spsur user guide
The user guide
Román Mínguez, University of Castilla-La Mancha (Spain)
Fernando A. López, Technical University of Cartagena (Spain)
Jesús Mur, University of Zaragoza, (Spain)
2022-10-29
Load packages:
library(spsur)
library(sf)
library(dplyr)1 Introduction
The purpose of this vignette is to show the functionalities of the new R-package spsur (Mínguez, López, and Mur 2022; López, Mı́nguez, and Mur 2020). In the vignette the reader will find information about the following aspects:
- Description of the dataset NCOVR, employed to illustrate the use of the package.
- Estimation of SUR models without spatial effects (the so called SUR-SIM models).
- Testing for various forms of spatial autocorrelation in the SUR-SIM model.
- In case of misspecification of the SUR-SIM model, estimation of SUR models with spatial effects, by ML or IV.
- Misspecification tests in a estimated spatial SUR model.
- Measures of spatial interaction for a spatial SUR model, the so-called Direct/Indirect/Total effects.
- Estimation of a spatial SUR panel data model, with 1 equation and several cross-sections.
- Generation of random data sets with the required spatial SUR structure.
2 The data set NCOVR
Throughout the vignette we use the dataset NCOVR (National Consortium on Violence Research) to illustrate the capabilities of the package. The same data were employed by Baller et al. (2001) to analyze the incidence of homicides rates in the US counties. The dataset can be freely dowloaded from https://geodacenter.github.io/data-and-lab/ncovr/
There are three dimensions to consider in a SUR model: N, number of individuals, Tm, number of time periods and G, number of equations. Typically, in a spatial setting N will be large and Tm moderate with a small number of equations. NCOVR fits very well with the above description: it contains 3085 spatial units (counties), for 4 different cross-sections (1960, 1970, 1980, 1990) and 69 variables. According to Baller et al. (2001), the main variables of interest are:
- HR: homicide rate per 100000 inhabitants
- RD: resource deprivation
- PS: population structure
- MA: median age
- DV: divorce rate (% males over 14 divorced)
- UE: unemployment rate
- SOUTH: dummy variable for Southern counties (South = 1)
spsur deals with datasets from large to very small, especially in what respects to the number of spatial units. In this case, the dimensions of NCOVR make it a large dataset.
First, we can read the NCOVR dataset as a simple feature (sf) object (named NCOVR.sf),
data(NCOVR, package = "spsur")The first three observations in NCOVR appear below:
NCOVR <- st_drop_geometry(NCOVR.sf)
knitr::kable(
head((NCOVR[1:3, 1:6])),
caption = 'First observations of NCOVR dataset' )| NAME | STATE_NAME | FIPS | SOUTH | HR60 | HR70 |
|---|---|---|---|---|---|
| Lake of the Woods | Minnesota | 27077 | 0 | 0.000000 | 0.000000 |
| Ferry | Washington | 53019 | 0 | 0.000000 | 0.000000 |
| Stevens | Washington | 53065 | 0 | 1.863863 | 1.915158 |
Whereas the geometry of the USA counties is shown in Figure 1:
plot(st_geometry(NCOVR.sf))
Figure 1: Geometry of the USA counties
Following Baller et al. (2001), we consider a W matrix based on the criterion of 10 nearest-neighbourhood, which is inmediate to obtain using the spdep package, Bivand, Pebesma, and Gómez-Rubio (2013). The resulting weighting matrix can be obtained using lwncovr object. Note that this matrix is non-symmetric.
# Obtain coordinates of centroids
co <- sf::st_coordinates(sf::st_centroid(NCOVR.sf))
lwncovr <- spdep::nb2listw(spdep::knn2nb(spdep::knearneigh(co, k = 10,longlat = TRUE)))Figure 2 shows the Quantile map for the Homicide Rate in the year 1960. Apparently there exist a strong structure of positive spatial autocorrelation. We observe a big cluster of the type High-High in the South East and South West counties, which evolves to Low-Low as we move to the North.
q <- quantile(NCOVR.sf$HR60)
NCOVR.sf$qm <- (NCOVR.sf$HR60 > q[2]) + (NCOVR.sf$HR60 > q[3]) +(NCOVR.sf$HR60 >= q[4]) + 1
plot(NCOVR.sf["qm"], pal = c("#FFFEDE","#FFDFA2", "#FFA93F", "#D5610D"),
main = "")
Figure 2: Quantile Map; Variable HR60
Baller et al. (2001) specify a single linear model to explain the case of HR in the year 1960 with the results shown in the Table below, identical to those in Table 1 of Baller et al. (2001).
\[\begin{equation} HR_{60} = \beta_{0}+\beta_{1}RD_{60}+\beta_{2}PS_{60}+\beta_{3}MA_{60}+\beta_{4}DV_{60} +\beta_{5}UE_{60} +\beta_{6}SOUTH+\epsilon_{60} \tag{2.1} \end{equation}\]
formula_60 <- HR60 ~ RD60 + PS60 + MA60 + DV60 + UE60 + SOUTH
lm_60 <- lm(formula = formula_60, data = NCOVR.sf)
summary(lm_60)#>
#> Call:
#> lm(formula = formula_60, data = NCOVR.sf)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -14.026 -2.217 -0.635 1.393 88.312
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 8.12591 0.63465 12.804 < 2e-16 ***
#> RD60 1.79824 0.12341 14.571 < 2e-16 ***
#> PS60 0.35871 0.09216 3.892 0.000101 ***
#> MA60 -0.23047 0.01932 -11.931 < 2e-16 ***
#> DV60 1.16002 0.09483 12.233 < 2e-16 ***
#> UE60 -0.06195 0.03515 -1.762 0.078138 .
#> SOUTH 2.63862 0.23325 11.312 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 4.743 on 3078 degrees of freedom
#> Multiple R-squared: 0.2966, Adjusted R-squared: 0.2952
#> F-statistic: 216.3 on 6 and 3078 DF, p-value: < 2.2e-16Figure 2 points to the existence of cross-sectional dependence in the spatial distribution of HR60 but the model in (2.1) does not contain elements of spatial interaction. The consequence is a scattermap of the LS residuals, Figure 3, with a strong structure of clusters, similar to that of HR60 in Figure 2.
q <- quantile(lm_60$residuals)
NCOVR.sf$qmr <- (lm_60$residuals > q[2]) + (lm_60$residuals > q[3]) + (lm_60$residuals >= q[4]) + 1
plot(NCOVR.sf["qmr"], pal = c("#FFFEDE","#FFDFA2", "#FFA93F", "#D5610D"), main = "")
Figure 3: Quantile Map of LS residuals
The battery of LM tests for omitted spatial structure, supplied by spdep Bivand, Pebesma, and Gómez-Rubio (2013), confirms the misspecification of (2.1). According to these results, the most adequate specification is a spatial lag model. Similar results are obtained for the years 1970, 1980 and 1990, which we omit for reasons of space.
print(spdep::lm.LMtests(model=lm_60,listw = lwncovr,test="all"))#>
#> Lagrange multiplier diagnostics for spatial dependence
#>
#> data:
#> model: lm(formula = formula_60, data = NCOVR.sf)
#> weights: lwncovr
#>
#> LMerr = 208.12, df = 1, p-value < 2.2e-16
#>
#>
#> Lagrange multiplier diagnostics for spatial dependence
#>
#> data:
#> model: lm(formula = formula_60, data = NCOVR.sf)
#> weights: lwncovr
#>
#> LMlag = 232.1, df = 1, p-value < 2.2e-16
#>
#>
#> Lagrange multiplier diagnostics for spatial dependence
#>
#> data:
#> model: lm(formula = formula_60, data = NCOVR.sf)
#> weights: lwncovr
#>
#> RLMerr = 1.1565, df = 1, p-value = 0.2822
#>
#>
#> Lagrange multiplier diagnostics for spatial dependence
#>
#> data:
#> model: lm(formula = formula_60, data = NCOVR.sf)
#> weights: lwncovr
#>
#> RLMlag = 25.138, df = 1, p-value = 5.337e-07
#>
#>
#> Lagrange multiplier diagnostics for spatial dependence
#>
#> data:
#> model: lm(formula = formula_60, data = NCOVR.sf)
#> weights: lwncovr
#>
#> SARMA = 233.25, df = 2, p-value < 2.2e-16The previous discussion is an example of the traditional approach in spatial econometrics, where we process the data cross-section by cross-section, perhaps in a panel framework. We stop here this analysis in favour of a multivariate approach using SUR models, which is the purpose of spsur.
3 A single SUR model
We begin by introducing dependence in the errors of the same county across time, as in expression (3.1) below. Note that time dependence in the errors of different counties is not allowed. To simplify we consider only two cross-sections, 1960 and 1970.
\[\begin{equation} HR_{60} = \beta_{10}+\beta_{11}RD_{60}+\beta_{12}PS_{60}+\beta_{13}MA_{60}+\beta_{14}DV_{60} +\beta_{15}UE_{60} +\beta_{16}SOUTH+\epsilon_{60}\\ HR_{70} = \beta_{20}+\beta_{21}RD_{70}+\beta_{22}PS_{70}+\beta_{23}MA_{70}+\beta_{24}DV_{70} +\beta_{25}UE_{70} +\beta_{26}SOUTH+\epsilon_{70} \\ Corr(\epsilon_{60},\epsilon_{70}) \neq 0 \\ \tag{3.1} \end{equation}\]
The package Formula is used to build the model in spsur
formula_sur <- HR60 | HR70 ~ RD60 + PS60 + MA60 + DV60 + UE60 + SOUTH | RD70 + PS70 + MA70 + DV70 + UE70 + SOUTHThe function spsurml() estimates a pure SUR model without spatial effects, which we call SUR-SIM. The key argument to estimate this model is type = “sim”, which declares the type of spatial specification desired by the user.
sur.sim <- spsurml(formula = formula_sur, type = "sim", data = NCOVR.sf)#> Initial point:
#> log_lik: -18908.66
#> Iteration: 1 log_lik: -18908.64
#> Iteration: 2 log_lik: -18908.64
#> Time to fit the model: 0.13 seconds
#> Time to compute covariances: 0.03 secondssummary(sur.sim)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, type = "sim")
#>
#>
#> Spatial SUR model type: sim
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 7.934600 0.628650 12.6216 < 2.2e-16 ***
#> RD60_1 1.704134 0.122195 13.9461 < 2.2e-16 ***
#> PS60_1 0.336554 0.091928 3.6610 0.0002533 ***
#> MA60_1 -0.222449 0.019157 -11.6119 < 2.2e-16 ***
#> DV60_1 1.111576 0.093760 11.8555 < 2.2e-16 ***
#> UE60_1 -0.063549 0.034323 -1.8515 0.0641472 .
#> SOUTH_1 2.764987 0.232276 11.9039 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.2964
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 8.457736 0.760008 11.1285 < 2.2e-16 ***
#> RD70_2 2.775541 0.147823 18.7761 < 2.2e-16 ***
#> PS70_2 0.772452 0.116483 6.6315 3.606e-11 ***
#> MA70_2 -0.186988 0.022532 -8.2988 < 2.2e-16 ***
#> DV70_2 1.162703 0.103180 11.2687 < 2.2e-16 ***
#> UE70_2 -0.223708 0.048739 -4.5899 4.522e-06 ***
#> SOUTH_2 3.768599 0.284478 13.2474 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3609
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 22.459968 7.288907
#> 7.288907 34.553637
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2616441
#> 0.2616441 1.0000000
#>
#> R-sq. pooled: 0.3503
#> Breusch-Pagan: 211.2 p-value: (7.54e-48)The signs of the variables are as expected and, more interesting, the SUR estimation detects a strong positive dependence between the errors for the year 1960 and those of the year 1970, 0.2616 according to the data in the Correlation Matrix of inter-equation residuals. The Breusch-Pagan diagonality test that appears at the end of the Table confirms that this dependence is statistically significant and should not be avoided.
