Package 'PanelCount' reference manual

Title:	Random Effects and/or Sample Selection Models for Panel Count Data
Description:	A high performance package implementing random effects and/or sample selection models for panel count data. The details of the models are discussed in Peng and Van den Bulte (2023) <doi:10.2139/ssrn.2702053>.
Authors:	Jing Peng
Maintainer:	Jing Peng <[email protected]>
License:	MIT + file LICENSE
Version:	2.0.1
Built:	2024-10-14 06:37:10 UTC
Source:	CRAN

Panel Count Models with Random Effects and/or Sample Selection

Description

A high performance package for estimating panel count models with random effects and/or sample selection.

Functions

ProbitRE: Probit model with random effects on individuals

PoissonRE: Poisson model with random effects on individuals

PLN_RE: Poisson Lognormal model with random effects on individuals

ProbitRE_PoissonRE: PoissonRE and ProbitRE model with correlated random effects on individuals

ProbitRE_PLNRE: PLN_RE and ProbitRE model with correlated random effects on individual level and correlated error terms on individual-time level

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/>

A Poisson Lognormal Model with Random Effects

Description

Estimate a Poisson model with random effects at the individual and individual-time levels.

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

Notations:

$x_{it}$ : variables influencing the selection decision $y_{it}$ , which could be a mixture of time-variant variables, time-invariant variables, and time dummies
$v_i$ : individual level random effect
$\epsilon_{it}$ : individual-time level random effect

$v_i$ and $\epsilon_{it}$ can both account for overdispersion.

Usage

PLN_RE(
  formula,
  data,
  id.name,
  par = NULL,
  sigma = NULL,
  gamma = NULL,
  method = "BFGS",
  adaptiveLL = TRUE,
  stopUpdate = FALSE,
  se_type = c("BHHH", "Hessian")[1],
  H = 12,
  psnH = 12,
  reltol = sqrt(.Machine$double.eps),
  verbose = 0
)
PLN_RE(
  formula,
  data,
  id.name,
  par = NULL,
  sigma = NULL,
  gamma = NULL,
  method = "BFGS",
  adaptiveLL = TRUE,
  stopUpdate = FALSE,
  se_type = c("BHHH", "Hessian")[1],
  H = 12,
  psnH = 12,
  reltol = sqrt(.Machine$double.eps),
  verbose = 0
)

Arguments

`formula`	Formula of the model
`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.
`par`	Starting values for estimates. Default to estimates of Poisson RE model.
`sigma`	Starting value for sigma. Defaults to 1 and will be ignored if par is provided.
`gamma`	Starting value for gamma. Defaults to 1 and will be ignored if par is provided.
`method`	Optimization method used by optim. Defaults to 'BFGS'.
`adaptiveLL`	Whether to use Adaptive Gaussian Quadrature. Defaults to TRUE because it is more reliable (though slower) for long panels.
`stopUpdate`	Whether to disable update of Adaptive Gaussian Quadrature parameters. Defaults to FALSE.
`se_type`	Report Hessian or BHHH standard errors. Defaults to BHHH.
`H`	Number of Quadrature points used for numerical integration using the Gaussian-Hermite Quadrature method. Defaults to 20.
`psnH`	Number of Quadrature points for Poisson 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

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
var_hessian: Inverse of negative Hessian matrix (the second order derivative of likelihood 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

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
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/

Examples


# Use the simulated dataset, in which the true coefficient of x is 1.
# Estimated coefficient is biased due to omission of self-selection
data(sim)
res = PLN_RE(y~x, data=sim[!is.na(sim$y), ], id.name='id', verbose=-1)
res$estimates

# Use the simulated dataset, in which the true coefficient of x is 1.
# Estimated coefficient is biased due to omission of self-selection
data(sim)
res = PLN_RE(y~x, data=sim[!is.na(sim$y), ], id.name='id', verbose=-1)
res$estimates

A Poisson Model with Random Effects

Description

Estimate a Poisson model with random effects at the individual level.

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

Notations:

$x_{it}$ : variables influencing the outcome $y_{it}$ , which could be a mixture of time-variant variables, time-invariant variables, and time dummies
$v_i$ : individual level random effect

Usage

PoissonRE(
  formula,
  data,
  id.name,
  par = NULL,
  sigma = NULL,
  method = "BFGS",
  stopUpdate = FALSE,
  se_type = c("Hessian", "BHHH")[1],
  H = 20,
  reltol = sqrt(.Machine$double.eps),
  verbose = 0
)
PoissonRE(
  formula,
  data,
  id.name,
  par = NULL,
  sigma = NULL,
  method = "BFGS",
  stopUpdate = FALSE,
  se_type = c("Hessian", "BHHH")[1],
  H = 20,
  reltol = sqrt(.Machine$double.eps),
  verbose = 0
)

