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Spatial Econometrics Lecture 5: Single-source model of spatial regression. Combining GIS and regional analysis (5) Spatial Econometrics 1 / 47

Outline 1 Linear model vs SAR/SLM (Spatial Lag) Linear model SAR (Spatial Lag, SLM) 2 Model SEM (Spatial Error) SEM model with global error dependence SEM model with local error dependence 3 SLX model 4 Combining point GIS data with regional statistics Example: location of Biedronka markets Homework (5) Spatial Econometrics 2 / 47

Plan prezentacji 1 Linear model vs SAR/SLM (Spatial Lag) 2 Model SEM (Spatial Error) 3 SLX model 4 Combining point GIS data with regional statistics (5) Spatial Econometrics 3 / 47

Linear model Linear regression model specication The well-known linear regression model: y = Xβ + ε Its parameters can be estimated in an unbiased, consistent and ecient way via Ordinary Least Squares (OLS) method. Appropriate, when spatial links in y are fully (implicitly) captured through the spatial autocorrelation of regressors included in X (spatial clustering of X). (5) Spatial Econometrics 4 / 47

Linear model Flow of impacts in the linear model (5) Spatial Econometrics 5 / 47

SAR (Spatial Lag, SLM) Flow of impacts in SAR model (5) Spatial Econometrics 6 / 47

SAR (Spatial Lag, SLM) SAR model relation to other models (5) Spatial Econometrics 7 / 47

SAR (Spatial Lag, SLM) SAR model relation to other models (5) Spatial Econometrics 8 / 47

SAR (Spatial Lag, SLM) SAR model specication Spatial autoregression with additional regressors. y = ρwy + Xβ + ε Without any explanatory variables X in the model, it would be identical with pure SAR. In this model, we do not assume any spatial clustering of the causes, but spatial interactions in outcomes (spatial global spillovers, spatial spillovers). Problem with OLS estimation: endogeneity (like in pure SAR). (5) Spatial Econometrics 9 / 47

SAR (Spatial Lag, SLM) Consequences of omitting spatial structure SAR (1) True data generating process: y = ρwy + Xβ + ε Estimated linear model omitting Wy (method OLS): y = Xβ KMNK + ε According to the general principles of econometrics, omitting a variable results in the estimation bias of β, that converges to the product of: (true) parameter of the skipped variable slope of the regression of the skipped variable on the included variables In our case: plimˆβ KMNK = β + ρ Cov(Wy,X) Var(X) (5) Spatial Econometrics 10 / 47

SAR (Spatial Lag, SLM) Consequences of omitting spatial structure SAR (2) Can Cov (Wy, X) possibly be 0? If the true data generating process is SAR, then... y = (I ρw) 1 Xβ + (I ρw) 1 ε y = Xβ + ρwxβ + ρ 2 W 2 Xβ +... + ε + ρwε + ρ 2 W 2 ε +... Wy = WXβ + ρw 2 Xβ + ρ 2 W 3 Xβ +... + Wε + ρw 2 ε + ρ 2 W 3 ε +... Thus (skipping the components related to ε, which as we know are uncorrelated to X): plimˆβ KMNK β = ρ Cov(WXβ,X) + ρ Cov(ρW2 Xβ,X) + ρ Cov(ρ2 W 3 Xβ,X) +... = Var(X) Var(X) Var(X) ρ [ ( = Var(X) Cov (WX, X) + ρ Cov W 2 X, X ) + ρ 2 Cov ( W 3 X, X ) +... ] β Even if X is not spatially autocorrelated and Cov (WX, X) = 0, further components cannot be equal to zero. W 2 and further powers of W are not any more matrices with zero diagonal elements. Interpretation: W 2 is the matrix of connections to neighbours of the neighbours. But the neighbour of your neighbour is i.a. You! (And You're always correlated with yourself.) (5) Spatial Econometrics 11 / 47

SAR (Spatial Lag, SLM) Spatial OLS (1) If the omission of spatial lag makes the OLS estimator biased, we should include it. Potentially easy to do: if W is predetermined, one can construct the spatial lag variable Wy upfront and estimate the SAR model y = ρwy + Xβ + ε with OLS (this method is referred to as Spatial OLS): y = [ Wy X ] [ ρ β ] + ε From OLS properties, we know that: ([ ]) E = ˆρˆβ [ ] ρ ( [ ] T [ ] ) 1 ( [ ] ) T + Wy X Wy X E Wy X ε β (5) Spatial Econometrics 12 / 47

