied. Consider an infinite number of spatial units that are linearly arranged. The distance of each spatial unit to its first left- and right-hand neighbour is d; to its second left- and right-hand neighbour, the distance is 2d; and so on. When W is an inverse distance matrix and the off-diagonal elements of W are of the form 1/d ij, where d ij is the distance between two spatial units i and j, each row sum is 2 ( d + 2d + + 3d...), representing a series that is not finite. This is perhaps the reason why some empirical applications introduce a cut-off point d* such that w ij =0 if d ij >d*. However, since the ratio 2 ( d + 2d + + 3d...)/N 0 as N goes to infinity, condition (b) is satisfied, which implies that an inverse distance matrix without a cut-off point does not necessarily have to be excluded in an empirical study for reasons of consistency. 35

36 The opposite situation occurs when all cross-sectional units are assumed to be neighbours of each other and are given equal weights. In that case all off-diagonal elements of the spatial weights matrix are w ij =1. Since the row and column sums are N-1, these sums diverge to infinity as N goes to infinity. In contrast to the previous case, however, (N-1)/N 1 instead of 0 as N goes to infinity. This implies that a row-normalized spatial weights matrix with equal weights, w ij =1/(N-1), must be excluded for reasons of consistency. The OLS estimator seems to be consistent in this case, but one needs to be cautious since there are other potential pitfalls when using the single group interaction matrix. Kelejian and Prucha (2002) and Lee (2004) point out that the spatial lag ρρρρyy tt in this specific case is proportional to the unit vector (ιι NN ) as N goes to infinity, and thus collinear with the intercept in the model. 36

37 The SLX approach Halleck Vega and Elhorst (SLX model, 2015, JRS; 2017, PEE): 1. Only in the SLX, SDEM, SDM and GNS models can the spatial spillover effects take any value. The SLX model is the simplest one in this family of spatial econometric models. 2.In the SLX model (JRS) W can easily be parameterized (ww iiii = 1 dd γγ ). Further note that there are K spatial lags WX, and only one WY and only one Wu, so it makes sense to focus on WX variables first. The estimation of this model also does not cause severe additional econometric problems (no endogeneity, no regularity conditions), provided that the explanatory variables X are exogenous and the functional form of spatial weights matrix W is known and exogenous. 37

38 Parameterizing W in the SLX model Inverse distance: ww iiii = 1 dd iiii γγ Negative exponential: ww iiii = exp ( δδdd iiii ) Gravity type of function: ww iiii = PP γγ1 ii PP γγ2 jj, dd γγ3 iiii where P measures the size of units i and j in terms of population and/or gross product. Preferably, theory should be the driving force behind W, the gravity type of model is such a theory. The spatial weights matrix of every exogenous spatial lag W k X k may 1 also be modeled as ww iiiiii = γγ dd iiii kk ; why should the distance decay effect be the same for every explanatory variable. 38

39 Table 3 SLX model estimation results explaining cigarette demand and the parameterization of W BC ID (γ=1) (1) (2) (3) Price (-24.77) (-25.28) (-24.43) Income (10.38) (13.73) (15.39) W Price (-0.34) (3.08) (-2.95) W Income (-2.80) (-6.63) (-4.76) γ distance (16.48) ID ED (4) (-29.58) (15.44) (2.08) (1.80) (9.99) ID γ k s (7) (-24.49) (15.76) (1.81) (-5.21) γ distance price in col.(7) and γ own population in col. (8) (8.86) γ distance income in col.(7) and γ population neighbors in col.(8) (17.70) ID Gravity (8) (-23.03) (15.16) (0.87) (-4.97) (11.50) (-0.41) R LogL LM test WY, W LM test Wu, W LM test WY, W=BC LM test Wu, W=BC Prob. SDM Prob. SDEM Notes: t-statistics in parentheses; coefficient estimates of WX variables in the SLX represent spillover effects. W matrix similar to the one used to model exogenous spatial lags WX (2.63) 39

40 (Robust) LM-tests To test whether the spatial lag model or the spatial error model is more appropriate to describe the data than a model without any spatial interaction effects (OLS model), Lagrange Multiplier (LM) tests have been developed. They test for a spatially lagged dependent variable or for spatial error autocorrelation. The robust LM-tests test for a spatially lagged dependent variable in the local presence of spatial error autocorrelation or for spatial error autocorrelation in the local presence of a spatially lagged dependent variable. These tests are based on the residuals of the OLS model with or without spatial and/or time-period fixed effects and follow a chi-squared distribution with one degree of freedom, which has a critical value of 3.84 at the 5% significance level. In the previous table, these tests are applied to the SLX model, testing whether the SLX model should be extended to SDM or SDEM, but whether these tests have any power in this setting has not been investigated up to now (SEA has paper with this issue under review). 40

