Title: Partial Maximum Likelihood Estimation of Spatial Probit Models
Version: 1.0.1
Description: Estimate spatial autoregressive nonlinear probit models with and without autoregressive disturbances using partial maximum likelihood estimation. Estimation and inference regarding marginal effects is also possible. For more details see Bille and Leorato (2020) <doi:10.1080/07474938.2019.1682314>.
License: GPL (≥ 3)
Encoding: UTF-8
RoxygenNote: 7.3.2
Suggests: testthat (≥ 3.0.0)
Config/testthat/edition: 3
Imports: abind, Matrix, matrixcalc, maxLik, methods, minqa, mvtnorm, numDeriv, qrng, spatialreg, spdep, stats, utils
Depends: R (≥ 3.5)
LazyData: true
URL: https://github.com/d-spinelli/pmlsp
BugReports: https://github.com/d-spinelli/pmlsp/issues
NeedsCompilation: no
Packaged: 2026-01-27 14:20:09 UTC; Ospite
Author: Daniele Spinelli [aut, cre], Anna Gloria Bille' [aut], Samantha Leorato [aut]
Maintainer: Daniele Spinelli <daniele.spinelli@unimib.it>
Repository: CRAN
Date/Publication: 2026-01-31 19:00:02 UTC

pmlsp: Partial Maximum Likelihood Estimation of Spatial Probit Models

Description

Estimate spatial autoregressive nonlinear probit models with and without autoregressive disturbances using partial maximum likelihood estimation. Estimation and inference regarding marginal effects is also possible. For more details see Bille and Leorato (2020) doi:10.1080/07474938.2019.1682314.

Author(s)

Maintainer: Daniele Spinelli daniele.spinelli@unimib.it

Authors:

See Also

Useful links:

Average Marginal Effects for Spatial Probit Models

Description

Computes average direct, indirect, and total marginal effects from an object of class "pmlsprobit" using numerical derivatives.

Usage

ape(mod, variables = NULL)

Arguments

mod

An object of class "pmlsprobit".

variables

Character vector specifying the names of covariates for which marginal effects are computed. This option should be used only when the model contains nonlinear terms in the specification X \beta.

Value

A named list with three elements corresponding to the average direct, indirect, and total marginal effects for each covariate.

See Also

Partial maximum likelihood estimation is implemented in pmlsbp.

Examples


data(oldcol, package = "spdep")
dat <- COL.OLD
dat$y <- as.numeric(dat$CRIME > 35)

listw <- spdep::nb2listw(COL.nb, style = "W")

set.seed(857489)
mod <- pmlsbp(
  y ~ HOVAL + INC, data = dat, W = listw,
  model = "SAR", grouping = 7,
  mvtnorm_control = list(
    M = 25e2, sim_type = "mc",
    tol = .Machine$double.eps, fast = FALSE
  ),
  na.action = na.omit, spectral = TRUE,
  iterlim = 1e5
)

ape(mod)

Coefficient Estimates for Partial Maximum Likelihood Spatial Probit Models

Description

Extracts the estimated regression coefficients from an object of class "pmlsprobit".

Usage

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

Arguments

object

An object of class "pmlsprobit".

...

Additional arguments (currently unused).

Value

A named numeric vector containing the coefficient estimates.

See Also

Partial maximum likelihood estimation is implemented in pmlsbp.

Eurostat employment data

Description

The eurostat data frame has 312 row and 10 variables. Unit of analysis: 312 regions in Europe. Additionally, the eurostat.nb includes the neighbour list based on a consider a k-nearest neighbour approach with k = 11

Usage

data(eurostat)

Format

A data frame with 312 rows and 10 variables:

nuts_id

Region identifier

country_id

Country identfier

employment

Employement rate in percentage

gdp

GDP per capita

isced_02

percentage of residents aged 25-64 with secondary school achievement or lower

isced_34

percentage of residents aged 25-64 with high school achievement or lower

isced_58

percentage of residents aged 25-64 with bachelor degree or higher

long

Longitude

lat

Latitude

Source

https://ec.europa.eu/eurostat/data/database

Fitting Spatial Probit Models by Partial Maximum Likelihood

Description

Fits spatial autoregressive probit models using partial maximum likelihood as proposed in Bille and Leorato (2020) doi:10.1080/07474938.2019.1682314. Both Spatial Autoregressive (SAR) probit models and Spatial Autoregressive Probit Models with Autoregressive Disturbances (SARAR) are supported.

