sbinaryGMM {spldv}

Estimation of SAR for binary dependent models using GMM

Description

Estimation of SAR model for binary dependent variables (either Probit or Logit), using one- or two-step GMM estimator. The type of model supported has the following structure:

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".

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,
  ...
)

## 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. The variables in the second part of formula must also appear in the first part. This rules out situations in which one of the regressors can be specified only in lagged form.

data

the data of class data.frame.

listw

object. An object of class listw, matrix, or Matrix.

nins

numerical. Order of instrumental-variable approximation; as default nins = 2, such that H = (Z, WZ, W^2Z) are used as instruments.

link

string. The assumption of the distribution of the error term; it can be either link = "probit" (the default) or link = "logit".

winitial

string. A string indicating the initial moment-weighting matrix \Psi; it can be either winitial = "optimal" (the default) or winitial = "identity".

s.matrix

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

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

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

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.

tol.solve

Tolerance for solve().

...

additional arguments passed to maxLik.

vce

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

string. Only valid if vce = "ml". It indicates the algorithm used to compute the Hessian matrix of the RIS estimator. The defult is "bhhh".

R

numeric. 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

the number of digits

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.

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)