R/package-pspatreg.R
In rominsal/pspatreg: Spatial and Spatio-Temporal Semiparametric Regression Models with Spatial Lags
#' @title pspatreg: A package to estimate and make inference for spatial
#' and spatio-temporal econometric regression models
#'
#' @docType package
#' @name pspatreg
#' @rdname pspatreg
#'
#' @description \pkg{pspatreg} offers the user a collection of functions to
#' estimate and make inference of geoadditive spatial or spatio-temporal
#' semiparametric regression models of type \emph{ps-sim}, \emph{ps-sar},
#' \emph{ps-sem}, \emph{ps-sarar}, \emph{ps-sdm}, \emph{ps-sdem} or
#' \emph{ps-slx}. These type of specifications are very general and they can
#' include parametric and non-parametric covariates, spatial or
#' spatio-temporal non-parametric trends and spatial lags of the dependent and
#' independent variables and/or the noise of the model. The non-parametric
#' terms (either trends or covariates) are modeled using P-Splines. The
#' non-parametric trend can be decomposed in an ANOVA way including main and
#' interactions effects of 2nd and 3rd order. The estimation method can be
#' restricted maximum likelihood (REML) or maximum likelihood (ML).
#'
#' @details
#' Some functionalities that have been included in \pkg{pspatreg}
#' package are:
#'
#' @section 1. Estimation of the semiparametric regression model:
#' \pkg{pspatreg}
#' allows the estimation of geoadditive spatial or spatio-temporal
#' semiparametric regression models which could include: \itemize{ \item An
#' spatial or spatio-temporal trend, that is, a geoadditive model either for
#' cross-section data or for panel data. This trend can be decomposed in main
#' and interaction functions in an ANOVA way. The spatial (or spatio-temporal)
#' trend gather the potential spatial heterogeneity of the data. \item
#' Parametric covariates as usual in regression models. \item Non-parametric
#' covariates in which the functional relationship is estimated from the data.
#' Both the trends and non-parametric covariates are modelled using P-splines.
#' \item Spatial dependence adding spatial lags of the dependent and
#' independent variables as usual in spatial econometric models. These models
#' gather the potential spatial spillovers. } Once specified, the whole model
#' can be estimated using either restricted maximum-likelihood (REML) or
#' maximum likelihood (ML). The spatial econometric specifications allowed in
#' \pkg{pspatreg} are the following ones: \itemize{ \item \emph{ps-sim}:
#' geoadditive semiparametric model without spatial effects (in addition to
#' the spatial or spatio-temporal trend, if it is included). \deqn{ y =
#' f(s_1,s_2,\tau_{t}) y + X \beta + + \sum_{i=1}^k g(z_i) + \epsilon } where:
#' \itemize{ \item \eqn{f(s_1,s_2,\tau_t)} is a smooth spatio-temporal trend
#' of the spatial coordinates \eqn{s1,s_2} and of the temporal coordinates
#' \eqn{\tau_t}. \item \eqn{X} is a matrix including values of parametric
#' covariates. \item \eqn{g(z_i)} are non-parametric smooth functions of the
#' covariates \eqn{z_i}. \item \eqn{\epsilon ~ N(0,R)} where \eqn{ R =
#' \sigma^2 I_T} if errors are uncorrelated or it follows an AR(1) temporal
#' autoregressive structure for serially correlated errors. } \item
#' \emph{ps-slx}: geoadditive semiparametric model with spatial lags of the
#' regresors (either parametric or non-parametric): \deqn{ y =
#' f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i
#' =1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) +
#' \epsilon } where: \itemize{ \item \eqn{W_N} is the spatial weights matrix.
