probit_linear {endogeneity}

Recursive Probit-Linear Model

Description

Estimate probit and linear models with bivariate normally distributed error terms.

First stage (Probit):

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

Second stage (Linear):

y_i = \boldsymbol{\beta}'\mathbf{x_i} + {\gamma}m_i + \sigma v_i

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

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_linear(
  form_probit,
  form_linear,
  data = NULL,
  par = NULL,
  method = "BFGS",
  init = c("zero", "unif", "norm", "default")[4],
  verbose = 0
)

Arguments

form_probit

Formula for the probit model

form_linear

Formula for the linear model

data

Input data, a data frame

par

Starting values for estimates

method

Optimization algorithm. Default is BFGS

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

• 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

Peng, Jing. (2023) Identification of Causal Mechanisms from Randomized Experiments: A Framework for Endogenous Mediation Analysis. Information Systems Research, 34(1):67-84. Available at https://doi.org/10.1287/isre.2022.1113

See Also

Other endogeneity: bilinear(), biprobit_latent(), biprobit_partial(), biprobit(), linear_probit(), pln_linear(), pln_probit(), probit_linearRE(), probit_linear_latent(), probit_linear_partial()

Examples

library(MASS)
N = 2000
rho = -0.5
set.seed(1)

x = rbinom(N, 1, 0.5)
z = rnorm(N)

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

m = as.numeric(1 + x + z + e1 > 0)
y = 1 + x + z + m + e2

est = probit_linear(m~x+z, y~x+z+m)
print(est$estimates, digits=3)