Figure 4 shows the quantile maps of the SUR residuals corresponding to the years 1960 and 1970. The maps reveal that time dependence was only part of the problem. Indeed, there remain a strong spatial structure in the SUR residuals which is apparently stable across time.
res <- residuals(sur.sim)
q <- quantile(res[[1]])
NCOVR.sf$SUR_residuals_1960 <- (res[[1]] > q[2]) + (res[[1]] > q[3]) + (res[[1]] >= q[4]) + 1
q <- quantile(res[[2]])
NCOVR.sf$SUR_residuals_1970 <- (res[[2]]>q[1]) + (res[[2]]>q[3]) + (res[[2]] >= q[4]) + 1
plot(NCOVR.sf[c("SUR_residuals_1960","SUR_residuals_1970")],
pal = c("#FFFEDE","#FFDFA2", "#FFA93F", "#D5610D"))
Figure 4: Quantile Map of LS residuals for SUR-SIM model
The battery of Lagrange Multipliers that appear below are obtained with the function lmtestspsur(). The output is a list of 5 elements each one of the class htest. They are the traditional LM misspecification tests (Anselin and Bera 1998) for omitted spatial elements, adapted to a SUR framework by (Mur, López, and Herrera 2010). All the Multipliers are statistically significant, both the raw and robust LMs as well as the joint LM for SARMA processes.
lmtest.sur <- lmtestspsur(formula = formula_sur,
listw = lwncovr, data = NCOVR.sf)
print(lmtest.sur)#> [[1]]
#>
#> LM-SUR-SLM
#>
#> data: NCOVR.sf
#> LM-stat = 408.46, df = 2, p-value < 2.2e-16
#>
#>
#> [[2]]
#>
#> LM-SUR-SEM
#>
#> data: NCOVR.sf
#> LM-stat = 382.82, df = 2, p-value < 2.2e-16
#>
#>
#> [[3]]
#>
#> LM*-SUR-SLM
#>
#> data: NCOVR.sf
#> LM-stat = 37.779, df = 2, p-value = 6.256e-09
#>
#>
#> [[4]]
#>
#> LM*-SUR-SEM
#>
#> data: NCOVR.sf
#> LM-stat = 12.144, df = 2, p-value = 0.002306
#>
#>
#> [[5]]
#>
#> LM-SUR-SARAR
#>
#> data: NCOVR.sf
#> LM-stat = 420.6, df = 4, p-value < 2.2e-16The conclusion is that the simple SUR model of expression (3.1) is misspecified due of the omission of relevant spatial elements. The problem for the user is the identification of the elements of spatial structure that are missing from the equation. In fact, all the Multipliers are statistically significant, which would point to a SUR-SARAR specification.
As a way of example, the next Section estimates all the possible spatial configurations of a SUR model by using the function spsurml().
4 Estimation of spatial SUR models
As said, spsur is an R package whose main objective is to estimate and test SUR models with a given spatial structure. It is assumed that the same spatial structure appears in all time periods and equations. The catalog of spatial specification that this package can process are
- SUR-SLX. SUR model with lags of the X variables.
- SUR-SLM. SUR model with lags of the endogenous variables.
- SUR-SEM. SUR model with spatial errors.
- SUR-SDM. SUR model with lags of the endogenous and of the X variables, or Spatial Durbin.
- SUR-SDEM. SUR model with with lags of the X variables and spatial errors.
- SUR-SARAR. SUR model with lags of the of the endogenous variables and spatial errors.
- SUR-GNM. SUR model with lags of the of the endogenous variables, lags of the X variables and lags spatial errors. This specification nest the rest of models.
The common estimation procedure for the seven cases is maximum-likelihood, ML, whereas the SUR-SLM and SUR-SDM can also be estimated by Three Stage-Least-Squares, 3SLS. Bivand and Piras (2015) offer an excellent revision of estimation methods in spatial econometrics, whereas López, Mı́nguez, and Mur (2020) focus specifically in the comparison of ML against 3SLS. First, we discuss the maximum likelihood approach and finish the Section with a few comments in relation to 3SLS.
4.1 SUR-SLX model
The SUR-SLX model includes spatial lags of of the X variables, assumed to be exogenous. The user should specify the argument type = “slx” in the function spsurml()
sur.slx <- spsurml(formula = formula_sur, listw = lwncovr,
type = "slx",data = NCOVR.sf)#> Initial point:
#> log_lik: -18855.23
#> Iteration: 1 log_lik: -18855.21
#> Iteration: 2 log_lik: -18855.21
#> Time to fit the model: 0.07 seconds
#> Time to compute covariances: 0.22 secondssummary(sur.slx)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "slx")
#>
#>
#> Spatial SUR model type: slx
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 9.401023 0.853874 11.0099 < 2.2e-16 ***
#> RD60_1 1.389561 0.198873 6.9872 3.101e-12 ***
#> PS60_1 0.355592 0.139128 2.5559 0.0106166 *
#> MA60_1 -0.166326 0.029978 -5.5482 3.007e-08 ***
#> DV60_1 0.678313 0.127574 5.3170 1.092e-07 ***
#> UE60_1 0.070046 0.045620 1.5354 0.1247268
#> SOUTH_1 -0.526364 0.854111 -0.6163 0.5377384
#> lag.RD60_1 0.586938 0.258467 2.2708 0.0231911 *
#> lag.PS60_1 0.178879 0.188822 0.9473 0.3435032
#> lag.MA60_1 -0.105026 0.039828 -2.6370 0.0083851 **
#> lag.DV60_1 1.026468 0.185926 5.5208 3.512e-08 ***
#> lag.UE60_1 -0.356145 0.070416 -5.0577 4.366e-07 ***
#> lag.SOUTH_1 3.210360 0.913474 3.5145 0.0004438 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3111
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 12.580067 1.089163 11.5502 < 2.2e-16 ***
#> RD70_2 2.524805 0.221782 11.3842 < 2.2e-16 ***
#> PS70_2 0.645347 0.171796 3.7565 0.0001739 ***
#> MA70_2 -0.086883 0.030887 -2.8129 0.0049246 **
#> DV70_2 1.108540 0.134863 8.2197 2.466e-16 ***
#> UE70_2 -0.079101 0.067060 -1.1796 0.2382249
#> SOUTH_2 -0.472159 1.062209 -0.4445 0.6566924
#> lag.RD70_2 0.208400 0.306689 0.6795 0.4968369
#> lag.PS70_2 0.194587 0.234474 0.8299 0.4066346
#> lag.MA70_2 -0.229987 0.045301 -5.0769 3.950e-07 ***
#> lag.DV70_2 0.275648 0.199942 1.3786 0.1680566
#> lag.UE70_2 -0.332812 0.097684 -3.4070 0.0006610 ***
#> lag.SOUTH_2 4.368685 1.145674 3.8132 0.0001385 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3729
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.990566 6.872353
#> 6.872353 33.903587
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2516891
#> 0.2516891 1.0000000
#>
#> R-sq. pooled: 0.363
#> Breusch-Pagan: 195.4 p-value: (2.08e-44)By default, type = “slx” implies that lags of all the X variables appearing in the equations are included as additional regressors. However, `spsurml()´allows the user to decide the lags of which X variables should be added. In this case, the user activates the argument Durbin = . Suppose, by example, that we only want to include the spatial lag of \(RD_{60}\) in the fist equation of (3.1) and the spatial lags of the \(DV_{70}\) and \(MA_{70}\) in the second equation. Then, we would write:
formulaD <- ~ DV60 | MA70 + UE70
sur.slx2 <- spsurml(formula = formula_sur, listw = lwncovr,
type = "slx", Durbin = formulaD,
data = NCOVR.sf)#> Initial point:
#> log_lik: -18884.89
#> Iteration: 1 log_lik: -18884.85
#> Iteration: 2 log_lik: -18884.85
#> Time to fit the model: 0.08 seconds
#> Time to compute covariances: 0.01 secondssummary(sur.slx2)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "slx", Durbin = formulaD)
#>
#>
#> Spatial SUR model type: slx
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 7.450241 0.646059 11.5318 < 2.2e-16 ***
#> RD60_1 1.817856 0.124964 14.5470 < 2.2e-16 ***
#> PS60_1 0.431101 0.094683 4.5531 5.388e-06 ***
#> MA60_1 -0.224789 0.019122 -11.7552 < 2.2e-16 ***
#> DV60_1 0.794215 0.123873 6.4115 1.548e-10 ***
#> UE60_1 -0.081720 0.034482 -2.3699 0.01782 *
#> SOUTH_1 2.720772 0.231840 11.7356 < 2.2e-16 ***
#> lag.DV60_1 0.671427 0.171205 3.9218 8.886e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.2999
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 12.537289 1.066144 11.7595 < 2.2e-16 ***
#> RD70_2 2.703698 0.148068 18.2598 < 2.2e-16 ***
#> PS70_2 0.753463 0.115901 6.5009 8.612e-11 ***
#> MA70_2 -0.093525 0.029255 -3.1969 0.001396 **
#> DV70_2 1.195748 0.103137 11.5938 < 2.2e-16 ***
#> UE70_2 -0.099544 0.065857 -1.5115 0.130711
#> SOUTH_2 3.727709 0.283142 13.1655 < 2.2e-16 ***
#> lag.MA70_2 -0.213519 0.041963 -5.0882 3.721e-07 ***
#> lag.UE70_2 -0.248043 0.089825 -2.7614 0.005773 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3684
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 22.346575 7.151706
#> 7.151706 34.145145
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2589045
#> 0.2589045 1.0000000
#>
#> R-sq. pooled: 0.3562
#> Breusch-Pagan: 206.8 p-value: (6.87e-47)4.2 SUR-SLM model
The specification of the SLM-SUR model that we want to estimate is standard,
\[\begin{equation} HR_{60} = \rho_{60}WHR_{60} + \beta_{10}+\beta_{11}RD_{60}+\beta_{12}PS_{60}+\beta_{13}MA_{60}+\beta_{14}DV_{60} +\beta_{15}UE_{60} +\beta_{16}SOUTH+\epsilon_{60} \\ HR_{70} = \rho_{60}WHR_{70} + \beta_{20}+\beta_{21}RD_{70}+\beta_{22}PS_{70}+\beta_{23}MA_{70}+\beta_{24}DV_{70} +\beta_{25}UE_{70} +\beta_{26}SOUTH+\epsilon_{70}\\ Corr(\epsilon_{60},\epsilon_{70}) \neq 0 \tag{4.1} \end{equation}\]
which can be estimated by maximum likelihood using the argument type = “slm” in the function spsurml().
A severe handicap of ML is the evaluation of the Jacobian term, which is highly time consuming. We offer some flexibility to the user with the arguments method = and control= imported from spatialreg, Bivand, Pebesma, and Gómez-Rubio (2013). The argument method = is used to select the method to evaluate the Jacobian. The options are eigen, Matrix, LU, Chebyshev and MC. The default is eigen and Matrix required the W matrix to be symmetric. control= adds a list of functionalities to expedite the calculus, which can be consulted in the documentation of spatialreg.
control <- list(fdHess = TRUE)
sur.slm <- spsurml(formula = formula_sur, listw = lwncovr,
method = "LU", type = "slm", control = control,
data = NCOVR.sf)#> sparse matrix LU decomposition
#> Initial point: log_lik: -18771.55 rhos: 0.335 0.376
#> Iteration: 1 log_lik: -18761.49 rhos: 0.346 0.393
#> Iteration: 2 log_lik: -18761.48 rhos: 0.347 0.393
#> Iteration: 3 log_lik: -18761.48 rhos: 0.347 0.393
#> Time to fit the model: 8.66 seconds
#> Computing numerical covariances...