#' \item \eqn{I_T} is an identity matrix of order \eqn{T} (\emph{T = 1} for
#' pure spatial data). } \item \emph{ps-sar}: geoadditive semiparametric model
#' with spatial lag of the dependent variable \deqn{ y = (\rho*W_{N} \otimes
#' I_T) y + f(s_1,s_2,\tau_{t}) + X \beta + \sum_{i =1}^k g(z_i) + \epsilon }
#' \item \emph{ps-sem}: geoadditive semiparametric model with a spatial lag of
#' the noise of the model \deqn{ y = f(s_1,s_2,\tau_{t}) + X \beta + \sum_{i
#' =1}^k g(z_i) + u } \deqn{ u = (\delta*W_{N} \otimes I_T) u + \epsilon }
#' \item \emph{ps-sdm}: geoadditive semiparametric model with spatial lags of
#' the endogenous variable and of the regressors (spatial durbin model) \deqn{
#' y = (\rho*W_{N} \otimes I_T) y + f(s_1,s_2,\tau_{t}) + X \beta + (W_{N}
#' \otimes I_T) X \theta + \sum_{i = 1}^k g(z_i) + \sum_{i = 1}^k
#' g((\gamma_i*W_{N} \otimes I_T) z_i) + \epsilon } \item \emph{ps-sdem}:
#' geoadditive semiparametric model with spatial errors and spatial lags of
#' the endogenous variable and of the regressors \deqn{ y =
#' f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i =
#' 1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) + u }
#' \deqn{ u = (\delta*W_{N} \otimes I_T) u + \epsilon } \item \emph{ps-sarar}:
#' geoadditive semiparametric model with a spatial lag for: both dependent
#' variable and errors \deqn{ y = (\rho*W_{N} \otimes I_T) y +
#' f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i =
#' 1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) + u }
#' \deqn{ u = (\delta*W_{N} \otimes I_T) u + \epsilon } }
#'
#' @section 2. Plot of the spatial and spatio-temporal trends:
#' Once estimated
#' the geoadditive semiparametric model, some functions of \pkg{pspatreg} are
#' suited to make plots of the spatial or spatio-temporal trends. These
#' functions, named \code{\link{plot_sp2d}} and \code{\link{plot_sp3d}}, can deal
#' either with `sf` objects or `dataframe` objects including spatial coordinates
#' (see the examples of the functions).
#' The function \code{\link{plot_sptime}} allows
#' to examine temporal trends for each spatial unit. Eventually, it is also
#' possible to get the plots on nonparametric covariates using
#' \code{\link{plot_terms}}.
#'
#' @section 3. Impacts and spatial spillovers:
#' It is very common in spatial econometrics to
#' evaluate the multiplier impacts that a change in the value of a regressor,
#' in a point in the space, has on the explained variable. The \pkg{pspatreg}
#' package allows the computation and inference of spatial impacts (direct,
#' indirect and total) either for parametric covariates or nonparametric
#' covariates (in the last case, the output are impact functions). The function
#' named \code{\link{impactspar}} compute the impacts for parametric
#' covariates in the usual way using simulation. On the other hand, the
#' function \code{\link{impactsnopar}} allows the computation of impact
#' functions for nonparametric covariates. For parametric covariates,
#' the method to compute the impacts is the same than the
#' exposed in LeSage and Page (2009). For nonparametric covariates the
#' method is described in the help of the function \code{\link{impactsnopar}}.
#' Both impact functions have dedicated methods to
#' get a summary, for the parametric covariates, and
#' plots, for the nonparametric covariates, of the
#' direct, indirect and total impacts.
#'
#' @section 4. Additional methods: The package \pkg{pspatreg} provides the usual
#' methods to extract information of the fitted models. The methods included
#' are:
#' \itemize{
#' \item \code{\link{anova}}: provides tables of fitted
#' `pspatreg` models including information criteria (AIC and BIC),
#' log-likelihood and degrees of freedom of each fitted model.
#' Also allows to perform LR tests between nested models.
#' \item \code{\link{print}} method 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.
#' \item \code{\link{summary}} method displays the results of
#' the estimation for spatial and spatio-temporal trends,
#' parametric and nonparametric covariates and spatial parameters.
#' \item \code{\link{coef}} extractor function of the parametric and
#' spatial coefficientes.
#' \item \code{\link{fitted}} extractor function of the fitted values.
#' \item \code{\link{logLik}} extractor function of the log-likelihood.
#' \item \code{\link{residuals}} extractor function of the residuals.
#' \item \code{\link{vcov}} extractor function of the covariance matrix
#' of the estimated parameters. The argument \code{bayesian}
#' (default = `TRUE`) allows to choose between sandwich
#' (frequentist) or bayesian method to compute the variances and
#' covariances. See Fahrmeir et al. (2021) for details.
