Package 'spldv'

Get Model Summaries for use with "mtable" for objects of class bingmm

Description

A generic function to collect coefficients and summary statistics from a bingmm object. It is used in mtable

Usage

## S3 method for class 'bingmm'
getSummary(obj, alpha = 0.05, ...)

Arguments

obj	a bingmm object,
alpha	level of the confidence intervals,
...	further arguments,

Details

For more details see package memisc.

Value

A list with an array with coefficient estimates and a vector containing the model summary statistics.

---

Get Model Summaries for use with "mtable" for objects of class binlgmm

Description

A generic function to collect coefficients and summary statistics from a binlgmm object. It is used in mtable

Usage

## S3 method for class 'binlgmm'
getSummary(obj, alpha = 0.05, ...)

Arguments

obj	a binlgmm object,
alpha	level of the confidence intervals,
...	further arguments,

Details

For more details see package memisc.

Value

A list with an array with coefficient estimates and a vector containing the model summary statistics.

---

Get Model Summaries for use with "mtable" for objects of class binris

Description

A generic function to collect coefficients and summary statistics from a binris object. It is used in mtable

Usage

## S3 method for class 'binris'
getSummary(obj, alpha = 0.05, ...)

Arguments

obj	a binris object,
alpha	level of the confidence intervals,
...	further arguments,

Details

For more details see package memisc.

Value

A list with an array with coefficient estimates and a vector containing the model summary statistics.

---

Compute Marginal Effects for Spatial Binary Models

Description

Computes marginal effects (direct, indirect, total, from-region, and cumulative) from a spatial binary response model. Supports inference via the Delta method or Monte Carlo simulation.

Usage

## S3 method for class 'bingmm'
impacts(
  obj,
  data = NULL,
  variable = NULL,
  from.unit = 1,
  dydx = c("exact", "numeric"),
  change = NULL,
  vce = c("robust", "efficient", "ml"),
  type = c("delta", "mc"),
  result = c("summary", "from.region", "cumulative"),
  vcov = NULL,
  het = TRUE,
  empirical = FALSE,
  approximation = FALSE,
  R = 50,
  pw = 5,
  draws = NULL,
  Q = 3,
  tol = 1e-06,
  verbose = FALSE,
  ...
)

## S3 method for class 'binlgmm'
impacts(
  obj,
  data = NULL,
  variable = NULL,
  from.unit = 1,
  dydx = c("exact", "numeric"),
  change = NULL,
  vce = c("robust", "efficient", "ml"),
  type = c("delta", "mc"),
  result = c("summary", "from.region", "cumulative"),
  vcov = NULL,
  het = TRUE,
  empirical = FALSE,
  approximation = FALSE,
  R = 50,
  pw = 5,
  draws = NULL,
  Q = 3,
  tol = 1e-06,
  verbose = FALSE,
  ...
)

## S3 method for class 'binris'
impacts(
  obj,
  data = NULL,
  variable = NULL,
  from.unit = 1,
  dydx = c("exact", "numeric"),
  change = NULL,
  vce = c("robust", "efficient", "ml"),
  type = c("delta", "mc"),
  result = c("summary", "from.region", "cumulative"),
  vcov = NULL,
  het = TRUE,
  empirical = FALSE,
  approximation = FALSE,
  R = 50,
  pw = 5,
  draws = NULL,
  Q = 3,
  tol = 1e-06,
  verbose = FALSE,
  ...
)

## S3 method for class 'impacts.bingmm'
print(x, ...)

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

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

Arguments

obj	An object of class bingmm, binlgmm or binris.
data	Optional dataset used for computing partial effects.
variable	Variable name (string) for which marginal effects are computed.
from.unit	Integer specifying the spatial unit (row) for computing impacts.
dydx	Character: type of derivative approximation to use ("exact" or "numeric").
change	For numeric variables, a vector indicating the lower and upper value to compute the partial change.
vce	A string indicating what kind of variance-covariance matrix of the estimate should be computed when using effect.bingmm. For the one-step GMM estimator, the options are "robust" and "ml". For the two-step GMM estimator, the options are "robust", "efficient" and "ml". The option "vce = ml" is an exploratory method that evaluates the VC of the RIS estimator using the GMM estimates.
type	String indicating which method is used to compute the standard errors of the marginal effects. If "mc", then the Monte Carlo approximation is used. If "delta", then the Delta Method is used.
result	String indicating the output format: "summary" presents the average direct, indirect or total effects; "from.region" display the total effect of the variable in "variable" form spatial unit indicated in "from.unit"; if "cumulative" the cumulative effect up to power order indicated in "Q" for the variable in "variable" form spatial unit indicated in "from.unit" is returned.
vcov	Optional user-supplied variance-covariance matrix.
het	Logical. If TRUE (the default), then the heteroskedasticity is taken into account when computing the marginal effects.
empirical	Logical. Argument passed to mvrnorm (default FALSE): if TRUE, the coefficients and their covariance matrix specify the empirical not population mean and covariance matrix.
approximation	Logical. If TRUE then $(I - \lambda W)^{-1}$ is approximated as $I + \lambda W + \lambda^2 W^2 + \lambda^3 W^3 + ... +\lambda^q W^q$. The default is FALSE.
R	Numerical. Indicates the number of draws used in the Monte Carlo approximation if type = "mc".
pw	numeric. The power used for the approximation $I + \lambda W + \lambda^2 W^2 + \lambda^3 W^3 + ... +\lambda^q W^q$. The default is 5.
draws	Optional matrix of parameter draws for Monte Carlo simulation.
Q	Number of columns for cumulative effects (if applicable).
tol	Numerical. Argument passed to mvrnorm: tolerance (relative to largest variance) for numerical lack of positive-definiteness in the coefficient covariance matrix.
verbose	Logical. Display progress.
...	further arguments. Ignored.
x	An object of class impacts.bingmm for print methods.
object	An object of class impacts.bingmm for summary methods.
digits	the number of digits.

