stsls: Generalized spatial two stage least squares
In spdep: Spatial Dependence: Weighting Schemes, Statistics and Models
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
The function fits a spatial lag model by two stage least squares, with the option of adjusting the results for heteroskedasticity.
Usage
1
2
Arguments
formula
a symbolic description of the model to be fit. The details
of model specification are given for lm()
data
an optional data frame containing the variables in the model.
By default the variables are taken from the environment which the function
is called.
listw
a listw object created for example by nb2listw
zero.policy
default NULL, use global option value; if TRUE assign zero to the lagged value of zones without
neighbours, if FALSE (default) assign NA - causing lagsarlm() to terminate with an error
na.action
a function (default na.fail), can also be na.omit or na.exclude with consequences for residuals and fitted values - in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.
robust
default FALSE, if TRUE, apply a heteroskedasticity correction to the coefficients covariances
HC
default NULL, if robust is TRUE, assigned “HC0”, may take values “HC0” or “HC1” for White estimates or MacKinnon-White estimates respectively
legacy
the argument chooses between two implementations of the robustness correction: default FALSE - use the estimate of Omega only in the White consistent estimator of the variance-covariance matrix, if TRUE, use the original implementation which runs a GLS using the estimate of Omega, and yields different coefficient estimates as well - see example below
W2X
default TRUE, if FALSE only WX are used as instruments in the spatial two stage least squares; until release 0.4-60, only WX were used - see example below
Details
The fitting implementation fits a spatial lag model:
y = rho W y + X beta + e
by using spatially lagged X variables as instruments for the spatially lagged dependent variable.
Value
an object of class "stsls" containing:
coefficients
coefficient estimates
var
coefficient covariance matrix
sse
sum of squared errors
residuals
model residuals
df
degrees of freedom
Author(s)
Luc Anselin, Gianfranco Piras and Roger Bivand
References
Kelejian, H.H. and I.R. Prucha (1998). A generalized spatial two
stage least squares procedure for estimating a spatial autoregressive
model with autoregressive disturbances. Journal of Real Estate
Finance and Economics 17, 99-121.
Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. http://www.jstatsoft.org/v63/i18/.
See Also
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
data(oldcol)
COL.lag.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb))
summary(COL.lag.eig, correlation=TRUE)
COL.lag.stsls <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb))
summary(COL.lag.stsls, correlation=TRUE)
COL.lag.stslsW <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb), W2X=FALSE)
summary(COL.lag.stslsW, correlation=TRUE)
COL.lag.stslsR <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb),
robust=TRUE, W2X=FALSE)
summary(COL.lag.stslsR, correlation=TRUE)
COL.lag.stslsRl <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb),
robust=TRUE, legacy=TRUE, W2X=FALSE)
summary(COL.lag.stslsRl, correlation=TRUE)
data(boston)
gp2a <- stsls(log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + I(RM^2) +
AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT),
data=boston.c, nb2listw(boston.soi))
summary(gp2a)
Example output
Loading required package: sp
Loading required package: Matrix
Call:
lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb))
Residuals:
Min 1Q Median 3Q Max
-37.68585 -5.35636 0.05421 6.02013 23.20555
Type: lag
Coefficients: (asymptotic standard errors)
Estimate Std. Error z value Pr(>|z|)
(Intercept) 45.079251 7.177347 6.2808 3.369e-10
INC -1.031616 0.305143 -3.3808 0.0007229
HOVAL -0.265926 0.088499 -3.0049 0.0026570
Rho: 0.43102, LR test value: 9.9736, p-value: 0.001588
Asymptotic standard error: 0.11768