#' }
#'
#' @section Datasets:
#' \pkg{pspatreg} includes a spatio-temporal panel database
#' including observations of unemployment, economic variables
#' and spatial coordinates (centroids) for 103 Italian provinces
#' in the period 1996-2019.
#' This database is provided in RData format and can be loaded
#' using the command \code{data(unemp_it, package = "pspatreg")}.
#' The database also includes a \emph{W} spatial neighborhood matrix
#' of the Italian provinces (computed using queen criterium).
#' Furthermore, a map of Italian provinces is also included as an sf object.
#' This map can be used to plot spatial and spatio-temporal trends estimated
#' for each province. Some examples of spatial and spatio-temporal
#' fitted trends are included in the help of the main function of
#' \pkg{pspatreg} package (see especially \code{?pspatfit}).
#' See Minguez, Basile and Durban (2020) for additional details about
#' this database.
#' \cr
#' source: Italian National Institute of Statistics (ISTAT)
#' \emph{https://www.istat.it}
#' \cr \cr
#'
#' For the spatial pure case, the examples included use the
#' household database \code{ames} included in \pkg{AmesHousing} package.
#' See the help of \code{?AmesHousing::make_ames} for an explanation of the
#' variables included in this database.
#' Examples of hedonic models including geoadditive spatial econometric
#' regressions are included in the examples of \pkg{pspatreg} package.
#'
#' @references
#' \itemize{
#' \item Basile, R.; Durban, M.; Minguez, R.; Montero, J. M.; and
#' Mur, J. (2014). Modeling regional economic dynamics: Spatial
#' dependence, spatial heterogeneity and nonlinearities.
#' \emph{Journal of Economic Dynamics and Control}, (48), 229-245.
#' <doi:10.1016/j.jedc.2014.06.011>
#'
#' \item Eilers, P. and Marx, B. (1996). Flexible Smoothing with
#' B-Splines and Penalties. \emph{Statistical Science}, (11), 89-121.
#'
#' \item Eilers, P. and Marx, B. (2021). \emph{Practical Smoothing.
#' The Joys of P-Splines}. Cambridge University Press.
#'
#' \item Fahrmeir, L.; Kneib, T.; Lang, S.; and Marx, B. (2021).
#' \emph{Regression. Models, Methods and Applications (2nd Ed.)}.
#' Springer.
#'
#' \item Lee, D. and Durban, M. (2011). P-Spline ANOVA Type Interaction
#' Models for Spatio-Temporal Smoothing. \emph{Statistical Modelling},
#' (11), 49-69. <doi:10.1177/1471082X1001100104>
#'
#' \item Lee, D. J., Durban, M., and Eilers, P. (2013). Efficient
#' two-dimensional smoothing with P-spline ANOVA mixed models
#' and nested bases. \emph{Computational Statistics & Data Analysis},
#' (61), 22-37. <doi:10.1016/j.csda.2012.11.013>
#'
#' \item LeSage, J. and Pace, K. (2009). \emph{Introduction to
#' Spatial Econometrics}. CRC Press, Boca Raton.
#'
#' \item Minguez, R.; Basile, R. and Durban, M. (2020). An Alternative
#' Semiparametric Model for Spatial Panel Data. \emph{Statistical Methods and Applications},
#' (29), 669-708. <doi: 10.1007/s10260-019-00492-8>
#'
#' \item Montero, J., Minguez, R., and Durban, M. (2012). SAR models
#' with nonparametric spatial trends: A P-Spline approach.
#' \emph{Estadistica Espanola}, (54:177), 89-111.
#'
#' \item Rodriguez-Alvarez, M. X.; Kneib, T.; Durban, M.; Lee, D.J.
#' and Eilers, P. (2015). Fast smoothing parameter separation
#' in multidimensional generalized P-splines: the SAP algorithm.
#' \emph{Statistics and Computing} 25 (5), 941-957.
#' <doi:10.1007/s11222-014-9464-2>
#'
#' \item Wood, S.N. (2017). \emph{Generalized Additive Models.
#' An Introduction with \code{R}} (second edition). CRC Press, Boca Raton.
