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Spatial Seemingly Unrelated Regression Models.

Description

spsur offers the user a collection of functions to estimate Spatial Seemingly Unrelated Regression (SUR) models by maximum likelihood or three-stage least squares, using spatial instrumental variables. Moreover,spsur obtains a collection of misspecificationtests for omitted or wrongly specified spatial structure. The user will find spatial models more popular in applied research such as the SUR-SLX, SUR-SLM, SUR-SEM, SUR-SDM, SUR-SDEM SUR-SARAR and SUR-GNM plus the spatially independent SUR, or SUR-SIM.

Details

Some functionalities that have been included inspsur package are:

1. Testing for spatial effects

The functionlmtestspsur provides a collection of Lagrange Multipliers, LM, for testing different forms of spatial dependence in SUR models. They are extended versions ofthe well-known LM tests for omitted lags of the explained variable in the right hand side of the equation, LM-SLM, the LM tests for omitted spatial errors, LM-SEM, the join test of omitted spatial lags and spatial errors, LM-SARAR, and the robust version of the firttwo Lagrange Multipliers, LM*-SLM and LM*-SEM.
These tests can be applied to models always with a SUR nature. Roughly, we may distinguish two situations:

• Datasets with a single equationG=1, for different time periodsTm>1 and a certain number of spatial units in the cross-sectional dimension,N. This is what we callspatial panel datasets. In this case, the SUR structure appears in form of (intra) serial dependence in the errors of each spatial unit.

• Datasets with a several equationsG>1, different time periodsTm>1 and a certain number of spatial units,N. The SUR structure appears, as usual, because the errorsof the spatial units for different equations are contemporaneously correlated.

2. Estimation of the Spatial SUR models

As indicated above,spsur package may work with a list of different spatial specifications.They are the following:

• SUR-SIM: SUR model without spatial effects

  y_{tg} = X_{tg} \beta_{g} + \epsilon_{tg}

• SUR-SLX: SUR model with spatial lags of the regresors

  y_{tg} = X_{tg} \beta_{g} + WX_{tg} \theta_{g} + \epsilon_{tg}

• SUR-SLM: SUR model with spatial lags of the endogenous

  y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + \epsilon_{tg}

• SUR-SEM: SUR model with spatial errors

  y_{tg} = X_{tg} \beta_{g} + u_{tg}

  u_{tg} = \lambda_{g} Wu_{tg} + \epsilon_{tg}

• SUR-SDM: SUR model with spatial lags of the endogenous variable and of the regressors or Spatial Durbin model

  y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + WX_{tg} \theta_{g} + \epsilon_{tg}

• SUR-SDEM: SUR model with spatial errors and spatial lags of the endogenous variable and of the regressors

  y_{tg} = X_{tg} \beta_{g} + WX_{tg} \theta_{g} + u_{tg}

  u_{tg} = \lambda_{g} W u_{tg} + \epsilon_{tg}

• SUR-SARAR: Spatial lag model with spatial errors

  y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + u_{tg}

  u_{tg} = \lambda_{g} W u_{tg} + \epsilon_{tg}

• SUR-GNM: SUR model with spatial lags of the explained variables, regressors and spatial errors

  y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + WX_{tg} \theta_{g} + u_{tg}

  u_{tg} = \lambda_{g} W u_{tg} + \epsilon_{tg}

wherey_{tg},u_{tg} and\epsilon_{tg} are(Nx1) vectors;X_{tg} is a matrix of regressors of order(NxP);\rho_{g} and\lambda_{g} are parameters of spatial dependence andW is the(NxN) spatial weighting matrix.

These specifications can be estimated by maximum-likelihood methods, using the functionspsurml. Moroever, the models that include spatial lags of the explainedvariables in the right hand side of the equations, and the errors are assumed to be spatially incorrelated (namely, the SUR-SLM and the SUR-SDM), can also be estimated usingthree-stage least-squares,spsur3sls, using spatial instrumental variable to correct for the problem ofendogeneity present in these cases.

3. Diagnostic tests

Testing for inconsistencies or misspecifications in the results of an estimated (SUR) model should be a primary task for the user.spsur focuses, especifically, on two main question such as omittedor wrongly specified spatial structure and the existence of structural breaks or relevant restrictionsin the parameters of the model. In this sense, the user will find:

1. Marginal tests
   The Marginal Multipliers test for omitted or wrongly specified spatial structure in the equations. They are routinely part of the output of the maximum-likelihood estimation, shown byspsurml. In particular, the LM(\rho|\lambda) tests for omitted spatial lags in a model specified with spatial errors (SUR-SEM; SUR-SDEM). The LM(\lambda|\rho) tests for omitted spatial error in a model specified with spatial lagsof the explained variable (SUR-SLM; SUR-SDM).

2. Coefficients stability tests
   spsur includes two functions designed to test for linear restrictions on the\beta coefficients of the models and on the spatial coefficients (\rhos and\lambdas terms). The function for the first case iswald_betas andwald_deltas that of the second case. The user has ample flexibility to define different forms of linear restrictions, so that it is possible, for example,to test for their time constancy to identify structural breaks.

4. Marginal effects

In recent years, since the publication of LeSage and Pace (2009), it has become popular inspatial econometrics to evaluate the multiplier effects that a change in the value of a regressor, in a point in the space, has on the explained variable.spsur includes a function,impactspsur, that computes these effects. Specifically,impactspsur obtainsthe average, over theN spatial units andTm time periods, of such a change on the contemporaneous value of the explained variable located in the same point as the modified variable. This is the so-calledAverage Direct effect. TheAverage Indirect effect measure the proportion of the impact that spills-over to other locations. The sum of the two effects is theAverage Total effect.
These estimates are complemented with a measure of statistical significance, following the randomization approach suggested by LeSage and Pace (2009).

5. Additional functionalities

A particular feature ofspsur is that the package allows to obtain simulated datasets with a SUR nature and the spatial structure decided by the user. This is the purpose of the functiondgp_spsur. The function can be inserted into a more general code to solve, for example, Monte Carlo studies related to these type of models or, simply, to show some of the stylized characteristics of a SUR model with certain spatial structure.

Datasets

spsur includes three different datasets: spc, NCOVR and spain.covid. These sets are used to illustrate the capabilities of different functions. Briefly, their main characteristics are the following

• Thespc dataset (Spatial Phillips-Curve) is a classical dataset taken from Anselin (1988, p. 203), of small dimensions.

• TheNCOVR dataset comprises Homicides and a list of selected socio-economic variables for continental U.S. counties in four decennial census years: 1960, 1970, 1980 and 1990. It is freely available fromhttps://geodacenter.github.io/data-and-lab/ncovr/.NCOVR is a typical spatial panel dataset(G=1).

• Thespain.covid dataset comprises Within and Exit mobility index together with the weeklly incidence COVID-19 at Spain provinces from February 21 to May 21 2020.

Author(s)

References

• Breusch T, Pagan A (1980). The Lagrange multiplier testand its applications to model specification in econometrics.Review of Economic Studies 47: 239-254.

• LeSage, J., and Pace, R. K. (2009).Introduction tospatial econometrics. Chapman and Hall/CRC.

• Lopez, F.A., Mur, J., and Angulo, A. (2014). Spatial modelselection strategies in a SUR framework. The case of regionalproductivity in EU.Annals of Regional Science,53(1), 197-220.<doi:10.1007/s00168-014-0624-2>

• Lopez, F.A., Martinez-Ortiz, P.J., and Cegarra-Navarro, J.G.(2017). Spatial spillovers in public expenditure on a municipallevel in Spain.Annals of Regional Science, 58(1), 39-65.<doi:10.1007/s00168-016-0780-7>

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43.<doi:10.18637/jss.v104.i11>

• Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatialeffects in seemingly unrelated regressions.Spatial EconomicAnalysis, 5(4), 399-440.<doi:10.1080/17421772.2010.516443>

See Also

Useful links:

• https://CRAN.R-project.org/package=spsur

• Report bugs athttps://github.com/rominsal/spsur/issues

Homicides in U.S. counties

Description

Homicides and selected socio-economic characteristics for continentalU.S. counties. Data for four decennial census years: 1960, 1970, 1980and 1990.

Usage

NCOVR.sf

Format

An spatial feature (sf) object with 3085 rows and 41 variables:

NAME

County coded as a name (factor)

STATE_NAME

state fips code (factor)

FIPS

state fips code (factor)

SOUTH

dummy variable for Southern counties (South = 1)

HR60, HR70, HR80, HR90

homicide rate per 100,000 (1960, 1970,1980, 1990)

HC60, HC70, HC80, HC90

homicide count, three year average centeredon 1960, 1970, 1980, 1990

PO60, PO70, PO80, PO90

county population, 1960, 1970, 1980, 1990

RD60, RD70, RD80, RD90

resource deprivation 1960, 1970, 1980,1990 (principal component, see Codebook for details)

PS60, PS70, PS80, PS90

population structure 1960, 1970, 1980,1990 (principal component, see Codebook for details)

UE60, UE70, UE80, UE90

unemployment rate 1960, 1970, 1980, 1990

DV60, DV70, DV80, DV90

divorce rate 1960, 1970, 1980, 1990(% males over 14 divorced)

MA60, MA70, MA80, MA90

median age 1960, 1970, 1980, 1990

FP59, FP69, FP79, FP89

% families below poverty 1960, 1970, 1980,1990 (see Codebook for details)

geometry

Multipolygon geometry of the spatial feature object

Source

S. Messner, L. Anselin, D. Hawkins, G. Deane, S. Tolnay, R. Baller(2000). An Atlas of the Spatial Patterning of County-LevelHomicide, 1960-1990. Pittsburgh, PA, National Consortium onViolence Research (NCOVR)https://geodacenter.github.io/data-and-lab/ncovr/

References

• Baller, R., L. Anselin, S. Messner, G. Deane and D. Hawkins(2001). Structural covariates of US county homicide rates:incorporating spatial effects.Criminology 39, 561-590.