41 Bayesian comparison approach To choose between SDM and SDEM, and thus between a global or local spillover model, as well as to choose between different potential specifications of W, a Bayesian comparison approach may be applied. This approach determines the Bayesian posterior model probabilities of the SDM and SDEM specifications given a particular W matrix, as well as the Bayesian posterior model probabilities of different W matrices given a particular model specification. These probabilities are based on the log marginal likelihood of a model obtained by integrating out all parameters of the model over the entire parameter space on which they are defined. If the log marginal likelihood value of one model or of one W is higher than that of another model or another W, the Bayesian posterior model probability is also higher. Whereas the popular likelihood ratio, Wald and/or Lagrange multiplier statistics compare the 41

42 performance of one model against another model based on specific parameter estimates within the parameter space, the Bayesian approach compares the performance of one model against another model, in this case SDM against SDEM, on their entire parameter space. This is the main strength of this approach. Inferences drawn on the log marginal likelihood function values for the SDM and SDEM model are further justified because they have the same set of explanatory variables, X and WX, and are based on the same uniform prior for ρ and λ. This prior takes the form p(ρ)=p(λ)=1/d, where D=1/ω max -1/ω min and ω max and ω min represent respectively the largest and the smallest (negative) eigenvalue of the matrix W. This prior requires no subjective information on the part of the practitioner as it relies on the parameter space (1/ω min, 1/ω max ) on which ρ and λ are defined, where ω max =1 if W is row-normalized. Note: only available in Matlab. 42

43 We find that the Bayesian posterior model probability for SDEM when W is specified as the parameterized distance matrix is (W in error term is same parameterized distance matrix binary contiguity matrix). The LM tests more point into the direction of SDM, but this might be due to low power and does not hold over the entire parameter space. Note: Bayesian comparison approach has been applied successfully in: Firmino Costa da Silva D., Elhorst J.P., Neto Silveira R.d.M. (2017), Urban and Rural Population Growth in a Spatial Panel of Municipalities, Regional Studies 51(6):

44 Yesilyurt M.E., Elhorst J.P. (2017) Impacts of neighboring countries on military expenditures: A dynamic spatial panel approach. Journal of Peace Research, forthcoming. Table II. Simultaneous Bayesian comparison of model specifications and spatial weight matrices Data Model W1 W2 W3 Enemy W1 + Enemy W1 + Superpower COW Static model COW Dynamic model WB/SIPRI Static model WB/SIPRI Dynamic model W1 + Dominanc e W1 + Enemy + Superpowers s SAR SDM SEM SDEM SAR SDM SEM SDEM SAR SDM SEM SDEM SAR SDM SEM SDEM The highest probability in each row is in bold and the probabilities in each block sum to 1. Source: Own calculations, based on LeSage (2014, 2015). Row total 44

45 Extension of SLX to SDM or SDEM SDM: YY = ρρρρρρ + ααιι NN + XXXX + WWWWWW + εε Spatial spillovers: (II ρρρρ) 1 (II NN ββ + WWWW) global, W sparse SDEM: YY = ααιι NN + XXXX + WWWWWW + uu and uu = λλλλλλ + εε Spatial spillovers: θ local, W dense - Classic and robust LM tests proposed by Anselin (1988) and Anselin et al. (1996) to test for the extension with either WY or Wu: unclear whether they are very powerful in the presence of WX variables, since this has not been investigated (only if WX is not included). 45

46 - Estimation of the GNS model (WY+WX+Wu) is also not of much help due to overfitting. - LeSage (2014): Bayesian comparison approach: simultaneously selecting appropriate model specification and spatial weights matrix. This approach is promising, but has only been developed for exogenous Ws and not for parameterized Ws. Furthermore, it is assumed that W of WY, WX and Wu is the same everywhere, but if W of WX and WY (or Wu) is parameterized by 1/dd γγ, we also expect γγ WWWW γγ WWWW ( γγ WWWW ). 46