Details

Let the model be:

$y^*= X\beta + WX\gamma + \lambda W y^* + \epsilon = Z\delta + \lambda Wy^{*} + \epsilon$

where $y = 1$ if $y^*>0$ and 0 otherwise; $\epsilon \sim N(0, 1)$ if link = "probit" or $\epsilon \sim L(0, \pi^2/3)$ if link = "logit". The RIS estimator assumes that $\epsilon \sim N(0, 1)$.

The marginal effects respect to variable $x_r$ can be computed as

$diag(f(a))D^{-1}_{\lambda}A^{-1}_{\lambda}\left(I_n\beta_r + W\gamma_r\right) = C_r(\theta)$

where $f()$ is the pdf, which depends on the assumption of the error terms; $diag$ is the operator that creates a $n \times n$ diagonal matrix; $A_{\lambda}= (I -\lambda W)$; and $D_{\lambda}$ is a diagonal matrix whose elements represent the square root of the diagonal elements of the variance-covariance matrix of $u = A_{\lambda}^{-1}\epsilon$.

We implement these three summary measures: (1) The average total effects, $ATE_r = n^{-1}i_n'C_{r}i_n$, (2) The average direct effects, $ADE_r = n^{-1}tr(C_{r})$, and (3) the average indirect effects, $ATE_r - ADE_r$.

The standard errors of the average total, direct and indirect effects can be estimated using either Monte Carlo (MC) approximation, which takes into account the sampling distribution of $\theta$, or Delta Method.

Value

An object of class impacts.bingmm.

Author(s)

Mauricio Sarrias and Gianfranco Piras.

See Also

sbinaryGMM, sbinaryLGMM.

Examples

# Data set
data(oldcol, package = "spdep")

# Create dependent (dummy) variable
COL.OLD$CRIMED <- as.numeric(COL.OLD$CRIME > 35)

# Two-step (Probit) GMM estimator
ts <- sbinaryGMM(CRIMED ~ INC * HOVAL + factor(CP) | HOVAL,
                link = "probit", 
                listw = spdep::nb2listw(COL.nb, style = "W"), 
                data = COL.OLD, 
                type = "twostep")
                
# Summary marginal effects using Delta Method
summary(impacts(ts, type = "delta"))

# Summary marginal effects using MC with 100 draws
summary(impacts(ts, type = "mc", R = 100))

# Marginal effects using efficient VC matrix
summary(impacts(ts, type = "delta", vce = "efficient"))

# Marginal effects using efficient VC matrix and ignoring the heteroskedasticity
summary(impacts(ts, type = "delta", vce = "efficient", het = FALSE))

# Obtain total impact when INC changes in region 20
imp <- impacts(ts, type = "delta", result = "from.region", from.unit = 20, variable = "INC")
head(imp$out, 6)

# Cumulative impact when INC changes in region 20 up to power = 3
cum <- impacts(ts, type = "delta", result = "cumulative", from.unit = 20, variable = "INC", Q = 3)
head(cum$out$`Q:3`, 6)

# Marginal effects using  RIS estimator
ris_sar <- sbinaryRis(CRIMED ~ INC + HOVAL, data = COL.OLD,
                      R = 50, 
                      listw = spdep::nb2listw(COL.nb, style = "W"))
summary(impacts(ris_sar, method = "delta"))
summary(impacts(ris_sar, method = "mc", R = 100))

---

Make instruments for spatial models

Description

Make instruments for spatial models

Usage

make.instruments(listw, x, q)

Arguments

listw	object. An object of class matrix, or Matrix.
x	variable(s) to be lagged
q	number of lags

Author(s)

Mauricio Sarrias and Gianfranco Piras.

---

Predictions for Spatial Binary GMM Models

Description

Computes predicted probabilities for spatial binary response models estimated via GMM. Supports both probit and logit links, accounts for spatial heteroskedasticity, and optionally returns standard errors using the Delta method with robust or efficient variance estimators.

Usage

## S3 method for class 'bingmm'
predict(
  object,
  newdata,
  Sinv = NULL,
  het = TRUE,
  approximation = FALSE,
  pw = 5,
  ses = FALSE,
  vce = c("robust", "efficient", "ml"),
  theta = NULL,
  ...
)

Arguments

object	An object of class bingmm returned by a spatial binary GMM estimation function.
newdata	An optional data frame in which to look for variables with which to predict. If omitted, the original data used to fit the model is used.
Sinv	Optional user-supplied spatial multiplier matrix $S = (I - \lambda W)^{-1}$. If NULL, it is computed using the spatial weight matrix.
het	Logical. If TRUE, assumes a heteroskedastic error structure with spatially varying variances.
approximation	Logical. If TRUE, uses power-series approximation to compute the inverse spatial matrix.
pw	Integer. Power-order to use when approximation = TRUE.
ses	Logical. If TRUE, standard errors of the predictions are computed using the Delta method.
vce	Type of variance-covariance estimator: "robust" (default), "efficient", or "ml".
theta	Optional parameter vector (including lambda) to use for prediction instead of the estimated one.
...	Additional arguments (currently unused).