Spatial weight matrix for South-West Ohio Counties to estimateSpatial Phillips-Curve

Description

A spatial weight matrix row-standardized based on first ordercontiguity criterium.

Usage

Wspc

Format

A row-standardized squared matrix with 25 rows and columns.The rows and columns follow the same order than provinces included inspc data frame.

Source

Anselin (1988, p. 207)

References

• Anselin, L. (1988).Spatial Econometrics:Methods and Models. Springer Science & Business Media.

Generation of a random dataset with a spatial SUR structure.

Description

The purpose of the functiondgp_spsur is to generate a random dataset with the dimensions and spatial structure decided by the user. This function may be useful in pure simulation experiments or with the aim of showing specific properties and characteristicsof a spatial SUR dataset and inferential procedures related to them.

The user ofdgp_spsur should think in terms of a Monte Carlo experiment. The arguments of the function specify the dimensions of the dataset to be generated, the spatial mechanism underlying the data, the intensity of the SUR structure among the equations and the values of the parameters to be used to obtain the simulated data, which includes the error terms, the regressors and the explained variables.

Usage

dgp_spsur(Sigma, Tm = 1, G, N, Betas, Thetas = NULL,                  rho = NULL, lambda = NULL, p = NULL, listw = NULL,                  X = NULL, type = "matrix", pdfU = "nvrnorm",                  pdfX = "nvrnorm")

Arguments

Details

The purpose of the functiondgp_spsur is to generate random datasets, of a SUR nature, with the spatial structure decided by the user. The function requires certain information to be supplied externally because, in fact,dgp_spsur constitutes a Data GenerationProcess, DGP. The following aspects should be addressed:

• The user must define the dimensions of the dataset, that is, number of equations,G, number of time periods,Tm, and number of cross-sectional units,N.

• The user must choose the type of spatial structure desired for the model from among the list of candidates of "sim", "slx", "slm", "sem", "sdm", "sdem" or "sarar". The default is the "sim" specification which does not have spatial structure. The decision is made implicitly, just omitting the specification of the spatial parameters which are not involved in the model (i.e., in a "slm"there are no\lambda parameters but appear\rho parameters; in a "sdem" model there are\lambda and\theta parameters but no\rho coefficients).

• If the user needs a model with spatial structure, a(NxN) weighting matrix,W, should be chosen.

• The next step builds the equations of the SUR model. In this case, the user must specify the number of regressors that intervene in each equation and the coefficients,\beta parameters,associated with each regressor. Thefirst question is solved through the argumentp which, if a scalar, indicates that the same number of regressors should appear in all the equationsof the model; if the user seeks for a model with different number of regressors in theG equations, the argumentp must be a(1xG) row vector with the required information. It must be remembered thatdgp_spsur assumes that anintercept appears in all equations of the model.

  Thesecond part of the problem posited above is solved through the argumentBetas, which is a row vector of order(1xp) with the information required for this set of coefficients.

• The user must specify, also, the values of the spatial parameters corresponding to the chosen specification; we are referring to the\rho_{g},\lambda_{g} and\theta_{g},forg=1, ..., G and k=1,..., K_{g} parameters. This is done thought the argumentsrho,lambda andtheta. The firs two,rho andlambda, work asK: if they are scalar, the same value will be used in theG equations of the SUR model; if they are(1xG) row vectors, a different value will be assigned for each equation.

  Moreover,Theta works like the argumentBetas. The user must define a row vector of order1xPTheta showing these values. It is worth to remember that in no case the intercept will appear among the lagged regressors.

• With the argumenttype the user take the decision of the output format. See Value section.

• Finally, the user must decide which values of the regressors and of the error terms are to be used in the simulation. The regressors can be uploaded from an external matrix generated previously by theuser. This is the argumentX. It is the responsibility of the user to check that the dimensions of the external matrix are consistent with the dataset required for the SUR model. A second possibilityimplies the regressors to be generated randomly by the functiondgp_spsur.In this case, the user must select the probability distribution function from which the corresponding data (of the regressors and the error terms) are to be drawn.
  

dgp_spsur provides two multivariate distribution functions, namely, the Normal and the log-Normal for the errors (the second should be taken as a clear departure from the standard assumption ofnormality). In both cases, random matrices of order(TmNxG) are obtained from a multivariate normal distribution, with a mean value of zero and the covariance matrix specified in the argumentSigma; then, this matrix is exponentiated for the log-Normal case. Roughly, the same procedure applies for drawing the values of the regressor. There are two distribution functions available, thenormal and the uniform in the intervalU[0,1]; the regressors are always independent.

Value

The default output ("matrix") is a list with a vectorY of order(TmNGx1) with the values generated for the explained variable in the G equations of the SUR and a matrixXX of order ((TmNGxsum(p)), with the valuesgenerated for the regressors of the SUR, including an intercept for each equation.

In case of Tm = 1 or G = 1 several alternatives output can be select:

• If the user selecttype = "df" the output is a data frame where eachcolumn is a variable.

• If the user selecttype = "panel" the output is a data frame in panel format including two factors. The first factor point out the observation of the individual and the second the equation for different Tm or G.

• Finally, iftype = "all" is select the output is a list including all alternatives format.

Author(s)

References

• Lopez, F. A., Minguez, R., Mur, J. (2020). ML versus IV estimates of spatial SUR models: evidence from the case of Airbnb in Madrid urban area.The Annals of Regional Science, 64(2), 313-347.<doi:10.1007/s00168-019-00914-1>

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43. <doi:10.18637/jss.v104.i11>

See Also

spsurml,spsur3sls,spsurtime

Examples

## VIP: The output of the whole set of the examples can be examined ## by executing demo(demo_dgp_spsur, package="spsur")################################################### PANEL DATA (Tm = 1 or G = 1)              ###################################################################################################### Example 1: DGP SLM model. G equations################################################rm(list = ls()) # Clean memoryTm <- 1 # Number of time periodsG <- 3 # Number of equationsN <- 200 # Number of spatial elementsp <- 3 # Number of independent variablesSigma <- matrix(0.3, ncol = G, nrow = G)diag(Sigma) <- 1Betas <- c(1, 2, 3, 1, -1, 0.5, 1, -0.5, 2)rho <- 0.5 # level of spatial dependencelambda <- 0.0 # spatial autocorrelation error term = 0##  random coordinatesco <- cbind(runif(N,0,1),runif(N,0,1))lw <- spdep::nb2listw(spdep::knn2nb(spdep::knearneigh(co, k = 5,                                                   longlat = FALSE)))DGP <- dgp_spsur(Sigma = Sigma, Betas = Betas,                 rho = rho, lambda = lambda, Tm = Tm,                 G = G, N = N, p = p, listw = lw)SLM <- spsurml(X = DGP$X, Y = DGP$Y, Tm = Tm, N = N, G = G,                p = c(3, 3, 3), listw = lw, type = "slm") summary(SLM)################################################### MULTI-DIMENSIONAL PANEL DATA G>1 and Tm>1 ##################################################rm(list = ls()) # Clean memoryTm <- 10 # Number of time periodsG <- 3 # Number of equationsN <- 100 # Number of spatial elementsp <- 3 # Number of independent variablesSigma <- matrix(0.5, ncol = G, nrow = G)diag(Sigma) <- 1Betas <- rep(1:3, G)rho <- c(0.5, 0.1, 0.8)lambda <- 0.0 # spatial autocorrelation error term = 0## random coordinatesco <- cbind(runif(N,0,1),runif(N,0,1))lw <- spdep::nb2listw(spdep::knn2nb(spdep::knearneigh(co, k = 5,                                                   longlat = FALSE)))DGP4 <- dgp_spsur(Sigma = Sigma, Betas = Betas, rho = rho,                   lambda = lambda, Tm = Tm, G = G, N = N, p = p,                   listw = lw)SLM4  <- spsurml(Y = DGP4$Y, X = DGP4$X, G = G, N = N, Tm = Tm,                 p = p, listw = lw, type = "slm")summary(SLM4)

Direct, indirect and total effects estimated for a spatial SUR model

Description

This function is a wrapper forimpacts methodused inspatialreg package. Nevertheless, in this case the same method is used for bothlagsarlm andlmSLX objects. For details of implementation, see the documentation ofimpacts function inspatialreg package.
The function obtains the multiplier effects, on the explained variable, of a change in a regressor for the model that has been estimated. For reasons given below, this function only applies to models with an autoregressive structure ("slm", "sdm", "sarar" and "gnm") or with spatial lags of the regressors ("slx", "sdem").
The measurement of the multiplier effects is a bit more complicated than in a pure time series context because, due to the spatial structure of the model, part of the impacts spills over non uniformly over the space. Using the notation introduced by LeSage and Pace (2009) we distinguish between:

• Average Direct effects: The average over theN spatial units andTm time periods of the effect of a unitary change in the value of a explanatory variable on the contemporaneous value of the corresponding explained variable, located in the same point of the intervened regressor. This calculus is solved for all the regressors that appear in theG equations of the model.