47 Table 4 Beyond the SLX model with parameterized W: SDEM and SDM Point estimates SDEM ID+λW BC u (1) Price (-23.03) Income (15.16) W ID (γ WX ) Price (0.87) W ID (γ WX ) Income (-4.97) SDM(a) ID+ρW BC Y (2) (-23.13) (13.35) (4.68) (-16.04) W BC u (4.58) (2) W BC Y (3) W ID (γ WY )Y (5.19) γ WX (21.36) (12.61) SDM(b) ID+ ρw ID (γ WY )Y (3) (-24.60) (16.27) (1.30) (-17.54) (-3.78) (15.43) γ WY (0.971) Direct effects (b1) (b2) Price (-23.03) Income (15.16) Spillover effects (-23.60) (13.66) (-23.87) (16.18) Price (0.87) (4.43) (4.32) Income (-4.97) (-4.37) (-4.40) R LogL (-22.77) (15.36) (0.24) (-0.29) Notes: t-statistics are reported in parentheses. W of WX variables parameterized by w ij =1/d ij γ with γ estimated. 1) SDEM=SLX extended with Wu where W is binary contiguity matrix. 2) SDM=SLX extended with WY where W is binary contiguity matrix. 3) SDM=SLX extended with WY where W is parameterized inverse distance matrix, w ij =1/d ij γ, and different γ parameters for WY and WX (see Section 4 for details). 47

48 SDEM Suppose W of WX variables is misspecified and that W* is the true matrix. This misspecification (W-W*)X is transmitted to the error term, as a result of which VVVVVV(εε) σσ 2 II. Instead, we have VVVVVV(uu) = σσ 2 [(II λλλλ) TT (II λλλλ)] 1 with VV WW, which only affects the efficiency of the parameter estimates but not their consistency. V=BC matrix may therefore suffice. A Hausman test can be used for comparing SLX and SDEM estimates LeSage and Pace (2009, p. 62). Rejection of the null hypothesis of equality in SLX and SDEM coefficient estimates can be useful in diagnosing the presence of omitted variables that are correlated with variables included in the model: (i) No rejection and spatial autocorrelation coefficient is not significant. Extension of the SLX model may be left aside. (ii) No rejection but spatial autocorrelation coefficient is significant. SDEM re-specification of the SLX model leads to an efficiency gain. (iii) Rejection, points to serious misspecification problems due to omission of relevant explanatory variables. 48

49 SDM W parameterized Perfect solution problem (Halleck Vega and Elhorst, 2015): If ww iiii = 1 dd iiii γ, the solution ρ = 1, γ = 0, α = YY YY NN, and ββ = 00 in the SAR model (YY = ρwwww + αιι NN + XXXX + εε) would perfectly fit the dependent variable in the SAR model, as well as the SEM and SDM models (together with θθ = 00). Formally, this perfect solution has been excluded by conditions (a) and (b). However, computer software working with real data and fixed sample sizes, NN = NN, cannot simply rule out these parameter values for ρ and γ, as a result of which it often converges to the perfect solution computationally. Figure 1. Log-likelihood function for the decay parameter γ WY using cigarette demand data set ,3 0,6 0,9 1,2 1,5 1,8 2,1 2,4 2,7 3 3,3 3,6 3,9 4,2 4,5 4,8 5,1 5,4 5,7 6 6,3 6,6 6,9 7,2 7,5 7,8 8,1 8,4 8,7 9 9,3 9,6 9,9 Note: Values of γ WY are in the interval (0, 10] using increments of

50 To demonstrate the importance of the uncertainty about specifying the appropriate W of WY, we also simulated the direct and spillover effects of SDM(b), but then also accounting for the variance in γ WY rather than treating it as fixed. These effects estimates are reported under (b2) in column 3 of Table 4. The direct effects are again relatively constant. The key difference is observed in the spillover impacts (both for the price and income variables), which are small and not different from zero when the distance decay parameter is not taken as fixed. Therefore, what is observed from this application to state-level cigarette demand is that there are significant spillover effects, but conditional on the exogeneity of W. That is, we can only claim there is evidence of spillovers (e.g., bootlegging behavior) if we are willing to accept that W is exogenous. The fact that the log-likelihood function for γ WY (Figure 1) is not strictly concave and rather flat points to identification problems since there is not a clear unique global maximum. It thus seems that spatial econometric models containing an endogenous spatial lag (i.e. global spillovers) are more difficult to identify. 50