#> Time to compute covariances: 2.15 secondssummary(sur.slm)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "slm", method = "LU", control = control)
#>
#>
#> Spatial SUR model type: slm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 5.607673 0.611299 9.1734 < 2.2e-16 ***
#> RD60_1 1.334419 0.118843 11.2284 < 2.2e-16 ***
#> PS60_1 0.274531 0.089152 3.0794 0.002084 **
#> MA60_1 -0.166839 0.018620 -8.9602 < 2.2e-16 ***
#> DV60_1 0.879403 0.091237 9.6386 < 2.2e-16 ***
#> UE60_1 -0.030445 0.033560 -0.9072 0.364341
#> SOUTH_1 1.506056 0.225406 6.6815 2.572e-11 ***
#> rho_1 0.346695 0.028107 12.3350 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3396
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 5.157484 0.729818 7.0668 1.761e-12 ***
#> RD70_2 2.201353 0.141953 15.5076 < 2.2e-16 ***
#> PS70_2 0.648528 0.111524 5.8151 6.362e-09 ***
#> MA70_2 -0.121851 0.021629 -5.6337 1.842e-08 ***
#> DV70_2 0.987470 0.099146 9.9598 < 2.2e-16 ***
#> UE70_2 -0.168980 0.047050 -3.5915 0.0003314 ***
#> SOUTH_2 1.616949 0.272648 5.9305 3.184e-09 ***
#> rho_2 0.393272 0.026744 14.7053 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4154
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.096585 5.361957
#> 5.361957 31.640805
#> Correlation Matrix of inter-equation residuals:
#> 1.000000 0.207536
#> 0.207536 1.000000
#>
#> R-sq. pooled: 0.3996
#> Breusch-Pagan: 132.9 p-value: (9.63e-31)The plot() function allows the user to plot both beta and spatial coefficients for all equations of the spsur model. The argument viewplot is used to choose between interactive or non-interactive plots.
# Interactive plot
plot(sur.slm)
The print() function is used to print short tables including the values of beta and spatial coefficients as well as p-values of significance test for each coefficient. This can be used as an alternative to summary() when a brief output is needed. The NA values indicate that the coefficient has not been fitted in the corresponding equation.
print(sur.slm)#> coeff_1 pval_1 coeff_2 pval_2
#> (Intercept) 5.6077 0.000 5.1575 0
#> RD60 1.3344 0.000 NA NA
#> PS60 0.2745 0.002 NA NA
#> MA60 -0.1668 0.000 NA NA
#> DV60 0.8794 0.000 NA NA
#> UE60 -0.0304 0.364 NA NA
#> SOUTH 1.5061 0.000 1.6169 0
#> RD70 NA NA 2.2014 0
#> PS70 NA NA 0.6485 0
#> MA70 NA NA -0.1219 0
#> DV70 NA NA 0.9875 0
#> UE70 NA NA -0.1690 0
#> rho 0.3467 0.000 0.3933 04.3 SUR-SEM model
The corresponding specification is the following:
\[\begin{equation} HR_{60} = \beta_{10}+\beta_{11}RD_{60}+\beta_{12}PS_{60}+\beta_{13}MA_{60}+\beta_{14}DV_{60} +\beta_{15}UE_{60} +\beta_{16}SOUTH+u_{60} \ ; \\ u_{60}=\lambda_{60}W u_{60}+\epsilon_{60}\\ \\ HR_{70} = \beta_{20}+\beta_{21}RD_{70}+\beta_{22}PS_{70}+\beta_{23}MA_{70}+\beta_{24}DV_{70} +\beta_{25}UE_{70} +\beta_{26}SOUTH+u_{70} ; \\ u_{70}=\lambda_{70}W u_{70}+\epsilon_{70} \\ \\ Corr(\epsilon_{60},\epsilon_{70}) \neq 0 \tag{4.2} \end{equation}\]
The model is estimated by ML, with method = “LU” and calculating the analytical Hessian (option fdHess = TRUE in the argument control) in place of using the numerical approximation. The results appear below.
sur.sem <- spsurml(formula = formula_sur, listw = lwncovr,
method = "LU", type = "sem",
control = control, data = NCOVR.sf)#> sparse matrix LU decomposition
#> Initial point: log_lik: -18779.5 lambdas: 0.334 0.409
#> Iteration: 1 log_lik: -18769.69 lambdas: 0.347 0.43
#> Iteration: 2 log_lik: -18769.67 lambdas: 0.348 0.43
#> Iteration: 3 log_lik: -18769.67 lambdas: 0.348 0.43
#> Time to fit the model: 11.62 seconds
#> Computing numerical covariances...
#> Time to compute covariances: 2.82 secondssummary(sur.sem)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "sem", method = "LU", control = control)
#>
#>
#> Spatial SUR model type: sem
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 7.5049435 0.6128151 12.2467 < 2.2e-16 ***
#> RD60_1 1.6575939 0.1191372 13.9133 < 2.2e-16 ***
#> PS60_1 0.3199694 0.0893774 3.5800 0.0003463 ***
#> MA60_1 -0.2057071 0.0186664 -11.0202 < 2.2e-16 ***
#> DV60_1 0.9407399 0.0914624 10.2855 < 2.2e-16 ***
#> UE60_1 0.0026013 0.0336394 0.0773 0.9383649
#> SOUTH_1 2.6240053 0.2259739 11.6120 < 2.2e-16 ***
#> lambda_1 0.3476493 0.0303349 11.4604 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3369
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 6.735502 0.728709 9.2431 < 2.2e-16 ***
#> RD70_2 2.804145 0.141737 19.7841 < 2.2e-16 ***
#> PS70_2 0.731634 0.111361 6.5699 5.447e-11 ***
#> MA70_2 -0.132338 0.021596 -6.1279 9.459e-10 ***
#> DV70_2 1.111776 0.098994 11.2307 < 2.2e-16 ***
#> UE70_2 -0.146819 0.046974 -3.1255 0.001783 **
#> SOUTH_2 3.458503 0.272244 12.7037 < 2.2e-16 ***
#> lambda_2 0.430480 0.028888 14.9016 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4178
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.20416 5.39982
#> 5.39982 31.54910
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2087734
#> 0.2087734 1.0000000
#>
#> R-sq. pooled: 0.4
#> Breusch-Pagan: 134.5 p-value: (4.33e-31)# Non-interactive plot
if (require(gridExtra)) {
pl <- plot(sur.sem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
4.4 SUR-SDEM model
The specification of a spatial Durbin model with spatial errors and a SUR framework is the following
\[\begin{equation} HR_{60} = \beta_{10}+\beta_{11}RD_{60}+\beta_{12}PS_{60}+\beta_{13}MA_{60}+\beta_{14}DV_{60} +\beta_{15}UE_{60} +\beta_{16}SOUTH+u_{60} +\theta_{1}WDV_{60}\ ; \\ u_{60}=\lambda_{60}W u_{60}+\epsilon_{60}\\ \\ HR_{70} = \beta_{20}+\beta_{21}RD_{70}+\beta_{22}PS_{70}+\beta_{23}MA_{70}+\beta_{24}DV_{70} +\beta_{25}UE_{70} +\beta_{26}SOUTH+u_{70}+\theta_{2}WMA_{70}+\theta_{3}WUE_{70} ; \\ u_{70}=\lambda_{70}W u_{70}+\epsilon_{70} \\ \\ Corr(\epsilon_{60},\epsilon_{70}) \neq 0 \tag{4.3} \end{equation}\]
Note that the specificaction above, in spite of its name, does not include lags of the endogenous variables; only lags of the X variables combined with spatial errors. The model is estimated by ML, with method = “LU” and fdHess = TRUE in the argument control. To shorten the output, we use the argument Durbin= to restrict the spatial lags of the X variables that we want to see in the SUR equations. The results appear below.
formulaD <- ~ DV60 | MA70 + UE70
sur.sdem <- spsurml(formula = formula_sur, listw = lwncovr,
method = "LU", type = "sdem", Durbin = formulaD,
control = control, data = NCOVR.sf)#> sparse matrix LU decomposition
#> Initial point: log_lik: -18764.09 lambdas: 0.324 0.393
#> Iteration: 1 log_lik: -18755.77 lambdas: 0.337 0.411
#> Iteration: 2 log_lik: -18755.76 lambdas: 0.338 0.411
#> Iteration: 3 log_lik: -18755.76 lambdas: 0.338 0.411
#> Time to fit the model: 12.02 seconds
#> Computing numerical covariances...
#> Time to compute covariances: 2.8 secondssummary(sur.sdem)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "sdem", Durbin = formulaD, method = "LU", control = control)
#>
#>
#> Spatial SUR model type: sdem
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 6.758640 0.630562 10.7184 < 2.2e-16 ***
#> RD60_1 1.773521 0.122008 14.5362 < 2.2e-16 ***
#> PS60_1 0.415781 0.092226 4.5083 6.656e-06 ***
#> MA60_1 -0.207591 0.018652 -11.1298 < 2.2e-16 ***
#> DV60_1 0.763530 0.121488 6.2848 3.508e-10 ***
#> UE60_1 -0.014819 0.033820 -0.4382 0.6612829
#> SOUTH_1 2.627198 0.225838 11.6331 < 2.2e-16 ***
#> lag.DV60_1 0.648362 0.168515 3.8475 0.0001205 ***
#> lambda_1 0.337788 0.030430 11.1004 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3382
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 12.121123 1.029051 11.7789 < 2.2e-16 ***
#> RD70_2 2.747320 0.142525 19.2761 < 2.2e-16 ***
#> PS70_2 0.710438 0.111236 6.3868 1.818e-10 ***
#> MA70_2 -0.095328 0.028288 -3.3699 0.0007566 ***
#> DV70_2 1.127622 0.099328 11.3525 < 2.2e-16 ***
#> UE70_2 -0.094322 0.063879 -1.4766 0.1398408
#> SOUTH_2 3.392350 0.272004 12.4717 < 2.2e-16 ***
#> lag.MA70_2 -0.192663 0.040742 -4.7289 2.308e-06 ***
#> lag.UE70_2 -0.217846 0.087162 -2.4993 0.0124689 *
#> lambda_2 0.411429 0.029221 14.0797 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.42
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.154911 5.381894
#> 5.381894 31.421125
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2087462
#> 0.2087462 1.0000000
#>
#> R-sq. pooled: 0.4018
#> Breusch-Pagan: 134.4 p-value: (4.4e-31)# Non-interactive plot
if (require(gridExtra)) {
pl <- plot(sur.sdem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
4.5 SUR-SDM model
To estimate a full spatial Durbin model set the argument type to type = “sdm”. Moreover, the same as in other previous cases, we can restrict the list of lags of the X variables that we want to appear in the equations, by using the argument Durbin =.
formulaD <- ~ RD60 + PS60 + MA60 + DV60 | RD70 + PS70 + DV70
sur.sdm <- spsurml(formula = formula_sur, listw = lwncovr,
type = "sdm", method = "LU",
Durbin = formulaD, data = NCOVR.sf)#> sparse matrix LU decomposition
#> Initial point: log_lik: -18758.26 rhos: 0.325 0.411
#> Iteration: 1 log_lik: -18747.79 rhos: 0.338 0.431
#> Iteration: 2 log_lik: -18747.77 rhos: 0.339 0.431
#> Iteration: 3 log_lik: -18747.77 rhos: 0.339 0.431
#> Time to fit the model: 9.85 seconds
#> Time to compute covariances: 21.21 secondssummary(sur.sdm)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "sdm", Durbin = formulaD, method = "LU")
#>
#>
#> Spatial SUR model type: sdm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 5.199036 0.823754 6.3114 2.958e-10 ***
#> RD60_1 1.530960 0.193838 7.8982 3.329e-15 ***
#> PS60_1 0.404736 0.135836 2.9796 0.002898 **
#> MA60_1 -0.175412 0.029214 -6.0044 2.030e-09 ***
#> DV60_1 0.699457 0.125303 5.5821 2.478e-08 ***
#> UE60_1 -0.037883 0.033872 -1.1184 0.263422
#> SOUTH_1 1.625112 0.263591 6.1653 7.483e-10 ***
#> lag.RD60_1 -0.208383 0.254629 -0.8184 0.413171
#> lag.PS60_1 -0.134430 0.181435 -0.7409 0.458767
#> lag.MA60_1 0.012782 0.038836 0.3291 0.742061
#> lag.DV60_1 0.342637 0.183159 1.8707 0.061433 .
