probit_linear_partial: Recursive Probit-Linear Model with Partially Observed First...

In endogeneity: Recursive Two-Stage Models to Address Endogeneity

Recursive Probit-Linear Model with Partially Observed First Stage

Description

Partially observed version of the Probit-Linear Model.

First stage (Probit, m_i is partially observed):

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.

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 a small proportion of observations (e.g., 10~20%) can significantly improve the identification of the model.

Usage

probit_linear_partial(
  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 partially observed
form_linear	Formula for the second stage linear 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 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

• 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()

Examples

library(MASS)
N = 1000
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)

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

endogeneity documentation built on Aug. 21, 2023, 9:11 a.m.