Usage

pmlsbp(
  formula, data, model = "SAR", grouping = 2,
  W = NULL,
  zero.policy = spatialreg::get.ZeroPolicyOption(),
  M = NULL, formula_xlag = NULL, W2 = NULL,
  method_inv = "solve",
  start = NULL, subset = NULL, na.action = na.fail,
  qu = Inf, iterlim = 1000,
  mvtnorm_control = list(M = 1e4, sim_type = "qmc",
                         tol = .Machine$double.eps, fast = FALSE),
  finalHessian = ifelse(method == "bhhh", method, TRUE),
  method = "bhhh", print.level = 2,
  vce.type = "asy",
  Conley = list(coords = NULL, LM = 2),
  nBoot = 1e3, spectral = FALSE,
  verbose = TRUE ,
  tol.solve = .Machine$double.eps,
  version = 0, ...
)

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

Arguments

formula

An object of class "formula" specifying the model.

data

An optional data frame containing the variables in the model.

model

A character string specifying the model type; either "SAR" or "SARAR".

grouping

An integer defining the number of observations to include in the tuples used for estimation.

W

A spatial weights object of class "listw", for example created by nb2listw.

zero.policy

Logical; if TRUE, assigns zero to lagged values for observations without neighbors.

M

A spatial weights object of class "listw" used for the disturbance process; relevant only if model = "SARAR".

formula_xlag

An optional "formula" specifying covariates to be spatially lagged.

W2

A spatial weights object used to construct spatially lagged covariates; relevant only if formula_xlag is not NULL.

method_inv

Character string specifying the method used to invert I - \rho W; one of "solve", "chol", or "fast".

start

Numeric vector of starting values.

subset

Optional vector specifying a subset of observations.

na.action

A function specifying how missing values are handled.

qu

Integer used only if method_inv = "fast".

iterlim

Integer specifying the maximum number of iterations.

mvtnorm_control

A list of control parameters for multivariate normal probability calculations; see lpmvnorm.

finalHessian

Logical or character specifying how the final Hessian matrix is computed; see maxLik.

method

Maximization method passed to maxLik.

print.level

Integer controlling the amount of diagnostic output.

vce.type

Character specifying the variance–covariance estimator; one of "asy", "bootstrap", or "mConley".

Conley

A list specifying options for the modified Conley estimator.

nBoot

Integer specifying the number of bootstrap replications.

spectral

Logical; if TRUE, uses spectral normalization of W.

verbose

Logical; if TRUE, diagnostic messages are printed during model estimation. The default is TRUE. Setting verbose = FALSE suppresses intermediate output.

tol.solve

Numeric tolerance used when inverting matrices.

version

Integer controlling the computation of bivariate normal integrals when grouping = 2.

...

Additional arguments passed to maxLik.

x

a "pmlsprobit" object

digits

integer, used for number formatting the decimal digits

Details

The SARAR model is defined as

y^* = \rho W y + X \beta + W_2 Z \gamma + u

with

u = \lambda M u + \epsilon,

where \epsilon \sim N(0,\sigma^2) and the observed binary outcome is y = 1(y^* > 0). The SAR model is obtained by setting \lambda = 0.

Value

An object of class "pmlsprobit", which is a list containing:

beta

Named vector of regression coefficients.

call

The matched function call.

code

Convergence code from maxLik.

estimate

Named vector of parameter estimates.

f

List of matrices used for prediction.

maximum

Value of the partial log-likelihood.

message

Convergence message from maxLik.

model

Model frame used in estimation.

model.type

Either "SAR" or "SARAR".

rho

Estimated spatial autoregressive parameter.

slx

Logical indicating the presence of spatially lagged covariates.

start

Starting values used in estimation.

terms

Terms object.

vcov

Variance–covariance matrix of the estimates.

W

Spatial weights matrix W.

lambda

Estimated disturbance parameter (SARAR only).

M

Spatial weights matrix M (SARAR only).

W2

Spatial weights matrix W_2 (if slx = TRUE).