#> rho_1 0.338837 0.032761 10.3427 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3404
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 5.529591 0.779831 7.0908 1.483e-12 ***
#> RD70_2 2.686495 0.210748 12.7474 < 2.2e-16 ***
#> PS70_2 0.569334 0.161980 3.5148 0.0004432 ***
#> MA70_2 -0.128224 0.021944 -5.8433 5.380e-09 ***
#> DV70_2 1.158369 0.128896 8.9869 < 2.2e-16 ***
#> UE70_2 -0.143309 0.047553 -3.0137 0.0025915 **
#> SOUTH_2 2.072420 0.330277 6.2748 3.740e-10 ***
#> lag.RD70_2 -1.088314 0.291587 -3.7324 0.0001914 ***
#> lag.PS70_2 -0.016707 0.217749 -0.0767 0.9388449
#> lag.DV70_2 -0.485267 0.185833 -2.6113 0.0090415 **
#> rho_2 0.431471 0.029678 14.5384 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4219
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.071758 5.326328
#> 5.326328 31.298884
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2074021
#> 0.2074021 1.0000000
#>
#> R-sq. pooled: 0.4039
#> Breusch-Pagan: 132.7 p-value: (1.05e-30)
#> LMM: 124.48 p-value: (6.63e-29)# Non-interactive plot
if (require(gridExtra)) {
pl <- plot(sur.sdm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
4.6 SUR-SARAR model
The key argument to build a SARAR model from a single SUR specification, like that in of (3.1), is type = “sarar”.
sur.sarar <- spsurml(formula = formula_sur, listw = lwncovr,
type = "sarar", method = "LU",
control= control,
data = NCOVR.sf)#> sparse matrix LU decomposition
#> Initial point: log_lik: -18751.87 rhos: 0.648 0.362 lambdas: -0.623 0.041
#> Iteration: 1 log_lik: -18724.82 rhos: 0.731 0.46 lambdas: -0.833 -0.091
#> Iteration: 2 log_lik: -18722.65 rhos: 0.752 0.478 lambdas: -0.896 -0.121
#> Iteration: 3 log_lik: -18722.44 rhos: 0.758 0.483 lambdas: -0.915 -0.129
#> Iteration: 4 log_lik: -18722.42 rhos: 0.76 0.485 lambdas: -0.92 -0.131
#> Iteration: 5 log_lik: -18722.42 rhos: 0.76 0.485 lambdas: -0.922 -0.132
#> Iteration: 6 log_lik: -18722.42 rhos: 0.761 0.485 lambdas: -0.922 -0.132
#> Time to fit the model: 247.33 seconds
#> Computing numerical covariances...
#> Time to compute covariances: 16.03 secondssummary(sur.sarar)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "sarar", method = "LU", control = control)
#>
#>
#> Spatial SUR model type: sarar
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 2.479266 0.566256 4.3783 1.216e-05 ***
#> RD60_1 0.680964 0.110086 6.1858 6.578e-10 ***
#> PS60_1 0.154374 0.082587 1.8692 0.06164 .
#> MA60_1 -0.080503 0.017248 -4.6673 3.116e-06 ***
#> DV60_1 0.520855 0.084514 6.1630 7.591e-10 ***
#> UE60_1 -0.035781 0.031084 -1.1511 0.24973
#> SOUTH_1 0.325702 0.208805 1.5598 0.11885
#> rho_1 0.760536 0.021699 35.0500 < 2.2e-16 ***
#> lambda_1 -0.922128 0.066464 -13.8742 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4405
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 4.530158 0.725704 6.2424 4.596e-10 ***
#> RD70_2 1.975305 0.141153 13.9941 < 2.2e-16 ***
#> PS70_2 0.603130 0.110901 5.4384 5.582e-08 ***
#> MA70_2 -0.109621 0.021507 -5.0970 3.554e-07 ***
#> DV70_2 0.924493 0.098586 9.3775 < 2.2e-16 ***
#> UE70_2 -0.169193 0.046781 -3.6167 0.0003008 ***
#> SOUTH_2 1.240679 0.271121 4.5761 4.830e-06 ***
#> rho_2 0.485140 0.068908 7.0404 2.126e-12 ***
#> lambda_2 -0.132170 0.122005 -1.0833 0.2787105
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4221
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 18.104351 4.966429
#> 4.966429 31.289101
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2086682
#> 0.2086682 1.0000000
#>
#> R-sq. pooled: 0.4395
#> Breusch-Pagan: 134.3 p-value: (4.63e-31)# Non-interactive plot
if (require(gridExtra)) {
pl <- plot(sur.sarar, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
4.7 SUR-GNM model
This is the most general specification of a spatial SUR model. It needs to be chosen as type = “gnm”.
sur.gnm <- spsurml(formula = formula_sur, listw = lwncovr,
type = "gnm", method = "LU",
control= control,
data = NCOVR.sf)#> sparse matrix LU decomposition
#> Initial point: log_lik: -18720.21 rhos: -0.69 0.542 lambdas: 0.685 -0.239
#> Iteration: 1 log_lik: -18687.14 rhos: -0.917 0.645 lambdas: 0.773 -0.426
#> Iteration: 2 log_lik: -18683.28 rhos: -0.992 0.669 lambdas: 0.797 -0.476
#> Iteration: 3 log_lik: -18682.89 rhos: -1 0.675 lambdas: 0.802 -0.49
#> Iteration: 4 log_lik: -18682.88 rhos: -1 0.677 lambdas: 0.802 -0.494
#> Iteration: 5 log_lik: -18682.88 rhos: -1 0.678 lambdas: 0.802 -0.495
#> Time to fit the model: 194.16 seconds
#> Computing numerical covariances...
#> Time to compute covariances: 16.59 secondssummary(sur.gnm)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "gnm", method = "LU", control = control)
#>
#>
#> Spatial SUR model type: gnm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 17.637671 0.762287 23.1378 < 2.2e-16 ***
#> RD60_1 1.558080 0.178061 8.7503 < 2.2e-16 ***
#> PS60_1 0.376947 0.124103 3.0374 0.002397 **
#> MA60_1 -0.180938 0.026815 -6.7476 1.641e-11 ***
#> DV60_1 0.746906 0.114355 6.5315 7.035e-11 ***
#> UE60_1 0.046907 0.041110 1.1410 0.253915
#> SOUTH_1 0.523163 0.760957 0.6875 0.491790
#> lag.RD60_1 2.403661 0.231207 10.3961 < 2.2e-16 ***
#> lag.PS60_1 0.154981 0.168389 0.9204 0.357415
#> lag.MA60_1 -0.323422 0.035598 -9.0855 < 2.2e-16 ***
#> lag.DV60_1 2.115808 0.166330 12.7206 < 2.2e-16 ***
#> lag.UE60_1 -0.286290 0.063278 -4.5243 6.173e-06 ***
#> lag.SOUTH_1 3.652172 0.813929 4.4871 7.352e-06 ***
#> rho_1 -1.000000 0.064391 -15.5300 < 2.2e-16 ***
#> lambda_1 0.801956 0.020253 39.5978 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4652
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 4.388868 1.013975 4.3284 1.526e-05 ***
#> RD70_2 2.570551 0.207109 12.4116 < 2.2e-16 ***
#> PS70_2 0.651371 0.159736 4.0778 4.604e-05 ***
#> MA70_2 -0.077525 0.028820 -2.6900 0.0071643 **
#> DV70_2 1.064836 0.126071 8.4463 < 2.2e-16 ***
#> UE70_2 -0.048163 0.063024 -0.7642 0.4447766
#> SOUTH_2 0.014559 0.986988 0.0148 0.9882313
#> lag.RD70_2 -1.722619 0.286084 -6.0214 1.829e-09 ***
#> lag.PS70_2 -0.387912 0.217997 -1.7794 0.0752177 .
#> lag.MA70_2 -0.033303 0.042225 -0.7887 0.4303147
#> lag.DV70_2 -0.629841 0.186557 -3.3761 0.0007397 ***
#> lag.UE70_2 -0.096449 0.091567 -1.0533 0.2922402
#> lag.SOUTH_2 1.233415 1.064744 1.1584 0.2467401
#> rho_2 0.677561 0.035178 19.2607 < 2.2e-16 ***
#> lambda_2 -0.494567 0.083217 -5.9431 2.951e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4614
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 17.454940 4.444949
#> 4.444949 29.269587
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.1966524
#> 0.1966524 1.0000000
#>
#> R-sq. pooled: 0.4725
#> Breusch-Pagan: 119.3 p-value: (8.99e-28)# Non-interactive plot
if (require(gridExtra)) {
pl <- plot(sur.gnm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
4.8 3SLS estimation of SUR-SLM and SUR-SDM models
We finish this Section by discussing briefly how to estimate a SUR-SLM model using a 3SLS approach (the same applies for the case of a SUR-SDM). The advantage of 3SLS over ML is time computing, because the first consists of a quasi-linear algorithm whereas the second may be highly non-linear and time consuming.
First is the obtaining of the spatial instruments using the information in the X variables, assumed to be exogenous. The matrix of instruments is \(H=[X,WX,W^2X,W^3X,...,W^rX]\), with r=2 for the SUR-SLM case and r=3 in case of a SUR-SDM model. This defaults can be changed by using the argument “maxlagW=” to the desired integer.
sur.slm.3sls <- spsur3sls(formula = formula_sur,
listw = lwncovr,
type = "slm", maxlagW = 3,
data = NCOVR.sf)#> Time to fit the model: 0.27 secondssummary(sur.slm.3sls)#> Call:
#> spsur3sls(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "slm", maxlagW = 3)
#>
#>
#> Spatial SUR model type: slm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 3.6991431 0.8086769 4.5743 4.872e-06 ***
#> RD60_1 1.0126646 0.1479221 6.8459 8.335e-12 ***
#> PS60_1 0.2207921 0.0920915 2.3975 0.01654 *
#> MA60_1 -0.1192853 0.0227998 -5.2319 1.734e-07 ***
#> DV60_1 0.6794240 0.1069872 6.3505 2.299e-10 ***
#> UE60_1 -0.0025294 0.0348838 -0.0725 0.94220
#> SOUTH_1 0.6305568 0.3411165 1.8485 0.06458 .
#> rho_1 0.6055468 0.0723971 8.3642 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.344
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 4.611480 0.926852 4.9754 6.688e-07 ***
#> RD70_2 2.078946 0.178233 11.6642 < 2.2e-16 ***
#> PS70_2 0.621482 0.117762 5.2774 1.355e-07 ***
#> MA70_2 -0.110960 0.024743 -4.4846 7.438e-06 ***
#> DV70_2 0.934966 0.108098 8.6493 < 2.2e-16 ***
#> UE70_2 -0.146392 0.049925 -2.9322 0.003378 **
#> SOUTH_2 1.356495 0.436969 3.1043 0.001916 **
#> rho_2 0.450279 0.063105 7.1353 1.076e-12 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4186
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.99787 6.87556
#> 6.87556 33.99110
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.1933044
#> 0.1933044 1.0000000
#>
#> R-sq. pooled: 0.4029
#> Breusch-Pagan: 195 p-value: (2.52e-44)As usual, we can activate the argument “Durbin =” to restrict the lags of the X variables in the equations of the SUR-SDM model.
formulaD <- ~ RD60 |PS70 + DV70
sur.sdm.3sls <- spsur3sls(formula = formula_sur,
listw = lwncovr, type = "sdm",
Durbin = formulaD, data = NCOVR.sf)#> Time to fit the model: 0.35 secondssummary(sur.sdm.3sls)#> Call:
#> spsur3sls(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "sdm", Durbin = formulaD)
#>
#>
#> Spatial SUR model type: sdm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 3.4624694 0.8270454 4.1866 2.872e-05 ***
#> RD60_1 1.2449345 0.1901092 6.5485 6.282e-11 ***
#> PS60_1 0.2397027 0.0923574 2.5954 0.009471 **
#> MA60_1 -0.1161461 0.0229283 -5.0656 4.189e-07 ***
#> DV60_1 0.6324323 0.1100932 5.7445 9.660e-09 ***
#> UE60_1 -0.0012328 0.0348684 -0.0354 0.971796
#> SOUTH_1 0.6419150 0.3405172 1.8851 0.059461 .
#> lag.RD60_1 -0.4485605 0.2513252 -1.7848 0.074346 .