• Average Indirect effects: The average over theN spatial units andTm time periods of the effects 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 intervened regressor. This calculus is solved for all the regressors that appear in theG equations of the model.

• Average total effects: The sum of Direct and Indirect effects.

The information on the three estimated effects is supplement with an indirect measure of statistical significance obtained from the randomization approach introduced in LeSage and Pace (2009).

Usage

impactspsur (obj, ..., tr = NULL, R = NULL, listw = NULL,                       evalues = NULL,tol = 1e-06,                       empirical = FALSE, Q = NULL)

Arguments

Details

LeSage and Pace (2009) adapt the classical notion of'economic multiplier' to the problem of measuring the impact that a unitary change in the value of a regressor, produced in a certain point in space, has on the explained variable. The question is interesting because, due to the spatial structure of the model, the impacts of such change spill non uniformly over the space. In fact, the reaction of the explained variable depends on its relative location in relation to the point of intervention.
To simplify matters, LeSage and Pace (2009) propose to obtain aggregated multipliers for each regressor, just averaging theN^{2} impacts that results from intervening the value of each regressor on each of theN points in Space, on the explained variable, measured also in each of theN points in space. This aggregated average is the so-calledTotal effect.
Part of this impact will be absorved by the explained variable located in the same point of the regressor whose value has been changed (for example, the k-th regresor in the g-th equation, in the n-th spatial unit) or,in other words, we expect that[d y_{tgn}]/[d x_{ktgn}] ne 0. The aggregated average for theN points in space (n=1,2,...,N) andTm time periods is the so-calledDirect effect.The difference between theTotal effect and theDirect effect measures the portion of the impact on the explained variable that leakes to other points in space,[d y_{tgn}]/[d x_{ktgm}] for n ne m;this is theIndirect effect.

impactspsur obtains the three multipliers together with an indirect measure of statistical significance, according to the randomization approach described in Lesage and Pace (2009). Briefly, they suggest to obtain a sequence ofnsim random matrices of order(NTmxG) from a multivariate normal distribution N(0;Sigma), beingSigma the estimated covariance matrix of theG equations in the SUR model. These random matrices, combined with the observed values of the regressors and the estimated values of the parameters of the corresponding spatial SUR model, are used to obtain simulated values of the explained variables. Then, for each one of thensim experiments, the SUR model is estimated, and the effectsare evaluated. The functionimpactspsur obtains the standard deviations of thensim estimated effects in the randomization procedure, which are used to test the significance of the estimated effects for the original data.

Finally, let us note that this is a SUR model where theG equations are connected only through the error terms. This means that if we intervene a regressor in equationg, in any point is space, only the explained variable of the same equationg should react. The impacts do not spill over equations. Moreover, the impact of a regressor, intervened in the spatial unitn, will cross the borders of this spatial unit only if in the right hand side of the equation there are spatial lags of the explained variables or of the regressors. In other words, theIndirect effect is zero for the "sim" and "sem" models.impactspsur produces no output for these two models.Lastly, it is clear that all the impacts are contemporaneous because the equations in the SUR model have no time dynamics.

Value

A list ofG objects either of classlagImpact orWXImpact.

For each of the G objects of the list, if no simulation is carried out the object returned is a list with:

and a matchingQres list attribute ifQ was given.

On the other hand, for each of the G objects of the list, if simulation is carried out the object returned is a list with:

Author(s)

References

• Bivand, R.S. and Piras G. (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics.Journal of Statistical Software, 63(18), 1-36. <doi:10.18637/jss.v063.i18>

• LeSage, J., and Pace, R. K. (2009).Introduction to spatialeconometrics. Chapman and Hall/CRC.

• Lopez, F.A., Mur, J., and Angulo, A. (2014). Spatial modelselection strategies in a SUR framework. The case of regionalproductivity in EU.Annals of Regional Science, 53(1), 197-220.<doi:10.1007/s00168-014-0624-2>

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43. <doi:10.18637/jss.v104.i11>

• Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatialeffects in seemingly unrelated regressions.Spatial Economic Analysis, 5(4), 399-440.<doi:10.1080/17421772.2010.516443>

See Also

impacts,spsurml,spsur3sls

Examples

## VIP: The output of the whole set of the examples can be examined ## by executing demo(demo_impactspsur, package="spsur")################################################## PURE CROSS SECTIONAL DATA(G>1; Tm=1) ######################################################### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203) rm(list = ls()) # Clean memory data(spc) lwspc <- spdep::mat2listw(Wspc, style = "W") Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA## For SLM, SDM and SARAR models the output is a list of "lagImpact" objects## See spatialreg::impacts for details. spcsur_slm <-spsurml(formula = Tformula, data = spc,                       type = "slm", listw = lwspc) summary(spcsur_slm) impacts_slm <- impactspsur(spcsur_slm, listw = lwspc, R = 1000)## Impacts equation 1 summary(impacts_slm[[1]], zstats = TRUE, short = TRUE)## Impacts equation 2 summary(impacts_slm[[2]], zstats = TRUE, short = TRUE)## For SLX and SDEM models the output is a list of "WXImpact" objects## See spatialreg::impacts for details.## A SUR-SLX model spcsur_slx <-spsurml(formula = Tformula, data = spc,                       type = "slx", listw = lwspc) summary(spcsur_slx) impacts_slx <- impactspsur(spcsur_slx, listw = lwspc) summary(impacts_slx[[1]], zstats = TRUE, short = TRUE) summary(impacts_slx[[2]], zstats = TRUE, short = TRUE)## A SUR-SDM model spcsur_sdm <-spsurml(formula = Tformula, data = spc,                       type = "sdm", listw = lwspc) impacts_sdm <- impactspsur(spcsur_sdm, listw = lwspc, R = 1000) summary(impacts_sdm[[1]], zstats = TRUE, short = TRUE) summary(impacts_sdm[[2]], zstats = TRUE, short = TRUE)## A SUR-SDM model with different spatial lags in each equation TformulaD <- ~ UN83 + NMR83 + SMSA | UN80 spcsur_sdm2 <-spsurml(formula = Tformula, data = spc, type = "sdm",                        listw = lwspc, Durbin = TformulaD) summary(spcsur_sdm2)                        impacts_sdm2 <- impactspsur(spcsur_sdm2, listw = lwspc, R = 1000) summary(impacts_sdm2[[1]], zstats = TRUE, short = TRUE) summary(impacts_sdm2[[2]], zstats = TRUE, short = TRUE) ## A SUR-SLX model with different spatial lags in each equation spcsur_slx2 <-spsurml(formula = Tformula, data = spc,                       type = "slx", listw = lwspc, Durbin = TformulaD) summary(spcsur_slx2) impacts_slx2 <- impactspsur(spcsur_slx2, listw = lwspc) summary(impacts_slx2[[1]], zstats = TRUE, short = TRUE) summary(impacts_slx2[[2]], zstats = TRUE, short = TRUE) # ##################################### ######## G=1; Tm>1               #### ##################################### rm(list = ls()) # Clean memory data(NCOVR, package="spsur") nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)### Some regions with no links... lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE) Tformula <- HR80  | HR90 ~ PS80 + UE80 | PS90 + UE90### A SUR-SLM model NCOVRSUR_slm <-spsurml(formula = Tformula, data = NCOVR.sf,                         type = "slm", listw = lwncovr,                         method = "Matrix", zero.policy = TRUE,                         control = list(fdHess = TRUE)) summary(NCOVRSUR_slm)### Use of trW to compute. Wncovr <- Matrix::Matrix(spdep::listw2mat(lwncovr)) trwncovr <- spatialreg::trW(Wncovr, type = "MC") impacts_NCOVRSUR_slm <- impactspsur(NCOVRSUR_slm, tr = trwncovr,                                 R = 1000) summary(impacts_NCOVRSUR_slm[[1]], zstats = TRUE, short = TRUE) summary(impacts_NCOVRSUR_slm[[2]], zstats = TRUE, short = TRUE)

Testing for the presence of spatial effects in Seemingly Unrelated Regressions

Description

The functionspsurml reports a collection of Lagrange Multipliers designed to test for the presence of different forms of spatial dependence in aSUR model of the "sim" type. That is, the approach of this function is from'specific to general'. As said, the model of the null hypothesis is the "sim" model whereas the model of the alternative depends on the effect whose omission we want to test.

The collection of Lagrange Multipliers obtained bylmtestspsur are standard in the literature and take into account the multivariate nature of theSUR model. As a limitation, note that each Multiplier tests for the omission of the same spatial effects in all the cross-sections of theG equations.

Usage

lmtestspsur(...)## S3 method for class 'formula'lmtestspsur(  formula,  data,  listw,  na.action,  time = NULL,  Tm = 1,  zero.policy = NULL,  R = NULL,  b = NULL,  ...)## Default S3 method:lmtestspsur(Y, X, G, N, Tm = 1, listw, p, R = NULL, b = NULL, ...)

Arguments

Details

lmtestspsur tests for the omission of spatial effects in the "sim" version of theSUR model:

y_{tg} = X_{tg} \beta_{g} + u_{tg}

E[u_{tg}u_{th}']= \sigma_{gh}I_{N} \quad E[u_{tg}u_{sh}']= 0 \mbox{ if } t ne s

wherey_{tg} andu_{tg} are(Nx1) vectors, corresponding to the g-th equation and time period t;X_{tg} is the matrix of exogenous variables, of order(Nxp_{g}). Moreover,\beta_{g} is an unknown(p_{g}x1) vector of coefficients and\sigma_{gh}I_{N} the covariance between equationsg andh,being\sigma_{gh} and scalar andI_{N} the identity matrix of orden N.

