R: Recursive Probit-LinearRE Model

probit_linearRE {endogeneity}

R Documentation

Recursive Probit-LinearRE Model

Description

A panel extension of the probit_linear model. The first stage is a probit model at the individual level. The second stage is a panel linear model at the individual-time level with individual-level random effects. The random effect is correlated with the error term in the first stage.

First stage (Probit):

m_i=1(\boldsymbol{\alpha}'\mathbf{w_i}+u_i>0)

Second stage (Panel linear model with individual-level random effects):

y_{it} = \boldsymbol{\beta}'\mathbf{x_{it}} + {\gamma}m_i + \lambda v_i +\sigma \epsilon_{it}

Endogeneity structure: u_i and v_i are bivariate normally distributed with a correlation of \rho.

This model uses Adaptive Gaussian Quadrature to overcome numerical challenges with long panels. w and x can be the same set of variables. Identification can be weak if w are not good predictors of m. This model still works if the first-stage dependent variable is not a regressor in the second stage.

Usage

probit_linearRE(
  form_probit,
  form_linear,
  id,
  data = NULL,
  par = NULL,
  method = "BFGS",
  H = 20,
  stopUpdate = F,
  init = c("zero", "unif", "norm", "default")[4],
  verbose = 0
)

Arguments

`form_probit`	Formula for the probit model at the individual level
`form_linear`	Formula for the linear model at the individual-time level
`id`	group id, character if data supplied or numerical vector if data not supplied
`data`	Input data, must be a data.table object
`par`	Starting values for estimates
`method`	Optimization algorithm. Default is BFGS
`H`	Number of quadrature points
`stopUpdate`	Adaptive Gaussian Quadrature disabled if TRUE
`init`	Initialization method
`verbose`	A integer indicating how much output to display during the estimation process. <0 - No ouput 0 - Basic output (model estimates) 1 - Moderate output, basic ouput + parameter and likelihood in each iteration 2 - Extensive output, moderate output + gradient values on each call

Value

A list containing the results of the estimated model, some of which are inherited from the return of maxLik

estimates: Model estimates with 95% confidence intervals
estimate or par: Point estimates
variance_type: covariance matrix used to calculate standard errors. Either BHHH or Hessian.
var: covariance matrix
se: standard errors
var_bhhh: BHHH covariance matrix, inverse of the outer product of gradient at the maximum
se_bhhh: BHHH standard errors
gradient: Gradient function at maximum
hessian: Hessian matrix at maximum
gtHg: g'H^-1g, where H^-1 is simply the covariance matrix. A value close to zero (e.g., <1e-3 or 1e-6) indicates good convergence.
LL or maximum: Likelihood
AIC: AIC
BIC: BIC
n_obs: Number of observations
n_par: Number of parameters
time: Time takes to estimate the model
LR_stat: Likelihood ratio test statistic for \rho=0
LR_p: p-value of likelihood ratio test
iterations: number of iterations taken to converge
message: Message regarding convergence status.

Note that the list inherits all the components in the output of maxLik. See the documentation of maxLik for more details.

References

Chen, H., Peng, J., Li, H., & Shankar, R. (2022). Impact of Refund Policy on Sales of Paid Information Services: The Moderating Role of Product Characteristics. Available at SSRN: https://ssrn.com/abstract=4114972.

Examples

library(MASS)
library(data.table)
N = 500
period = 5
obs = N*period
rho = -0.5
set.seed(100)

e = mvrnorm(N, mu=c(0,0), Sigma=matrix(c(1,rho,rho,1), nrow=2))
e1 = e[,1]
e2 = e[,2]

t = rep(1:period, N)
id = rep(1:N, each=period)
w = rnorm(N)
m = as.numeric(1+w+e1>0)
m_long = rep(m, each=period)

x = rnorm(obs)
y = 1 + x + m_long + rep(e2, each=period) + rnorm(obs)

dt = data.table(y, x, id, t, m=rep(m, each=period), w=rep(w, each=period))

est = probit_linearRE(m~w, y~x+m, 'id', dt)
print(est$estimates, digits=3)

[Package endogeneity version 2.1.3 Index]