#' }
#'
#' @author
#' \tabular{ll}{
#' Roman Minguez \tab \email{roman.minguez@@uclm.es} \cr
#' Roberto Basile \tab \email{roberto.basile@@univaq.it} \cr Maria Durban \tab
#' \email{mdurban@@est-econ.uc3m.es} \cr Gonzalo Espana-Heredia \tab
#' \email{gehllanza@@gmail.com} \cr
#' }
#'
#' @importFrom AmesHousing make_ames
#' @importFrom dplyr left_join
#' @importFrom fields image.plot
#' @importFrom ggplot2 ggplot geom_line ggtitle labs aes xlim ylim
#' @importFrom ggplot2 geom_histogram
#' @importFrom graphics image contour matplot title points
#' @importFrom graphics par abline lines
#' @importFrom grDevices heat.colors hcl.colors
#' @importFrom MBA mba.surf
#' @importFrom MASS ginv mvrnorm
#' @importFrom Matrix bandSparse bdiag crossprod determinant
#' @importFrom Matrix diag Diagonal kronecker Matrix
#' @importFrom Matrix rowSums solve t tcrossprod
#' @importFrom methods as
#' @importFrom minqa bobyqa
#' @importFrom numDeriv hessian
#' @importFrom plm plm pdata.frame Between Within index
#' @importFrom Rdpack reprompt
#' @importFrom sf st_as_sf st_drop_geometry st_coordinates
#' @importFrom sf st_geometry_type
#' @importFrom spatialreg get.ZeroPolicyOption create_WX
#' @importFrom spatialreg can.be.simmed jacobianSetup do_ldet
#' @importFrom spatialreg intImpacts lmSLX invIrW trW
#' @importFrom spdep listw2mat mat2listw nb2listw
#' @importFrom spdep tri2nb graph2nb soi.graph is.symmetric.nb
#' @importFrom splines spline.des
#' @importFrom stats var sd loess predict vcov
#' @importFrom stats model.response as.formula .getXlevels
#' @importFrom stats pchisq pnorm pt rnorm qnorm
#' @importFrom stats coefficients fitted residuals printCoefmat
#' @importFrom stats model.frame model.matrix terms lag
#' @importFrom stats anova coef formula logLik AIC BIC
#' @importFrom stats na.action napredict update
#' @importFrom stringr str_detect str_replace str_split str_extract str_length
NULL
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#' @title pspatreg: A package to estimate and make inference for spatial
#' and spatio-temporal econometric regression models
#'
#' @docType package
#' @name pspatreg
#' @rdname pspatreg
#'
#' @description \pkg{pspatreg} offers the user a collection of functions to
#' estimate and make inference of geoadditive spatial or spatio-temporal
#' semiparametric regression models of type \emph{ps-sim}, \emph{ps-sar},
#' \emph{ps-sem}, \emph{ps-sarar}, \emph{ps-sdm}, \emph{ps-sdem} or
#' \emph{ps-slx}. These type of specifications are very general and they can
#' include parametric and non-parametric covariates, spatial or
#' spatio-temporal non-parametric trends and spatial lags of the dependent and
#' independent variables and/or the noise of the model. The non-parametric
#' terms (either trends or covariates) are modeled using P-Splines. The
#' non-parametric trend can be decomposed in an ANOVA way including main and
#' interactions effects of 2nd and 3rd order. The estimation method can be
#' restricted maximum likelihood (REML) or maximum likelihood (ML).
#'
#' @details
#' Some functionalities that have been included in \pkg{pspatreg}
#' package are:
#'
#' @section 1. Estimation of the semiparametric regression model:
#' \pkg{pspatreg}
#' allows the estimation of geoadditive spatial or spatio-temporal
#' semiparametric regression models which could include: \itemize{ \item An
#' spatial or spatio-temporal trend, that is, a geoadditive model either for
#' cross-section data or for panel data. This trend can be decomposed in main
#' and interaction functions in an ANOVA way. The spatial (or spatio-temporal)
#' trend gather the potential spatial heterogeneity of the data. \item
#' Parametric covariates as usual in regression models. \item Non-parametric
#' covariates in which the functional relationship is estimated from the data.
#' Both the trends and non-parametric covariates are modelled using P-splines.