Details

The function computes predicted probabilities $\hat{p}_i = F(a_i)$ where $a_i$ is a spatially filtered linear index. In the presence of heteroskedasticity (het = TRUE), the normalization involves the row-wise standard deviation of the spatial multiplier. When ses = TRUE, standard errors are computed using the analytical Jacobian of the prediction function with respect to the parameters and the estimated variance-covariance matrix.

Value

A numeric vector of predicted probabilities if ses = FALSE. If ses = TRUE, returns a matrix with:

p_hat

Predicted probabilities.

Std. error

Standard errors of the predictions.

z value

Z-statistics.

Pr(> z)

Two-sided p-values.

Author(s)

Mauricio Sarrias and Gianfranco Piras.

See Also

sbinaryGMM, vcov.bingmm

Examples

# Data set
data(oldcol, package = "spdep")

# Create dependent (dummy) variable
COL.OLD$CRIMED <- as.numeric(COL.OLD$CRIME > 35)

# Two-step (Probit) GMM estimator
ts <- sbinaryGMM(CRIMED ~ INC*HOVAL + factor(CP),
                link = "probit", 
                listw = spdep::nb2listw(COL.nb, style = "W"), 
                data = COL.OLD, 
                type = "twostep",
                verbose = FALSE)
                
# Compute just the predicted probability
p_hat <- predict(ts)

# Compute predicted probability and standard errors
res <- predict(ts, ses = TRUE, het = TRUE, vce = "efficient")
head(res)

---

Predictions for Spatial Binary LGMM Models

Description

Computes predicted probabilities for spatial binary response models estimated via LGMM. Supports both probit and logit links, accounts for spatial heteroskedasticity, and optionally returns standard errors using the Delta method.

Usage

## S3 method for class 'binlgmm'
predict(
  object,
  newdata,
  Sinv = NULL,
  het = TRUE,
  approximation = FALSE,
  pw = 5,
  ses = FALSE,
  theta = NULL,
  ...
)

Arguments

object	An object of class binlgmm returned by a spatial binary GMM estimation function.
newdata	An optional data frame in which to look for variables with which to predict. If omitted, the original data used to fit the model is used.
Sinv	Optional user-supplied spatial multiplier matrix $S = (I - \lambda W)^{-1}$. If NULL, it is computed using the spatial weight matrix.
het	Logical. If TRUE, assumes a heteroskedastic error structure with spatially varying variances.
approximation	Logical. If TRUE, uses power-series approximation to compute the inverse spatial matrix.
pw	Integer. Power-order to use when approximation = TRUE.
ses	Logical. If TRUE, standard errors of the predictions are computed using the Delta method.
theta	Optional parameter vector (including lambda) to use for prediction instead of the estimated one.
...	Additional arguments (currently unused).

Details

The function computes predicted probabilities $\hat{p}_i = F(a_i)$ where $a_i$ is a spatially filtered linear index. In the presence of heteroskedasticity (het = TRUE), the normalization involves the row-wise standard deviation of the spatial multiplier. When ses = TRUE, standard errors are computed using the analytical Jacobian of the prediction function with respect to the parameters and the estimated variance-covariance matrix.

Value

A numeric vector of predicted probabilities if ses = FALSE. If ses = TRUE, returns a matrix with:

p_hat

Predicted probabilities.

Std. error

Standard errors of the predictions.

z value

Z-statistics.

Pr(> z)

Two-sided p-values.

Author(s)

Mauricio Sarrias and Gianfranco Piras.

See Also

sbinaryLGMM, vcov.binlgmm

Examples

# Data set
data(oldcol, package = "spdep")

# Create dependent (dummy) variable
COL.OLD$CRIMED <- as.numeric(COL.OLD$CRIME > 35)

# Estimate the model
lgmm <- sbinaryLGMM(CRIMED ~ INC + HOVAL | INC,
                    link  = "probit",
                    listw = spdep::nb2listw(COL.nb, style = "W"),
                    nins  = 3,
                    data  = COL.OLD)

# Predicted probabilities with SES
out <- predict(lgmm, ses = TRUE)
head(out, 5)

---

Predictions for Spatial Binary RIS Models

Description

Computes predicted probabilities for spatial binary response models estimated via RIS. Supports probit models and accounts for spatial heteroskedasticity, and optionally returns standard errors using the Delta method.

Usage

## S3 method for class 'binris'
predict(
  object,
  newdata,
  Sinv = NULL,
  het = TRUE,
  approximation = FALSE,
  pw = 5,
  ses = FALSE,
  theta = NULL,
  ...
)

Arguments

object	An object of class binris.
newdata	An optional data frame in which to look for variables with which to predict. If omitted, the original data used to fit the model is used.
Sinv	Optional user-supplied spatial multiplier matrix $S = (I - \lambda W)^{-1}$. If NULL, it is computed using the spatial weight matrix.
het	Logical. If TRUE, assumes a heteroskedastic error structure with spatially varying variances.
approximation	Logical. If TRUE, uses power-series approximation to compute the inverse spatial matrix.
pw	Integer. Power-order to use when approximation = TRUE.
ses	Logical. If TRUE, standard errors of the predictions are computed using the Delta method.
theta	Optional parameter vector (including lambda) to use for prediction instead of the estimated one.
...	Additional arguments (currently unused).

