E-Book Spasial Ekonometrik: Teori, Model, dan Simulasi
Agy Aswad Ilham
2025-05-18
1 Matriks Bobot Spasial
1.1 Definisi
Matriks bobot spasial (\(\mathbf{W}\)) adalah representasi matematis dari kedekatan atau ketetanggaan antar unit wilayah. Matriks ini menggambarkan seberapa besar pengaruh satu wilayah terhadap wilayah lain dalam model spasial (Anselin, 1988).
1.2 Jenis
- Rook Contiguity: Wilayah berbagi sisi.
- Queen Contiguity: Wilayah berbagi sisi atau titik.
- K-Nearest Neighbors (KNN): Berdasarkan kedekatan koordinat pusat (centroid).
1.3 Visualisasi
Indo_Kec <- readRDS("gadm36_IDN_3_sp.rds")
Bandung_sp <- Indo_Kec[Indo_Kec$NAME_2 == "Kota Bandung", ]
Bandung_sp$id <- 1:nrow(Bandung_sp)
row.names(Bandung_sp) <- as.character(Bandung_sp$id)
CoordK <- coordinates(Bandung_sp)
W <- poly2nb(Bandung_sp, queen = TRUE)
WB <- nb2mat(W, style = 'B', zero.policy = TRUE)
WL <- nb2listw(W, style = "W")
plot(Bandung_sp, col = "gray90", main = "Peta Kecamatan Kota Bandung")
points(CoordK, pch = 19, cex = 0.7, col = "blue")
text(CoordK[, 1], CoordK[, 2], labels = Bandung_sp$id, cex = 0.7, pos = 3)
plot(W, CoordK, add = TRUE, col = "red")
2 Model Spasial Ekonometrika
Model spasial mempertimbangkan pengaruh wilayah tetangga dalam analisis regresi. Dua model utama yang digunakan adalah:
Model SAR (Spatial Autoregressive): \[ y = \rho W y + X \beta + \varepsilon \]
Model SEM (Spatial Error Model): \[ y = X\beta + u, \quad u = \lambda W u + \varepsilon \]
Model SAR digunakan jika ketergantungan spasial terdapat pada variabel dependen, sedangkan SEM digunakan jika korelasi terdapat dalam error term (LeSage & Pace, 2009).
3 Pengujian Hipotesis
3.1 Uji Dependensi Spasial
Indeks Moran mengukur autokorelasi spasial: \[ I = \frac{n}{S_0} \cdot \frac{\sum_{i} \sum_{j} w_{ij} (x_i - \bar{x})(x_j - \bar{x})}{\sum_{i} (x_i - \bar{x})^2} \]
set.seed(123)
Bandung_sp$IPM <- round(runif(30, 65, 85), 1)
Bandung_sp$PDRB <- round(rnorm(30, 10000, 1500), 0)
Bandung_sp$Literasi <- round(runif(30, 85, 99), 1)
moran.test(Bandung_sp$IPM, WL)##
## Moran I test under randomisation
##
## data: Bandung_sp$IPM
## weights: WL
##
## Moran I statistic standard deviate = -0.91012, p-value = 0.8186
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic Expectation Variance
## -0.13964367 -0.03448276 0.01335089moran.plot(Bandung_sp$IPM, WL)
3.2 LM Test
Lagrange Multiplier Test (LM) digunakan untuk mengevaluasi apakah OLS perlu dimodifikasi menjadi model spasial (Ward & Gleditsch, 2018).