Arguments

`formula`	Formula of the model
`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.
`par`	Starting values for estimates. Default to estimates of Poisson Model
`sigma`	Starting value for sigma. Defaults to 1 and will be ignored if par is provided.
`method`	Optimization method used by optim. Defaults to 'BFGS'.
`stopUpdate`	Whether to disable update of Adaptive Gaussian Quadrature parameters. Defaults to FALSE.
`se_type`	Report Hessian or BHHH standard errors. Defaults to Hessian.
`H`	Number of Quadrature points used for numerical integration using the Gaussian-Hermite Quadrature method. Defaults to 20.
`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

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
var_hessian: Inverse of negative Hessian matrix (the second order derivative of likelihood 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

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
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/

Examples

# Use the simulated dataset, in which the true coefficient of x is 1.
# Estimated coefficient is biased primarily due to omission of self-selection
data(sim)
res = PoissonRE(y~x, data=sim[!is.na(sim$y), ], id.name='id', verbose=-1)
res$estimates
# Use the simulated dataset, in which the true coefficient of x is 1.
# Estimated coefficient is biased primarily due to omission of self-selection
data(sim)
res = PoissonRE(y~x, data=sim[!is.na(sim$y), ], id.name='id', verbose=-1)
res$estimates

Predictions of ProbitRE_PLNRE model on new sample

Description

Predictions of ProbitRE_PLNRE model on new sample. Please make sure the factor variables in the test data do not have levels not shown in the training data.

Usage

predict_ProbitRE_PLNRE(
  par,
  sel_form,
  out_form,
  data,
  offset_w_name = NULL,
  offset_x_name = NULL
)
predict_ProbitRE_PLNRE(
  par,
  sel_form,
  out_form,
  data,
  offset_w_name = NULL,
  offset_x_name = NULL
)

Arguments

`par`	Model estimates
`sel_form`	Formula for selection equation, a Probit model with random effects
`out_form`	Formula for outcome equation, a Poisson Lognormal model with random effects
`data`	Input data, a data.frame object
`offset_w_name`	Offset variables in selection equation, if any.
`offset_x_name`	Offset variables in outcome equation, if any.

Value

A list with three sets of predictions

prob: Predicted probability to participate
outcome: Predicted potential outcome
actual_outcome: Predicted actual outcome

Predictions of ProbitRE_PoissonRE model on new sample

Description

Predictions of ProbitRE_PoissonRE model on new sample. Please make sure the factor variables in the test data do not have levels not shown in the training data.

Usage

predict_ProbitRE_PoissonRE(
  par,
  sel_form,
  out_form,
  data,
  offset_w_name = NULL,
  offset_x_name = NULL
)
predict_ProbitRE_PoissonRE(
  par,
  sel_form,
  out_form,
  data,
  offset_w_name = NULL,
  offset_x_name = NULL
)

Arguments

`par`	Model estimates
`sel_form`	Formula for selection equation, a Probit model with random effects
`out_form`	Formula for outcome equation, a Poisson Lognormal model with random effects
`data`	Input data, a data.frame object
`offset_w_name`	Offset variables in selection equation, if any.
`offset_x_name`	Offset variables in outcome equation, if any.

Value

A list with three sets of predictions

prob: Predicted probability to participate
outcome: Predicted potential outcome
actual_outcome: Predicted actual outcome

A Probit Model with Random Effects

Description

Estimate a Probit model with random effects at the individual level.

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

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
$u_i$ : individual level random effect
$\xi_{it}$ : error term

Usage

ProbitRE(
  formula,
  data,
  id.name,
  par = NULL,
  delta = NULL,
  method = "BFGS",
  se_type = c("Hessian", "BHHH")[1],
  H = 20,
  reltol = sqrt(.Machine$double.eps),
  verbose = 0
)
ProbitRE(
  formula,
  data,
  id.name,
  par = NULL,
  delta = NULL,
  method = "BFGS",
  se_type = c("Hessian", "BHHH")[1],
  H = 20,
  reltol = sqrt(.Machine$double.eps),
  verbose = 0
)

Arguments

`formula`	Formula of the model
`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.
`par`	Starting values for estimates. Default to estimates of Probit model.
`delta`	Starting value for delta. Defaults to 1 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 Hessian.
`H`	Number of Quadrature points used for numerical integration using the Gaussian-Hermite Quadrature method. Defaults to 20.
`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

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
var_hessian: Inverse of negative Hessian matrix (the second order derivative of likelihood 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.
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
var_hessian Inverse of negative Hessian matrix (the second order derivative of likelihood at the maximum)
se_bhhh BHHH standard errors
g graident function at maximum
LL likelihood
AIC AIC
BIC BIC
n_obs Number of observations
counts 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.
time Time takes to estimate the model
message A character string giving any additional information returned by the optimizer, or NULL.
convergence 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

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
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/

Examples

# Use the simulated dataset, in which the true coefficients of x and w are 1.
data(sim)
res = ProbitRE(z~x+w, data=sim, id.name='id', verbose=-1)
res$estimates
# Use the simulated dataset, in which the true coefficients of x and w are 1.
data(sim)
res = ProbitRE(z~x+w, data=sim, id.name='id', verbose=-1)
res$estimates

Poisson Lognormal Model with Random Effects and 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 (PLN_RE - Poisson Lognormal model with individual-time level random effects):

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

Correlation (self-selection at both individual and individual-time level):

$u_i$ and $v_i$ are bivariate normally distributed with a correlation of $\rho$ .
$\xi_{it}$ and $\epsilon_{it}$ are bivariate normally distributed with a correlation of $\tau$ .