Details

The function computes predicted probabilities $\hat{p}_i = F(a_i)$ where $a_i$ is a spatially filtered linear index. In the presence of heteroskedasticity (het = TRUE), the normalization involves the row-wise standard deviation of the spatial multiplier. When ses = TRUE, standard errors are computed using the analytical Jacobian of the prediction function with respect to the parameters and the estimated variance-covariance matrix.

Value

A numeric vector of predicted probabilities if ses = FALSE. If ses = TRUE, returns a matrix with:

p_hat

Predicted probabilities.

Std. error

Standard errors of the predictions.

z value

Z-statistics.

Pr(> z)

Two-sided p-values.

Author(s)

Mauricio Sarrias and Gianfranco Piras.

See Also

sbinaryRis

Examples

data(oldcol, package = "spdep")
# Create dependent (dummy) variable
COL.OLD$CRIMED <- as.numeric(COL.OLD$CRIME > 35)

# Estimate the model
ris_sar <- sbinaryRis(CRIMED ~ INC + HOVAL, data = COL.OLD,
                      R = 50,
                      listw = spdep::nb2listw(COL.nb, style = "W"))
                      
# Predicted probabilities with SES
out <- predict(ris_sar, ses = TRUE)
head(out, 5)

---

GMM Estimation for Binary Spatial Autoregressive Models.

Description

Estimates a Spatial Autoregressive (SAR) model for binary outcomes using the Generalized Method of Moments (GMM). The model supports both Probit and Logit links, with one-step or two-step GMM estimation. The structural equation is:

$y^*= X\beta + WX\gamma + \lambda W y^* + \epsilon = Z\delta + \lambda Wy^{*} + \epsilon,$

where $y = 1$ if $y^*>0$ and 0 otherwise. The error term $\varepsilon$ follows a standard normal distribution (link = "probit") or logistic distribution (link = "logit").

Usage

sbinaryGMM(
  formula,
  data,
  listw = NULL,
  nins = 2,
  link = c("probit", "logit"),
  winitial = c("optimal", "identity"),
  s.matrix = c("robust", "iid"),
  type = c("onestep", "twostep"),
  gradient = TRUE,
  start = NULL,
  cons.opt = FALSE,
  approximation = FALSE,
  verbose = TRUE,
  print.init = FALSE,
  pw = 5,
  tol.solve = .Machine$double.eps,
  fastmom = FALSE,
  ...
)

## S3 method for class 'bingmm'
coef(object, ...)

## S3 method for class 'bingmm'
vcov(
  object,
  vce = c("robust", "efficient", "ml"),
  method = "bhhh",
  R = 1000,
  tol.solve = .Machine$double.eps,
  ...
)

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

## S3 method for class 'bingmm'
summary(
  object,
  vce = c("robust", "efficient", "ml"),
  method = "bhhh",
  R = 1000,
  tol.solve = .Machine$double.eps,
  ...
)

## S3 method for class 'summary.bingmm'
print(x, digits = max(5, getOption("digits") - 3), ...)

Arguments

formula	A symbolic description of the model of the form y ~ x | wx where y is the binary dependent variable, x are the independent variables. The variables after | are those variables that enter spatially lagged: $WX$.
data	A data.frame containing the variables in the model.
listw	A spatial weights matrix of class listw, matrix, or Matrix.
nins	Integer. Order of spatial instruments to generate; default is nins = 2, such that $H = (Z, WZ, W^2Z)$ are used as instruments.
link	Character string; distributional assumption on the error term. Either "probit" (default) or "logit".
winitial	Character string. A string indicating the initial moment-weighting matrix $\Psi$; it can be either winitial = "optimal" (the default, $(H'H/n)^{-1}$) or winitial = "identity".
s.matrix	Character string. Only valid of type = "twostep" is used. This is a string indicating the type of variance-covariance matrix $\hat{S}$ to be used in the second-step procedure; it can be s.matrix = "robust" (the default) or s.matrix = "iid".
type	Character string. A string indicating whether the one-step (type = "onestep"), or two-step GMM (type = "twostep") should be computed.
gradient	Logical. Only for testing procedures. Should the analytic gradient be used in the GMM optimization procedure? TRUE as default. If FALSE, then the numerical gradient is used.
start	Vector of optional starting values. If not NULL, the user must provide a vector of initial parameters for the optimization procedure. When start = NULL, sbinaryGMM uses the traditional Probit or Logit estimates as initial values for the parameters, and the correlation between $y$ and $Wy$ as initial value for $\lambda$.
cons.opt	Logical. Should a constrained optimization procedure for $\lambda$ be used? FALSE as default.
approximation	Logical. If TRUE, then $(I - \lambda W)^{-1}$ is approximated as $I + \lambda W + \lambda^2 W^2 + \lambda^3 W^3 + ... +\lambda^q W^q$. The default is FALSE.
verbose	Logical. If TRUE, the code reports messages and some values during optimization.
print.init	Logical. If TRUE the initial parameters used in the optimization of the first step are printed.
pw	Integer. The power used for the approximation $I + \lambda W + \lambda^2 W^2 + \lambda^3 W^3 + ... +\lambda^q W^q$. The default is 5.
tol.solve	Tolerance for solve().
fastmom	Logical. Use faster approximate gradient for $\lambda$? Default is FALSE.
...	additional arguments passed to maxLik.
vce	Character string. A string indicating what kind of standard errors should be computed when using summary. For the one-step GMM estimator, the options are "robust" and "ml". For the two-step GMM estimator, the options are "robust", "efficient" and "ml". The option "vce = ml" is an exploratory method that evaluates the VC of the RIS estimator using the GMM estimates.
method	Character string. Only valid if vce = "ml". It indicates the algorithm used to compute the Hessian matrix of the RIS estimator. The default is "bhhh".
R	Integer. Only valid if vce = "ml". It indicates the number of draws used to compute the simulated probability in the RIS estimator.
x, object	an object of class bingmm.
digits	Integer. The number of digits for summary methods.