#> rho_1 0.6553687 0.0791900 8.2759 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3443
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 4.835165 0.934961 5.1715 2.395e-07 ***
#> RD70_2 1.917663 0.193597 9.9054 < 2.2e-16 ***
#> PS70_2 0.420037 0.164394 2.5551 0.010641 *
#> MA70_2 -0.105585 0.025258 -4.1803 2.951e-05 ***
#> DV70_2 1.167850 0.132806 8.7936 < 2.2e-16 ***
#> UE70_2 -0.119046 0.051768 -2.2996 0.021504 *
#> SOUTH_2 1.116616 0.446322 2.5018 0.012382 *
#> lag.PS70_2 0.215349 0.214306 1.0049 0.314999
#> lag.DV70_2 -0.551847 0.201028 -2.7451 0.006066 **
#> rho_2 0.507463 0.068120 7.4495 1.066e-13 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4222
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.933463 6.773914
#> 6.773914 33.894781
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.1870956
#> 0.1870956 1.0000000
#>
#> R-sq. pooled: 0.4053
#> Breusch-Pagan: 190.4 p-value: (2.58e-43)In the example above we have used the default in the argument “maxlagW=”, 3 for a SUR-SDM.
5 Testing for the specification
spsur includes a collection of test which may help the user to improve the specification. Some of them appear routinely as part of the output of the estimation of spatial SUR models whereas others require the intervention of the user. Below we discuss what we consider the most interest tests for the user.
5.1 Routine tests
spsur deals with SUR models applied in a spatial setting and, thus, is important to control for the SUR nature of the system of equations and for the symptoms of spatial misspecification. In the first case we are refering to the well known Breusch-Pagan LM test (Breusch and Pagan (1980)) of diagonality of the covariance matrix of the SUR residuals. This test appears in the last line of the table of results for any estimated SUR model. Furthermore, The value of the statistic can be recovered using the command print().
print(sur.sim$BP)#> [1] 211.1929print(sur.slm$BP)#> [1] 132.8747print(sur.slm.3sls$BP)#> [1] 195.0408The second issue is solved by using the Marginal Lagrange Multipliers, \(LM(\lambda|\rho)\) and \(LM(\rho|\lambda)\) (see López, Mur, and Angulo (2014)), which tests for omitted spatial structure, conditional to the estimated spatial SUR model . That is, the SUR-SLM and SUR-SDM models include spatial lags of the endogenous variables in the right hand side of the equations, but we may wonder if we need also of spatial errors. This is the purpose of the \(LM(\lambda|\rho)\) Multiplier. Moreover, in the SUR-SEM and SUR-SDEM models, we may wonder if spatial errors are enough or we would need the introduction of lags of the endogeneous variables, which is the purpose of the \(LM(\rho|\lambda)\) Multiplier.
The value of this test appears at the bottom of the table of estimation results for any of the four SUR models, indicated under the heading of LMM (the Marginal Multipliers makes no sense for the SUR-SLX and SUR-SARAR cases). This test can be viewed as a general measure of misspecification in what respects the spatial structure of the model. Finaly, note that the Marginal Multipliers are obtained, for reasons of calculus, only for the ML algorithm and under the option fdHess = FALSE which is the default (you may change it to fdHess = TRUE, using the argument control=); in fact, we need the analytical Hessian.
By way of example, in the case of the SUR-SLM, the Marginal Multiplier (\(LM(\lambda|\rho)\) in this case) in the Table below clearly indicates that the spatial lags of the endogeous variables are not enough to model the spatial structure present in the data. The LMM test (last line) appears with a value of 30.856 which is highly significant. On other words, the user should consider a SUR-SARAR specification, better than just a SUR-SLM.
control <- list(fdHess = FALSE)
sur.slm1 <- spsurml(formula = formula_sur, listw = lwncovr,
method = "LU", type = "slm",
data = NCOVR.sf)#> sparse matrix LU decomposition
#> Initial point: log_lik: -18771.55 rhos: 0.335 0.376
#> Iteration: 1 log_lik: -18761.49 rhos: 0.346 0.393
#> Iteration: 2 log_lik: -18761.48 rhos: 0.347 0.393
#> Iteration: 3 log_lik: -18761.48 rhos: 0.347 0.393
#> Time to fit the model: 8.59 seconds
#> Time to compute covariances: 15.02 secondssummary(sur.slm1)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "slm", method = "LU")
#>
#>
#> Spatial SUR model type: slm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 5.607673 0.645042 8.6935 < 2.2e-16 ***
#> RD60_1 1.334419 0.124778 10.6943 < 2.2e-16 ***
#> PS60_1 0.274531 0.089300 3.0742 0.00212 **
#> MA60_1 -0.166839 0.019299 -8.6449 < 2.2e-16 ***
#> DV60_1 0.879403 0.093120 9.4437 < 2.2e-16 ***
#> UE60_1 -0.030445 0.033560 -0.9072 0.36435
#> SOUTH_1 1.506056 0.245815 6.1268 9.522e-10 ***
#> rho_1 0.346695 0.030621 11.3223 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3396
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 5.157484 0.753684 6.8430 8.504e-12 ***
#> RD70_2 2.201353 0.151750 14.5064 < 2.2e-16 ***
#> PS70_2 0.648528 0.112107 5.7849 7.612e-09 ***
#> MA70_2 -0.121851 0.021874 -5.5705 2.648e-08 ***
#> DV70_2 0.987470 0.100386 9.8367 < 2.2e-16 ***
#> UE70_2 -0.168980 0.047095 -3.5880 0.0003358 ***
#> SOUTH_2 1.616949 0.304425 5.3115 1.126e-07 ***
#> rho_2 0.393272 0.028399 13.8482 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4154
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.096585 5.361957
#> 5.361957 31.640805
#> Correlation Matrix of inter-equation residuals:
#> 1.000000 0.207536
#> 0.207536 1.000000
#>
#> R-sq. pooled: 0.3996
#> Breusch-Pagan: 132.9 p-value: (9.63e-31)
#> LMM: 27.477 p-value: (1.59e-07)5.2 Linear restrictions on the \(\beta\) parameters
spsur allows to tests linear restrictions on the \(\beta\) parameters of a SUR model that has been previously estimated. The restrictions may refer to parameters of the same or of different equations and are solved through a Wald test, asymptotically distributed as a \(\chi^2\) with degrees of freedom equal to the number of restrictions involved in the test. In this sense, the function wald_betas() offers a great flexibility; it requires the especification of three arguments:
- The matrix of restrictions: A \((r \times K)\) matrix, being r the number of restrictions a K the total number of \(\beta\) parameters in the models; R1 in the example below.
- The value of the restrictions under the null hypothesis. This is a a \((r \times 1)\) vector; b1 in the example below.
- The model where we wish to evaluate these restrictions
For example, suppose that we want to test if the parameters associated with variable SOUTH in the two equations of the model (4.1) are equal; so that \[\begin{equation} H_0:\beta_{16} = \beta_{26} \\ H_A:\beta_{16} \neq \beta_{26} \tag{5.1} \end{equation}\]
We should remind that there are six \(\beta\) coefficients in each of the two equation of the SUR-SLM and that the SOUTH variable appears in position six and twelve, respectively. Then we write
R1 <- matrix(c(0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, -1), nrow = 1)
b1 <- matrix(0, ncol = 1)
waldbeta1 <- wald_betas(sur.slm, R1, b1)
print(waldbeta1)#>
#> Wald test on beta parameters
#>
#> data: NCOVR.sf
#> Wald test = 0.12143, df = 1, p-value = 0.7275The test confirms our suspicious that both coefficients are not statistically different. Therefore, the user would be surely interested in the restricted version where the two coefficients are indeed the same. That is
\[\begin{equation} HR_{60} = \beta_{10}+\beta_{11}RD_{60}+\beta_{12}PS_{60}+\beta_{13}MA_{60}+\beta_{14}DV_{60} +\beta_{15}UE_{60} +\beta_{16}SOUTH+\epsilon_{60}\\ HR_{70} = \beta_{20}+\beta_{21}RD_{70}+\beta_{22}PS_{70}+\beta_{23}MA_{70}+\beta_{24}DV_{70} +\beta_{25}UE_{70} +\beta_{16}SOUTH+\epsilon_{70} \\ Corr(\epsilon_{60},\epsilon_{70}) \neq 0 \tag{5.2} \end{equation}\]
spsur allows easily for the estimation of a linearly restricted model, in the \(\beta\) coefficients, just by adding two new arguments “R=” and “b= to the respective spsurml()function. For example, in the case of the SUR-SLM model analyzed so far, we obtain
R1 <- matrix(c(0,0,0,0,0,0,1,
0,0,0,0,0,0,-1), nrow=1)
b1 <- matrix(0, ncol = 1)
sur.slm.r <- spsurml(formula = formula_sur, listw = lwncovr,
type = "slm", R = R1, b = b1,
control = control, data = NCOVR.sf)#> neighbourhood matrix eigenvalues
#> Computing eigenvalues ...
#>
#> Initial point: log_lik: -18771.73 rhos: 0.332 0.381
#> Iteration: 1 log_lik: -18761.54 rhos: 0.344 0.396
#> Iteration: 2 log_lik: -18761.53 rhos: 0.345 0.396
#> Iteration: 3 log_lik: -18761.53 rhos: 0.345 0.396
#> Time to fit the model: 3.28 seconds
#> Time to compute covariances: 13.44 secondssummary(sur.slm.r)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "slm", R = R1, b = b1, control = control)
#>
#>
#> Spatial SUR model type: slm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 5.591069 0.643010 8.6951 < 2.2e-16 ***
#> RD60_1 1.325351 0.121683 10.8918 < 2.2e-16 ***
#> PS60_1 0.271923 0.088913 3.0583 0.002236 **
#> MA60_1 -0.166639 0.019291 -8.6382 < 2.2e-16 ***
#> DV60_1 0.879342 0.093132 9.4419 < 2.2e-16 ***
#> UE60_1 -0.030164 0.033527 -0.8997 0.368308
#> SOUTH_1 1.546569 0.206496 7.4896 7.882e-14 ***
#> rho_1 0.344663 0.029622 11.6352 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3394
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 5.167209 0.752040 6.8709 7.007e-12 ***
#> RD70_2 2.215108 0.146012 15.1707 < 2.2e-16 ***
#> PS70_2 0.652578 0.111367 5.8597 4.877e-09 ***
#> MA70_2 -0.121543 0.021865 -5.5588 2.831e-08 ***
#> DV70_2 0.988142 0.100342 9.8477 < 2.2e-16 ***
#> UE70_2 -0.170805 0.046532 -3.6707 0.0002439 ***
#> SOUTH_2 1.546569 0.206496 7.4896 7.882e-14 ***
#> rho_2 0.396384 0.026442 14.9909 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4157
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.101489 5.357079
#> 5.357079 31.628040
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2073649
#> 0.2073649 1.0000000
#>
#> R-sq. pooled: 0.3997
#> Breusch-Pagan: 132.7 p-value: (1.08e-30)
#> LMM: 20.6 p-value: (5.66e-06)The same technique applies for the other spatial SUR models estimated by spsurml().
More complex hypotheses can be tested; for example, now including two restrictions:
\[\begin{equation} H_0:\beta_{10}=\beta_{20} \ and \ \beta_{16}=\beta_{26} \\ \ H_A:\beta_{10} \neq \beta_{20} \ \text{and/or} \ \beta_{16} \neq \beta_{26} \end{equation}\]
# equal SOUTH coefficient
R21 <- matrix(c(0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, -1), nrow = 1)
# equal intercept
R22 <- matrix(c(1, 0, 0, 0, 0, 0, 0,
-1, 0, 0, 0, 0, 0, 0), nrow = 1)
R2 <- rbind(R21, R22)
b2 <- matrix(c(0, 0), ncol = 1)
waldbeta2 <- wald_betas(sur.slm, R1, b1)
print(waldbeta2)#>
#> Wald test on beta parameters
#>
#> data: NCOVR.sf
#> Wald test = 0.12143, df = 1, p-value = 0.7275The corresponding restricted model can be estimated as before
sur.slm.r2 <- spsurml(formula = formula_sur, listw = lwncovr,
type = "slm", R = R2, b = b2,
control = control, data = NCOVR.sf)#> neighbourhood matrix eigenvalues
#> Computing eigenvalues ...