51 Conclusion The sign, magnitude, and significance level of the spillover effects are sensitive to both the specification of W and the spatial econometric model specification; the SLX model helps to test different non-parameterized and parameterized specifications of W against each other. The claim made in many empirical studies that their results are robust to the specification of W should thus be more sufficiently substantiated. It might be that these studies mainly focus on the direct effects rather than the spatial spillover effects, which are often the main issue in spatial econometric studies. Test statistics that help to find out whether the SLX model should be further extended to SDM or SDEM are still in their infancy. ****************************************************** 51

52 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 Y t =ρwy t +X t β+wx t θ+μ+α t ι N +u t Space-time data 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 52

53 Reasoning to consider fixed effects/random effects models The standard reasoning behind spatial specific effects is that they control for all space-specific time-invariant variables whose omission could bias the estimates in a typical cross-sectional study (Baltagi, 2005). The spatial specific effects may be treated as fixed effects or as random effects. In the fixed effects model, a dummy variable is introduced for each spatial unit, while in the random effects model, μ i is treated as a random variable that is independently and identically distributed with zero mean and variance σ μ. Furthermore, it is assumed that the random variables μ i and ε it are independent of each other. 53

54 To test the assumption of zero correlation between the random effects μ i and the explanatory variables, the Hausman specification test might be used (Baltagi 2005, pp ). The hypothesis being tested is H 0 : h=0, where h = d [var(d)] T 1 d = βˆ FE βˆ RE var(d) = σˆ d σ 2 T 1 2 *T * 1 RE (X X ) ˆ FE (X X ) Note the reverse sequence with which d and var(d) are calculated. This test statistic has a chisquared distribution with K degrees of freedom (the number of explanatory variables in the model, excluding the constant term). Hausman's specification test can also be used when the model is extended to include spatial error autocorrelation or a spatially lagged dependent variable. Since the SAR model has one additional explanatory variable, the test statistic for this T T T T model, which should be calculated by d = [ βˆ ˆ] ρ FE [ˆ β ˆ] ρ RE, has a chi-squared distribution with K+1 degrees of freedom. A similar change applies to WX variables. 54

55 Fixed effects versus random effects specification (short explanation) 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 specification. In addition, experience shows that the random effects is generally rejected in favor of the fixed effects model using the Hausman test. Longer explanation: slides

56 The random effects model is quite popular among spatial econometricians, which can be explained by three reasons. (1) It may be considered as a compromise solution to the all or nothing way of utilizing the cross-sectional component of the data. Panel data models with controls for spatial fixed effects only utilize the time-series component of the data, whereas these models without such controls employ both time-series and cross-sectional components. The parameter φ in random effects models, which can take values on the interval [0,1], may be used to estimate the weight that may be attached to the cross-sectional component of the data. If this weight equals 0, the random effects model reduces to the fixed effects model; if it goes to 1, it converges to its counterpart without controls for spatial fixed effects. (2) The random effects model avoids the loss of degrees of freedom incurred in the fixed effects model associated with a relatively large N. Besides, the spatial fixed effects can only be estimated consistently when T is sufficiently large, because the number of observations available for the estimation of each μ i is T. Recall that the inconsistency of μ i is not transmitted to the estimator of the slope coefficients β, since it is not a function of the estimated μ i. In other words, the incidental parameters problem does not matter when β are the coefficients of interest and the spatial fixed effects μ i are not, which is the case in most empirical studies. (3) It avoids the problem that the coefficients of time-invariant variables or variables that only vary a little cannot be estimated. 56

57 Despite its popularity, the question whether the random effects model is also an appropriate specification is often left unanswered. Three conditions should be satisfied before the random effects model may be implemented: (1) The number of units should potentially be able to go to infinity. (2) The units of observation should be representative of a larger population. (3) The traditional assumption of zero correlation between the random effects μ i and the explanatory variables needs to be made, which in general is particularly restrictive. Whether these conditions are satisfied in spatial research is at least controversial. 57

58 There are two types of asymptotics that are commonly used in the context of spatial observations: (a) The infill asymptotic structure, where the sampling region remains bounded as N. In this case more units of information come from observations taken from between those already observed; and (b) The increasing domain asymptotic structure, where the sampling region grows as N. In this case there is a minimum distance separating any two spatial units for all N. According to Lahiri (2003), there are also two types of sampling designs: (a) The stochastic design where the spatial units are randomly drawn; and (b) The fixed design where the spatial units lie on a nonrandom field, possibly irregularly spaced. The spatial econometric literature mainly focuses on increasing domain asymptotics under the fixed sample design (Cressie 1993, p. 100; Griffith and Lagona 1998; Lahiri 2003). 58