Details

The data generating process is:

$y^*= X\beta + WX\gamma + \lambda W y^* + \epsilon = Z\delta + \lambda Wy^{*} + \epsilon$

where $y = 1$ if $y^*>0$ and 0 otherwise; $\epsilon \sim N(0, 1)$ if link = "probit" or $\epsilon \sim L(0, \pi^2/3)$ if link = "logit". The general GMM estimator minimizes:

$J(\theta) = g'(\theta)\hat{\Psi} g(\theta),$

where $\theta = (\beta, \gamma, \lambda)$ and

$g = n^{-1}H'v,$

where $v$ is the generalized residuals. Let $Z = (X, WX)$, then the instrument matrix $H$ contains the linearly independent columns of $H = (Z, WZ, ..., W^qZ)$. The one-step GMM estimator minimizes $J(\theta)$ setting either $\hat{\Psi} = I_p$ if winitial = "identity" or $\hat{\Psi} = (H'H/n)^{-1}$ if winitial = "optimal". The two-step GMM estimator uses an additional step to achieve higher efficiency by computing the variance-covariance matrix of the moments $\hat{S}$ to weight the sample moments. This matrix is computed using the residuals or generalized residuals from the first-step, which are consistent. This matrix is computed as $\hat{S} = n^{-1}\sum_{i = 1}^n h_i(f^2/(F(1 - F)))h_i'$ if s.matrix = "robust" or $\hat{S} = n^{-1}\sum_{i = 1}^n \hat{v}_ih_ih_i'$, where $\hat{v}$ are the first-step generalized residuals.

Value

An object of class “bingmm”, a list with elements:

coefficients	the estimated coefficients,
call	the matched call,
callF	the full matched call,
X	the X matrix, which contains also WX if the second part of the formula is used,
H	the H matrix of instruments used,
y	the dependent variable,
listw	the spatial weight matrix,
link	the string indicating the distribution of the error term,
Psi	the moment-weighting matrix used in the last round,
type	type of model that was fitted,
s.matrix	the type of S matrix used in the second round,
winitial	the moment-weighting matrix used for the first step procedure,
opt	object of class maxLik,
approximation	a logical value indicating whether approximation was used to compute the inverse matrix,
pw	the powers for the approximation,
formula	the formula,
data	the data,
contrastsX	the contrasts used in the first part of the formula,
contrastsD	the contrasts used in the second part of the formula,
Xlevels	a record of the levels of the factors used in fitting,
fastmom	whether the model was fitted using the faster gradient for $\lambda$.

Author(s)

Mauricio Sarrias and Gianfranco Piras.

References

Pinkse, J., & Slade, M. E. (1998). Contracting in space: An application of spatial statistics to discrete-choice models. Journal of Econometrics, 85(1), 125-154.

Fleming, M. M. (2004). Techniques for estimating spatially dependent discrete choice models. In Advances in spatial econometrics (pp. 145-168). Springer, Berlin, Heidelberg.

Klier, T., & McMillen, D. P. (2008). Clustering of auto supplier plants in the United States: generalized method of moments spatial logit for large samples. Journal of Business & Economic Statistics, 26(4), 460-471.

LeSage, J. P., Kelley Pace, R., Lam, N., Campanella, R., & Liu, X. (2011). New Orleans business recovery in the aftermath of Hurricane Katrina. Journal of the Royal Statistical Society: Series A (Statistics in Society), 174(4), 1007-1027.

Piras, G., & Sarrias, M. (2023). One or Two-Step? Evaluating GMM Efficiency for Spatial Binary Probit Models. Journal of choice modelling, 48, 100432.

Piras, G,. & Sarrias, M. (2023). GMM Estimators for Binary Spatial Models in R. Journal of Statistical Software, 107(8), 1-33.

See Also

sbinaryLGMM, impacts.bingmm.