z-value: 3.6626, p-value: 0.00024962
Wald statistic: 13.415, p-value: 0.00024962
Log likelihood: -182.3904 for lag model
ML residual variance (sigma squared): 95.494, (sigma: 9.7721)
Number of observations: 49
Number of parameters estimated: 5
AIC: 374.78, (AIC for lm: 382.75)
LM test for residual autocorrelation
test value: 0.31955, p-value: 0.57188
Correlation of coefficients
sigma rho (Intercept) INC
rho -0.14
(Intercept) 0.12 -0.83
INC -0.05 0.35 -0.61
HOVAL -0.01 0.08 -0.25 -0.44
Call:
stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb))
Residuals:
Min 1Q Median 3Q Max
-37.86437 -5.65096 -0.13669 6.23315 22.90823
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Rho 0.454567 0.185118 2.4555 0.014067
(Intercept) 43.793442 10.952229 3.9986 6.372e-05
INC -1.000716 0.383858 -2.6070 0.009134
HOVAL -0.265489 0.091852 -2.8904 0.003847
Residual variance (sigma squared): 103.44, (sigma: 10.171)
Correlation of Coefficients:
[,1] [,2] [,3]
[1,] -0.92
[2,] 0.63 -0.76
[3,] 0.04 -0.16 -0.36
Call:
stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb),
W2X = FALSE)
Residuals:
Min 1Q Median 3Q Max
-37.785778 -5.442414 -0.052649 6.170104 23.039123
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Rho 0.444202 0.189141 2.3485 0.018848
(Intercept) 44.359512 11.157079 3.9759 7.011e-05
INC -1.014319 0.387469 -2.6178 0.008850
HOVAL -0.265681 0.091954 -2.8893 0.003861
Residual variance (sigma squared): 103.67, (sigma: 10.182)
Correlation of Coefficients:
[,1] [,2] [,3]
[1,] -0.93
[2,] 0.64 -0.77
[3,] 0.04 -0.16 -0.36
Call:
stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb),
robust = TRUE, W2X = FALSE)
Residuals:
Min 1Q Median 3Q Max
-37.785778 -5.442414 -0.052649 6.170104 23.039123
Coefficients:
Estimate HC0 std. Error z value Pr(>|z|)
Rho 0.44420 0.13748 3.2310 0.001234
(Intercept) 44.35951 7.67306 5.7812 7.417e-09
INC -1.01432 0.44113 -2.2993 0.021486
HOVAL -0.26568 0.17353 -1.5311 0.125752
Residual variance (sigma squared): 103.67, (sigma: 10.182)
Correlation of Coefficients:
[,1] [,2] [,3]
[1,] -0.90
[2,] 0.15 -0.28
[3,] 0.24 -0.24 -0.83
Call:
stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb),
robust = TRUE, legacy = TRUE, W2X = FALSE)
Residuals:
Min 1Q Median 3Q Max
-38.654607 -5.141303 -0.065221 5.864384 23.671589
Coefficients:
Estimate HC0 std. Error z value Pr(>|z|)
Rho 0.40138 0.13554 2.9613 0.003064
(Intercept) 47.37696 7.49975 6.3171 2.664e-10
INC -1.15183 0.43490 -2.6485 0.008085
HOVAL -0.25047 0.17333 -1.4450 0.148461
Asymptotic robust residual variance: 96.446, (sigma: 9.8207)
Correlation of Coefficients:
Rho (Intercept) INC
(Intercept) -0.89
INC 0.12 -0.26
HOVAL 0.25 -0.26 -0.83
Call:stsls(formula = log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) +
I(RM^2) + AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B +
log(LSTAT), data = boston.c, listw = nb2listw(boston.soi))
Residuals:
Min 1Q Median 3Q Max
-0.5356002 -0.0758562 -0.0045074 0.0719613 0.7128012
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Rho 4.5925e-01 3.8485e-02 11.9330 < 2.2e-16
(Intercept) 2.4025e+00 2.1710e-01 11.0661 < 2.2e-16
CRIM -7.3557e-03 1.0345e-03 -7.1100 1.160e-12
ZN 3.6435e-04 3.9311e-04 0.9268 0.3540112
INDUS 1.1992e-03 1.8365e-03 0.6530 0.5137794
CHAS1 1.1929e-02 2.6632e-02 0.4479 0.6542202
I(NOX^2) -2.8874e-01 9.2546e-02 -3.1199 0.0018091
I(RM^2) 6.6991e-03 1.0192e-03 6.5728 4.938e-11
AGE -2.5810e-04 4.0940e-04 -0.6304 0.5284073
log(DIS) -1.6043e-01 2.6107e-02 -6.1451 7.993e-10
log(RAD) 7.1704e-02 1.4926e-02 4.8038 1.557e-06
TAX -3.6857e-04 9.5315e-05 -3.8668 0.0001103
PTRATIO -1.2957e-02 4.1334e-03 -3.1347 0.0017203
B 2.8845e-04 8.0266e-05 3.5937 0.0003261
log(LSTAT) -2.3984e-01 2.2470e-02 -10.6740 < 2.2e-16
Residual variance (sigma squared): 0.020054, (sigma: 0.14161)
spdep documentation built on Aug. 19, 2017, 3:01 a.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
The function fits a spatial lag model by two stage least squares, with the option of adjusting the results for heteroskedasticity.