#' \item Spatial dependence adding spatial lags of the dependent and
#' independent variables as usual in spatial econometric models. These models
#' gather the potential spatial spillovers. } Once specified, the whole model
#' can be estimated using either restricted maximum-likelihood (REML) or
#' maximum likelihood (ML). The spatial econometric specifications allowed in
#' \pkg{pspatreg} are the following ones: \itemize{ \item \emph{ps-sim}:
#' geoadditive semiparametric model without spatial effects (in addition to
#' the spatial or spatio-temporal trend, if it is included). \deqn{ y =
#' f(s_1,s_2,\tau_{t}) y + X \beta + + \sum_{i=1}^k g(z_i) + \epsilon } where:
#' \itemize{ \item \eqn{f(s_1,s_2,\tau_t)} is a smooth spatio-temporal trend
#' of the spatial coordinates \eqn{s1,s_2} and of the temporal coordinates
#' \eqn{\tau_t}. \item \eqn{X} is a matrix including values of parametric
#' covariates. \item \eqn{g(z_i)} are non-parametric smooth functions of the
#' covariates \eqn{z_i}. \item \eqn{\epsilon ~ N(0,R)} where \eqn{ R =
#' \sigma^2 I_T} if errors are uncorrelated or it follows an AR(1) temporal
#' autoregressive structure for serially correlated errors. } \item
#' \emph{ps-slx}: geoadditive semiparametric model with spatial lags of the
#' regresors (either parametric or non-parametric): \deqn{ y =
#' f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i
#' =1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) +
#' \epsilon } where: \itemize{ \item \eqn{W_N} is the spatial weights matrix.
#' \item \eqn{I_T} is an identity matrix of order \eqn{T} (\emph{T = 1} for
#' pure spatial data). } \item \emph{ps-sar}: geoadditive semiparametric model
#' with spatial lag of the dependent variable \deqn{ y = (\rho*W_{N} \otimes
#' I_T) y + f(s_1,s_2,\tau_{t}) + X \beta + \sum_{i =1}^k g(z_i) + \epsilon }
#' \item \emph{ps-sem}: geoadditive semiparametric model with a spatial lag of
#' the noise of the model \deqn{ y = f(s_1,s_2,\tau_{t}) + X \beta + \sum_{i
#' =1}^k g(z_i) + u } \deqn{ u = (\delta*W_{N} \otimes I_T) u + \epsilon }
#' \item \emph{ps-sdm}: geoadditive semiparametric model with spatial lags of
#' the endogenous variable and of the regressors (spatial durbin model) \deqn{
#' y = (\rho*W_{N} \otimes I_T) y + f(s_1,s_2,\tau_{t}) + X \beta + (W_{N}
#' \otimes I_T) X \theta + \sum_{i = 1}^k g(z_i) + \sum_{i = 1}^k
#' g((\gamma_i*W_{N} \otimes I_T) z_i) + \epsilon } \item \emph{ps-sdem}:
#' geoadditive semiparametric model with spatial errors and spatial lags of
#' the endogenous variable and of the regressors \deqn{ y =
#' f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i =
#' 1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) + u }
#' \deqn{ u = (\delta*W_{N} \otimes I_T) u + \epsilon } \item \emph{ps-sarar}:
#' geoadditive semiparametric model with a spatial lag for: both dependent
#' variable and errors \deqn{ y = (\rho*W_{N} \otimes I_T) y +
#' f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i =
#' 1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) + u }
#' \deqn{ u = (\delta*W_{N} \otimes I_T) u + \epsilon } }
#'
#' @section 2. Plot of the spatial and spatio-temporal trends:
#' Once estimated
#' the geoadditive semiparametric model, some functions of \pkg{pspatreg} are
#' suited to make plots of the spatial or spatio-temporal trends. These
#' functions, named \code{\link{plot_sp2d}} and \code{\link{plot_sp3d}}, can deal
#' either with `sf` objects or `dataframe` objects including spatial coordinates
#' (see the examples of the functions).
#' The function \code{\link{plot_sptime}} allows
#' to examine temporal trends for each spatial unit. Eventually, it is also
#' possible to get the plots on nonparametric covariates using
#' \code{\link{plot_terms}}.