The Lagrange Multipliers reported by this function are the followings:

• LM-SUR-LAG: Tests for the omission of a spatial lag of the explained variable in the right hand side of the "sim" equation. The model of the alternative is:
  

  y_{tg} = \rho_{g}Wy_{tg} + X_{tg} \beta_{g} + u_{tg}

  The null and alternative hypotheses are:

  H_{0}: \rho_{g}=0 (forall g) vsH_{A}: \rho_{g} ne 0 (exist g)

• LM-SUR-ERR: Tests for the omission of spatial dependence in the equation of the errors of the "sim" model. The model of the alternative is:

  y_{tg} = X_{tg} \beta_{g} + u_{tg};u_{tg}= \lambda_{g}Wu_{tg}+\epsilon_{tg}

  The null and alternative hypotheses are:

  H_{0}: \lambda_{g}=0 (forall g) vsH_{A}: \lambda_{g} ne 0 (exist g)

• LM-SUR-SARAR: Tests for the simultaneous omission of a spatial lag of the explained variable in the right hand side of the "sim" equation and spatial dependence in the equation of the errors. The model of the alternative is:

  y_{tg} = \rho_{g}Wy_{tg}+X_{tg} \beta_{g} + u_{tg};u_{tg}= \lambda_{g}Wu_{tg}+\epsilon_{tg}

  The null and alternative hypotheses are:

  H_{0}: \rho_{g}=\lambda_{g}=0 (forall g) vsH_{A}: \rho_{g} ne 0 or \lambda_{g} ne 0 (exist g)

• LM*-SUR-SLM andLM*-SUR-SEM: These two test are the robustifyed version of the original, raw Multipliers,LM-SUR-SLM andLM-SUR-SEM, which can be severely oversized if the respective alternative hypothesis is misspeficied (this would be the case if, for example, we are testing for omitted lags of the explained variable whereas the problem is that there is spatial dependence in the errors, or viceversa). The null and alternative hypotheses of both test are totally analogous to theirtwin non robust Multipliers.

Value

A list ofhtest objects each one including the Waldstatistic, the corresponding p-value and the degrees offreedom.

Author(s)

References

• Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatialeffects in seemingly unrelated regressions.Spatial Economic Analysis, 5(4), 399-440.<doi:10.1080/17421772.2010.516443>

• Lopez, F.A., Mur, J., and Angulo, A. (2014). Spatial modelselection strategies in a SUR framework. The case of regionalproductivity in EU.Annals of Regional Science, 53(1),197-220.<doi:10.1007/s00168-014-0624-2>

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43. <doi:10.18637/jss.v104.i11>#'

• Anselin, L. (1988) A test for spatial autocorrelation in seemingly unrelated regressionsEconomics Letters 28(4), 335-341.<doi:10.1016/0165-1765(88)90009-2>

• Anselin, L. (1988)Spatial econometrics: methods and models Chap. 9 Dordrecht

• Anselin, L. (2016) Estimation and Testing in the Spatial Seemingly Unrelated Regression (SUR).Geoda Center for Geospatial Analysis and Computation, Arizona State University. Working Paper 2016-01.<doi:10.13140/RG.2.2.15925.40163>

See Also

spsurml,anova

Examples

######################################################### CROSS SECTION DATA (G>1; Tm=1) # ############################################################ Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)rm(list = ls()) # Clean memorydata("spc")Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSAlwspc <- spdep::mat2listw(Wspc, style = "W")lmtestspsur(formula = Tformula, data = spc, listw = lwspc)## VIP: The output of the whole set of the examples can be examined ## by executing demo(demo_lmtestspsur, package="spsur")######################################################### PANEL DATA (G>1; Tm>1)          ############################################################# Example 2: Homicides & Socio-Economics (1960-90)# Homicides and selected socio-economic characteristics for# continental U.S. counties.# Data for four decennial census years: 1960, 1970, 1980 and 1990.# https://geodacenter.github.io/data-and-lab/ncovr/data(NCOVR, package="spsur")nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)### Some regions with no links...lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE)### With different number of exogenous variables in each equationTformula <- HR70 | HR80  | HR90 ~ PS70 + UE70 | PS80 + UE80 +RD80 |            PS90 + UE90 + RD90 + PO90lmtestspsur(formula = Tformula, data = NCOVR.sf,             listw = lwncovr)########################################################################## PANEL DATA: TEMPORAL CORRELATIONS (G=1; Tm>1) ############################################################################## Example 3: NCOVR in panel data formYear <- as.numeric(kronecker(c(1960,1970,1980,1990),                    matrix(1,nrow = dim(NCOVR.sf)[1])))HR <- c(NCOVR.sf$HR60,NCOVR.sf$HR70,NCOVR.sf$HR80,NCOVR.sf$HR90)PS <- c(NCOVR.sf$PS60,NCOVR.sf$PS70,NCOVR.sf$PS80,NCOVR.sf$PS90)UE <- c(NCOVR.sf$UE60,NCOVR.sf$UE70,NCOVR.sf$UE80,NCOVR.sf$UE90)NCOVRpanel <- as.data.frame(cbind(Year,HR,PS,UE))Tformula <- HR ~ PS + UElmtestspsur(formula = Tformula, data = NCOVRpanel, time = Year, listw = lwncovr)

Likelihood ratio for testing homogeneity constraints on beta coefficients of the SUR equations.

Description

Functionlr_betas obtains a Likelihood Ratio test, LR in what follows,with the purpose of testing if some of the\beta coefficients in theG equations of theSUR model are equal. This function has a straightforward application, especially whenG=1,to the case of testing for the existence of structural breaks in the\beta parameters.

The function can test for the homogeneity of only one coefficient, of a few of themor even the homogeneity of all the slope terms. The testing procedure implies,first,the estimation of both a constrained and a unconstrained model and,second, the comparisonof the log-likelihoods to compute the LR statistics.

@usage lr_betas (obj, R, b)

Usage

lr_betas(obj, R, b)

Arguments

Value

Object ofhtest including the LRstatistic, the corresponding p-value, the degrees offreedom and the values of the sample estimates.

Author(s)

References

• Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatialeffects in seemingly unrelated regressions.Spatial Economic Analysis, 5(4), 399-440.<doi:10.1080/17421772.2010.516443>

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43. <doi:10.18637/jss.v104.i11>

See Also

spsurml,spsurtime,wald_betas

Examples

## VIP: The output of the whole set of the examples can be examined ## by executing demo(demo_lr_betas, package="spsur")#' ######################################################### CROSS SECTION DATA (G>1; Tm=1)  ############################################################# Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)rm(list = ls()) # Clean memorydata(spc)lwspc <- spdep::mat2listw(Wspc, style = "W")Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA### H0: equal beta for SMSA in both equations.R <- matrix(c(0,0,0,1,0,0,0,-1), nrow=1)b <- matrix(0, ncol=1)spcsur.slm <- spsurml(formula = Tformula, data = spc,                       type = "slm", listw = lwspc)summary(spcsur.slm)lr_betas(spcsur.slm, R = R, b = b)### Estimate restricted SUR-SLM modelspcsur.slmr <- spsurml(formula = Tformula, data = spc,                       type = "slm", listw = lwspc,                      R = R, b = b)summary(spcsur.slmr)

Methods for class spsur

Description

Theanova() function provides tables of fitted spsur models including information criteria (AIC and BIC), log-likelihood and degrees of freedom of each fitted model. The argumentlrtest allows to perform LR tests between nested models.Theplot() function allows the user to plot both beta and spatialcoefficients for all equations of the spsur model. The argumentviewplot is used to choose between interactive or non-interactiveplots. Theprint() function is used to print short tables including the values of beta andspatial coefficients as well as p-values of significance test for each coefficient. This can be used as an alternative tosummary.spsur when a brief output is needed. The rest of methods works in the usual way.

Usage

## S3 method for class 'spsur'anova(object, ..., lrtest = TRUE)## S3 method for class 'spsur'coef(object, ...)## S3 method for class 'spsur'fitted(object, ...)## S3 method for class 'spsur'logLik(object, ...)## S3 method for class 'spsur'residuals(object, ...)## S3 method for class 'spsur'vcov(object, ...)## S3 method for class 'spsur'print(x, digits = max(3L, getOption("digits") - 3L), ...)## S3 method for class 'spsur'plot(x, ci = 0.95, viewplot = TRUE, ...)

Arguments

Author(s)

Examples

rm(list = ls()) # Clean memorydata(spc)Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSAspcsur.sim <-spsurml(formula = Tformula, data = spc, type = "sim")## Print Table       print(spcsur.sim)spcsur.slm <-spsurml(formula = Tformula, data = spc, type = "slm",                      listw = Wspc)# ANOVA table and LR test for nested models:anova(spcsur.sim, spcsur.slm)## Plot spatial and beta coefficients# Interactive plotplot(spcsur.slm)# Non-interactive plotif (require(gridExtra)) {  pl <- plot(spcsur.slm, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                pl$pldeltas, nrow = 3)}

Print method for objects of class summary.spsur.

Description

Print method for objects of class summary.spsur.