Usage
1 2 |
Arguments
formula |
a symbolic description of the model to be fit. The details
of model specification are given for |
data |
an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called. |
listw |
a |
zero.policy |
default NULL, use global option value; if TRUE assign zero to the lagged value of zones without
neighbours, if FALSE (default) assign NA - causing |
na.action |
a function (default |
robust |
default FALSE, if TRUE, apply a heteroskedasticity correction to the coefficients covariances |
HC |
default NULL, if |
legacy |
the argument chooses between two implementations of the robustness correction: default FALSE - use the estimate of Omega only in the White consistent estimator of the variance-covariance matrix, if TRUE, use the original implementation which runs a GLS using the estimate of Omega, and yields different coefficient estimates as well - see example below |
W2X |
default TRUE, if FALSE only WX are used as instruments in the spatial two stage least squares; until release 0.4-60, only WX were used - see example below |
Details
The fitting implementation fits a spatial lag model:
y = rho W y + X beta + e
by using spatially lagged X variables as instruments for the spatially lagged dependent variable.
Value
an object of class "stsls" containing:
coefficients |
coefficient estimates |
var |
coefficient covariance matrix |
sse |
sum of squared errors |
residuals |
model residuals |
df |
degrees of freedom |
Author(s)
Luc Anselin, Gianfranco Piras and Roger Bivand
References
Kelejian, H.H. and I.R. Prucha (1998). A generalized spatial two stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances. Journal of Real Estate Finance and Economics 17, 99-121.
Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. http://www.jstatsoft.org/v63/i18/.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | data(oldcol)
COL.lag.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb))
summary(COL.lag.eig, correlation=TRUE)
COL.lag.stsls <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb))
summary(COL.lag.stsls, correlation=TRUE)
COL.lag.stslsW <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb), W2X=FALSE)
summary(COL.lag.stslsW, correlation=TRUE)
COL.lag.stslsR <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb),
robust=TRUE, W2X=FALSE)
summary(COL.lag.stslsR, correlation=TRUE)
COL.lag.stslsRl <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb),
robust=TRUE, legacy=TRUE, W2X=FALSE)
summary(COL.lag.stslsRl, correlation=TRUE)
data(boston)
gp2a <- stsls(log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + I(RM^2) +
AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT),
data=boston.c, nb2listw(boston.soi))
summary(gp2a)
|
Example output
Loading required package: sp
Loading required package: Matrix
Call:
lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb))
Residuals:
Min 1Q Median 3Q Max
-37.68585 -5.35636 0.05421 6.02013 23.20555
Type: lag
Coefficients: (asymptotic standard errors)
Estimate Std. Error z value Pr(>|z|)
(Intercept) 45.079251 7.177347 6.2808 3.369e-10
INC -1.031616 0.305143 -3.3808 0.0007229
HOVAL -0.265926 0.088499 -3.0049 0.0026570
Rho: 0.43102, LR test value: 9.9736, p-value: 0.001588
Asymptotic standard error: 0.11768
z-value: 3.6626, p-value: 0.00024962
Wald statistic: 13.415, p-value: 0.00024962
Log likelihood: -182.3904 for lag model
ML residual variance (sigma squared): 95.494, (sigma: 9.7721)
Number of observations: 49
Number of parameters estimated: 5
AIC: 374.78, (AIC for lm: 382.75)