59 Although the number of spatial units under the fixed sample design can potentially go to infinity, this design is incompatible with the increasing domain asymptotic structure. If there is a minimum distance separating spatial units and the researcher wants to collect data for a certain type of spatial units within a particular study area, there will be an upper bound on the number of spatial units. Furthermore, when data on all spatial units within a study area are collected it is questionable whether they are still representative of a larger population. For a given set of regions, such as all counties of a state or all regions in a country, the population may be said to be sampled exhaustively (Nerlove and Balestra 1996, p. 4), and the individual spatial units have characteristics that actually set them apart from a larger population (Anselin 1988, p. 51). In other words, if the data happen to be a random sample of the population, unconditional inference about the population necessitates estimation with random effects. If, however, the objective is limited to making conditional inferences about the sample, then fixed effects should be specified. 59

60 In spatial research there is a prominent reason why investigators generally do not draw a limited sample of units from a particular study area, but rather work with cross-sectional or space-time data of adjacent spatial units located in unbroken study areas. This is because otherwise the spatial weights matrix cannot be defined and the impact of spatial interaction effects cannot be consistently estimated. Only when neighboring units are also part of the sample, it is possible to measure the impact of these neighboring units. In other words, this type of research just requires that the data covers the whole population, since it would break down when having a random sample of the population. In conclusion, we can say that the fixed effects model is generally more appropriate than the random effects model since spatial econometricians tend to work with space-time data of adjacent spatial units located in unbroken study areas, such as all counties of a state or all regions in a country. 60

61 Direct, indirect, and spatial spillover effects Cross-section or non-dynamic spatial panel data model Y t =ρwy t +X t β+wx t θ+μ+α t ι N +u t Reduced form: Y t =(I-ρW) -1 [X t β+wx t θ+μ+α t ι N +u t ] E(Y) x1k. E(Y) x Nk t E(y1) x1k =. E(yN ) x1k... E(y x. E(y x 1 Nk N Nk ) ) t = (I ρw) 1 βk w 21θ. w N1θ Direct effect: Mean diagonal element (or of different groups) Indirect effect: Mean row sum of off-diagonal elements Problem: Calculation of t-values of indirect effects (bootstrapping) k k w w 12 β. k N2 θ θ k k.... w w 1N 2N. β k θ θ k k 61

62 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) E(Y) x1k E(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*) E(Y) x1k E(Y) x Nk = [(1 τ)i ( ρ + η)w] 1 [ β Note: Available in Matlab (and R), long-term effects not available in Stata. Generally, it is hard to find significant spillovers since they depend on so many parameters (3 short term, 5 long term); many empirical studies do not recognize this. k I N + θ k W]. 62

63 Empirical illustration: Cigarette Demand in the US Table 1. Estimation results of cigarette demand using panel data models without spatial interaction effects Determinants (1) (2) (3) (4) Log(P) (-25.16) Log(Y) Pooled OLS Spatial fixed effects (10.85) Intercept (30.75) (-38.88) (-0.66) Time-period fixed effects (-22.66) (18.66) Spatial and time-period fixed effects (-25.63) (11.67) R LogL LM spatial lag LM spatial error robust LM spatial lag robust LM spatial error LR test spatial fixed effects: (2315.7, with 45 degrees of freedom [df], p < 0.01) LR test time-period fixed effects: (473.1, 29 df, p < 0.01) (robust) LM test (critical value 3.84): error model 63

64 To investigate the (null) hypothesis that the spatial fixed effects are jointly insignificant, one may perform a likelihood ratio (LR) test. The results (2315.7, with 46 degrees of freedom [df], p < 0.01) indicate that this hypothesis must be rejected. Similarly, the hypothesis that the time-period fixed effects are jointly insignificant must be rejected (473.1, 30 df, p < 0.01). These test results justify the extension of the model with spatial and time-period fixed effects. One of the main questions is which model best describes the data. One of the criterions that may be used for this purpose is the likelihood ratio (LR) test based on the log-likelihood function values of the different models. The LR test is based on minus two times the difference between the value of the log-likelihood function in the restricted model and the value of the loglikelihood function of the unrestricted model: -2*(logL restricted -logl unrestricted ). This test statistic has a chi-squared distribution with degrees of freedom equal to the number of restriction imposed. Robust LM tests point to spatial error model. Critical value for 1 degree of freedom is

