biprobit_latent {endogeneity} | R Documentation |
Recursive Bivariate Probit Model with Latent First Stage
Description
Estimate two probit models with bivariate normally distributed error terms, in which the dependent variable of the first stage model is unobserved.
First stage (Probit, is unobserved):
Second stage (Probit):
Endogeneity structure:
and
are bivariate normally distributed with a correlation of
.
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. is assumed to be positive to ensure that the model estimates are unique.
Usage
biprobit_latent(
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 unobserved. Use a formula like ~w to avoid specifying the dependent variable. |
form2 |
Formula for the second probit 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 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.
|
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:
, 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_partial()
,
biprobit()
,
linear_probit()
,
pln_linear()
,
pln_probit()
,
probit_linearRE()
,
probit_linear_latent()
,
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 = as.numeric(1 + x + z + m + e2 > 0)
est = biprobit(m~x+z, y~x+z+m)
print(est$estimates, digits=3)
est_latent = biprobit_latent(~x+z, y~x+z)
print(est_latent$estimates, digits=3)