Usage

## S3 method for class 'summary.spsur'print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

Author(s)

See Also

summary.spsur.

Examples

 # See examples for \code{\link{spsurml}} or # \code{\link{spsur3sls}} functions.

Within/Exit mobility index and incidence COVID-19 at Spain provinces

Description

Weekly within/exit mobility indices and COVID-19 incidence in Spain at provincial level. A total of 17 weeks from February 21 to May 21. Every week starts on a Friday and ends on the following Thursday. All travels are expressed with respect to the pre-COVID week of February 14th-20th, 2020 (week0). A value lower than 1indicate a reduction of the mobility. Upper values than 1 indicate an increase in the mobility with respectto the reference week (week0).

Usage

spain.covid

Format

A data frame with 850 rows and 11 variables:

province

Province name coded as a factor

indiv

Province coded as a number

time

Number of week afther pre-COVID (week0: February 14th-20th, 2020). time=1 (week1, February 21th-27th); time=2 (week2,February 28th-March 5th);....

Within

Mobility index within of the province. See details for a formal definition

Exits

Mobility index of exits of the province. See details for a formal definition

Emergence

Dummy variable. 1 if the Emergence State ("Estado de Alarma") is active in the week "time" but economicactivity of essential services are allowed. 0 in anoter case.The Emergence State was active in Spain from February 14th, 2020 to May 21th, 2020

EmergenceTotal

Dummy variable. 1 if the Emergence State ("Estado de Alarma") is active in the week "time" and economic activity of essential services are not allowed. 0 in anoter case

Old65

Percentage of population aged 65 and older in the province

Density

Inhabitants (in thousands) per km^2 in the province

Essential

Percentage of firms in the province "indiv" with essential activities (food, health) over the total of firms in the province "indiv" in the province (e.g., food, heath and some economic subsectors of industry and construction)

Incidence

Weekly incidence in the week "time-1" in logs

Details

• Mobility
  
  The mobility indices Within and Exits has been obtain as ratio of the total number of weekly travelsin reference to the total number of travels in the week of reference (week0).
  In particular,
  
  Within = \frac{\textnormal{Number of travels within the province 'indiv' in the week 'time'}} {\textnormal{Number of travels within the province 'indiv' in the reference week (week0)}}

Exits = \frac{ \textnormal{Number of travels with origin in the province 'indiv' and arrival to another province in the week 'time'} } { \textnormal{Number of travels with origin in the province 'indiv' and arrival to another province in the reference week (week0)} }
  
  A ‘travel’ is a displacement from an origin to a destination of at least 500m. A travel can have several stages.These stages of the same travel are calculated based on the duration of the intermediate stop. For example, if I move from origin A to destination B with a stop at a point C of long duration, itis considered two travels, but if the stop is short, it is considered a single travel. For example,I can go by train from Madrid to Alicante, and there takes a bus to Benidorm. The travel will be one(Madrid-Benidorm). If I do the same but the stop in Alicante is long (or an overnight stay, for example),two travels will be considered (Madrid-Alicante, and Alicante-Benidorm). Similarly, if I go by car fromMadrid to Alicante and stop for 15 minutes to take a coffee, it is also considered only one travel and not two.The travels considered in this study are always from 500m, due to the limitation of source data that is basedon mobile telephony and its antennas. But one travel can be 600 meters or 600km.
  
  

• Incidence
  
  Incidence = \log \left( \frac{ \textnormal{total diagnostic cases of COVID-19, PCR test in the week 'time' at the province 'indiv'} } {\textnormal{total population in the province 'indiv'}} \right)
  
  

• Essential activities
  
  Essential activities. Economic activities whose activities are essential for the population. By example, essential activities arehealthcare, food supply, State security, media and communication, refuse collection, management and public transport, etc.Non essential activities are by example, restaurants, hotels, hairdressers, etc. A full list in the Spanish official bulletin

Source

The National Statistics Institute
https://www.ine.es/en/index.htm
Instituto de Salud Carlos III
https://cnecovid.isciii.es/covid19/

Spain geometry

Description

A sf object with the Spanish geometry

Usage

spain.covid.sf

Format

A sf object with the Spanish geometry

PROVINCIA

province name coded as a factor

ID_INE

province coded as a number

A classical Spatial Phillips-Curve

Description

A data set from Anselin (1988, p. 203-2011) used to estimate a SpatialPhillips-Curve for 25 counties in South-West Ohio for two timeperiods (1981 and 1983).

Usage

spc

Format

A data frame with 25 rows and 10 variables:

COUNTY

County coded as a name.

WAGE83

Changes in wage rates for 1983.

UN83

Inverse unemployment rate in 1983.

NMR83

Net migration rate 1983.

SMSA

Dummy variable to identify counties defined asStandard Metropolitan Statistical Areas (SMSA = 1).

WAGE82

Changes in wage rates for 1982.

WAGE81

Changes in wage rates for 1981.

UN80

Inverse unemployment rate in 1980.

NMR80

Net migration rate 1983.

WAGE80

changes in wage rates.

geometry

geometry of sf object.

Source

Anselin (1988, p. 203-211)

References

• Anselin, L. (1988).Spatial Econometrics:Methods and Models. Springer Science & Business Media.

Three Stages Least Squares estimation,3sls, of spatial SUR models.

Description

The function estimates spatial SUR models using three stagesleast squares, where the instruments are obtained from the spatial lagsof theX variables, assumed to be exogenous. The number of equations, time periods and spatial units is not restricted. The user can choose between a Spatial Durbin Model or a Spatial Lag Model, as described below. The estimation procedure allows for the introduction of linear restrictions on the\beta parameters associated to the regressors.

Usage

spsur3sls (formula = NULL, data = NULL, na.action,                  R = NULL, b = NULL, listw = NULL,                   zero.policy = NULL, X= NULL, Y = NULL, G = NULL,                   N = NULL, Tm = NULL, p = NULL,                    type = "slm", Durbin = NULL, maxlagW = NULL,                  trace = TRUE)

Arguments

Details

spsur3sls can be used to estimate two groups of spatial models:

• "slm": SUR model with spatial lags of the endogenous in the right hand side of the equations

  y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + \epsilon_{tg}

• "sdm": SUR model of the Spatial Durbin type

  y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + WX_{tg} \theta_{g} + \epsilon_{tg}

wherey_{tg} and\epsilon_{tg} are(Nx1) vectors,corresponding to the g-th equation and time period t;X_{tg} is the matrix of regressors, of orderN \times p_g. Moreover,\rho_{g} is a spatial coefficient andW is a(NxN) spatial weighting matrix.

By default, the input of this function is an object created withFormula and a data frame. However,spsur3sls also allows for the direct specification of vectorY and matrixX, with the explained variables and regressors respectively, as inputs (these terms may be the result, for example, ofdgp_spsur).

spsur3sls is a Least-Squares procedure in three-stages designed to circumvent the endogeneity problems due to the presence of spatial lags of the explained variable in the right hand side of the equations do the SUR. The instruments are produced internally byspsur3sls using a sequence of spatial lags of theX variables, which are assumed to be exogenous. The user must define the number of (spatial) instruments to be used in the procedure, through the argumentmaxlagW (i.e. maxlagW = 3). Then, the collection of instruments generated is[WX_{tg}; W*WX_{tg}; W*W*WX_{tg}]. In the case of aSDM, the first lag of theX matrix already is in the equation and cannot be used as instrument. In the example above, the list of instruments for aSDM model would be[W^{2}X_{tg}; W^{3}X_{tg}].

Thefirst stage of the procedure consists in the least squares of theY variables on the set of instruments. From this estimation, the procedure retains the estimates ofYin the so-calledYls variables. In thesecond stage, theY variables that appear in the right hand side of the equation are substituted byYls and the SUR modelis estimated by Least Squares. Thethird stage improves the estimates of the second stage through a Feasible Generalized Least Squares estimation of the parameters of the model,using the residuals of thesecond stage to estimate theSigma matrix.

The argumentsR andb allows to introduce linear restrictions on thebeta coefficients of theG equations.spsur3sls, first, introduces the linear restrictions in the SUR model and builds, internally, the corresponding constrainedSUR model. Then, the function estimates the restricted model which is shown in the output. The function does not compute the unconstrained model nor test for the linear restrictions. The user may ask for the unconstrained estimation using anotherspsurmlestimation. Moreover, the functionwald_betas obtains the Wald test of a set of linear restrictions for an object created previously byspsurml orspsur3sls.

Value

Object ofspsur class with the output of the three-stages least-squares estimation of the specified spatial model.A list with:

Author(s)

References

• Anselin, L. (2016) Estimation and Testing in the Spatial Seemingly Unrelated Regression (SUR).Geoda Center for Geospatial Analysis and Computation, Arizona State University. Working Paper 2016-01.<doi:10.13140/RG.2.2.15925.40163>

• , Anselin, L. (1988).Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht, The Netherlands (p. 146).