65 Elhorst (2013, Handbook of Regional Science): Focus on SDM (SDEM left aside) To test the hypothesis whether the spatial Durbin model can be simplified to the spatial error model, H 0 : θ+λβ=0, one may perform a Wald or LR test. The results reported in the second column using the Wald test (8.18, with 2 degrees of freedom [df], p=0.017) or using the LR test (8.28, 2 df, p=0.016) indicate that this hypothesis must be rejected. Similarly, the hypothesis that the spatial Durbin model can be simplified to the spatial lag model, H 0 : θ=0, must be rejected (Wald test: 17.96, 2 df, p=0.000; LR test: 15.80, 2 df, p=0.000). This implies that both the spatial error model and the spatial lag model must be rejected in favor of the spatial Durbin model. 65

66 Estimation results of cigarette demand: spatial Durbin model specification with spatial and time-period specific effects Determinants (1) (2) Spatial and time-period fixed effects bias-corrected Random spatial effects, Fixed time-period effects W*Log(C) (8.25) (6.82) Log(P) (-24.36) (-24.91) Log(Y) (10.27) (10.71) W*Log(P) (1.13) (0.81) W*Log(Y) (-3.93) (-3.55) phi (6.81) σ (Pseudo) R (Pseudo) Corrected R LogL Wald test spatial lag (p=0.000) (p=0.001) LR test spatial lag (p=0.000) (p=0.000) Wald test spatial error 8.18 (p=0.017) 7.38 (p=0.025) LR test spatial error 8.28 (p=0.016) 7.27 (p=0.026) Corrected R 2 is R 2 without the contribution of fixed effects 66

67 The last column in the Table above reports the parameter estimates if we treat c i as a random variable rather than a set of fixed effects. Hausman's specification test can be used to test the random effects model against the fixed effects model. The results (30.61, 5 df, p<0.01) 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" ( 2 φ 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=0.087, with t-value of 6.81, which just as Hausman's specification test indicates that the fixed and random effects models are significantly different from each other. 67

68 Direct effect Log(P) (-24.73) (-23.93) Indirect effect Log(P) (-2.26) (-2.12) Total effect Log(P) (-11.31) (-11.26) Direct effect Log(Y) (10.45) (10.67) Indirect effect Log(Y) (-2.15) (-2.18) Total effect Log(Y) (4.61) (4.62) (-24.64) (-25.03) (-2.28) (-2.19) (-12.43) (-12.21) (10.68) (10.53) (-2.03) (-2.06) (5.45) (5.37) Notes: t-values in parentheses. Direct and indirect effects estimates: Left column (I-λW) -1 computed every draw, right column (I-λW) -1 calculated by Equation (12). No evidence of bootlegging effect or substitution effect! We found evidence in favour of spatial Durbin model with fixed effects, but not of the bootlegging effect. What happens if we add dynamics? 68

69 Table 1. Estimation results of cigarette demand using different model specifications Determinants (1) (2) Non-dynamic spatial Durbin model with fixed effects Dynamic spatial Durbin model with lag WY t-1 Intercept Log(C) (65.04) W*Log(C) (8.25) (2.00) W*Log(C) (-0.29) Log(P) (-24.36) (-13.19) Log(Y) (10.27) (4.16) W*Log(P) (1.13) (3.66) W*Log(Y) (-3.93) (-0.87) R LogL Notes: t-values in parentheses Dynamic spatial Durbin model outperforms its non-dynamic counterpart. 69

70 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) Short-term direct effect Log(P) Short-term indirect effect Log(P) Short-term direct effect Log(Y) Short-term indirect effect Log(Y) Long-term direct effect Log(P) Long-term indirect effect Log(P) Long-term direct effect Log(Y) Long-term indirect effect Log(Y) Notes: t-values in parentheses Non-dynamic spatial Durbin model with fixed effects Dynamic spatial Durbin model with lag WY t (-11.48) (3.49) (3.36) (-0.45) (-24.73) (-9.59) (-2.26) (0.98) (10.45) (3.55) (-2.15) (0.48) A non-dynamic spatial Durbin model cannot be used to calculate short-term effect estimates of the explanatory variables. 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 and the income elasticity to in nondynamic model. 70