#>
#> Initial point: log_lik: -18771.81 rhos: 0.334 0.378
#> Iteration: 1 log_lik: -18761.65 rhos: 0.347 0.392
#> Iteration: 2 log_lik: -18761.63 rhos: 0.348 0.393
#> Iteration: 3 log_lik: -18761.63 rhos: 0.348 0.393
#> Time to fit the model: 3.53 seconds
#> Time to compute covariances: 13.61 secondssummary(sur.slm.r2)#> Call:
#> spsurml(formula = formula_sur, data = NCOVR.sf, listw = lwncovr,
#> type = "slm", R = R2, b = b2, control = control)
#>
#>
#> Spatial SUR model type: slm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 5.423289 0.528372 10.2641 < 2.2e-16 ***
#> RD60_1 1.320225 0.121024 10.9088 < 2.2e-16 ***
#> PS60_1 0.271675 0.088905 3.0558 0.002254 **
#> MA60_1 -0.162014 0.016437 -9.8565 < 2.2e-16 ***
#> DV60_1 0.877817 0.093074 9.4314 < 2.2e-16 ***
#> UE60_1 -0.026588 0.032774 -0.8113 0.417247
#> SOUTH_1 1.550475 0.206478 7.5092 6.797e-14 ***
#> rho_1 0.347579 0.028640 12.1360 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3396
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 5.423289 0.528372 10.2641 < 2.2e-16 ***
#> RD70_2 2.223552 0.144979 15.3371 < 2.2e-16 ***
#> PS70_2 0.646027 0.110415 5.8509 5.140e-09 ***
#> MA70_2 -0.128197 0.016815 -7.6240 2.832e-14 ***
#> DV70_2 0.986949 0.100295 9.8405 < 2.2e-16 ***
#> UE70_2 -0.177353 0.044704 -3.9673 7.353e-05 ***
#> SOUTH_2 1.550475 0.206478 7.5092 6.797e-14 ***
#> rho_2 0.392920 0.025567 15.3681 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4154
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.096221 5.357976
#> 5.357976 31.641269
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2073822
#> 0.2073822 1.0000000
#>
#> R-sq. pooled: 0.3996
#> Breusch-Pagan: 132.7 p-value: (1.06e-30)
#> LMM: 19.908 p-value: (8.13e-06)Finally, the same technique applies for the 3SLS estimation algorithm. Assuming, for example, that we want to test that the intercepts of the two equations of the SUR-SLM estimated in Section 4.7 are the same
\[\begin{equation} H_0:\beta_{10}=\beta_{20} \\ H_A:\beta_{10} \neq \beta_{20} \end{equation}\]
The function,wald_betas(), and the arguments are the same. The only change is the estimated model that the user uploads, sur.slm.3sls in this occasion.
R3 <- matrix(c(1, 0, 0, 0, 0, 0, 0,
-1, 0, 0, 0, 0, 0, 0), nrow = 1)
b3 <- matrix(0, ncol = 1)
waldbeta3 <- wald_betas(sur.slm.3sls, R = R3, b = b3)
print(waldbeta3)#>
#> Wald test on beta parameters
#>
#> data: NCOVR.sf
#> Wald test = 0.69014, df = 1, p-value = 0.4061The estimation of the restricted model, by 3SLS produces exactly the same results as with the ML algorithm.
sur.slm.3sls.r <- spsur3sls(formula = formula_sur,
listw = lwncovr, type = "slm",
R = R3, b = b3, data = NCOVR.sf)#> Time to fit the model: 0.19 secondssummary(sur.slm.3sls.r)#> Call:
#> spsur3sls(formula = formula_sur, data = NCOVR.sf, R = R3, b = b3,
#> listw = lwncovr, type = "slm")
#>
#>
#> Spatial SUR model type: slm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 4.269568 0.677692 6.3002 3.179e-10 ***
#> RD60_1 0.992254 0.147207 6.7405 1.722e-11 ***
#> PS60_1 0.215403 0.092280 2.3342 0.01962 *
#> MA60_1 -0.135676 0.019616 -6.9165 5.096e-12 ***
#> DV60_1 0.654135 0.108243 6.0432 1.599e-09 ***
#> UE60_1 -0.015988 0.033849 -0.4723 0.63670
#> SOUTH_1 0.443863 0.362430 1.2247 0.22074
#> rho_1 0.633161 0.074501 8.4987 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3431
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 4.269568 0.677692 6.3002 3.179e-10 ***
#> RD70_2 2.191341 0.175413 12.4924 < 2.2e-16 ***
#> PS70_2 0.685723 0.118665 5.7786 7.900e-09 ***
#> MA70_2 -0.097262 0.019404 -5.0126 5.520e-07 ***
#> DV70_2 0.995512 0.109084 9.1261 < 2.2e-16 ***
#> UE70_2 -0.132163 0.047192 -2.8005 0.005118 **
#> SOUTH_2 1.921206 0.454551 4.2266 2.407e-05 ***
#> rho_2 0.363183 0.063745 5.6974 1.273e-08 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4126
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 22.059885 6.905177
#> 6.905177 34.062443
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.1933558
#> 0.1933558 1.0000000
#>
#> R-sq. pooled: 0.3987
#> Breusch-Pagan: 195.8 p-value: (1.76e-44)5.3 Linear restrictions on the spatial coefficients
With slight variations, a similar discussion applies to tests for linear restrictions on the spatial coefficients of a spatial SUR model, now using the function wald_deltas(). Continuing with the case of the SUR-SLM model of Section 4.2, we may test if the two \(\rho\) coefficients are equal
\[\begin{equation} H_0:\rho_1 = \rho_2 \\ H_A:\rho_1 \neq \rho_2 \end{equation}\]
wald_deltas() requires the same three arguments as before specifying the \((r \times D)\) R matrix of restrictions, being D the number of spatial coefficients in the SUR model, the \((r \times 1)\) b vector of values of the restrictions under the null hypothesis and the model to test the linear constraints. So, we would write
R1 <- matrix(c(1, -1), nrow = 1)
b1 <- matrix(0, ncol = 1)
wald_deltas(sur.slm, R = R1, b = b1)#>
#> Wald test on spatial delta parameters
#>
#> data: NCOVR.sf
#> Wald test = 1.5751, df = 1, p-value = 0.2095In the case of the SUR-SEM model of Section 4.3, we are interested in
\[\begin{equation} H_0:\lambda_1 = \lambda_2 \\ H_A:\lambda_1 \neq \lambda_2 \end{equation}\]
The code is very simple.
wald_deltas(sur.sem, R = R1, b = b1)#>
#> Wald test on spatial delta parameters
#>
#> data: NCOVR.sf
#> Wald test = 4.2004, df = 1, p-value = 0.04041In the case of a SUR-SARAR model, Section 4.6, we must consider that there are four spatial coefficients, for which we can test two equality constraints:
\[\begin{equation} H_0:\rho_1 = \rho_2 \ and \ \lambda_1 = \lambda_2 \\ H_A:\rho_1 \neq \rho_2 \ or \ \lambda_1 \neq \lambda_2 \end{equation}\]
The code is only a bit more complex
R3 <- matrix(c(1, -1, 0, 0, 0, 0, 1, -1), nrow = 2, ncol = 4,
byrow = TRUE)
b3 <- matrix(c(0, 0), ncol = 1)
wald_deltas(sur.sarar, R = R3, b = b3)#>
#> Wald test on spatial delta parameters
#>
#> data: NCOVR.sf
#> Wald test = 40.508, df = 2, p-value = 1.599e-09The same as before, wald_deltas() operates with ML or with 3SLS estimates.
Finally, note that if we have the ML estimation of a collection of models, some of which are nested in the others, it is immediate to obtain the corresponding Likelihood Ratios as an additional procedure to test linear constraints. Firts, let us note that the function logLik() shows the value of the estimated log-likelihood for the corresponding ML estimation
logLik(sur.sarar)#> 'log Lik.' -18722.42 (df=21)The anova() function shows information criteria (AIC and BIC), log-likelihood and degrees of freedom of each fitted model. When the argument lrtest = TRUE this function also computes the LR corresponding to nested SUR models. Under the null hypothesis, the statistics is distributed as a \(\chi^2\) with degrees of freedom equal to the number of restrictions involved in the nesting sequence.
anova(sur.sim, sur.slx, lrtest = TRUE)#> logLik df AIC BIC LRtest p.val
#> model 1: sim -18909 17 37851 37829
#> model 2: slx -18855 29 37768 37731 106.87 2.483e-17anova(sur.slm, sur.sarar, lrtest = TRUE)#> logLik df AIC BIC LRtest p.val
#> model 1: slm -18762 19 37561 37536
#> model 2: sarar -18722 21 37487 37459 78.111 1.0925e-17anova(sur.slm.r2, sur.slm, lrtest = TRUE)#> logLik df AIC BIC LRtest p.val
#> model 1: slm -18762 17 37557 37535
#> model 2: slm -18762 19 37561 37536 0.31287 0.85519The LR can be applied both in the case of linear restrictions on the \(\beta\) parameters and in the case of the spatial coefficients.
6 Impacts: Direct, Indirect and Total Effects
LeSage and Pace (2009) address the problem of measuring the impact that a unitary change in the value of a regressor, in a certain point in space, has on the explained variable. The question is of interest because the impact of such changes are, in general, non linear and may spillover non uniformly over the space. In fact, the reaction of the explained variable, in every point of space, depends on its relative location in relation to the point of intervention. Obviously, the location is measured according the corresponding W matrix. To simplify matters, they propose obtaining only aggregated multipliers for each regressor, by averaging the \(N^{2}\) impacts that result from intervening the value of one regressor on each of the N points in space, on the explained variable, measured also in each of the N points in space.
Using the notation introduced by LeSage and Pace (2009), we can distinguish three types of effects; namely
Average Direct Effects: Measure the impact average, over the N spatial units and Tm time periods, of a unitary change in the value of a explanatory variable on the contemporaneous value of the corresponding explained variable, located in the same spatial unit of the regressor. This calculus is solved for all the regressors that appear in the G equations of the model.
Average Indirect Effects: Measure the impact average, over the N spatial units and Tm time periods, of a unitary change in the value of a explanatory variable on the contemporaneous value of the corresponding explained variable, located in a different spatial unit that that of the regressor. Once again, this calculus is solved for all the regressors that appear in the G equations of the model.
Average Total Effects, which is simply the sum of Direct and Indirect Effects.
Note that these effects are exactly linear in a standard time series model. For example, assume the simple following case \(y_t=\beta_0+\beta_1x_t+u_t\) where \(y_t\), \(x_t\) and \(u_t\) are the endogenous variable, the exogenous variable and the error term, respectively, in period t. In that case, if we increase the value of the exogenous in period t by one unit, \(x_t+1\), the endogenous variable will also increase in \(\beta_1\) units in this period, \(y_t+\beta_1\). The chain of effects stops here. For the same reason, if a cross-sectional model lacks of substantial interaction mechanisms, there will be no spill-overs and the effects will be merely linear inside each spatial unit.
It is possible to obtain the three multipliers together with indirect measures of statistical significance, according to the randomization approach described in LeSage and Pace (2009). Briefly, they suggest to obtain a sequence of R random matrices of order (NTmxG) from a multivariate normal distribution \(N(0; \Sigma)\), being \(\Sigma\) the estimated covariance matrix of the G equations of the SUR model. These random matrices, combined with the observed values of the regressors and the estimated values of the parameters, are used to obtain simulated values of the explained variables. Then, for each one of the R experiments, the SUR model is estimated, and the effects are evaluated. Therefore, it is possible to obtain the standard deviations of the R estimated effects in the randomization procedure, which are used to test the significance of the estimated effects for the original data.
The spsur package implements the function impactspsur() to compute direct, indirect and total impacts either to models with an autoregressive structure (“slm”, “sdm” or “sarar”) or models with spatial lags of the regressors (“slx” or “sdem”). The function impactspsur() is a wrapper to method impacts available in spatialreg package (see Bivand and Piras (2015)) and both functions share the same arguments (see the help of any of them for details). The output of impactspsur() is a list of objects (one for each equation of the SUR model) either of class lagImpact(for “slm”, “sdm” or “sarar”) or class WXImpact (for “slx” or “sdem”). It is important to highlight than the previous simulation method (described in LeSage and Pace (2009)) it is used only to compute the variances of the impacts for models with spatial lags of the dependent variable (“slm”, “sdm” or “sarar”); for models with only spatial lags of exogeneous variables (“slx” or “sdem”) the dispersion measures can be obtained directly from the estimates without any need to simulate.
As a way of example, below appear the impacts estimated for the case of the SUR-SDM model of Section 4.5. To speed the computations, we previosly compute a vector of traces of W matrix (see help(spatialreg::trW) for details) which is provided as an argument to the function.