No return value

Functions

References

Bille', A. G., & Leorato, S. (2020). Partial ML estimation for spatial autoregressive nonlinear probit models with autoregressive disturbances. Econometric Reviews, 39(5), 437–475. doi:10.1080/07474938.2019.1682314

See Also

pmlsbp, maxLik

Examples


data(oldcol, package = "spdep")
dat <- COL.OLD
dat$y <- as.numeric(dat$CRIME > 35)

listw <- spdep::nb2listw(COL.nb, style = "W")

set.seed(857489)
mod <- pmlsbp(
  y ~ HOVAL + INC, data = dat, W = listw,
  model = "SAR", grouping = 7,
  na.action = na.omit, spectral = TRUE,
  iterlim = 1e5
)

Predicted Values and Marginal Effects for Spatial Probit Models

Description

Computes predicted probabilities or average marginal effects from an object of class "pmlsprobit".

Usage

## S3 method for class 'pmlsprobit'
predict(
  object,
  type = "prob",
  newdata = NULL,
  variables = NULL,
  delta.method = FALSE,
  ...
)

Arguments

object

An object of class "pmlsprobit".

type

Character string specifying the type of prediction. Use "prob" for predicted probabilities P(Y_i = 1), or "me" for direct, indirect, and total average marginal effects.

newdata

Optional data frame containing variables used for prediction. If omitted, the fitted values are used.

variables

Character vector of covariate names for which marginal effects are computed using numerical derivatives. Only used if type = "me".

delta.method

Logical; if TRUE, standard errors of the predictions are computed. Only used if type = "me".

...

Additional arguments (currently unused).

Value

If type = "prob", a numeric vector of predicted probabilities. If type = "me", a list of matrices containing individual direct, indirect, and total marginal effects for each covariate.

See Also

Partial maximum likelihood estimation is implemented in pmlsbp.

Examples


data(oldcol, package = "spdep")
dat <- COL.OLD
dat$y <- as.numeric(dat$CRIME > 35)

listw <- spdep::nb2listw(COL.nb, style = "W")

set.seed(857489)
mod <- pmlsbp(
  y ~ HOVAL + INC, data = dat, W = listw,
  model = "SAR", grouping = 7,
  mvtnorm_control = list(
    M = 25e2, sim_type = "mc",
    tol = .Machine$double.eps, fast = FALSE
  ),
  na.action = na.omit, spectral = TRUE,
  iterlim = 1e5
)

prob <- predict(mod)
head(prob)

Print method for summary.pmlsprobit objects

Description

S3 method to print objects of class summary.pmlsprobit.

Usage

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

Arguments

x

An object of class summary.pmlsprobit.

digits

Number of significant digits to print for coefficients.

...

Additional arguments (currently ignored).

Value

No return value

Summarizing Partial Maximum Likelihood Spatial Probit Models

Description

Provides a summary method for objects of class "pmlsprobit".

Usage

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

Arguments

object

An object of class "pmlsprobit".

...

Additional arguments (currently unused).

Value

An object of class "summary.pmlsprobit" containing:

estimate

Matrix of estimates, standard errors, z-values, and p-values.

rho

Estimated spatial autoregressive parameter.

lambda

Estimated spatial error parameter.

model.type

Type of spatial probit model fitted.

loglik

Log-likelihood at the optimum.

iteration

Number of iterations performed.

returnCode

Optimizer convergence code.

returnMessage

Message associated with the convergence code.

vcov

Variance–covariance matrix of the parameter estimates.

See Also

Partial maximum likelihood estimation is implemented in pmlsbp.

Examples


data(oldcol, package = "spdep")
dat <- COL.OLD
dat$y <- as.numeric(dat$CRIME > 35)

listw <- spdep::nb2listw(COL.nb, style = "W")

set.seed(857489)
mod <- pmlsbp(
  y ~ HOVAL + INC, data = dat, W = listw,
  model = "SAR", grouping = 7,
  na.action = na.omit, spectral = TRUE,
  iterlim = 1e5
)
summary(mod)

Variance–Covariance Matrix for Partial Maximum Likelihood Spatial Probit Models

Description

Extracts the variance–covariance matrix from an object of class "pmlsprobit".

Usage

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

Arguments

object

An object of class "pmlsprobit".

...

Additional arguments (currently unused).

Value

A numeric matrix containing the variance–covariance matrix of the parameter estimates.

See Also

Partial maximum likelihood estimation is implemented in pmlsbp.