biprobit_partial {endogeneity}

Recursive Bivariate Probit Model with Partially Observed First Stage

Description

Estimate two probit models with bivariate normally distributed error terms, in which the dependent variable of the first stage model is partially observed (or unobserved).

First stage (Probit, m_i is partially observed):

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

Second stage (Probit):

y_i = 1(\boldsymbol{\beta}'\mathbf{x_i} + {\gamma}m_i + \sigma v_i>0)

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

Unobserved m_i should be coded as NA. w and x can be the same set of variables. Identification can be weak if w are not good predictors of m. Observing m_i for 10%~20% of observations can significantly improve the identification of the model.

Usage

biprobit_partial(
  form1,
  form2,
  data = NULL,
  EM = FALSE,
  par = NULL,
  method = "BFGS",
  verbose = 0,
  maxIter = 500,
  tol = 1e-05,
  tol_LL = 1e-06
)

Arguments

form1

Formula for the first probit model, in which the dependent variable is partially observed.

form2

Formula for the second probit model, the partially observed dependent variable of the first stage is automatically added as a regressor in this model (do not add manually)

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 FLASE, but should change to TRUE is the model has convergence issues.

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. Prefix "1" means first stage variables.

• 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

• 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(), linear_probit(), pln_linear(), pln_probit(), probit_linearRE(), probit_linear_latent(), probit_linear_partial(), probit_linear()

Examples

library(MASS)
N = 5000
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 + 3*z + e1 > 0)
y = as.numeric(1 + x + z + m + e2 > 0)

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

# partially observed version of m
observed_pct = 0.2
m_p = m
m_p[sample(N, N*(1-observed_pct))] = NA
est_partial = biprobit_partial(m_p~x+z, y~x+z)
print(est_partial$estimates, digits=3)