71 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) Short-term direct effect Log(P) Short-term indirect effect Log(P) Short-term direct effect Log(Y) Short-term indirect effect Log(Y) Long-term direct effect Log(P) Long-term indirect effect Log(P) Long-term direct effect Log(Y) Long-term indirect effect Log(Y) Notes: t-values in parentheses Non-dynamic spatial Durbin model with fixed effects Dynamic spatial Durbin model with lag WY t (-11.48) (3.49) (3.36) (-0.45) (-24.73) (-9.59) (-2.26) (0.98) (10.45) (3.55) (-2.15) (0.48) The spatial spillover effects of both variables in the non-dynamic spatial Durbin model 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 -0.22). 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. 71

72 The negative price spillover effects 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 nonsmokers 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. 72

73 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) Short-term direct effect Log(P) Short-term indirect effect Log(P) Short-term direct effect Log(Y) Short-term indirect effect Log(Y) Long-term direct effect Log(P) Long-term indirect effect Log(P) Long-term direct effect Log(Y) Long-term indirect effect Log(Y) Non-dynamic spatial Durbin model with fixed effects Dynamic spatial Durbin model with lag WY t (-11.48) (3.49) (3.36) (-0.45) (-24.73) (-9.59) (-2.26) (0.98) (10.45) (3.55) (-2.15) (0.48) Notes: t-values in parentheses The short-term spatial spillover effect of a price increase turns out to be positive; the elasticity amounts to 0.16 and is highly significant (t-value 3.49). 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). 73

74 Effects estimates of cigarette demand using different model specifications Determinants (1) (2) Short-term direct effect Log(P) Short-term indirect effect Log(P) Short-term direct effect Log(Y) Short-term indirect effect Log(Y) Long-term direct effect Log(P) Long-term indirect effect Log(P) Long-term direct effect Log(Y) Long-term indirect effect Log(Y) Non-dynamic spatial Durbin model with fixed effects Dynamic spatial Durbin model with lag WY t (-11.48) (3.49) (3.36) (-0.45) (-24.73) (-9.59) (-2.26) (0.98) (10.45) (3.55) (-2.15) (0.48) Notes: t-values in parentheses Consistent with microeconomic theory, the short-term direct effects appear to be substantially smaller than the long-term direct effects; versus for the price variable and versus for the income variable. The long-term direct effects in the dynamic spatial Durbin model, on their turn, appear to be greater (in absolute value) than their counterparts in the non-dynamic spatial Durbin model; versus for the price variable and versus for the income variable. Apparently, the nondynamic model underestimates the long-term effects. 74

75 Determinants (1) (2) Non-dynamic Dynamic spatial Durbin model with fixed effects spatial Durbin model with lag WY t-1 Log(C) (65.04) W*Log(C) (8.25) (2.00) W*Log(C) (-0.29) Log(P) (-24.36) (-13.19) Log(Y) (10.27) (4.16) W*Log(P) (1.13) (3.66) W*Log(Y) (-3.93) (-0.87) 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 ( )= with 2 df) evidently justifies the extension of the model with dynamic effects. Conclusion: If dynamic rather than non-dynamic spatial econometric model is adopted and W is specified as binary contiguity matrix, empirical evidence is found in favor of the bootlegging effect. However, questions that remain are the theoretical motivation in favor of adding a spatial lag W*Log(C), and whether is W correctly specified. **************************************************************** 75

76 A regional unemployment model simultaneously accounting for serial dynamics, spatial dependence and common factors Solmaria Halleck Vega and J. Paul Elhorst Key words: Regional unemployment, strong and weak cross-sectional dependence, dynamic spatial panel models, the Netherlands JEL classification: C23, C33, C38, R23 Regional Science and Urban Economics 60 (2016)

77 Regional unemployment rates tend to be strongly correlated over time: Serial dynamics. Table 1. Regional unemployment correlations over time Year

78 Regional unemployment rates in the Netherlands, Percentage Year Groningen Drenthe Flevoland Utrecht Zuid-Holland Noord-Brabant Friesland Overijssel Gelderland Noord-Holland Zeeland Limburg Regional unemployment rates parallel the national unemployment rate: Strong cross-sectional dependence. 78

79 Regional unemployment rates are correlated across space: Weak cross-sectional dependence. 79