#'
#' @section 3. Impacts and spatial spillovers:
#' It is very common in spatial econometrics to
#' evaluate the multiplier impacts that a change in the value of a regressor,
#' in a point in the space, has on the explained variable. The \pkg{pspatreg}
#' package allows the computation and inference of spatial impacts (direct,
#' indirect and total) either for parametric covariates or nonparametric
#' covariates (in the last case, the output are impact functions). The function
#' named \code{\link{impactspar}} compute the impacts for parametric
#' covariates in the usual way using simulation. On the other hand, the
#' function \code{\link{impactsnopar}} allows the computation of impact
#' functions for nonparametric covariates. For parametric covariates,
#' the method to compute the impacts is the same than the
#' exposed in LeSage and Page (2009). For nonparametric covariates the
#' method is described in the help of the function \code{\link{impactsnopar}}.
#' Both impact functions have dedicated methods to
#' get a summary, for the parametric covariates, and
#' plots, for the nonparametric covariates, of the
#' direct, indirect and total impacts.
#'
#' @section 4. Additional methods: The package \pkg{pspatreg} provides the usual
#' methods to extract information of the fitted models. The methods included
#' are:
#' \itemize{
#' \item \code{\link{anova}}: provides tables of fitted
#' `pspatreg` models including information criteria (AIC and BIC),
#' log-likelihood and degrees of freedom of each fitted model.
#' Also allows to perform LR tests between nested models.
#' \item \code{\link{print}} method 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.
#' \item \code{\link{summary}} method displays the results of
#' the estimation for spatial and spatio-temporal trends,
#' parametric and nonparametric covariates and spatial parameters.
#' \item \code{\link{coef}} extractor function of the parametric and
#' spatial coefficientes.
#' \item \code{\link{fitted}} extractor function of the fitted values.
#' \item \code{\link{logLik}} extractor function of the log-likelihood.
#' \item \code{\link{residuals}} extractor function of the residuals.
#' \item \code{\link{vcov}} extractor function of the covariance matrix
#' of the estimated parameters. The argument \code{bayesian}
#' (default = `TRUE`) allows to choose between sandwich
#' (frequentist) or bayesian method to compute the variances and
#' covariances. See Fahrmeir et al. (2021) for details.
#' }
#'
#' @section Datasets:
#' \pkg{pspatreg} includes a spatio-temporal panel database
#' including observations of unemployment, economic variables
#' and spatial coordinates (centroids) for 103 Italian provinces
#' in the period 1996-2019.
#' This database is provided in RData format and can be loaded
#' using the command \code{data(unemp_it, package = "pspatreg")}.
#' The database also includes a \emph{W} spatial neighborhood matrix
#' of the Italian provinces (computed using queen criterium).
#' Furthermore, a map of Italian provinces is also included as an sf object.
#' This map can be used to plot spatial and spatio-temporal trends estimated
#' for each province. Some examples of spatial and spatio-temporal
#' fitted trends are included in the help of the main function of
#' \pkg{pspatreg} package (see especially \code{?pspatfit}).
#' See Minguez, Basile and Durban (2020) for additional details about
#' this database.
#' \cr
#' source: Italian National Institute of Statistics (ISTAT)
#' \emph{https://www.istat.it}
#' \cr \cr
#'
#' For the spatial pure case, the examples included use the
#' household database \code{ames} included in \pkg{AmesHousing} package.
#' See the help of \code{?AmesHousing::make_ames} for an explanation of the
#' variables included in this database.
#' Examples of hedonic models including geoadditive spatial econometric
#' regressions are included in the examples of \pkg{pspatreg} package.
#'
#' @references
#' \itemize{
#' \item Basile, R.; Durban, M.; Minguez, R.; Montero, J. M.; and
#' Mur, J. (2014). Modeling regional economic dynamics: Spatial
#' dependence, spatial heterogeneity and nonlinearities.
#' \emph{Journal of Economic Dynamics and Control}, (48), 229-245.
#' <doi:10.1016/j.jedc.2014.06.011>
#'
#' \item Eilers, P. and Marx, B. (1996). Flexible Smoothing with
#' B-Splines and Penalties. \emph{Statistical Science}, (11), 89-121.
#'
#' \item Eilers, P. and Marx, B. (2021). \emph{Practical Smoothing.
#' The Joys of P-Splines}. Cambridge University Press.
#'
#' \item Fahrmeir, L.; Kneib, T.; Lang, S.; and Marx, B. (2021).