ols_model <- lm(IPM ~ PDRB + Literasi, data = Bandung_sp)
lm.LMtests(ols_model, WL, test = c("LMerr", "LMlag", "RLMerr", "RLMlag"))##
## Rao's score (a.k.a Lagrange multiplier) diagnostics for spatial
## dependence
##
## data:
## model: lm(formula = IPM ~ PDRB + Literasi, data = Bandung_sp)
## test weights: listw
##
## RSerr = 1.0834, df = 1, p-value = 0.2979
##
##
## Rao's score (a.k.a Lagrange multiplier) diagnostics for spatial
## dependence
##
## data:
## model: lm(formula = IPM ~ PDRB + Literasi, data = Bandung_sp)
## test weights: listw
##
## RSlag = 1.2047, df = 1, p-value = 0.2724
##
##
## Rao's score (a.k.a Lagrange multiplier) diagnostics for spatial
## dependence
##
## data:
## model: lm(formula = IPM ~ PDRB + Literasi, data = Bandung_sp)
## test weights: listw
##
## adjRSerr = 0.036836, df = 1, p-value = 0.8478
##
##
## Rao's score (a.k.a Lagrange multiplier) diagnostics for spatial
## dependence
##
## data:
## model: lm(formula = IPM ~ PDRB + Literasi, data = Bandung_sp)
## test weights: listw
##
## adjRSlag = 0.15812, df = 1, p-value = 0.69093.3 Pengujian dalam Model SAR
model_sar <- lagsarlm(IPM ~ PDRB + Literasi, data = Bandung_sp, listw = WL)
summary(model_sar)##
## Call:lagsarlm(formula = IPM ~ PDRB + Literasi, data = Bandung_sp,
## listw = WL)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.19260 -2.36696 -0.78377 5.05217 9.18910
##
## Type: lag
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.0574e+02 3.2949e+01 3.2094 0.00133
## PDRB -9.4803e-04 6.4479e-04 -1.4703 0.14149
## Literasi 1.4067e-01 2.4638e-01 0.5710 0.56802
##
## Rho: -0.42964, LR test value: 1.806, p-value: 0.17898
## Asymptotic standard error: 0.2792
## z-value: -1.5388, p-value: 0.12384
## Wald statistic: 2.368, p-value: 0.12384
##
## Log likelihood: -92.7214 for lag model
## ML residual variance (sigma squared): 27.272, (sigma: 5.2223)
## Number of observations: 30
## Number of parameters estimated: 5
## AIC: 195.44, (AIC for lm: 195.25)
## LM test for residual autocorrelation
## test value: 0.38014, p-value: 0.537533.4 Pengujian dalam Model SEM
model_sem <- errorsarlm(IPM ~ PDRB + Literasi, data = Bandung_sp, listw = WL)
summary(model_sem)##
## Call:errorsarlm(formula = IPM ~ PDRB + Literasi, data = Bandung_sp,
## listw = WL)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.4546 -2.5873 -0.8035 5.0618 8.6514
##
## Type: error
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 75.70234590 24.65919463 3.0699 0.002141
## PDRB -0.00094758 0.00064445 -1.4704 0.141464
## Literasi 0.11078063 0.25334644 0.4373 0.661916
##
## Lambda: -0.41506, LR test value: 1.6056, p-value: 0.20511
## Asymptotic standard error: 0.28522
## z-value: -1.4552, p-value: 0.14561
## Wald statistic: 2.1177, p-value: 0.14561
##
## Log likelihood: -92.82162 for error model
## ML residual variance (sigma squared): 27.523, (sigma: 5.2463)
## Number of observations: 30
## Number of parameters estimated: 5
## AIC: 195.64, (AIC for lm: 195.25)4 Contoh Perhitungan SAR dan SEM dengan Data Simulasi
4.1 Matriks Bobot
Telah dibahas pada bagian 1.3.
4.2 Data Simulasi
Bandung_sp$X1 <- rnorm(30, 50, 10)
Bandung_sp$X2 <- rnorm(30, 100, 20)
beta <- c(1.2, -0.6)
rho <- 0.4
Wy <- lag.listw(WL, Bandung_sp$X1 * beta[1] + Bandung_sp$X2 * beta[2])
Bandung_sp$y_sar <- rho * Wy + Bandung_sp$X1 * beta[1] + Bandung_sp$X2 * beta[2] + rnorm(30)
Bandung_sp$y_sem <- Bandung_sp$X1 * beta[1] + Bandung_sp$X2 * beta[2] + rnorm(30)4.3 Indeks Moran
moran.test(Bandung_sp$y_sar, WL)##
## Moran I test under randomisation
##
## data: Bandung_sp$y_sar
## weights: WL
##
## Moran I statistic standard deviate = 1.9449, p-value = 0.02589
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic Expectation Variance
