linear_probit: Recursive Linear-Probit Model

Description

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

First stage (Linear): $$m_i=\boldsymbol{\alpha}'\mathbf{w_i}+\sigma u_i$$ Second stage (Probit): $$y_i = 1(\boldsymbol{\beta}'\mathbf{x_i} + {\gamma}m_i + v_i>0)$$ Endogeneity structure: $u_i$ and $v_i$ are bivariate normally distributed with a correlation of $\rho$.

The identification of this model requires an instrumental variable that appears in w but not x. This model still works if the first-stage dependent variable is not a regressor in the second stage.

Usage

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

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.

Arguments

form_linear

Formula for the linear model

form_probit

Formula for the probit 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

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

Examples

Run this code

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

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