Examples

# Data set
data(oldcol, package = "spdep")

# Create dependent (dummy) variable
COL.OLD$CRIMED <- as.numeric(COL.OLD$CRIME > 35)

# Two-step (Probit) GMM estimator
ts <- sbinaryGMM(CRIMED ~ INC + HOVAL,
                link = "probit", 
                listw = spdep::nb2listw(COL.nb, style = "W"), 
                data = COL.OLD, 
                type = "twostep",
                verbose = TRUE)
                
# Robust standard errors
summary(ts)
# Efficient standard errors
summary(ts, vce = "efficient")

# One-step (Probit) GMM estimator 
os <- sbinaryGMM(CRIMED ~ INC + HOVAL,
                link = "probit", 
                listw = spdep::nb2listw(COL.nb, style = "W"), 
                data = COL.OLD, 
                type = "onestep",
                verbose = TRUE)
summary(os)

# One-step (Logit) GMM estimator with identity matrix as initial weight matrix
os_l <- sbinaryGMM(CRIMED ~ INC + HOVAL,
                  link = "logit", 
                  listw = spdep::nb2listw(COL.nb, style = "W"), 
                  data = COL.OLD, 
                  type = "onestep",
                  winitial = "identity", 
                  verbose = TRUE)
summary(os_l)

# Two-step (Probit) GMM estimator with WX
ts_wx <- sbinaryGMM(CRIMED ~ INC + HOVAL| INC + HOVAL,
                   link = "probit", 
                   listw = spdep::nb2listw(COL.nb, style = "W"), 
                   data = COL.OLD, 
                   type = "twostep",
                   verbose = FALSE)
summary(ts_wx)

# Constrained two-step (Probit) GMM estimator 
ts_c <- sbinaryGMM(CRIMED ~ INC + HOVAL,
                  link = "probit", 
                  listw = spdep::nb2listw(COL.nb, style = "W"), 
                  data = COL.OLD, 
                  type = "twostep",
                  verbose = TRUE, 
                  cons.opt = TRUE)
summary(ts_c)

---

Estimation of SAR for binary models using Linearized GMM.

Description

Estimation of SAR model for binary dependent variables (either Probit or Logit), using Linearized GMM estimator suggested by Klier and McMillen (2008). The model is:

$y^*= X\beta + WX\gamma + \lambda W y^* + \epsilon = Z\delta + \lambda Wy^{*} + \epsilon,$

where $y = 1$ if $y^*>0$ and 0 otherwise; $\epsilon \sim N(0, 1)$ if link = "probit" or $\epsilon \sim L(0, \pi^2/3)$ link = "logit".

Usage

sbinaryLGMM(
  formula,
  data,
  listw = NULL,
  nins = 2,
  link = c("logit", "probit"),
  ...
)

## S3 method for class 'binlgmm'
coef(object, ...)

## S3 method for class 'binlgmm'
vcov(object, ...)

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

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

## S3 method for class 'summary.binlgmm'
print(x, digits = max(3, getOption("digits") - 2), ...)

Arguments

formula	A symbolic description of the model of the form y ~ x | wx where y is the binary dependent variable, x are the independent variables. The variables after | are those variables that enter spatially lagged: $WX$.
data	A data.frame containing the variables in the model.
listw	A spatial weights matrix of class listw, matrix, or Matrix.
nins	Integer. Order of spatial instruments to generate; default is nins = 2, such that $H = (Z, WZ, W^2Z)$ are used as instruments.
link	Character string; distributional assumption on the error term. Either "probit" (default) or "logit".
...	additional arguments.
x, object	an object of class binlgmm.
digits	Integer. The number of digits for summary methods.

Details

The steps for the linearized spatial Probit/Logit model are the following:

1. Estimate the model by standard Probit/Logit model, in which spatial autocorrelation and heteroskedasticity are ignored. The estimated values are $\beta_0$. Calculate the generalized residuals assuming that $\lambda = 0$ and the gradient terms $G_{\beta}$ and $G_{\lambda}$.

2. The second step is a two-stage least squares estimator of the linearized model. Thus regress $G_{\beta}$ and $G_{\lambda}$ on $H = (Z, WZ, W^2Z, ...., W^qZ)$ and obtain the predicted values $\hat{G}$. Then regress $u_0 + G_{\beta}'\hat{\beta}_0$ on $\hat{G}$. The coefficients are the estimated values of $\beta$ and $\lambda$.

The variance-covariance matrix can be computed using the traditional White-corrected coefficient covariance matrix from the last two-stage least squares estimator of the linearlized model.

Value

An object of class “bingmm”, a list with elements:

coefficients	the estimated coefficients,
call	the matched call,
X	the X matrix, which contains also WX if the second part of the formula is used,
H	the H matrix of instruments used,
y	the dependent variable,
listw	the spatial weight matrix,
link	the string indicating the distribution of the error term,
fit	an object of lm representing the T2SLS,
formula	the formula,
data	the data,
contrastsX	the contrasts used in the first part of the formula,
contrastsD	the contrasts used in the second part of the formula,
Xlevels	a record of the levels of the factors used in fitting.

Author(s)

Mauricio Sarrias and Gianfranco Piras.

References

Klier, T., & McMillen, D. P. (2008). Clustering of auto supplier plants in the United States: generalized method of moments spatial logit for large samples. Journal of Business & Economic Statistics, 26(4), 460-471.

Piras, G., & Sarrias, M. (2023). One or Two-Step? Evaluating GMM Efficiency for Spatial Binary Probit Models. Journal of choice modelling, 48, 100432.

Piras, G,. & Sarrias, M. (2023). GMM Estimators for Binary Spatial Models in R. Journal of Statistical Software, 107(8), 1-33.