SAR (Spatial Lag, SLM) Spatial OLS (2) In the linear regression model, we( assume that the error terms are [ ] ) T independent of regressors, i.e. E Wy X ε = 0, and this proves the unbiasedness of the OLS estimator in such a model. It holds that E ( X T ε ) = 0, but: [ ] { [W E (Wy) T ε = E (I ρw) 1 Xβ + W (I ρw) 1 ε ] } T ε = { [W = E (I ρw) 1 Xβ ] T [ ε + W (I ρw) 1 ε ] } T ε = = E {ε [ T W (I ρw) 1] } T ε 0 Our model is not the classical regression model, because observations depend on one another (y i depends on the neighbour y j and vice versa). Situation similar to the simultaneous equations models. (5) Spatial Econometrics 13 / 47

SAR (Spatial Lag, SLM) Spatial OLS (3) For simplication, consider the SAR model with 1 explanatory variable x: = = ([ ]) E = ˆρˆβ [ ] [ ρ 1 + ( β [ ] det Wy x T [ ] ) Wy x }{{} γ>0 usually>0 [ ] {}}{ ρ γx T x (Wy) T ε + β γx T (Wy) (Wy) T ε }{{} usually<0 x T x x T (Wy) (Wy) T x (Wy) T (Wy) ] (Wy) T ε x T ε }{{} =0 = So, the spatial OLS delivers biased estimates! (ρ usually upward biased, β downward biased). In the multivariate cases, the bias is concentrated on the parameters for variables X whose spatial patterns most resembles the spatial pattern of y. (5) Spatial Econometrics 14 / 47

SAR (Spatial Lag, SLM) Spatial 2SLS (1) The simultaneous equation bias in y = ρwy + Xβ + ε can be treated analogously to the case of endogenous regressors: i.e. use the instrumental variables method. This implementation is consistent, unbiased and is referred to as spatial 2-stage least squares (S2SLS). A valid instrumental variable is correlated with the problematic regressor (Wy), but uncorrelated with the error term (ε). Recall that for the SAR model: Wy = WXβ + ρw 2 Xβ + ρ 2 W 3 Xβ +... + Wε + ρw 2 ε + ρ 2 W 3 ε +... }{{} ideal instruments! Step 1: Linear regression of Wy on the matrix including exogenous variables and a certain number of instruments: Π = [ X WX W 2 X... ] (OLS). ( ) 1 Theoretical values: Ŵy = Π Π T Π Π T Wy }{{} P (5) Spatial Econometrics 15 / 47

SAR (Spatial Lag, SLM) Spatial 2SLS (2) Step 2: OLS estimation of the SAR model parameters after the replacement of Wy with [ ] ( Ŵy: [ ] T [ ] ) 1 [ ] T = Ŵy X Ŵy X Ŵy X y ˆρˆβ Spatial 2SLS (S2SLS) model <- stsls(y ~ x, listw = W) (5) Spatial Econometrics 16 / 47

SAR (Spatial Lag, SLM) Spatial ML (1) Variant 2: maximum likelihood method M M {}}{{}}{ y = (I ρw) 1 Xβ + (I ρw) 1 u, u N(0, σ 2 ) L (u) = ( ) N ( ) 1 2 σ 2 exp ut u 2π 2σ 2 By the change of variables theorem (multivariate case): L (y) = det L (y) = M 1 {( }} ){ u L [u (y)] y det ( M 1) ( ( ) ) N 1 2 σ 2 exp (y MXβ)T (M 1 ) T (M 1 )(y MXβ) 2π 2σ 2 ˆβ = arg maxl (y) β (5) Spatial Econometrics 17 / 47

SAR (Spatial Lag, SLM) Spatial ML (2) Standard errors evaluated on the basis of Hessian matrix at the maximum point of the likelihood function (typical for ML). If M = I, the likelihood function identical as in the linear model. ML for the SAR model in R model <- lagsarlm(y ~ x, listw = W) The same model is estimated when the formula argument in the function spautolm (pure SAR) is supplied with additional regressors. (5) Spatial Econometrics 18 / 47

SAR (Spatial Lag, SLM) Tests: linear model vs SAR (1) This illustration demonstrates the univariate case (θ scalar). (5) Spatial Econometrics 19 / 47

SAR (Spatial Lag, SLM) Tests: linear model vs SAR (2) LM ρ = N tr[(w T +W)W]+ 1 (WXˆβ) T ˆε T [I X(X T X)X T ](WXˆβ) ˆε χ 2 (1) ( ) ˆε T 2 Wy ˆε T ˆε H 0 : linear model (ρ = 0) H 1 : SAR (5) Spatial Econometrics 20 / 47

