Spatial Econometrics

I. Basic Models spdep

NOTE: the definitions of \(\lambda\) and \(\rho\) in this sheet and R are opposite

Concept

\[ y = \lambda W y + X \beta_{(1)} + WX\beta_{(2)} + u \qquad |\lambda|<1 \\ u = \rho W u + \epsilon \qquad\qquad\qquad\qquad\qquad |\rho|<1 \]

	Model	Condition	Estimator	Function
1	Pure spatial autoregressive	\(\beta=0; \quad \lambda, \rho=0\)	ML	spautolm()
2	Lagged independent variable model	\(\lambda=\rho=0\)
3	Spatial Error Model (SEM)	\(\lambda = 0; \quad \rho \neq 0\)	ML	errorsarlm()
			FGLS	GMerrorsar()
4	Spatial Lag Model (SLM)	\(\lambda \neq 0; \quad \rho = 0\)	ML	lagsarlm()
			2SLS	stsls()
5	Complete model (SARAR)	\(\lambda \neq 0; \quad \rho \neq 0\)	ML	sacsarlm()
			GS2SLS	gstsls()
			BFGS2SLS

1) Pure spatial autoregressive

• Condition: \(\quad \beta=0 \quad\) & \(\quad \lambda, \rho=0\)
  1. ML: Most Likelihood
     The procedure is deloped by Whittle (1954).
     PSA_ML <- spautolm(Y~X, data = df, listw = W)
     \[ (i)\qquad\qquad y = \lambda W y \qquad\qquad |\lambda|<1 \\ (ii)\qquad\quad y = \rho W u + \epsilon \qquad\quad |\rho|<1 \]

2) Lagged independent variable model

• Condition: \(\quad \lambda=\rho=0\) \[ y = X \beta_{(1)} + WX\beta_{(2)} + u \qquad |\lambda|<1 \]

3) Spatial Error Model (SEM)

• Condition: \(\quad \lambda = 0 \quad\) & \(\quad \rho \neq 0\) \[ y = X \beta_{(1)} + WX\beta_{(2)} + u \qquad |\lambda|<1 \\ u = \rho W u + \epsilon \quad\qquad\qquad\qquad |\rho|<1 \]
• SEM is referred by Anselin (1988) and Arbis (2006).
  1. ML: Most Likelihood
     Lee (2004) proves constant and asymptotically normal in a model.
     SEM_ML <- errorsarlm(Y~X, data = df, listw = W)
     
  2. FGLS: Feasible GLS
     The procedure is deloped by Kelejian and Prucha (1998).
     SEM_FGLS <- GMerrorsar(Y~X, data = df, listw = W)

4) Spatial Lag Model (SLM)

• Condition: \(\quad \lambda \neq 0 \quad\) & \(\quad \rho = 0\) \[ y = \lambda W y + X \beta_{(1)} + WX\beta_{(2)} + u \qquad |\lambda|<1 \]
• SLM is referred by Anselin (1988) and Arbis (2006).
  1. ML: Most Likelihood
     Assumption: normality of the residuals (BP test)
     SLM_ML <- lagsarlm(Y~X, data = df, listw = W)
     
  2. 2SLS: Two-Stage Least Squares
     No assumption of normality of the residuals (BP test)
     SLM_2SLS <- stsls(Y~X, data = df, listw = W)

5) Complete model (SARAR)

• Condition: \(\quad \lambda \neq 0 \quad\) & \(\quad \rho \neq 0\) \[ y = \lambda W y + X \beta_{(1)} + WX\beta_{(2)} + u \qquad |\lambda|<1 \\ u = \rho W u + \epsilon \qquad\qquad\qquad\qquad\qquad |\rho|<1 \]
• SARAR is referred as SARAR(1,1) by Kelejian and Prucha (1998) and as General Spatial Model by Anselin (1988).
  1. ML: Most Likelihood
     Limitation is there is currently no formal proof that the estimators possess the usual optimal large sample properties.
     Assumption: normality of the residuals (BP test)
     SARAR_ML <- sacsarlm(Y~X, data = df, listw = W)
     