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
$\epsilon_{it}$ : individual-time level random effect in the outcome equation

Usage

ProbitRE_PLNRE(
  sel_form,
  out_form,
  data,
  id.name,
  testData = NULL,
  par = NULL,
  disable_rho = FALSE,
  disable_tau = FALSE,
  delta = NULL,
  sigma = NULL,
  gamma = NULL,
  rho = NULL,
  tau = NULL,
  method = "BFGS",
  se_type = c("BHHH", "Hessian")[1],
  H = c(10, 10),
  psnH = 20,
  prbH = 20,
  plnreH = 20,
  reltol = sqrt(.Machine$double.eps),
  factr = 1e+07,
  verbose = 1,
  offset_w_name = NULL,
  offset_x_name = NULL
)
ProbitRE_PLNRE(
  sel_form,
  out_form,
  data,
  id.name,
  testData = NULL,
  par = NULL,
  disable_rho = FALSE,
  disable_tau = FALSE,
  delta = NULL,
  sigma = NULL,
  gamma = NULL,
  rho = NULL,
  tau = NULL,
  method = "BFGS",
  se_type = c("BHHH", "Hessian")[1],
  H = c(10, 10),
  psnH = 20,
  prbH = 20,
  plnreH = 20,
  reltol = sqrt(.Machine$double.eps),
  factr = 1e+07,
  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 Lognormal 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.
`disable_rho`	Whether to disable correlation at the individual level random effect. Defaults to FALSE.
`disable_tau`	Whether to disable correlation at the individual-time level random effect / error term. Defaults to FALSE.
`delta`	Starting value for delta. Will be ignored if par is provided.
`sigma`	Starting value for sigma. Will be ignored if par is provided.
`gamma`	Starting value for gamma. Will be ignored if par is provided.
`rho`	Starting value for rho. Defaults to 0 and will be ignored if par is provided.
`tau`	Starting value for tau. 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. Hessian matrix is extremely time-consuming to calculate numerically for large datasets.
`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 Quadrature points for Probit RE model
`plnreH`	Number of Quadrature points for PLN_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.
`factr`	L-BFGS-B method uses factr instead of reltol to control for precision. Default is 1e7, that is a tolerance of 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

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
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/

Examples


# Use the simulated dataset, in which the true coefficients of x and w are 1 in both stages.
# The model can recover the true parameters very well
data(sim)
res = ProbitRE_PLNRE(z~x+w, y~x, data=sim, id.name='id')
res$estimates

# Use the simulated dataset, in which the true coefficients of x and w are 1 in both stages.
# The model can recover the true parameters very well
data(sim)
res = ProbitRE_PLNRE(z~x+w, y~x, data=sim, id.name='id')
res$estimates

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

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
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/

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

# 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

Simulated dataset with self-selection at both individual and individual-time level

Description

A simulated dataset with 200 individuals and 10 periods. The true data generating process is the following:

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

$z_{it}=1(1+x_{it}+w_{it}+u_i+\xi_{it} > 0$

Outcome Equation (PLN_RE - Poisson Lognormal model with individual-time level random effects):

$E[y_{it}|x_{it},v_i,\epsilon_{it}] = exp(-1+x_{it} + v_i + \epsilon_{it})$

Correlation (self-selection at both individual and individual-time level):

$u_i$ and $v_i$ are bivariate normally distributed with a correlation of 0.25.
$\xi_{it}$ and $\epsilon_{it}$ are bivariate normally distributed with a correlation of 0.5.

Usage

sim
sim

Format

A simulated dataset with 200 individuals and 10 periods.

id: id, from 1-200
time: Time periods, from 1-10
z: Whether an individual is selected in a given period. Outcome is observed only when z=1
y: The outcome of an individual in a given period
x: A covariate influencing both z and y, with true effects being 1
w: A covariate influencing only z, with true effect being 1

Package 'PanelCount'

Help Index

Panel Count Models with Random Effects and/or Sample Selection

Description

Functions

References

A Poisson Lognormal Model with Random Effects

Description

Usage

Arguments

Value

References

See Also

Examples

A Poisson Model with Random Effects

Description

Usage

Arguments

Value

References

See Also

Examples

Predictions of ProbitRE_PLNRE model on new sample

Description

Usage

Arguments

Value

Predictions of ProbitRE_PoissonRE model on new sample

Description

Usage

Arguments

Value

A Probit Model with Random Effects

Description

Usage

Arguments

Value

References

See Also

Examples

Poisson Lognormal Model with Random Effects and Sample Selection

Description

Usage

Arguments

Value

References

See Also

Examples

Poisson RE model with Sample Selection

Description

Usage

Arguments

Value

References

See Also

Examples

Simulated dataset with self-selection at both individual and individual-time level

Description

Usage

Format