Plan prezentacji 1 Linear model vs SAR/SLM (Spatial Lag) 2 Model SEM (Spatial Error) 3 SLX model 4 Combining point GIS data with regional statistics (5) Spatial Econometrics 21 / 47

SEM model with global error dependence Flow of impacts in SEM model (5) Spatial Econometrics 22 / 47

SEM model with global error dependence SEM model relation to other models (5) Spatial Econometrics 23 / 47

SEM model with global error dependence SEM model relation to other models (5) Spatial Econometrics 24 / 47

SEM model with global error dependence SEM model specication It is not the dependent variable, but the error term, that is subject to spatial autocorrelation the dierence is analogous to the dierence between AR and MA models. y = Xβ + ε ε = λwε + u In the absence of regressors X, the model would be equivalent to (pure) SAR. Spatial clustering in unobservables (shocks). (5) Spatial Econometrics 25 / 47

SEM model with global error dependence SEM model estimation (1) OLS estimator is inecient (and the standard errors biased), because: y = Xβ + ε ε = λwε + u, czyli ε = (I λw) 1 u Var (ε) = E ( εε T ) = (I λw) 1 E ( uu T ) [ (I λw) 1] T = σ 2 (I λw) 1 [ (I λw) 1] T σ 2 I Variant 1: as usually with non-spherical errors, the solution is Generalised Least Squares estimation: ˆβ = ( X T Ω 1 X ) 1 X T Ω 1 y with given Ω = (I λw) 1 [ (I λw) 1] T W known, λ estimated based on errors derived from the consistent OLS estimation (details of the procedure: Kelejian and Prucha, 1998; Arbia, 2014). ) Var (ˆβ = ˆσ ( 2 X T Ω 1 X ) 1 (5) Spatial Econometrics 26 / 47

SEM model with global error dependence SEM model estimation (2) Spatial GLS in R model4 <- GMerrorsar(y ~ x, listw = W) (5) Spatial Econometrics 27 / 47

SEM model with global error dependence SEM model estimation (3) Variant 2: maximum likelihood method M {}}{ y = Xβ + (I λw) 1 u, u N(0, σ 2 ) L (u) = ( ) N ( ) 1 2 σ 2 exp ut u 2π 2σ 2 By the change of variables theorem (multivariate case): L (y) = det L (y) = M 1 {( }} ){ u L [u (y)] y det ( M 1) ( ( ) ) N 1 2 σ 2 exp (y Xβ)T (M 1 ) T (M 1 )(y Xβ) 2π 2σ 2 ˆβ = arg maxl (y) β (5) Spatial Econometrics 28 / 47

SEM model with global error dependence SEM model estimation (4) Standard errors evaluated on the basis of Hessian matrix at the maximum point of the likelihood function (typical for ML). If M = I, the likelihood function identical as in the linear model. ML for SEM model in R model <- errorsarlm(y ~ x, listw = W) The same model will also be estimated, if the formula in the function spautolm (pure SAR) supplied with regressors. (5) Spatial Econometrics 29 / 47

SEM model with global error dependence Both SAR and SEM collapse to pure SAR without regressors X SAR y = ρwy + Xβ + ε y = ρwy + ε y ρwy = ε (I ρw) y = ε y = (I ρw) 1 ε SEM y = Xβ + (I λw) 1 u β = 0 y = (I λw) 1 u (5) Spatial Econometrics 30 / 47

SEM model with global error dependence LM tests: linear model vs SEM LM λ = N 2 tr[(w T +W)W] H 0 : linear model (λ = 0) H 1 : SEM ( ) 2 ût Wû û T û χ2 (1) (5) Spatial Econometrics 31 / 47

SEM model with global error dependence Robust LM tests (1) In LM thests for SAR and SEM specications (respectively): 1 H 0 : linear model (ρ = 0), H 1 : SAR 2 H 0 : linear model (λ = 0), H 1 : SEM Problem: each pair of hypotheses leaves out of sight the alternative hypothesis from the other pair of the other test. Consequence: test 1 rejects H 0 even under false H 1 (but true H 1 from test 2). And vice versa. RLMlag and RLMerr Anselin et al. (1996) propose robust test statistics LMρ and LMλ, which by construction exclude the possibility that an incorrect process is captured by the alternative hypothesis (see Arbia, 2014). LMρ = LM ρλ LM λ LMλ = LM ρλ LM ρ (5) Spatial Econometrics 32 / 47