  2. GS2SLS: Generalized Spatial Two-Stage Least Squares
     An extension of 2SLS proposed by Kelejian and Prucha (1998).
     No assumption of normality of the residuals (BP test)
     Limitation is not asymptotically fully efficient.
     SARAR_GS2SLS <- gstsls(Y~X, data = df, listw = W)
     
  3. BFGS2SLS: Best Feasible GS2SLS
     An extension of GS2SLS proposed by Lee (2003), also known as LIV (Lee’s Instrumental Variable)
     No assumption of normality of the residuals (BP test)
     Limitation is numerically challenging in very large samples.
     
  • Kelejian et al. (2004) shows BFGS2SLS and its simplified version do not differ substantially in terms of efficiency from GS2SLS in small sample.

Model test

• Moran I test: SEM vs SLM
  • \(H_0\): No spatial autocorrelation in regression residual
    
  • but there is no alternative hypothesis considering to contrast the null of uncorrelation
  • Moran I test is proposed by Moran (1950).
    
  1. spdep::lm.morantest(OLS, listw = W)
     
  2. spdep::moran.test(REG$residuals, listw = W)
     Both have same Moran’s I statistic value, but different p-value.
     
• LM and RLM test
  • \(H_0\): \(\lambda=0\) for \(LM_{SLM}\), \(\rho=0\) for \(LM_{SEM}\)
    
  • LM stands for Lagrange Multiplier and its robust version is proposed by Anselin et al. (1996).
    • spdep::lm.LMtests(OLS, listw = W, test = "all")
      
• Wald test
  • \(H_0\): \(\lambda=\rho=0\)

Interpretation of parameters in spatial econometrics

• LeSage and Pace (2009) introduce the impact measures
  1. ADI: average direct impact
  2. AII: average indirect impact
  3. ATI: average total impact
  4. ATIT: average total impact to an observation
  5. ATIF: average total impact from an observation
  • spdep::impacts(SLM/SARAR, listw = W)

- Further material
- Arbia (2011) provides revivews for spatial panel analysis.

II. Panel Data splm

Concept

• Basic for panel data \[ y_{it} = \alpha + X_{it} \beta + u_{it} = \alpha + X_{it} \beta + (\mu_i + \epsilon_{it}) \] where \(\mu_i\) is individual error component and \(\epsilon_{it}\) is idiosysncratic error component.
  1. Pooling model
     Condition: \(\mu_i\) (individual error component) is same among all indivisuals i.
     OLS_pooled <- plm(Y~X, model = "pooling")
     
  2. Fixed model
     Condition: \(\mu_i\) (individual error component) is correlated with the regressors.
     OLS_within <- plm(Y~X, model = "within")
     
  3. Random model
     Condition: \(\mu_i\) (individual error component) is uncorrelated with the regressors.
     FGLS_random <- plm(Y~X, model = "random")

	Model	Estimator	Function	model =	spatial.error =	lag =
1	SEM-RE	ML	splm()	"random", fixed	"b"	FALSE
	KKP (SEM)	ML	splm()	"random", fixed	"kkp"	FALSE
		GM	spgm()	"random", fixed	TRUE	FALSE
2	SLM-RE	ML	splm()	"random", fixed	"none"	TRUE
		GM	spgm()	"random", fixed	FALSE	TRUE
3	SARAR	ML	splm()	"random", fixed	"b", "kkp"	TRUE
		GM	spgm()	"random", fixed	TRUE	TRUE

Note. 1) splm() and spgm() are used for ML and GM estimator, respectively. 2) listw = W and data = data.frame should be specified in each model. 3) model = c("random", "within") should be chosen according to Hausman test.