See Also

sbinaryGMM, impacts.bingmm.

Examples

# Data set
data(oldcol, package = "spdep")

# Create dependent (dummy) variable
COL.OLD$CRIMED <- as.numeric(COL.OLD$CRIME > 35)

# LGMM for probit using q = 3 for instruments
lgmm <- sbinaryLGMM(CRIMED ~ INC + HOVAL | INC,
                link  = "probit", 
                listw = spdep::nb2listw(COL.nb, style = "W"),
                nins  = 3, 
                data  = COL.OLD)
summary(lgmm)

---

Estimation of spatial probit model for binary outcomes using RIS (GHK) simulator

Description

Estimation of spatial probit model using RIS-normal (a.k.a GHK) simulator. The models can be the SAR or SEM probit model model. The SAR probit model has the following structure:

$y^*= X\beta + WX\gamma + \lambda W y^* + \epsilon = Z\delta + \lambda Wy^{*} + \epsilon = A_{\lambda}^{-1}Z\delta + u,$

where $y = 1$ if $y^*>0$ and 0 otherwise, $Z = (X, WX)$, $\delta = (\beta', \gamma')'$, $u = A_{\lambda}^{-1}\epsilon$ with $A_{\lambda} = (I - \lambda W)$, and $\epsilon \sim N(0, I)$. The SEM probit model has the following structure:

$y^* = X\beta + WX\gamma + u = Z\delta + u$

where $y = 1$ if $y^*>0$ and 0 otherwise, $Z = (X, WX)$, $\delta = (\beta', \gamma')'$, $u = \rho W u + \epsilon$ such that $u = A_{\rho}^{-1}\epsilon$, and $\epsilon \sim N(0, I)$,

Usage

sbinaryRis(
  formula,
  data,
  subset,
  na.action,
  listw = NULL,
  R = 20,
  model = c("SAR", "SEM"),
  varcov = c("invsigma", "sigma"),
  approximation = TRUE,
  pw = 5,
  start = NULL,
  Qneg = FALSE,
  print.init = FALSE,
  ...
)

## S3 method for class 'binris'
terms(x, ...)

## S3 method for class 'binris'
estfun(x, ...)

## S3 method for class 'binris'
bread(x, ...)

## S3 method for class 'binris'
df.residual(object, ...)

## S3 method for class 'binris'
vcov(object, ...)

## S3 method for class 'binris'
coef(object, ...)

## S3 method for class 'binris'
logLik(object, ...)

## S3 method for class 'binris'
print(x, ...)

## S3 method for class 'binris'
summary(object, eigentol = 1e-12, ...)

## S3 method for class 'summary.binris'
print(x, digits = max(3, getOption("digits") - 2), ...)

Arguments

formula	A symbolic description of the model of the form y ~ x | wx where y is the binary dependent variable, x are the independent variables. The variables after | are those variables that enter spatially lagged: $WX$.
data	A data.frame containing the variables in the model.
subset	An optional vector specifying a subset of observations to be used in the fitting process.
na.action	A function which indicates what should happen when the data contains NAs.
listw	Object. An object of class listw, matrix, or Matrix.
R	Integer. The number of draws used in RIS (GHK) simulator.
model	String. A string indicating which model to estimate. It can be "SAR" for the spatial autoregressive spatial model or "SEM" for the spatial error model.
varcov	String. A string indicating over which variance-covariance matrix to apply the Chokesly factorization.
approximation	Logical. If TRUE (the default) then $(I - \lambda W)^{-1}$ or $(I - \rho W)^{-1}$ is approximated as $I + \lambda W + \lambda^2 W^2 + \lambda^3 W^3 + ... +\lambda^q W^q$. If FALSE, then the inverse is computed without approximations.
pw	Integer. The power used for the approximation $I + \lambda W + \lambda^2 W^2 + \lambda^3 W^3 + ... +\lambda^q W^q$. The default is 5.
start	If not NULL, the user must provide a vector of initial parameters for the optimization procedure. When start = NULL, sbinaryRis uses the traditional Probit estimates as initial values for the parameters, and the correlation between $y$ and $Wy$ as initial value for $\lambda$ or $\rho$.
Qneg	Logical. Whether to construct the negative of the diagonal elements of $Q$. If Qneg = FALSE, then $q_{ii} = 2y_{i} - 1$. If Qneg = TRUE, then $q_{ii} = 1- 2y_{i}$.
print.init	Logical. If TRUE the initial parameters used in the optimization of the first step are printed.
...	additional arguments passed to maxLik.
x, object	An object of class bingmm.
eigentol	The standard errors are only calculated if the ratio of the smallest and largest eigenvalue of the Hessian matrix is less than eigentol. Otherwise the Hessian is treated as singular.
digits	The number of digits for summary methods.

Details

The models are estimated by simulating the probabilities using the RIS-normal (GHK) simulator. The aim is to evaluate the multivariate density function $P(\upsilon = Q u < s)$, where $Q$ is a diagonal matrix with entries $q_{ii} = 2y_i - 1$, and the $n\times 1$ vector $s$ depends on whether the model is SAR or SEM. If model = "SAR", then $s = QA_{\lambda}^{-1}Z\delta$ where $A_{\lambda} = (I - \lambda W)$; if model = "SEM", then $s = QZ\delta$.

