hspm (version 1.1)

spregimes: Estimation of spatial regimes models

Description

The function spregimes deals with the estimation of spatial regimes models. This is a general function that allows the estimation of various spatial specifications, including the spatial lag regimes model, the spatial error regimes model, and the spatial SARAR regimes model. Since the estimation is based on generalized method of moments (GMM), endogenous variables can be included. For further information on estimation, see details.

Usage

spregimes(
  formula,
  data = list(),
  model = c("sarar", "lag", "error", "ols"),
  listw,
  wy_rg = FALSE,
  weps_rg = FALSE,
  initial.value = NULL,
  rgv = NULL,
  het = FALSE,
  verbose = FALSE,
  control = list()
)
# S3 method for spregimes
coef(object, ...)
# S3 method for spregimes
vcov(object, ...)
# S3 method for spregimes
print(x, digits = max(3, getOption("digits") - 3), ...)
# S3 method for spregimes
summary(object, ...)
# S3 method for summary.spregimes
print(x, digits = max(5, getOption("digits") - 3), ...)
# S3 method for spregimes
residuals(object, ...)
# S3 method for spregimes
fitted(object, ...)

Value

An object of class ``spregimes''

Arguments

formula

a symbolic description of the model of the form y ~ x_f | x_v | wx | h_f | h_v | wh where y is the dependent variable, x_f are the regressors that do not vary by regimes, x_v are the regressors that vary by regimes, wx are the spatially lagged regressors, h_f are the instruments that do not vary by regimes, h_v are the instruments that vary by regimes, wh are the spatially lagged instruments.

data

the data of class data.frame.

model

should be one of c("sarar", "lag", "error", "ols")

listw

a spatial weighting matrix of class listw, matrix or Matrix

wy_rg

default wy_rg = FALSE, the lagged dependent variable does not vary by regime (see details)

weps_rg

default weps_rg = FALSE, if TRUE the spatial error term varies by regimes (see details)

initial.value

initial value for the spatial error parameter

rgv

an object of class formula to identify the regime variables

het

heteroskedastic variance-covariance matrix

verbose

print a trace of the optimization

control

select arguments for the optimization

object

an object of class spregimes

...

additional arguments

x

an object of class spregimes

digits

number of digits

Author

Gianfranco Piras and Mauricio Sarrias

Details

The function spregimes is a wrapper that allows the estimation of a general spatial regimes model. For convenience and without loss of generality, we assume the presence of only two regimes. In this case the general model can be written as: $$ \begin{aligned} y = & W\begin{bmatrix} y_1& 0 \\ 0 & y_2 \\ \end{bmatrix} \begin{bmatrix} \lambda_1 \\ \lambda_2 \\ \end{bmatrix} + \begin{bmatrix} X_1& 0 \\ 0 & X_2 \\ \end{bmatrix} \begin{bmatrix} \beta_1 \\ \beta_2 \\ \end{bmatrix} + X\beta + \begin{bmatrix} Y_1& 0 \\ 0 & Y_2 \\ \end{bmatrix} \begin{bmatrix} \pi_1 \\ \pi_2 \\ \end{bmatrix} + Y\pi + \\ & W\begin{bmatrix} X_1& 0 \\ 0 & X_2 \\ \end{bmatrix} \begin{bmatrix} \delta_1 \\ \delta_2 \\ \end{bmatrix}+ WX\delta+ W \begin{bmatrix} Y_1& 0 \\ 0 & Y_2 \\ \end{bmatrix} \begin{bmatrix} \theta_1 \\ \theta_2 \\ \end{bmatrix} + WY\theta + \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \end{bmatrix} \end{aligned} $$ where $$ \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \end{bmatrix} =W \begin{bmatrix} \varepsilon_1&0 \\ 0&\varepsilon_2 \\ \end{bmatrix} \begin{bmatrix} \rho_1 \\ \rho_2 \\ \end{bmatrix} +u \nonumber $$ The model includes the spatial lag of the dependent variable, the spatial lag of the regressors, the spatial lag of the errors and, possibly, additional endogenous variables. The function spregimes estimates all of the nested specifications deriving from this model. There are, however, some restrictions. For example, if weps_rg is set to TRUE, all the regressors in the model should also vary by regimes. The estimation of the different models relies heavily on code available from the package sphet.

1. For the spatial lag (or Durbin) regimes model (i.e, when \(\rho_1\) and \(\rho_2\) are zero), an instrumental variable procedure is adopted, where the matrix of instruments is formed by the spatial lags of the exogenous variables and the additional instruments included in the formula. A robust estimation of the variance-covariance matrix can be obtained by setting het = TRUE.

2. For the spatial error regime models (i.e, when \(\lambda_1\) and \(\lambda_2\) are zero), the spatial coefficient(s) are estimated with the GMM procedure described in Kelejian and Prucha (2010) and Drukker et al., (2013). The difference between Kelejian and Prucha (2010) and Drukker et al., (2013), is that the former assume heteroskedastic innovations (het = TRUE), while the latter does not (het = FALSE).

