pln {endogeneity}

Poisson Lognormal Model

Description

Estimate a Poisson model with a log-normally distributed heterogeneity term, which is also referred to as the Poisson-Normal model.

E[y_i|x_i,u_i]=exp(\boldsymbol{\alpha}'\mathbf{x_i}+\lambda u_i)

The estimates of this model are often similar to those of a negative binomial model.

Usage

pln(
  form,
  data = NULL,
  par = NULL,
  method = "BFGS",
  init = c("zero", "unif", "norm", "default")[4],
  H = 20,
  verbose = 0
)

Arguments

form

Formula

data

Input data, a data frame

par

Starting values for estimates

method

Optimization algorithm.

init

Initialization method

H

Number of quadrature points

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

• 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

• LR_stat: Likelihood ratio test statistic for the heterogeneity term \lambda=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

Examples

library(MASS)
N = 2000
set.seed(1)

# Works well when the variance of the normal term is not overly large
# When the variance is very large, it tends to be underestimated
x = rbinom(N, 1, 0.5)
z = rnorm(N)
y = rpois(N, exp(-1 + x + z + 0.5 * rnorm(N)))
est = pln(y~x+z)
print(est$estimates, digits=3)