1) Random effects

• SEM: Spatial Error Model

  1. SEM-RE: spatial error model with random effects \[ \epsilon_t = \rho W \epsilon_t + \eta_t \] SEM-RE considers only \(\epsilon_{it}\) (idiosysncratic error component).
     SEM_RE_ML <- spml(Y~X, listw = W, data = dada.frame, model = "random", spatial.error = "b", lag = FALSE)

  2. KPP \[ u = \mu + \epsilon = \rho (I_T*W)u + \eta \] KKP considers both \(\mu_i\) (individual error component) and \(\epsilon_{it}\) (idiosysncratic error component).
     KKP is proposed by Kapoor et al. (2007).
     KKP_ML <- spml(Y~X, listw = W, data = dada.frame, model = "random", spatial.error = "kkp", lag = FALSE)
KKP_GM <- spgm(Y~X, listw = W, data = dada.frame, model = "random", spatial.error = TRUE, lag = FALSE)
     

• SLM: Spatial Lag Model
  1. SLM-RE: spatial lag model with random effects SLM_RE_ML <- spml(Y~X, listw = W, data = dada.frame, model = "random", spatial.error = "none", lag = TRUE)
SLM_RE_GM <- spgm(Y~X, listw = W, data = dada.frame, model = "random", spatial.error = FALSE, lag = TRUE)

2) Fixed effects

• SEM:
  1. SEM-FE
     SEM_FE_ML <- spml(Y~X, listw = W, data = dada.frame, model = "within", spatial.error = "b", lag = FALSE)
     
  2. KPP-FE KKP_FE_ML <- spml(Y~X, listw = W, data = dada.frame, model = "within", spatial.error = "kkp", lag = FALSE)
KPP_FE_GM <- spgm(Y~X, listw = W, data = dada.frame, model = "within", spatial.error = TRUE, lag = FALSE)
     
• SLM:
  1. SLM-FE
     SLM_FE_ML <- spml(Y~X, listw = W, data = dada.frame, model = "within", spatial.error = "none", lag = TRUE)
SLM_FE_GM <- spgm(Y~X, listw = W, data = dada.frame, model = "within", spatial.error = FALSE, lag = TRUE)

Model test and estimator

• Hausan test: Randon vs Fixed
  • Hausan test is proposed by Hausman (1978) and extended for spatial panels by Lee and Yu (2012).
    
  • \(H_0\): Random model is true
    splm::sphtest(Y~X, data = data.frame, listw = W, spatial.model = c("error", "lag" "sarar"), method = c("ML", GM")
    When "ML" is chosed for method, errors = c("BSL", "KPP") can be specified.
• Estimator for spatial panel data: ML vs GM
  1. ML: Maximum Likelihood
     ML for SEM and SLM for spatial panels is proposed by Elhorst (2003).
     \(+\) most efficient when all distributional assumptions are met
     \(-\) large computational demand
     
  2. GM: Generalized Moments
     \(+\) less computational demand
     \(+\) relaxed normality assumption compared with GM -> more robust
     
     
• Further material
  • Baltagi and Pesaran (2007), Lee and Yu (2010), and Lee and Yu (2011) provide revivews for spatial panel analysis.
  • Millo and Piras (2012) provides further detail in the splm package for spatial econometrics.

III. Advanced

	Type	Model	Estimator	Package	Function
1	Heteroscedastic model	FGS2SLS		sphet	gstslshet()
		Spatial HAC		sphet	stslshac()
2	Descrete model	a-Spatial		McSpatial	glm()
		Spatial Probit	ML	McSpatial	spprobitml()
			GMM	McSpatial	gmmprobit()
			LGMM	McSpatial	spprobit()
3	Non-stationary	GWR		spgwr	gwr()

1) Heteroscedastic model sphet

• Parametric
  1. FGS2SLS: Feasible 2SLS
     Modifying SARAR, the procedure is proposed by Kelejian and Prucha (2010).
     Hetero_SARAR <- gstslshet(Y~X, listw = W)
     
• Non-parametric
  2. Spatial HAC: heteroscadasticity and autocorrelation
     the procedure is proposed by Kelejian and Prucha (2007).
     Spatial_HAC <- stslshac(Y~X, listw = W, distance = D, type = "Epanechnikov")

2) Discrete model McSpatial

• a-spatial logia and probit
  1. Probit
     Probit <- glm(Y~X, family = binomial(link="probit"))
     