3. For the SARAR regimes model, the estimation procedure alternates a series of IV and GMM steps. The variance-covariance can be estimated assuming that the innovations are homoskedastic (het = FALSE) as well as heteroskedastic (het = TRUE).

References

Arraiz, I. and Drukker, M.D. and Kelejian, H.H. and Prucha, I.R. (2010) A spatial Cliff-Ord-type Model with Heteroskedastic Innovations: Small and Large Sample Results, Journal of Regional Sciences, 50, pages 592--614.

Drukker, D.M. and Egger, P. and Prucha, I.R. (2013) On Two-step Estimation of a Spatial Auto regressive Model with Autoregressive Disturbances and Endogenous Regressors, Econometric Review, 32, pages 686--733.

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

Gianfranco Piras (2010). sphet: Spatial Models with Heteroskedastic Innovations in R. Journal of Statistical Software, 35(1), 1-21. tools:::Rd_expr_doi("10.18637/jss.v035.i01").

Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. tools:::Rd_expr_doi("10.18637/jss.v063.i18").

Gianfranco Piras, Paolo Postiglione (2022). A deeper look at impacts in spatial Durbin model with sphet. Geographical Analysis, 54(3), 664-684. https://onlinelibrary.wiley.com/doi/10.1111/gean.12318

Luc Anselin, Sergio J. Rey (2014). Modern Spatial Econometrics in Practice: A Guide to GeoDa, GeoDaSpace and PySal. GeoDa Press LLC.

Examples

data("natreg")
data("ws_6")

form <-  HR90  ~ 0 | MA90 + PS90 +
RD90 + UE90 | 0 | 0 | MA90 + PS90 +
RD90 + FH90 + FP89 + GI89 | 0

form1 <-  HR90  ~ MA90 -1 |  PS90 +
RD90 + UE90 | 0 | MA90 -1 |  PS90 +
RD90 + FH90 + FP89 + GI89 | 0

form2 <-  HR90  ~ MA90 -1 |  PS90 +
RD90 + UE90 | MA90 | MA90 -1 |  PS90 +
RD90 + FH90 + FP89 + GI89 | 0

form3 <-  HR90  ~ MA90 -1 |  PS90 +
RD90 + UE90 | MA90 | MA90 -1 |  PS90 +
RD90 + FH90 + FP89 + GI89 | GI89

form4 <-  HR90  ~ MA90 -1 |  PS90 +
RD90 + UE90 | MA90 + RD90 | MA90 -1 |  PS90 +
RD90 + FH90 + FP89 + GI89 | GI89


split  <- ~ REGIONS

###################################################
# Linear model with regimes and lagged regressors #
###################################################
mod <- spregimes(formula = form2, data = natreg,
rgv = split, listw = ws_6, model = "ols")
summary(mod)

mod1 <- spregimes(formula = form3, data = natreg,
rgv = split, listw = ws_6, model = "ols")
summary(mod1)

mod2 <- spregimes(formula = form4, data = natreg,
rgv = split, listw = ws_6, model = "ols")
summary(mod2)


###############################
# Spatial Error regimes model #
###############################
mod <- spregimes(formula = form, data = natreg,
rgv = split, listw = ws_6, model = "error", het = TRUE)
summary(mod)
mod1 <- spregimes(formula = form, data = natreg,
rgv = split, listw = ws_6, model = "error",
weps_rg = TRUE, het = TRUE)
summary(mod1)
mod2 <- spregimes(formula = form1, data = natreg,
rgv = split, listw = ws_6, model = "error", het = TRUE)
summary(mod2)

###############################
#  Spatial Lag regimes model  #
###############################
mod4 <- spregimes(formula = form, data = natreg,
rgv = split, listw = ws_6, model = "lag",
het = TRUE, wy_rg = TRUE)
summary(mod4)
mod5 <- spregimes(formula = form1, data = natreg,
rgv = split, listw = ws_6, model = "lag",
het = TRUE, wy_rg = TRUE)
summary(mod5)

###############################
# Spatial SARAR regimes model #
###############################
mod6 <- spregimes(formula = form, data = natreg,
rgv = split, listw = ws_6, model = "sarar",
het = TRUE, wy_rg = TRUE, weps_rg = TRUE)
summary(mod6)
mod7 <- spregimes(formula = form, data = natreg,
rgv = split, listw = ws_6, model = "sarar",
het = TRUE, wy_rg = FALSE, weps_rg = FALSE)
summary(mod7)
mod8 <- spregimes(formula = form1, data = natreg,
rgv = split, listw = ws_6, model = "sarar",
het = TRUE, wy_rg = TRUE, weps_rg = FALSE)
summary(mod8)