#' \emph{Regression. Models, Methods and Applications (2nd Ed.)}.
#' Springer.
#'
#' \item Lee, D. and Durban, M. (2011). P-Spline ANOVA Type Interaction
#' Models for Spatio-Temporal Smoothing. \emph{Statistical Modelling},
#' (11), 49-69. <doi:10.1177/1471082X1001100104>
#'
#' \item Lee, D. J., Durban, M., and Eilers, P. (2013). Efficient
#' two-dimensional smoothing with P-spline ANOVA mixed models
#' and nested bases. \emph{Computational Statistics & Data Analysis},
#' (61), 22-37. <doi:10.1016/j.csda.2012.11.013>
#'
#' \item LeSage, J. and Pace, K. (2009). \emph{Introduction to
#' Spatial Econometrics}. CRC Press, Boca Raton.
#'
#' \item Minguez, R.; Basile, R. and Durban, M. (2020). An Alternative
#' Semiparametric Model for Spatial Panel Data. \emph{Statistical Methods and Applications},
#' (29), 669-708. <doi: 10.1007/s10260-019-00492-8>
#'
#' \item Montero, J., Minguez, R., and Durban, M. (2012). SAR models
#' with nonparametric spatial trends: A P-Spline approach.
#' \emph{Estadistica Espanola}, (54:177), 89-111.
#'
#' \item Rodriguez-Alvarez, M. X.; Kneib, T.; Durban, M.; Lee, D.J.
#' and Eilers, P. (2015). Fast smoothing parameter separation
#' in multidimensional generalized P-splines: the SAP algorithm.
#' \emph{Statistics and Computing} 25 (5), 941-957.
#' <doi:10.1007/s11222-014-9464-2>
#'
#' \item Wood, S.N. (2017). \emph{Generalized Additive Models.
#' An Introduction with \code{R}} (second edition). CRC Press, Boca Raton.
#' }
#'
#' @author
#' \tabular{ll}{
#' Roman Minguez \tab \email{roman.minguez@@uclm.es} \cr
#' Roberto Basile \tab \email{roberto.basile@@univaq.it} \cr Maria Durban \tab
#' \email{mdurban@@est-econ.uc3m.es} \cr Gonzalo Espana-Heredia \tab
#' \email{gehllanza@@gmail.com} \cr
#' }
#'
#' @importFrom AmesHousing make_ames
#' @importFrom dplyr left_join
#' @importFrom fields image.plot
#' @importFrom ggplot2 ggplot geom_line ggtitle labs aes xlim ylim
#' @importFrom ggplot2 geom_histogram
#' @importFrom graphics image contour matplot title points
#' @importFrom graphics par abline lines
#' @importFrom grDevices heat.colors hcl.colors
#' @importFrom MBA mba.surf
#' @importFrom MASS ginv mvrnorm
#' @importFrom Matrix bandSparse bdiag crossprod determinant
#' @importFrom Matrix diag Diagonal kronecker Matrix
#' @importFrom Matrix rowSums solve t tcrossprod
#' @importFrom methods as
#' @importFrom minqa bobyqa
#' @importFrom numDeriv hessian
#' @importFrom plm plm pdata.frame Between Within index
#' @importFrom Rdpack reprompt
#' @importFrom sf st_as_sf st_drop_geometry st_coordinates
#' @importFrom sf st_geometry_type
#' @importFrom spatialreg get.ZeroPolicyOption create_WX
#' @importFrom spatialreg can.be.simmed jacobianSetup do_ldet
#' @importFrom spatialreg intImpacts lmSLX invIrW trW
#' @importFrom spdep listw2mat mat2listw nb2listw
#' @importFrom spdep tri2nb graph2nb soi.graph is.symmetric.nb
#' @importFrom splines spline.des
#' @importFrom stats var sd loess predict vcov
#' @importFrom stats model.response as.formula .getXlevels
#' @importFrom stats pchisq pnorm pt rnorm qnorm
#' @importFrom stats coefficients fitted residuals printCoefmat
#' @importFrom stats model.frame model.matrix terms lag
#' @importFrom stats anova coef formula logLik AIC BIC
#' @importFrom stats na.action napredict update
#' @importFrom stringr str_detect str_replace str_split str_extract str_length
NULL
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