• Anselin, L., Le Gallo, J., Hubert J. (2008) Spatial Panel Econometrics. InThe econometrics of panel data. Fundamentals and recent developments in theory and practice. (Chap 19, p. 653)

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43.<doi:10.18637/jss.v104.i11>

• Lopez, F. A., Minguez, R., Mur, J. (2020). ML versus IV estimates of spatial SUR models: evidence from the case of Airbnb in Madrid urban area.The Annals of Regional Science, 64(2), 313-347.<doi:10.1007/s00168-019-00914-1>

See Also

spsurml,stsls,wald_betas

Examples

######################################################### CLASSIC PANEL DATA (G=1; Tm>1)  ############################################################# Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)## A SUR model without spatial effectsrm(list = ls()) # Clean memorydata(spc)lwspc <- spdep::mat2listw(Wspc, style = "W")Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA## A SUR-SLM model (3SLS Estimation)spcsur_slm_3sls <-spsur3sls(formula = Tformula, data = spc,                            type = "slm", listw = lwspc)summary(spcsur_slm_3sls)print(spcsur_slm_3sls)if (require(gridExtra)) {  pl <- plot(spcsur_slm_3sls, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                pl$pldeltas, nrow = 3)}## VIP: The output of the whole set of the examples can be examined ## by executing demo(demo_spsur3sls, package="spsur")## A SUR-SDM model (3SLS Estimation)spcsur_sdm_3sls <- spsur3sls(formula = Tformula, data = spc,                             type = "sdm", listw = lwspc)summary(spcsur_sdm_3sls)if (require(gridExtra)) {  pl <- plot(spcsur_sdm_3sls, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                pl$pldeltas, nrow = 3)}rm(spcsur_sdm_3sls)## A SUR-SDM model with different spatial lags in each equation TformulaD <-  ~ UN83 + NMR83 + SMSA | UN80 + NMR80   spcsur_sdm2_3sls <-spsur3sls(formula = Tformula, data = spc,                             type = "sdm", listw = lwspc,                             Durbin = TformulaD) summary(spcsur_sdm2_3sls)if (require(gridExtra)) {  pl <- plot(spcsur_sdm2_3sls, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                pl$pldeltas, nrow = 3)}rm(spcsur_sdm2_3sls)####################################################  MULTI-DIMENSIONAL PANEL DATA (G>1; Tm>1) ######################################################## Example 3: Homicides + Socio-Economics (1960-90)# Homicides and selected socio-economic characteristics for continental# U.S. counties.# Data for four decennial census years: 1960, 1970, 1980 and 1990.# https://geodacenter.github.io/data-and-lab/ncovr/rm(list = ls()) # Clean memorydata(NCOVR, package = "spsur")nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)## Some regions with no links...lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE)Tformula <- HR80  | HR90 ~ PS80 + UE80 | PS90 + UE90## A SUR-SLM modelNCOVRSUR_slm_3sls <- spsur3sls(formula = Tformula, data = NCOVR.sf,                                type = "slm", zero.policy = TRUE,                               listw = lwncovr, trace = FALSE)summary(NCOVRSUR_slm_3sls)if (require(gridExtra)) {  pl <- plot(NCOVRSUR_slm_3sls, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                pl$pldeltas, nrow = 3)}rm(NCOVRSUR_slm_3sls)

General Spatial 3SLS for systems of spatial equations.

Description

The function estimates spatial SUR models using general spatial three stages least squares. This is a system instrumental variable procedure which also include GMM estimation when there is spatial correlations in the errors.The procedure allows for additional endogenous regressors inaddition to spatial lags of the dependent variable. It could be applied to "slm", "sdm", "sem" and "sarar" spatial models. Furthermore, for non-spatial models including endogenous regressors ("iv"), it could be used to estimate using instrumental variables and Feasible Generalized Least Squares.

Usage

spsurgs3sls(formula = NULL, data = NULL, na.action,                  listw = NULL, zero.policy = NULL,                   type = "slm", Durbin = FALSE,                  endog = NULL, instruments = NULL,                  lag.instr = FALSE, initial.value = 0.2,                   het = FALSE, trace = TRUE)

Arguments

Details

spsurg3sls generalize thespreg functionto multiequational spatial SUR models. The methodology to estimatespatial SUR models by Generalized 3SLS follows the steps outlined inKelejian and Piras (pp. 304-305). The summary of the algorithm isthe next one:

• Estimate each equation by 2SLS and obtain the estimated residuals\hat{u}_j for each equation.

• If the model includes a spatial lag for the errors. (that is, it is a SEM/SARAR model), apply GMM to obtain the spatial parameters\hat{\lambda}_j for the residualsin each equation. In this case thespreg function is used as a wrapper for the GMM estimation. If themodel does not include a spatial lag for the errors (that is, it is a "sim", "iv", "slm" or "sdm" model), then\hat{\lambda}_j = 0

• Compute

  \hat{v}_j = \hat{u}_j-\hat{\lambda}_j W \hat{u}_j

  and the covariances

  \hat{\sigma}_{i,j} = N^{-1}\hat{v}_i\hat{v}_j

  . Build\hat{Sigma}=\lbrace \hat{\sigma_{i,j}} \rbrace

• Compute

  y_j^* = y_j - \hat{\lambda}_j W y_j

  and

  X_j^* = X_j- \hat{\lambda}_j W X_j

  Compute

  \hat{X}_j^* = H_j(H_j^T H_j)^{-1} H_j^T X_j^*

  whereH_j is the matrix including all the instruments andthe exogenous regressors for each equation. That is,\hat{X}_j^* is the projection ofX_j^* usingthe instruments matrixH_j.

• Compute, in a multiequational way, the Feasible Generalized Least Squares estimation using the new variables\hat{y}_j^*,\hat{X}_j^* and\hat{Sigma}. This is the 3sls step.

Value

Object ofspsur class with the output of the three-stages least-squares estimation of the specified spatial model.A list with:

Author(s)

References

• Kelejian, H. H. and Piras, G. (2017). Spatial Econometrics. Academic Press.

• Kelejian, H.H. and Prucha, I.R. (2010).Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances.Journal of Econometrics, 157, pp. 53-67.

• Kelejian, H.H. and Prucha, I.R. (1999). A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model.International Economic Review,40, pp. 509-533.

• Kelejian, H.H. and Prucha, I.R. (1998).A Generalized Spatial Two Stage Least Square Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances.Journal of Real Estate Finance and Economics, 17, pp. 99–121.

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43.<doi:10.18637/jss.v104.i11>

• Piras, G. (2010). sphet: SpatialModels with Heteroskedastic Innovations in R.Journal of Statistical Software, 35(1), pp. 1-21.https://www.jstatsoft.org/v35/i01/. -

See Also

spreg,spsur3sls,stsls,spsurml

Examples

#### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)rm(list = ls()) # Clean memorydata(spc)lwspc <- spdep::mat2listw(Wspc, style = "W")## No endogenous regressors  Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA## Endogenous regressors and InstrumentsTformula2 <- WAGE83 | WAGE81 ~  NMR83 | NMR80 ## Endogenous regressors: UN83 , UN80## Instrumental variable: SMSA## A IV model with endogenous regressors only in first equationspciv <- spsurgs3sls(formula = Tformula2, data = spc,                      type = "iv", listw = lwspc,                      endog = ~ UN83 | .,                       instruments = ~ SMSA | .)summary(spciv)print(spciv)########################################################################### A SLM model with endogenous regressors spcslm <- spsurgs3sls(formula = Tformula2, data = spc,                              endog = ~ UN83 | .,                               instruments = ~ SMSA |.,                              type = "slm",                               listw = lwspc)summary(spcslm)print(spcslm)                           impacts_spcslm <- impactspsur(spcslm, listw = lwspc, R = 1000)summary(impacts_spcslm[[1]], zstats = TRUE, short = TRUE)summary(impacts_spcslm[[2]], zstats = TRUE, short = TRUE)########################################################################### A SDM model with endogenous regressors spcsdm <- spsurgs3sls(formula = Tformula2, data = spc,                   endog = ~ UN83 | UN80,                    instruments = ~ SMSA | SMSA,                   type = "sdm", listw = lwspc,                   Durbin =  ~  NMR83 | NMR80)summary(spcsdm)## Durbin only in one equationspcsdm2 <- spsurgs3sls(formula = Tformula2, data = spc,                    endog = ~ UN83 | UN80,                     instruments = ~ SMSA | SMSA,                    type = "sdm", listw = lwspc,                    Durbin =  ~  NMR83 | .)summary(spcsdm2)########################################################################### A SEM model with endogenous regressors spcsem <- spsurgs3sls(formula = Tformula2, data = spc,                      endog = ~ UN83 | UN80,                       instruments = ~ SMSA | SMSA,                      type = "sem", listw = lwspc)summary(spcsem)print(spcsem)                           ########################################################################### A SARAR model with endogenous regressors spcsarar <- spsurgs3sls(formula = Tformula2, data = spc,                        endog = ~ UN83 | UN80,                         instruments = ~ SMSA | SMSA,                        type = "sarar", listw = lwspc)summary(spcsarar)print(spcsarar)                           impacts_spcsarar <- impactspsur(spcsarar, listw = lwspc, R = 1000)summary(impacts_spcsarar[[1]], zstats = TRUE, short = TRUE)summary(impacts_spcsarar[[2]], zstats = TRUE, short = TRUE)

Maximum likelihood estimation of spatial SUR model.

Description

This function estimates spatial SUR models using maximum-likelihood methods.The number of equations, time periods and cross-sectional units is not restricted.The user can choose between different spatial specifications as described below.The estimation procedure allows for the introduction of linear restrictions on the\beta parameters associated to the regressors.

