ProbitRE_PoissonRE {PanelCount}

Poisson RE model with Sample Selection

Description

Estimates the following two-stage model

Selection equation (ProbitRE - Probit model with individual level random effects):

z_{it}=1(\boldsymbol{\alpha}\mathbf{w_{it}}'+\delta u_i+\xi_{it} > 0)

Outcome Equation (PoissonRE - Poisson with individual level random effects):

E[y_{it}|x_{it},v_i] = exp(\boldsymbol{\beta}\mathbf{x_{it}}' + \sigma v_i)

Correlation (self-selection at individual level):

• u_i and v_i are bivariate normally distributed with a correlation of \rho.

Notations:

• w_{it}: variables influencing the selection decision z_{it}, which could be a mixture of time-variant variables, time-invariant variables, and time dummies

• x_{it}: variables influencing the outcome y_{it}, which could be a mixture of time-variant variables, time-invariant variables, and time dummies

• u_i: individual level random effect in the selection equation

• v_i: individual level random effect in the outcome equation

• \xi_{it}: error term in the selection equation

Usage

ProbitRE_PoissonRE(
  sel_form,
  out_form,
  data,
  id.name,
  testData = NULL,
  par = NULL,
  delta = NULL,
  sigma = NULL,
  rho = NULL,
  method = "BFGS",
  se_type = c("BHHH", "Hessian")[1],
  H = c(10, 10),
  psnH = 20,
  prbH = 20,
  reltol = sqrt(.Machine$double.eps),
  verbose = 1,
  offset_w_name = NULL,
  offset_x_name = NULL
)

Arguments

sel_form

Formula for selection equation, a Probit model with random effects

out_form

Formula for outcome equation, a Poisson model with random effects

data

Input data, a data.frame object

id.name

The name of the column representing id. Data will be sorted by id to improve estimation speed.

testData

Test data for prediction, a data.frame object

par

Starting values for estimates. Default to estimates of standalone selection and outcome models.

delta

Starting value for delta. Will be ignored if par is provided.

sigma

Starting value for sigma. Will be ignored if par is provided.

rho

Starting value for rho. Defaults to 0 and will be ignored if par is provided.

method

Optimization method used by optim. Defaults to 'BFGS'.

se_type

Report Hessian or BHHH standard errors. Defaults to BHHH.

H

A integer vector of length 2, specifying the number of points for inner and outer Quadratures

psnH

Number of Quadrature points for Poisson RE model

prbH

Number of Quddrature points for Probit RE model

reltol

Relative convergence tolerance. The algorithm stops if it is unable to reduce the value by a factor of reltol * (abs(val) + reltol) at a step. Defaults to sqrt(.Machine$double.eps), typically about 1e-8.

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

offset_w_name

An offset variable whose coefficient is assumed to be 1 in the selection equation

offset_x_name

An offset variable whose coefficient is assumed to be 1 in the outcome equation

Value

A list containing the results of the estimated model, some of which are inherited from the return of optim

• estimates: Model estimates with 95% confidence intervals

• par: Point estimates

• var_bhhh: BHHH covariance matrix, inverse of the outer product of gradient at the maximum

• se_bhhh: BHHH standard errors

• g: Gradient function at maximum

• gtHg: g'H^-1g, where H^-1 is approximated by var_bhhh. A value close to zero (e.g., <1e-3 or 1e-6) indicates good convergence.

• LL: Likelihood

• AIC: AIC

• BIC: BIC

• n_obs: Number of observations

• time: Time takes to estimate the model

• partial: Average partial effect at the population level

• paritalAvgObs: Partial effect for an individual with average characteristics

• predict: A list with predicted participation probability (prob), predicted potential outcome (outcome), and predicted actual outcome (actual_outcome).

• counts: From optim. A two-element integer vector giving the number of calls to fn and gr respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to fn to compute a finite-difference approximation to the gradient.

• message: From optim. A character string giving any additional information returned by the optimizer, or NULL.

• convergence: From optim. An integer code. 0 indicates successful completion. Note that the list inherits all the complements in the output of optim. See the documentation of optim for more details.

References

1. Peng, J., & Van den Bulte, C. (2023). Participation vs. Effectiveness in Sponsored Tweet Campaigns: A Quality-Quantity Conundrum. Management Science (forthcoming). Available at SSRN: https://www.ssrn.com/abstract=2702053

2. Peng, J., & Van den Bulte, C. (2015). How to Better Target and Incent Paid Endorsers in Social Advertising Campaigns: A Field Experiment. 2015 International Conference on Information Systems. https://aisel.aisnet.org/icis2015/proceedings/SocialMedia/24/

See Also

Other PanelCount: PLN_RE(), PoissonRE(), ProbitRE_PLNRE(), ProbitRE()

Examples

# Use the simulated dataset, in which the true coefficients of x and w are 1 in both stages.
# The simulated dataset includes self-selection at both individual and individual-time level,
# but this model only considers self-selection at the individual level.
data(sim)
res = ProbitRE_PoissonRE(z~x+w, y~x, data=sim, id.name='id')
res$estimates