Let $\Sigma_{\upsilon} = QVar(u)Q'$ be the variance-covariance model of the transformed model. If model = "SAR" $\Sigma_{\upsilon} = Q\Sigma_{\lambda}Q'$, where $\Sigma_{\lambda} = (A_{\lambda}'A_{\lambda})^{-1}$. If model = "SEM", then $\Sigma_{\upsilon} = Q\Sigma_{\rho}Q'$, where $\Sigma_{\rho} = (A_{\rho}'A_{\rho})^{-1}$.

Since $\Sigma_{\upsilon}$ is positive definite, there exists a Cholesky decomposition such that $C'C = \Sigma_{\upsilon}^{-1}$, where $C$ is the upper triangular Cholesky matrix and $\Sigma_{\upsilon}^{-1}$ is the precision matrix. Let $B = C^{-1}$. Then the random vector $Qu$ can be replaced by $Qu = C^{-1}\eta = B\eta$, where $\eta$ is a vector of standard normal variables. Then, the upper limit of the integration becomes $Qu = B \eta < s$, which can be written as $\eta < B^{-1}s = \nu$.

The RIS simulator is implemented by drawing a large number $R$ of random vector $\eta$ and computing $\eta_{rj}$ recursively for $j = 1, \ldots, n$. The parameters are estimated using the simulated maximum likelihood (SML) function:

$\ln \tilde{L}(\theta) = \ln \left(\frac{1}{R}\sum_{r = 1}^R \tilde{p}_r\right),$

where:

$\tilde{p}_r = \prod_{j = 1}^n \Phi(\hat{\eta}_{jr}),$

and $\Phi(\cdot)$ is the univariate CDF of the standard normal density.

By default, sbinaryRis compute the SML using the Cholesky transformation on $\Sigma_{\upsilon}^{-1}$, varcov = "invsigma". The transformation can also be applied to $\Sigma_{\upsilon}$ using varcov = "sigma", which is slower than the previous option.

This estimator can take several minutes for large datasets. Thus, by default the inverse matrices $A_{\lambda}^{-1}$ and $A_{\rho}^{-1}$ are approximated using the Leontief expansion.

Value

An object of class "binris", a list with elements:

varcov	the matrix over which the Chokesly factorization is applied,
model	type of model that was fitted,
Qneg	matrix Q used,
approximation	a logical value indicating whether approximation was used to compute the inverse matrix,
pw	the powers for the approximation,
call	the matched call,
X	the X matrix, which contains also WX if the second part of the formula is used,
y	the dependent variable,
listw	the spatial weight matrix,
formula	the formula,
R	number of draws,
mf	model frame,
data	the data,
contrastsX	the contrasts used in the first part of the formula,
contrastsD	the contrasts used in the second part of the formula,
Xlevels	a record of the levels of the factors used in fitting.

Author(s)

Mauricio Sarrias and Gianfranco Piras.

References

Beron, K. J., & Vijverberg, W. P. (2004). Probit in a spatial context: a Monte Carlo analysis. In Advances in spatial econometrics: methodology, tools and applications (pp. 169-195). Berlin, Heidelberg: Springer Berlin Heidelberg.

Fleming, M. M. (2004). Techniques for estimating spatially dependent discrete choice models. In Advances in spatial econometrics (pp. 145-168). Springer, Berlin, Heidelberg.

Pace, R. K., & LeSage, J. P. (2016). Fast simulated maximum likelihood estimation of the spatial probit model capable of handling large samples. In Spatial Econometrics: Qualitative and Limited Dependent Variables (pp. 3-34). Emerald Group Publishing Limited.

Piras, G., & Sarrias, M. (2023). One or Two-Step? Evaluating GMM Efficiency for Spatial Binary Probit Models. Journal of choice modelling, 48, 100432.

See Also

sbinaryGMM, impacts.binris.

Examples

data(oldcol, package = "spdep")

# Create dependent (dummy) variable
COL.OLD$CRIMED <- as.numeric(COL.OLD$CRIME > 35)

# Estimate SAR probit model using RIS simulator using Sigma_v^{-1}
ris_sar <- sbinaryRis(CRIMED ~ INC + HOVAL, 
                     data = COL.OLD,
                     R = 50, 
                     listw = spdep::nb2listw(COL.nb, style = "W"), 
                     print.level = 2)
summary(ris_sar)

# Estimate SAR probit model using RIS simulator using Sigma_v
ris_sar_2  <- sbinaryRis(CRIMED ~ INC + HOVAL, 
                        data = COL.OLD,
                        R = 50, 
                        listw = spdep::nb2listw(COL.nb, style = "W"), 
                        varcov = "sigma", 
                        print.level = 2)
summary(ris_sar_2)

# Estimate SDM probit model using RIS simulator
ris_sdm <- sbinaryRis(CRIMED ~ INC + HOVAL | INC + HOVAL, 
                     data = COL.OLD,
                     R = 50, 
                     listw = spdep::nb2listw(COL.nb, style = "W"), 
                     print.level = 2)
summary(ris_sdm)

# Estimate SEM probit model using RIS simulator
ris_sem <- sbinaryRis(CRIMED ~ INC + HOVAL | INC, 
                     data = COL.OLD,
                     R = 50, 
                     listw = spdep::nb2listw(COL.nb, style = "W"), 
                     model = "SEM")
summary(ris_sem)

---