Usage

spsurml(formula = NULL, data = NULL, na.action,               listw = NULL, type = "sim", Durbin = NULL,               method = "eigen", zero.policy = NULL, interval = NULL,               trs = NULL, R = NULL, b = NULL, X = NULL, Y = NULL,                G = NULL, N = NULL, Tm = NULL,p = NULL,                control = list() )

Arguments

Details

The list of (spatial) models that can be estimated with thespsurml function are:

• "sim": SUR model with no spatial effects

  y_{tg} = X_{tg} \beta_{g} + \epsilon_{tg}

• "slx": SUR model with spatial lags of the regressors

  y_{tg} = X_{tg} \beta_{g} + WX_{tg} \theta_{g} + \epsilon_{tg}

• "slm": SUR model with spatial lags of the explained variables

  y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + \epsilon_{tg}

• "sem": SUR model with spatial errors

  y_{tg} = X_{tg} \beta_{g} + u_{tg}

  u_{tg} = \lambda_{g} Wu_{tg} + \epsilon_{tg}

• "sdm": SUR model of the Spatial Durbin type

  y_{tg} = \rho_{g} Wy_{tg} + X_{tt} \beta_{g} + WX_{tg} \theta_{g} + \epsilon_{tg}

• "sdem": SUR model with spatial lags of the regressors and spatial errors

  y_{tg} = X_{tg} \beta_{g} + WX_{tg} \theta_{g} + u_{tg}

  u_{tg} = \lambda_{g} W u_{tg} + \epsilon_{tg}

• "sarar": SUR model with spatial lags of the explained variables and spatialerrors

  y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + u_{tg}

  u_{tg} = \lambda_{g} W u_{tg} + \epsilon_{tg}

• "gnm": SUR model with spatial lags of the explained variables, regressors and spatial errors

  y_{tg} = \rho_{g} Wy_{tg} + X_{tg} \beta_{g} + WX_{tg} \theta_{g} + u_{tg}

  u_{tg} = \lambda_{g} W u_{tg} + \epsilon_{tg}

Value

Object ofspsur class with the output of the maximum-likelihood estimation of the specified spatial SUR model. A list with:

Control arguments

Author(s)

References

• Anselin, L. (1988).Spatial econometrics: methods and models. Dordrecht: Kluwer

• Bivand, R.S. and Piras G. (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics.Journal of Statistical Software, 63(18), 1-36. <doi:10.18637/jss.v063.i18>

• Bivand, R. S., Hauke, J., and Kossowski, T. (2013). Computing the Jacobian in Gaussian spatial autoregressive models: An illustrated comparison of available methods. Geographical Analysis, 45(2), 150-179. <doi:10.1111/gean.12008>

• Breusch T., Pagan A. (1980). The Lagrange multiplier test and itsapplications to model specification in econometrics.Rev Econ Stud 47: 239-254

• Cliff, A.D. and Ord, J.K. (1981).Spatial processes: Models and applications, Pion.

• LeSage J and Pace, R.K. (2009).Introduction to Spatial Econometrics. CRC Press, Boca Raton.

• Lopez, F.A., Mur, J., and Angulo, A. (2014). Spatial modelselection strategies in a SUR framework. The case of regionalproductivity in EU.Annals of Regional Science, 53(1), 197-220.<doi:10.1007/s00168-014-0624-2>

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43.<doi:10.18637/jss.v104.i11>

• Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatialeffects in seemingly unrelated regressions.Spatial Economic Analysis, 5(4), 399-440.<doi:10.1080/17421772.2010.516443>

• Ord, J.K. (1975). Estimation methods for models of spatial interaction,Journal of the American Statistical Association, 70, 120-126.

See Also

spsur3sls,lagsarlm,lmtestspsur,wald_betas,lr_betas

Examples

######################################################### CROSS SECTION DATA (G>1; Tm=1) ############################################################# Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)rm(list = ls()) # Clean memorydata(spc)Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSAspcsur.sim <- spsurml(formula = Tformula, data = spc, type = "sim")summary(spcsur.sim)# All the coefficients in a single table.print(spcsur.sim)# Plot of the coefficients of each equation in different graphsplot(spcsur.sim) ## A SUR-SLX model ## (listw argument can be either a matrix or a listw object )spcsur.slx <- spsurml(formula = Tformula, data = spc, type = "slx",   listw = Wspc, Durbin = TRUE)summary(spcsur.slx)# All the coefficients in a single table.print(spcsur.slx)# Plot of the coefficients in a single graphif (require(gridExtra)) {  pl <- plot(spcsur.slx, viewplot = FALSE)  grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                         nrow = 2)} ## VIP: The output of the whole set of the examples can be examined ## by executing demo(demo_spsurml, package="spsur")  ### A SUR-SLM modelspcsur.slm <- spsurml(formula = Tformula, data = spc, type = "slm",                      listw = Wspc)summary(spcsur.slm)if (require(gridExtra)) {  pl <- plot(spcsur.slm, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                pl$pldeltas, nrow = 3)}### A SUR-SDM modelspcsur.sdm <- spsurml(formula = Tformula, data = spc, type = "sdm",                      listw = Wspc)summary(spcsur.sdm)print(spcsur.sdm)if (require(gridExtra)) {  pl <- plot(spcsur.sdm, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                pl$pldeltas, nrow = 3)}## A SUR-SDM model with different spatial lags in each equationTformulaD <- ~ UN83 + NMR83 + SMSA | UN80spcsur.sdm2 <- spsurml(formula = Tformula, data = spc, type = "sdm",                       listw = Wspc, Durbin = TformulaD)summary(spcsur.sdm2)if (require(gridExtra)) {  pl <- plot(spcsur.sdm2, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                pl$pldeltas, nrow = 3)}###########################################################  CLASSIC PANEL DATA G=1; Tm>1   ################################################################ Example 2: Homicides + Socio-Economics (1960-90)## Homicides and selected socio-economic characteristics for continental## U.S. counties.## Data for four decennial census years: 1960, 1970, 1980 and 1990.## \url{https://geodacenter.github.io/data-and-lab/ncovr/}### It usually requires 1-2 minutes maximum...rm(list = ls()) # Clean memory### Read NCOVR.sf objectdata(NCOVR, package = "spsur")nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)### Some regions with no links...lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE)Tformula <- HR80  | HR90 ~ PS80 + UE80 | PS90 + UE90### A SUR-SIM modelNCOVRSUR.sim <- spsurml(formula = Tformula, data = NCOVR.sf, type = "sim")summary(NCOVRSUR.sim)if (require(gridExtra)) {  pl <- plot(NCOVRSUR.sim, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], nrow = 3)}### A SUR-SLX modelNCOVRSUR.slx <- spsurml(formula = Tformula, data = NCOVR.sf, type = "slx",                        listw = lwncovr, zero.policy = TRUE)print(NCOVRSUR.slx)if (require(gridExtra)) {  pl <- plot(NCOVRSUR.slx, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], nrow = 2)}### A SUR-SLM model### method = "Matrix" (Cholesky) instead of "eigen"### (fdHess = TRUE to compute numerical covariances )NCOVRSUR.slm <- spsurml(formula = Tformula, data = NCOVR.sf,                        type = "slm", listw = lwncovr, method = "Matrix",                        zero.policy = TRUE, control = list(fdHess = TRUE))summary(NCOVRSUR.slm)if (require(gridExtra)) {  pl <- plot(NCOVRSUR.slm, viewplot = FALSE)   grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],                pl$pldeltas, nrow = 3)}# LR test for nested modelsanova(NCOVRSUR.sim, NCOVRSUR.slm)

Estimation of SUR models for simple spatial panels (G=1).

Description

This function estimates SUR models for simple spatial panel datasets.spsurtime is restricted, specifically, to cases where there is only one equation,G=1, and a varying number of spatial units,N, and time periods,Tm. The SUR structure appearsin form of serial dependence among the error terms corresponding to the same spatial unit.Note that it is assumed that all spatial units share a common pattern of serial dependence.

The user can choose between different types of spatial specifications, as described below, and the estimation algorithms allow for the introduction of linear restrictions on the\beta parametersassociated to the regressors. The spatial panels with SUR structure can be estimated by maximum-likelihood methods or three-stages least squares procedures, using spatial instrumental variables.

Usage

spsurtime (formula, data, time, na.action,                  listw = NULL, type = "sim", Durbin = NULL,                   method = "eigen", fit_method = "ml", maxlagW = NULL,                  zero.policy = NULL, interval = NULL, trs = NULL,                  R = NULL, b = NULL, demean = FALSE, control = list() )

Arguments

Details

Functionspsurtime only admits a formula, created withFormula and a dataset of classdata.frame ormatrix. That is, the data cannot be uploaded using data matricesY andX provided for other functions in this package.
The argumenttime selects the variable, in thedata.frame, associated to the time dimension in the panel dataset. Thenspsurtime operates as in Anselin (1988), that is,each cross-section is treated as if it were an equation in a SUR model, which now hasTm 'equations' andN individuals.
The SUR structure appears because there is serial dependence in the errors of each individual in the panel. The serial dependence in the errors is not parameterized, but estimated non-parametrically in theSigma covariance matrix returned by the function. An important constraint to mention is that the serial dependence assumed to be the same for all individuals in the sample. Serial dependence among individuals is excluded from Anselin approach.

Value

Anspsur object with the output of the maximum-likelihood or three-stages least-squares estimation of the spatial panel SUR model.

Author(s)

References

• Anselin, L. (1988). Spatial econometrics: methods and models. Dordrecht,Kluwer Academic Publishers.

