probit_linear_latent {endogeneity}

Recursive Probit-Linear Model with Latent First Stage

Description

Latent version of the Probit-Linear Model.

First stage (Probit, m_i^* is unobserved):

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. The identification of this model is generally weak, especially if w are not good predictors of m. \gamma is assumed to be positive to ensure that the model estimates are unique.

Usage

probit_linear_latent(
  form_probit,
  form_linear,
  data = NULL,
  EM = TRUE,
  par = NULL,
  method = "BFGS",
  verbose = 0,
  maxIter = 500,
  tol = 1e-06,
  tol_LL = 1e-08
)

Arguments

form_probit

Formula for the first-stage probit model, in which the dependent variable is latent

form_linear

Formula for the second stage linear model. The latent dependent variable of the first stage is automatically added as a regressor in this model

data

Input data, a data frame

EM

Whether to maximize likelihood use the Expectation-Maximization (EM) algorithm, which is slower but more robust. Defaults to TRUE.

par

Starting values for estimates

method

Optimization algorithm. Default is BFGS

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

maxIter

max iterations for EM algorithm

tol

tolerance for convergence of EM algorithm

tol_LL

tolerance for convergence of likelihood

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

• 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

• iter: 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_partial(), probit_linear()

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)

est_latent = probit_linear_latent(~x+z, y~x+z)
print(est_latent$estimates, digits=3)