Wncovr <- Matrix::Matrix(spdep::list2mat(lwncovr))
trWncovr <- spatialreg::trW(Wncovr, type = "MC")
sur.sdm.impacts <- impactspsur(sur.slm, tr = trWncovr, R = 1000)
class(sur.sdm.impacts[[1]])#> [1] "lagImpact"summary(sur.sdm.impacts[[1]], zstats = TRUE, short = TRUE)#> Impact measures (lag, trace):
#> Direct Indirect Total
#> RD60_1 1.35213975 0.69042771 2.04256745
#> PS60_1 0.27817647 0.14204208 0.42021855
#> MA60_1 -0.16905446 -0.08632235 -0.25537681
#> DV60_1 0.89108131 0.45500269 1.34608401
#> UE60_1 -0.03084949 -0.01575232 -0.04660181
#> SOUTH_1 1.52605595 0.77923255 2.30528850
#> ========================================================
#> Simulation results ( variance matrix):
#> ========================================================
#> Simulated standard errors
#> Direct Indirect Total
#> RD60_1 0.11703831 0.10448274 0.19901068
#> PS60_1 0.08940017 0.04840224 0.13563805
#> MA60_1 0.01871509 0.01412591 0.03019991
#> DV60_1 0.09273968 0.07466478 0.15383048
#> UE60_1 0.03282561 0.01681956 0.04950094
#> SOUTH_1 0.22889434 0.14842343 0.35699832
#>
#> Simulated z-values:
#> Direct Indirect Total
#> RD60_1 11.4815668 6.5829830 10.208453
#> PS60_1 3.0716621 2.9009683 3.059764
#> MA60_1 -9.0599755 -6.1415086 -8.487197
#> DV60_1 9.6135076 6.1138683 8.763180
#> UE60_1 -0.9354442 -0.9319543 -0.936984
#> SOUTH_1 6.6935936 5.2795053 6.486664
#>
#> Simulated p-values:
#> Direct Indirect Total
#> RD60_1 < 2.22e-16 4.6110e-11 < 2.22e-16
#> PS60_1 0.0021287 0.0037201 0.0022151
#> MA60_1 < 2.22e-16 8.1741e-10 < 2.22e-16
#> DV60_1 < 2.22e-16 9.7245e-10 < 2.22e-16
#> UE60_1 0.3495594 0.3513601 0.3487668
#> SOUTH_1 2.1775e-11 1.2953e-07 8.7758e-11summary(sur.sdm.impacts[[2]], zstats = TRUE, short = TRUE)#> Impact measures (lag, trace):
#> Direct Indirect Total
#> RD70_2 2.2404848 1.38775098 3.6282358
#> PS70_2 0.6600567 0.40883756 1.0688942
#> MA70_2 -0.1240170 -0.07681585 -0.2008329
#> DV70_2 1.0050239 0.62250940 1.6275333
#> UE70_2 -0.1719837 -0.10652629 -0.2785100
#> SOUTH_2 1.6456920 1.01933775 2.6650298
#> ========================================================
#> Simulation results ( variance matrix):
#> ========================================================
#> Simulated standard errors
#> Direct Indirect Total
#> RD70_2 0.14643676 0.17648186 0.28280140
#> PS70_2 0.11364985 0.08239305 0.18802458
#> MA70_2 0.02225844 0.01624519 0.03715816
#> DV70_2 0.09969887 0.09373538 0.17813204
#> UE70_2 0.04851033 0.03250015 0.07962182
#> SOUTH_2 0.26315022 0.19921014 0.44205944
#>
#> Simulated z-values:
#> Direct Indirect Total
#> RD70_2 15.271393 7.876525 12.822981
#> PS70_2 5.870699 5.028814 5.752138
#> MA70_2 -5.516492 -4.698497 -5.358620
#> DV70_2 10.041132 6.640642 9.114321
#> UE70_2 -3.532639 -3.275616 -3.489338
#> SOUTH_2 6.253998 5.135172 6.037015
#>
#> Simulated p-values:
#> Direct Indirect Total
#> RD70_2 < 2.22e-16 3.3307e-15 < 2.22e-16
#> PS70_2 4.3396e-09 4.9352e-07 8.8122e-09
#> MA70_2 3.4583e-08 2.6208e-06 8.3860e-08
#> DV70_2 < 2.22e-16 3.1232e-11 < 2.22e-16
#> UE70_2 0.00041143 0.0010543 0.00048422
#> SOUTH_2 4.0008e-10 2.8189e-07 1.5699e-09Finally, an example of the impacts estimated for the case of the SUR-SLX model of Section 4.1 is given below (remark: in this case it is necessary to use print instead of summary method to check the output),
sur.slx.impacts <- impactspsur(sur.slx, listw = lwncovr)
print(sur.slx.impacts)#> [[1]]
#> Impact measures (SLX, estimable):
#> Direct Indirect Total
#> RD60_1 1.38956078 0.5869375 1.9764983
#> PS60_1 0.35559176 0.1788786 0.5344704
#> MA60_1 -0.16632571 -0.1050259 -0.2713517
#> DV60_1 0.67831290 1.0264676 1.7047805
#> UE60_1 0.07004619 -0.3561454 -0.2860992
#> SOUTH_1 -0.52636385 3.2103598 2.6839960
#>
#>
#> [[2]]
#> Impact measures (SLX, estimable):
#> Direct Indirect Total
#> RD70_2 2.52480544 0.2084002 2.7332057
#> PS70_2 0.64534716 0.1945871 0.8399343
#> MA70_2 -0.08688263 -0.2299874 -0.3168700
#> DV70_2 1.10853988 0.2756482 1.3841880
#> UE70_2 -0.07910077 -0.3328120 -0.4119128
#> SOUTH_2 -0.47215851 4.3686848 3.89652627 The Anselin’s approach to spatial panel SUR models and the function spsurtime()
Anselin (2016), see also Anselin (1988), reserves the term of spatial panel SUR models to the case of N individuals, or spatial units, Tm time periods and only 1 equation, G=1. Moreover, the errors corresponding to the same spatial unit are serially dependent but remain independent for different spatial units. For example, assuming a SLM structure:
\[\begin{equation} y_t=\rho W y_t+x_t\beta+u_t; t=1,2,..., Tm \\ E[u_t]=0; E[u_tu_s']=\sigma_{ts}I_N; \end{equation}\]
\(y_t\) and \(u_t\) are \((N\times 1)\) vectors corresponding to time period t, and \(x_t\) is a \((N\times k)\) matrix corresponding to the observations of the X variables in the same period. As said, the SUR structure appears because there is serial dependence in the error terms of each individual in the sample, as corresponds to the diagonality of the covariance matrix. The serial dependence is estimated non parametrically in the \((Tm \times Tm)\) \(\Sigma\) matrix of the SUR model. This serial dependence is assumed to be uniform for all individuals in the sample, which is an important constraint.
The function spsurtime() is directed to deal with this type of spatial panel SUR models. The syntax is very transparent
spsurtime (formula, data, time, listw = NULL, type = "sim", Durbin = NULL, method = "eigen",
fit_method = "ml", maxlagW = NULL, R = NULL, b = NULL, demean = FALSE, control = list())
type= decides the spatial structure required for the panel specification, where SUR-SIM is the default, and fit_method= the estimation algorithm, which can be ML, the default, or 3SLS (in which case the argument maxlagW = can be activated). R, b and Durbin can be used to restrict, as usual, the corresponding panel equation and time= identifies the time variable in the dataframe needed to structrure the panel data. Finally demean allows the user to demean the data, to remove potential unobserved effects, by substracting the corresponding individual sampling average from the data of each spatial unit in the sample. This transform would require the data be ordered, first, by time and then by individual. If this is not the case, the data are routinely sorted by spsurtime(), according to the argument time.
In this vignette we are illustrating the capabilities of spsur by using the dataset NCOVR.sf, which is not, strictly, a panel data set. So, let us in the first place to manipulate slightly this dataset to produce a panel data frame as required by the function spsurtime(). This can be done quite easily
N <- nrow(NCOVR.sf)
Tm <- 2
index_name <- rep(NCOVR.sf$NAME,Tm)
index_indiv <- rep(1:N, Tm)
index_time <- rep(1:Tm, each = N)
HR <- c(NCOVR.sf$HR60, NCOVR.sf$HR70)
RD <- c(NCOVR.sf$RD60, NCOVR.sf$RD70)
PS <- c(NCOVR.sf$PS60, NCOVR.sf$PS70)
MA <- c(NCOVR.sf$MA60, NCOVR.sf$MA70)
DV <- c(NCOVR.sf$DV60, NCOVR.sf$DV70)
UE <- c(NCOVR.sf$UE60, NCOVR.sf$UE70)
SOUTH <- c(NCOVR.sf$SOUTH, NCOVR.sf$SOUTH)
NCOVR.df <- data.frame(index_name, index_indiv, index_time, HR,
RD, PS, MA, DV, UE, SOUTH)Once we have the data in the format required it is immediate to estimate, for example, a SLM model where the error of each individual are serially correlated. The data are not demeaned. The classical expresion for the formula is used in the function spsurtime(). Moreover, use the argument fit_method to select ML or IV.
formula_sur_time <- HR ~ RD + PS + MA + DV + UE + SOUTH
sur.slm.time <- spsurtime(formula = formula_sur_time,
data = NCOVR.df,
listw = lwncovr,
time = NCOVR.df$index_time,
type = "slm", fit_method = "ml")#> neighbourhood matrix eigenvalues
#> Computing eigenvalues ...
#>
#> Initial point: log_lik: -18771.55 rhos: 0.335 0.376
#> Iteration: 1 log_lik: -18761.49 rhos: 0.346 0.393
#> Iteration: 2 log_lik: -18761.48 rhos: 0.347 0.393
#> Iteration: 3 log_lik: -18761.48 rhos: 0.347 0.393
#> Time to fit the model: 2.81 seconds
#> Time to compute covariances: 13.44 secondssummary(sur.slm.time)#> Call:
#> spsurtime(formula = formula_sur_time, data = NCOVR.df, time = NCOVR.df$index_time,
#> listw = lwncovr, type = "slm", fit_method = "ml")
#>
#>
#> Spatial SUR model type: slm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 5.607673 0.645042 8.6935 < 2.2e-16 ***
#> RD_1 1.334419 0.124778 10.6943 < 2.2e-16 ***
#> PS_1 0.274531 0.089300 3.0742 0.00212 **
#> MA_1 -0.166839 0.019299 -8.6449 < 2.2e-16 ***
#> DV_1 0.879403 0.093120 9.4437 < 2.2e-16 ***
#> UE_1 -0.030445 0.033560 -0.9072 0.36435
#> SOUTH_1 1.506056 0.245815 6.1268 9.522e-10 ***
#> rho_1 0.346695 0.030621 11.3223 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3396
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 5.157484 0.753684 6.8430 8.504e-12 ***
#> RD_2 2.201353 0.151750 14.5064 < 2.2e-16 ***
#> PS_2 0.648528 0.112107 5.7849 7.612e-09 ***
#> MA_2 -0.121851 0.021874 -5.5705 2.648e-08 ***
#> DV_2 0.987470 0.100386 9.8367 < 2.2e-16 ***
#> UE_2 -0.168980 0.047095 -3.5880 0.0003358 ***
#> SOUTH_2 1.616949 0.304425 5.3115 1.126e-07 ***
#> rho_2 0.393272 0.028399 13.8482 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4154
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 21.096585 5.361957
#> 5.361957 31.640805
#> Correlation Matrix of inter-equation residuals:
#> 1.000000 0.207536
#> 0.207536 1.000000
#>
#> R-sq. pooled: 0.3996
#> Breusch-Pagan: 132.9 p-value: (9.63e-31)
#> LMM: 27.477 p-value: (1.59e-07)According to our experience, demean should be used with caution, only in cases where the time span is sufficiently large and the X variables have enough time variability. Other case, demean can produce singularities and unexpected results. Finally, note that the argument demean can be used only with the function spsurtime(), when the user has panel data.
formula.panel <- HR | RD ~ PS + MA + SOUTH | UE + DV
sur.slm.panel.demean <- spsurml(formula = formula.panel,
listw = lwncovr, type = "slm",
method ="LU", Tm = 2,
data = NCOVR.df,
control = list(fdHess = TRUE))#> sparse matrix LU decomposition
#> Initial point: log_lik: -25677.85 rhos: 0.521 0.774
#> Iteration: 1 log_lik: -23831.13 rhos: 0.543 0.887
#> Iteration: 2 log_lik: -23814.76 rhos: 0.543 0.894
#> Iteration: 3 log_lik: -23814.73 rhos: 0.543 0.894
#> Iteration: 4 log_lik: -23814.73 rhos: 0.543 0.894
#> Time to fit the model: 15.89 seconds
#> Computing numerical covariances...
