bilinear {endogeneity}

Recusrive Bivariate Linear Model

Description

Estimate two linear models with bivariate normally distributed error terms.

First stage (Linear):

m_i=\boldsymbol{\alpha}'\mathbf{w_i}+\lambda u_i

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.

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

bilinear(form1, form2, data = NULL, par = NULL, method = "BFGS", verbose = 0)

Arguments

form1

Formula for the first linear model

form2

Formula for the second linear model

data

Input data, a data frame

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

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

• 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.

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: biprobit_latent(), 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 = -1 + x + z + e1
y = -1 + x + m + e2

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