SEM model with local error dependence Global vs local SEM model (1) The previously presented SEM model stipulated a global dependence between unobservables: The local SEM version: y = Xβ + ε ε = λwε + u y = Xβ + ε ε = λwu + u What is the dierence? Consider spatial multiplier matrices of y with respect to u in both cases: local SEM: y = Xβ + ε, ε = (I + λw)u M = y u = (I + λw) global SEM: y = Xβ + ε, ε = (I λw) 1 u M = y u = (I λw) 1 Algebraically, note that: multiplier SEM glob {}}{ (I λw) 1 = multiplier SEM loc {}}{ I + λw + λ 2 W 2 + λ 3 W 3 +... (5) Spatial Econometrics 33 / 47

SEM model with local error dependence Global vs local SEM model (2) Example: Canada, USA, Mexico; W = λ = 0.4; shock u = 1 occurs in Mexico. Spatial multiplierrs for local SEM: 0 0.5 0.5 0 I + 0.4 1 0 0 0 = 1 0 0 1 1 0.2 0.2 0.4 1 0 0.4 0 1 0 0 1 = 0.2 0 1 US CA 0 0.5 MX 0.5 1 0 0 1 0 0 y MX = 1, y US = 0.2, no eect for Canada. Shock in u aected y in the directly linked units. ; (5) Spatial Econometrics 34 / 47

SEM model with local error dependence Global vs local SEM model (3) Spatial multipliers for global SEM: 1 0 0.5 0.5 0 I 0.4 1 0 0 0 1 0 0 1 1 0.2 0.2 0.4 1 0 0.4 0 1 1.19 0.24 0.24 0.48 1.10 0.10 0.48 0.10 1.10 1 0 0 1 0.24 0.10 1.10 = y MX > 1, y US > 0.2, there is (weak but positive) eect for Canada The impulse spills over to the related units, and then to their own related units, etc. (including the feedback into the impulse region). (5) Spatial Econometrics 35 / 47

Plan prezentacji 1 Linear model vs SAR/SLM (Spatial Lag) 2 Model SEM (Spatial Error) 3 SLX model 4 Combining point GIS data with regional statistics (5) Spatial Econometrics 36 / 47

SLX model Flow of impacts in the SLX model (5) Spatial Econometrics 37 / 47

SLX model SLX model relation to other models (5) Spatial Econometrics 38 / 47

SLX model SLX model relation to other models (5) Spatial Econometrics 39 / 47

SLX model SLX model specication Direct impact of causes in the neighbourhood on the consequence in the observed region spatial spillovers:: y = Xβ + WXθ + ε Consistent, ecient and unbiased estimation with OLS. (5) Spatial Econometrics 40 / 47

Plan prezentacji 1 Linear model vs SAR/SLM (Spatial Lag) 2 Model SEM (Spatial Error) 3 SLX model 4 Combining point GIS data with regional statistics (5) Spatial Econometrics 41 / 47

Example: location of Biedronka markets GIS data about the markets Biedronka Source: poiplaza.com POI: points of interest (usually published for the users of car GPS navigation sets) Point data about location of individual markets in Poland. Longitude and latitude. Question: what regional criteria do the managers / owners of Biedronka use when locating their markets? In other words, is there a relationship between the number of Biedronka markets (per capita) and local socio-economic characteristics from Local Data Bank, e.g. on the level of poviats? (5) Spatial Econometrics 42 / 47

Example: location of Biedronka markets Aggregation of points on the predened map (5) Spatial Econometrics 43 / 47

Example: location of Biedronka markets Aggregation of points on the predened map Andrzej Function Torój ClassIntervals watch outinstitute the style of Econometrics parameter. Department So far, weofhave Applied been Econometrics (5) Spatial using Econometrics quantile division into classes of equal count for the purpose of presentation. 44 / 47

Example: location of Biedronka markets Estimates of 3 models Fit the following regression models of the number of Biedronka markets per 10 thousand residents on labour market statistics (unemployment, wages): linear SLM SEM SLX Compare the models as regards the signicance of variables, AIC criterion, log-likelihood value at maximum and the presence of unremoved spatial autocorrelation. Which model is the best? (5) Spatial Econometrics 45 / 47

Homework Exercise Derive the likelihood function for SEM model with local error dependence. Develop an R code for the estimation of such a model. (5) Spatial Econometrics 46 / 47

Homework Homework 5 Build a set of potential regressors (X) and merge it with the spatial dataset used in homework 1 (y). Using the spatial weight matrix W constructed in homework 2 estimate the pure SAR model for the variable from homework 1. Is this result in line with the testing results from homework 3? Estimate models SAR, SLX, SEM for the considered X, y and W. Visualise the residuals from these models on a map and test them for the presence of spatial autocorrelation in residuals. (5) Spatial Econometrics 47 / 47