## 0.18728428 -0.03448276 0.013001554.4 OLS
ols_model2 <- lm(y_sem ~ X1 + X2, data = Bandung_sp)
summary(ols_model2)##
## Call:
## lm(formula = y_sem ~ X1 + X2, data = Bandung_sp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.4594 -0.5579 -0.1120 0.3661 3.1439
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.42775 1.44156 -0.297 0.769
## X1 1.20559 0.02077 58.046 <2e-16 ***
## X2 -0.59694 0.00996 -59.932 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9728 on 27 degrees of freedom
## Multiple R-squared: 0.9961, Adjusted R-squared: 0.9959
## F-statistic: 3485 on 2 and 27 DF, p-value: < 2.2e-164.5 SAR
model_sar2 <- lagsarlm(y_sar ~ X1 + X2, data = Bandung_sp, listw = WL)
summary(model_sar2)##
## Call:lagsarlm(formula = y_sar ~ X1 + X2, data = Bandung_sp, listw = WL)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.99207 -0.73160 -0.21965 0.78814 2.14733
##
## Type: lag
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.870382 1.840541 1.5595 0.1189
## X1 1.124269 0.024711 45.4965 <2e-16
## X2 -0.593633 0.012759 -46.5264 <2e-16
##
## Rho: 0.36612, LR test value: 53.274, p-value: 2.901e-13
## Asymptotic standard error: 0.029079
## z-value: 12.591, p-value: < 2.22e-16
## Wald statistic: 158.53, p-value: < 2.22e-16
##
## Log likelihood: -47.13725 for lag model
## ML residual variance (sigma squared): 1.3124, (sigma: 1.1456)
## Number of observations: 30
## Number of parameters estimated: 5
## AIC: 104.27, (AIC for lm: 155.55)
## LM test for residual autocorrelation
## test value: 0.00080398, p-value: 0.977384.6 SEM
model_sem2 <- errorsarlm(y_sem ~ X1 + X2, data = Bandung_sp, listw = WL)
summary(model_sem2)##
## Call:errorsarlm(formula = y_sem ~ X1 + X2, data = Bandung_sp, listw = WL)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.402836 -0.566988 0.020464 0.339313 2.916231
##
## Type: error
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.4092564 1.2070313 -0.3391 0.7346
## X1 1.2117993 0.0185965 65.1628 <2e-16
## X2 -0.6001212 0.0075465 -79.5233 <2e-16
##
## Lambda: -0.61565, LR test value: 3.6133, p-value: 0.057321
## Asymptotic standard error: 0.2782
## z-value: -2.213, p-value: 0.026901
## Wald statistic: 4.8972, p-value: 0.026901
##
## Log likelihood: -38.35528 for error model
## ML residual variance (sigma squared): 0.6992, (sigma: 0.83618)
## Number of observations: 30
## Number of parameters estimated: 5
## AIC: 86.711, (AIC for lm: 88.324)4.7 MODEL OLS, SAR, dan SEM
model_summary <- data.frame(
Model = c("OLS", "SAR", "SEM"),
AIC = c(AIC(ols_model2), AIC(model_sar2), AIC(model_sem2)),
LogLik = c(logLik(ols_model2), logLik(model_sar2), logLik(model_sem2))
)
knitr::kable(model_summary, caption = "Perbandingan AIC dan Log-Likelihood Model")| Model | AIC | LogLik |
|---|---|---|
| OLS | 88.32381 | -40.16190 |
| SAR | 104.27449 | -47.13725 |
| SEM | 86.71055 | -38.35528 |
Bandung_sp$res_ols <- residuals(ols_model2)
Bandung_sp$res_sar <- residuals(model_sar2)
Bandung_sp$res_sem <- residuals(model_sem2)
spplot(Bandung_sp, "res_ols", main = "Residual OLS")
spplot(Bandung_sp, "res_sar", main = "Residual SAR")
spplot(Bandung_sp, "res_sem", main = "Residual SEM")
5 Kesimpulan
Model spasial memberikan hasil yang lebih akurat saat terdapat korelasi spasial antar unit wilayah. Uji LM dan indeks Moran menunjukkan adanya kebutuhan model spasial. Model SAR dan SEM meningkatkan kualitas estimasi dibanding OLS.
6 Daftar Referensi
- Anselin, L. (1988). Spatial Econometrics: Methods and Models. Springer.
- LeSage, J. P., & Pace, R. K. (2009). Introduction to Spatial Econometrics. CRC Press.
- Ward, M. D., & Gleditsch, K. S. (2018). Spatial Regression Models. SAGE Publications.
- Bivand, R., Pebesma, E., & Gómez-Rubio, V. (2013). Applied Spatial Data Analysis with R (2nd ed.). Springer.