  2. Logit
     Logit <- glm(Y~X, family = binomial(link="logit"))
     
• Spatial Probit
  1. ML (probit)
     ML cannot be found analytically and has computational demanding.
     Probit_ML <- spprobitml(Y~X, wmat = Wdash, stdprobit = FALSE)
     
  2. GMM (probit)
     GMM is proposed by Pinkse and Slade (1998).
     GMM has computational demanding in a large sample.
     rho = rho0 (\(|\rho_0|<1\))
     Probit_GMM <- gmmprobit(Y~X, wmat = Wdash, startrho = rho)
     
  3. LGMM (logit)
     LGMM is proposed by Klier and McMillen (2008).
     LGMM has less computational demanding but less accuracy.
     When \(\lambda<0.5\), there is no bias; otherwise, there is a upward biase.
     Lobit_LGMM <- spprobit(Y~X, wmat = Wdash)
     
  • To create Wdash, Spatial Probit model requires sepdep::nb2mat insted of spdep::nb2listw().
• Logit vs Probit
  • Probit is more popular than Logit.
    
  • Anselin (2002) shows the error term of the spatial Logit is analytically intractable.
    
  • but Smirnov (2010) shows Probit is not easy to be extended to more than two alternatives.

3) Non-stationary spgwr

• Scan statistics (parametric)
  
• LWR: locally weighted regression (non-parametric)
  • LWR can be seen as a scan statistic technique to perform a regression around a point of interest using only a limited number of training data.
    
  • LWR is produced by Clevel and Devlin (1988).
    
  • LWR applies kernel; thus, produces a smooth variation though the regression estimates is calculated separately.
• GWR: geographically weighted regression (non-parametric)
  • GWR is a particular case of LWR.
  • GWR applies geographical space as a selection criterion.
  • Articles based on GWR are publised by Brunsdon et al. (1996), Fotheringham et al. (1998), Fotheringham et al. (2002), Fotheringham et al. (2007), McMillen and McDonald (1997), and McMillen and McDonald (2004).
    • Calibrate bandwidth
      bw <- gwr.sel(y ~ X + Z, coords, gweight = gwr.Gauss, adapt = TRUE)
adapt can be specified as TRUE or FALSE according to cernel bandwidth or global bandwidth
      
    • Regression
      GWR <- gwr(Y~X, coords, adapt = bw, hatmatrix = TRUE)
      
    • Mapping
      plot(GWR$SDF, col = cols[findInterval(GWR$SDF$X, brks, all.inside=TRUE)])
      
    • Moran’s I test
      Null of no spatial autocorrelation in regression residual
      gwr.morantest(GWR, listw = W)

IV. Big data

under construction

Model test

The Model: REG <- lm(Y~X)

1) Homoscedasticity

• BP test
  \(H_0\): No homoscedasticity
  lmtest::bptest(REG)

2) Normality of the regression residual

• JB test
  \(H_0\): Normality distribution of regression residual
  tseries::jarque.bera.test(REG$residuals)

3) Spatial autoCorrelation for regression residuals

• Moran I test, proposed by by Moran (1950)
  \(H_0\): No spatial autocorrelation in regression residual
  [i] sepdep::lm.morantest(REG, listw = W)
  [ii] sepdep::moran.test(REG$residuals, listw = W)
  Both have same Moran’s I statistic value, but different p-value.
• LM and RLM test,
  \(H_0\): \(\lambda=0\) for \(LM_{SLM}\), \(\rho=0\) for \(LM_{SEM}\)
  spdep::lm.LMtests(OLS, listw = W, test = "all")

4) \(\lambda=\rho=0\)

• Wald test
  \(H_0\): \(\lambda=\rho=0\)

5) Randon vs Fixed for spatial panel

• Hausan test is proposed by Hausman (1978) and extended for spatial panels by Lee and Yu (2012).
  \(H_0\): Random model is true
  splm::sphtest(Y~X, data = data.frame, listw = W, spatial.model = c("error", "lag" "sarar"), method = c("ML", GM")
  When "ML" is chosed for method, errors = c("BSL", "KPP") can be specified.