LM test for residual autocorrelation
test value: 0.31955, p-value: 0.57188
Correlation of coefficients
sigma rho (Intercept) INC
rho -0.14
(Intercept) 0.12 -0.83
INC -0.05 0.35 -0.61
HOVAL -0.01 0.08 -0.25 -0.44
Call:
stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb))
Residuals:
Min 1Q Median 3Q Max
-37.86437 -5.65096 -0.13669 6.23315 22.90823
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Rho 0.454567 0.185118 2.4555 0.014067
(Intercept) 43.793442 10.952229 3.9986 6.372e-05
INC -1.000716 0.383858 -2.6070 0.009134
HOVAL -0.265489 0.091852 -2.8904 0.003847
Residual variance (sigma squared): 103.44, (sigma: 10.171)
Correlation of Coefficients:
[,1] [,2] [,3]
[1,] -0.92
[2,] 0.63 -0.76
[3,] 0.04 -0.16 -0.36
Call:
stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb),
W2X = FALSE)
Residuals:
Min 1Q Median 3Q Max
-37.785778 -5.442414 -0.052649 6.170104 23.039123
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Rho 0.444202 0.189141 2.3485 0.018848
(Intercept) 44.359512 11.157079 3.9759 7.011e-05
INC -1.014319 0.387469 -2.6178 0.008850
HOVAL -0.265681 0.091954 -2.8893 0.003861
Residual variance (sigma squared): 103.67, (sigma: 10.182)
Correlation of Coefficients:
[,1] [,2] [,3]
[1,] -0.93
[2,] 0.64 -0.77
[3,] 0.04 -0.16 -0.36
Call:
stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb),
robust = TRUE, W2X = FALSE)
Residuals:
Min 1Q Median 3Q Max
-37.785778 -5.442414 -0.052649 6.170104 23.039123
Coefficients:
Estimate HC0 std. Error z value Pr(>|z|)
Rho 0.44420 0.13748 3.2310 0.001234
(Intercept) 44.35951 7.67306 5.7812 7.417e-09
INC -1.01432 0.44113 -2.2993 0.021486
HOVAL -0.26568 0.17353 -1.5311 0.125752
Residual variance (sigma squared): 103.67, (sigma: 10.182)
Correlation of Coefficients:
[,1] [,2] [,3]
[1,] -0.90
[2,] 0.15 -0.28
[3,] 0.24 -0.24 -0.83
Call:
stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = nb2listw(COL.nb),
robust = TRUE, legacy = TRUE, W2X = FALSE)
Residuals:
Min 1Q Median 3Q Max
-38.654607 -5.141303 -0.065221 5.864384 23.671589
Coefficients:
Estimate HC0 std. Error z value Pr(>|z|)
Rho 0.40138 0.13554 2.9613 0.003064
(Intercept) 47.37696 7.49975 6.3171 2.664e-10
INC -1.15183 0.43490 -2.6485 0.008085
HOVAL -0.25047 0.17333 -1.4450 0.148461
Asymptotic robust residual variance: 96.446, (sigma: 9.8207)
Correlation of Coefficients:
Rho (Intercept) INC
(Intercept) -0.89
INC 0.12 -0.26
HOVAL 0.25 -0.26 -0.83
Call:stsls(formula = log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) +
I(RM^2) + AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B +
log(LSTAT), data = boston.c, listw = nb2listw(boston.soi))
Residuals:
Min 1Q Median 3Q Max
-0.5356002 -0.0758562 -0.0045074 0.0719613 0.7128012
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Rho 4.5925e-01 3.8485e-02 11.9330 < 2.2e-16
(Intercept) 2.4025e+00 2.1710e-01 11.0661 < 2.2e-16
CRIM -7.3557e-03 1.0345e-03 -7.1100 1.160e-12
ZN 3.6435e-04 3.9311e-04 0.9268 0.3540112
INDUS 1.1992e-03 1.8365e-03 0.6530 0.5137794
CHAS1 1.1929e-02 2.6632e-02 0.4479 0.6542202
I(NOX^2) -2.8874e-01 9.2546e-02 -3.1199 0.0018091
I(RM^2) 6.6991e-03 1.0192e-03 6.5728 4.938e-11
AGE -2.5810e-04 4.0940e-04 -0.6304 0.5284073
log(DIS) -1.6043e-01 2.6107e-02 -6.1451 7.993e-10
log(RAD) 7.1704e-02 1.4926e-02 4.8038 1.557e-06
TAX -3.6857e-04 9.5315e-05 -3.8668 0.0001103
PTRATIO -1.2957e-02 4.1334e-03 -3.1347 0.0017203
B 2.8845e-04 8.0266e-05 3.5937 0.0003261
log(LSTAT) -2.3984e-01 2.2470e-02 -10.6740 < 2.2e-16
Residual variance (sigma squared): 0.020054, (sigma: 0.14161)
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