#> Time to compute covariances: 2.18 secondssummary(sur.slm.panel.demean)#> Call:
#> spsurml(formula = formula.panel, data = NCOVR.df, listw = lwncovr,
#> type = "slm", method = "LU", Tm = 2, control = list(fdHess = TRUE))
#>
#>
#> Spatial SUR model type: slm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 4.696791 0.434835 10.8013 < 2.2e-16 ***
#> PS_1 0.316359 0.067318 4.6994 2.637e-06 ***
#> MA_1 -0.111253 0.014047 -7.9201 2.575e-15 ***
#> SOUTH_1 2.357344 0.134939 17.4697 < 2.2e-16 ***
#> rho_1 0.543272 0.016647 32.6349 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3584
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 -0.0290463 0.0174860 -1.6611 0.09671 .
#> UE_2 0.0327689 0.0025227 12.9897 < 2e-16 ***
#> DV_2 -0.0594530 0.0058044 -10.2428 < 2e-16 ***
#> rho_2 0.8939697 0.0060673 147.3426 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.7624
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 28.29602 0.5620700
#> 0.56207 0.2430182
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2143425
#> 0.2143425 1.0000000
#>
#> R-sq. pooled: 0.5231
#> Breusch-Pagan: 283.5 p-value: (1.32e-63)7.1 A general SUR model, with G>1
Finally, it is relatively simple to generalize the panel data framework of the previous Section to a case where the user has to work with more than one single equation. For example, suppose the case of the previous SLM panel data model, that now we want to estimate as a standard SUR system such as the following
\[\begin{equation} HR_{t} = \rho_{1} WHR_{t} +PS_{t} \beta_{1} + MA_{t} \beta_{2} + SOUTH_{t} \beta_{3} + u_{HRt}; \\ RD_{t} = \rho_{2} WRD_{t} +UE_{t} \beta_{4} + DV_{t} \beta_{5} + u_{RDt}; \\ \ t=60, 70 \\ Corr(u_{HRt},u_{RDt}) \neq \ 0 \ (\forall t) \end{equation}\]
We have G=2 equations, Tm=2 time periods and N=3085 spatial units. The ML estimate of this SLM-SUR model appears below.
formula.panel <- HR | RD ~ PS + MA + SOUTH | UE + DV
sur.slm.panel <- spsurml(formula = formula.panel,
listw = lwncovr, type = "slm",
method = "LU",
Tm = 2,
data = NCOVR.df,
control = list(fdHess = TRUE))#> sparse matrix LU decomposition
#> Initial point: log_lik: -25677.85 rhos: 0.521 0.774
#> Iteration: 1 log_lik: -23831.13 rhos: 0.543 0.887
#> Iteration: 2 log_lik: -23814.76 rhos: 0.543 0.894
#> Iteration: 3 log_lik: -23814.73 rhos: 0.543 0.894
#> Iteration: 4 log_lik: -23814.73 rhos: 0.543 0.894
#> Time to fit the model: 15.62 seconds
#> Computing numerical covariances...
#> Time to compute covariances: 2.38 secondssummary(sur.slm.panel)#> Call:
#> spsurml(formula = formula.panel, data = NCOVR.df, listw = lwncovr,
#> type = "slm", method = "LU", Tm = 2, control = list(fdHess = TRUE))
#>
#>
#> Spatial SUR model type: slm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_1 4.696791 0.434835 10.8013 < 2.2e-16 ***
#> PS_1 0.316359 0.067318 4.6994 2.637e-06 ***
#> MA_1 -0.111253 0.014047 -7.9201 2.575e-15 ***
#> SOUTH_1 2.357344 0.134939 17.4697 < 2.2e-16 ***
#> rho_1 0.543272 0.016647 32.6349 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3584
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept)_2 -0.0290463 0.0174860 -1.6611 0.09671 .
#> UE_2 0.0327689 0.0025227 12.9897 < 2e-16 ***
#> DV_2 -0.0594530 0.0058044 -10.2428 < 2e-16 ***
#> rho_2 0.8939697 0.0060673 147.3426 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.7624
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 28.29602 0.5620700
#> 0.56207 0.2430182
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2143425
#> 0.2143425 1.0000000
#>
#> R-sq. pooled: 0.5231
#> Breusch-Pagan: 283.5 p-value: (1.32e-63)8 Simulation of spatial SUR datasets
This section describes an additional functionality included in spsur, which offers the possibility of generate random data sets with the dimensions and spatial structure decided by the user. This is the purpose of the function dgp_spsur(), which has great flexibility to accomodate the necessities of the user. Our impression is that the function may be of special interest in two different situations: (i) as part of a larger simulation experiment where the user needs random data sets with specific features (SUR nature, spatial structure, etc.) or (ii) with the aim of showing specific properties of spatial SUR models and inferential procedures related to them.
dgp_spsur() refers to a Data Generating Process functionality where the user decides the dimensions of the required data set, specificies the values of the parameters that intervene in the corresponding SUR model and selects the distribution function from which the regressors and the random terms are to be drawn. The syntax of the function is the following:
dgp_spsur (Sigma, Tm = 1, G, N, Betas, Thetas = NULL, durbin = FALSE,
rho = NULL,lambda = NULL, p = NULL, listw = NULL,
X = NULL, pdfU = "nvrnorm", pdfX = "nvrnorm")
The dimensions of the data set are defined by the arguments Tm, N and G. Then the user specifies the SUR structure, begining by the covariance matrix among the residuals of the G equations, the core of the SUR model, with the argument Sigma. Moreover, a \((N\times N)\) spatial weighting matrix should be uploaded in the argument listw =.
Then, the user should fix the parameters that intervene in the equations of the SUR. This is the purpose of the arguments Betas, Thetas, rho and lambda which are defined through row vectors of the adequate dimensions. A fundamental piece of information is the argument p that defines the number of regressors (that is, X variables) that appear in each equation. We can distinguish two cases: (i)- if p is a scalar, every equation has the same number of regressors whereas (ii) if p is a \((1 \times G)\) vector the model has a different number of regressor in each equation. In both cases, Betas is a row vector of order \((1 \times G^*)\), where \(G^*\) is equal to \(pG\) in the first case and \(\Sigma^G_1p_g\) in the second case.
The type of spatial model needed by the user is unequivocally determined depending on the values given to these parameters. For example, if Thetas=NULL, rho=NULL but lambda is not NULL, the user is specifying a SUR-SEM model or a SUR-SLM model in the case of lambda=NULL, Thetas=NULL and rho different from NULL.
There are two possibilities to build the matrix X of regressors. In Monte Carlo experiments is very usual to maintain fixed the values of the regressors and repeatedly draw random matrices for the error terms, to simulate the explained variable. If this is the case, the user should upload the required X matrix in the argument X, which must be consistent with the dimensions of the SUR model. If X=NULL, which is the default, the user prefers to randomly draw this matrix using the functionalies in dgp_spsur(). This is the purpose of the argument pdfX. By default, dgp_spsur() uses a multivariate standard Normal distribution to draw the observations of the regressors; the alternative is a Uniform [0,1] distribution for each regressor. In both cases, the regressors are generated independently.
Finally, the argument dfU selects the multivariate probability distribution function that should be used to draw the values of the error terms. dgp_spsur() offers two possibilities, the multivariate Normal, which is the default, and the log-Normal distribution. In both cases, the covariance matrix matches the \(\Sigma\) matrix uploaded previously. The log-Normal should be taken as a clear departure from the assumption of normality.
The output of dgp_spsur()` is a vector, called Y, of order \((TmNG\times 1)\), with the values generated for the explained variable for the N individuals, Tm time periods and G equations of the spatial SUR model. If the argument X is set to NULL, the user will receive another matrix, called XX, of order \((TmNG\times \Sigma^G_1p_g)\), with the values generated for the regressors of the SUR.
The example below shows the random generation of 3 variables, with a SUR structure, for 500 individuals and 1 cross-section, using a SUR-SDM model. Each equation contains 3 regressors, as specified in p. Note that the argument listw includes a matrix instead of a listw object. In fact spsur accepts both formats.
Tm <- 1 # Number of time periods
G <- 3 # Number of equations
N <- 500 # Number of spatial elements
p <- 3 # Number of independent variables
Sigma <- matrix(0.3, ncol = G, nrow = G) # Matrix error correlation
diag(Sigma) <- 1
Betas <- c(1, 2, 3, 1, -1, 0.5, 1, -0.5, 2) # Beta coefficients
rho <- 0.5 # Level of sustantive spatial dependence
lambda <- 0.0 # spatial autocorrelation error term = 0
# random coordinates
co <- cbind(runif(N,0,1),runif(N,0,1))
lwdgp <- spdep::nb2listw(spdep::knn2nb(spdep::knearneigh(co, k = 5, longlat = FALSE)))
sur.dgp <- dgp_spsur(Sigma = Sigma, Betas = Betas, rho = rho,
lambda = lambda, Tm = Tm, G = G, N = N,
p = p, pdfU = "nvrnorm", listw = lwdgp)As said, the output of this function is a list with the data (explained variable and regressors) in matrix form. Next code estimates a SDM model for the simulated data:
sdm_dgp <- spsurml(Y = sur.dgp$Y, X = sur.dgp$X, type = "sdm", G = G,
Tm = Tm, N = N, p = 3, listw = lwdgp,
control = list(tol = 0.01, maxit = 200,
trace = FALSE))
summary(sdm_dgp)#> Call:
#> spsurml(listw = lwdgp, type = "sdm", X = sur.dgp$X, Y = sur.dgp$Y,
#> G = G, N = N, Tm = Tm, p = 3, control = list(tol = 0.01,
#> maxit = 200, trace = FALSE))
#>
#>
#> Spatial SUR model type: sdm
#>
#> Equation 1
#> Estimate Std. Error t value Pr(>|t|)
#> Intercep_1 0.959207 0.055699 17.221 < 2.2e-16 ***
#> X1_1 2.026348 0.046773 43.323 < 2.2e-16 ***
#> X1_2 3.067916 0.042537 72.124 < 2.2e-16 ***
#> rho_1 0.506645 0.015327 33.055 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.9441
#> Equation 2
#> Estimate Std. Error t value Pr(>|t|)
#> Intercep_2 1.121547 0.092200 12.164 < 2.2e-16 ***
#> X2_1 -1.079978 0.045058 -23.968 < 2.2e-16 ***
#> X2_2 0.473361 0.044973 10.526 < 2.2e-16 ***
#> rho_2 0.444652 0.040186 11.065 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.6164
#> Equation 3
#> Estimate Std. Error t value Pr(>|t|)
#> Intercep_3 0.997241 0.072371 13.780 < 2.2e-16 ***
#> X3_1 -0.501703 0.040879 -12.273 < 2.2e-16 ***
#> X3_2 2.042920 0.042384 48.200 < 2.2e-16 ***
#> rho_3 0.486816 0.025774 18.888 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.8453
#>
#> Variance-Covariance Matrix of inter-equation residuals:
#> 1.0668570 0.2677292 0.3455255
#> 0.2677292 1.0861678 0.2622235
#> 0.3455255 0.2622235 1.0375186
#> Correlation Matrix of inter-equation residuals:
#> 1.0000000 0.2487106 0.3284196
#> 0.2487106 1.0000000 0.2470160
#> 0.3284196 0.2470160 1.0000000
#>
#> R-sq. pooled: 0.8886
#> Breusch-Pagan: 115.4 p-value: (7.68e-25)
#> LMM: 2.8434 p-value: (0.416)Indeed, the ML estimates of the parameters are very close to the values specified in dgp_spsur().