• Lopez, F.A., Mur, J., and Angulo, A. (2014). Spatial modelselection strategies in a SUR framework. The case of regionalproductivity in EU.Annals of Regional Science, 53(1), 197-220.<doi:10.1007/s00168-014-0624-2>

• Lopez, F.A., Martinez-Ortiz, P.J., and Cegarra-Navarro, J.G. (2017).Spatial spillovers in public expenditure on a municipal level inSpain.Annals of Regional Science, 58(1), 39-65.<doi:10.1007/s00168-016-0780-7>

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43. <doi:10.18637/jss.v104.i11>

• Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatialeffects in seemingly unrelated regressions.Spatial Economic Analysis, 5(4), 399-440. <doi:10.1080/17421772.2010.516443>

See Also

spsurml,spsur3sls,wald_betas,wald_deltas,lmtestspsur,lr_betas

Examples

############################################ PANEL DATA (G=1; Tm>1) ######################################### Example 1:rm(list = ls()) # Clean memorydata(spc)lwspc <- spdep::mat2listw(Wspc, style = "W")N <- nrow(spc)Tm <- 2index_time <- rep(1:Tm, each = N)index_indiv <- rep(1:N, Tm)WAGE <- c(spc$WAGE83, spc$WAGE81)UN <- c(spc$UN83, spc$UN80)NMR <- c(spc$NMR83, spc$NMR80)SMSA <- c(spc$SMSA, spc$SMSA)pspc <- data.frame(index_indiv, index_time, WAGE, UN,                    NMR, SMSA)form_pspc <- WAGE ~ UN + NMR + SMSAform2_pspc <- WAGE | NMR ~ UN  | UN + SMSA# SLM pspc_slm <- spsurtime(formula = form_pspc, data = pspc,                       listw = lwspc,                      time = pspc$index_time,                       type = "slm", fit_method = "ml")summary(pspc_slm) pspc_slm2 <- spsurtime(formula = form2_pspc, data = pspc,                       listw = lwspc,                      time = pspc$index_time,                       type = "slm", fit_method = "ml")summary(pspc_slm2)## VIP: The output of the whole set of the examples can be examined ## by executing demo(demo_spsurtime, package="spsur") ### Example 2:rm(list = ls()) # Clean memory### Read NCOVR.sf objectdata(NCOVR, package="spsur")nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)### Some regions with no links...lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE)N <- nrow(NCOVR.sf)Tm <- 4index_time <- rep(1:Tm, each = N)index_indiv <- rep(1:N, Tm)pHR <- c(NCOVR.sf$HR60, NCOVR.sf$HR70, NCOVR.sf$HR80, NCOVR.sf$HR90)pPS <- c(NCOVR.sf$PS60, NCOVR.sf$PS70, NCOVR.sf$PS80, NCOVR.sf$PS90)pUE <- c(NCOVR.sf$UE60, NCOVR.sf$UE70, NCOVR.sf$UE80, NCOVR.sf$UE90)pNCOVR <- data.frame(indiv = index_indiv, time = index_time,                     HR = pHR, PS = pPS, UE = pUE)form_pHR <- HR ~ PS + UE## SIMpHR_sim <- spsurtime(formula = form_pHR, data = pNCOVR,                     time = pNCOVR$time, type = "sim", fit_method = "ml")summary(pHR_sim)## SLM by 3SLS. pHR_slm <- spsurtime(formula = form_pHR, data = pNCOVR, listw = lwncovr,                     time = pNCOVR$time, type = "slm",                      fit_method = "3sls")summary(pHR_slm)############################# Wald tests about betas in spatio-temporal models### H0: equal betas for PS in equations 1, 3 and 4.R <- matrix(0, nrow = 2, ncol = 12) ## nrow = number of restrictions ## ncol = number of beta parametersR[1, 2] <- 1; R[1, 8] <- -1 # PS beta coefficient in equations 1 equal to 3R[2, 2] <- 1; R[2, 11] <- -1 # PS beta coefficient in equations 1 equal to 4b <- matrix(0, nrow=2, ncol=1)wald_betas(pHR_sim , R = R , b = b) # SIM modelwald_betas(pHR_slm , R = R , b = b) # SLM model############################# Wald tests about spatial-parameters in############################# spatio-temporal models### H0: equal rhos in slm model for equations 1 and 2.R2 <- matrix(0, nrow = 1, ncol = 4)R2[1, 1] <- 1; R2[1, 2] <- -1b2 <- matrix(0, nrow = 1, ncol = 1)wald_deltas(pHR_slm, R = R2, b = b2)

Summary of estimated objects of classspsur.

Description

This function summarizes estimatedspsur objects. The tables in the outputinclude basic information for each equation. The report also shows other complementary resultscorresponding to the SUR model like the(GxG) covariance matrix of the residuals of theequations of the SUR, the estimated log-likelihood, the Breusch-Pagan diagonality test or the MarginalLagrange Multiplier, LMM, tests of spatial dependence.

Usage

## S3 method for class 'spsur'summary(object, ...)

Arguments

Value

An object of classsummary.spsur

Author(s)

See Also

print.summary.spsur;spsurml;spsur3sls.

Examples

 # See examples for \code{\link{spsurml}} or # \code{\link{spsur3sls}} functions.

Wald tests on thebeta coefficients

Description

The functionwald_betas can be seen as a complement to the restricted estimation procedures included in the functionsspsurml andspsur3sls.wald_betas obtains Wald tests for sets of linear restrictions on the coefficients\beta of the SUR model.The restrictions may involve coefficients of the same equation or coefficients from different equations. The function has great flexibility in this respect. Note thatwald_betas is more general thanlr_betas in the sense that the last functiononly allows to test for restrictions of homogeneity of subsets of\beta coefficients among the different equations in the SUR model, and in a maximum-likelihood framework.

In order to work withwald_betas, the model on which the linear restrictions are to be tested needs to exists as anspsur object. Using the information contained in the object,wald_betas obtains the corresponding Wald estatistic for the null hypotheses specified by the user through theR row vector andb column vector, used also inspsurml andspsur3sls. The function shows the value of the Wald teststatistics and its associated p-values.

Usage

wald_betas (obj , R , b)

Arguments

Value

Object ofhtest class including the Waldstatistic, the corresponding p-value, the degrees offreedom and the values of the sample estimates.

Author(s)

References

• Lopez, F.A., Mur, J., and Angulo, A. (2014). Spatial modelselection strategies in a SUR framework. The case of regionalproductivity in EU.Annals of Regional Science, 53(1), 197-220.<doi:10.1007/s00168-014-0624-2>

• Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatialeffects in seemingly unrelated regressions.Spatial Economic Analysis, 5(4), 399-440. <doi:10.1080/17421772.2010.516443>

• Anselin, L. (2016) Estimation and Testing in the Spatial Seemingly Unrelated Regression (SUR).Geoda Center for Geospatial Analysis and Computation, Arizona State University. Working Paper 2016-01.<doi:10.13140/RG.2.2.15925.40163>

• Minguez, R., Lopez, F.A. and Mur, J. (2022).spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models.Journal of Statistical Software, 104(11), 1–43. <doi:10.18637/jss.v104.i11>

See Also

spsurml,spsur3sls,lr_betas

Examples

## VIP: The output of the whole set of the examples can be examined ## by executing demo(demo_wald_betas, package="spsur")######################################################### CROSS SECTION DATA (G=1; Tm>1) ############################################################## Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)rm(list = ls()) # Clean memorydata(spc)lwspc <- spdep::mat2listw(Wspc, style = "W")Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA### Estimate SUR-SLM modelspcsur.slm <- spsurml(formula = Tformula, data = spc,                       type = "slm", listw = lwspc)summary(spcsur.slm)### H_0: equality between SMSA coefficients in both equations.R1 <- matrix(c(0,0,0,1,0,0,0,-1), nrow=1)b1 <- matrix(0, ncol=1)wald_betas(spcsur.slm, R = R1, b = b1)

Wald tests for spatial parameters coefficients.

Description

Functionwald_deltas obtains Wald tests for linear restrictions on the spatial coefficients of a SUR model that has been estimated previously through the functionspsurml. The restrictions can affect to coefficients of the same equation(i.e.,\lambda_{g}=\rho_{g} forall g) or can involve coefficients from different equations (i.e.,\lambda_{g}=\lambda_{h}). The function has great flexibility in this respect. Note thatwald_deltas only works in a maximum-likelihood framework.

In order to work withwald_betas, the model on which the linear restrictions are to be tested needs to exists as anspsur object. Using the information contained in the object,wald_deltas obtains the corresponding Wald statistic for the null hypotheses specified by the user through theR row vector andb column vector discussed, used also inspsurml. The function shows the resulting Wald test statistics and their corresponding p-values.

Usage

wald_deltas (obj , R , b)

Arguments

Value

Object ofhtest including the Waldstatistic, the corresponding p-value, the degrees offreedom and the values of the sample estimates.

Author(s)

See Also

spsurml,spsur3sls

Examples

######################################################### CROSS SECTION DATA (G>1; Tm=1) #########################################################rm(list = ls()) # Clean memorydata(spc, package = "spsur")lwspc <- spdep::mat2listw(Wspc, style = "W")Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA################################### Estimate SUR-SLM modelspcsur.slm <-spsurml(formula = Tformula, data = spc,                        type = "slm", listw = lwspc)summary(spcsur.slm)## H_0: equality of the lambda parameters of both equations.R1 <- matrix(c(1,-1), nrow=1)b1 <- matrix(0, ncol=1)wald_deltas(spcsur.slm, R = R1, b = b1)