grouped_plm {PAGFL}

Grouped Panel Data Model

Description

Estimate a grouped panel data model given an observed group structure. Slope parameters are homogeneous within groups but heterogeneous across groups. This function supports both static and dynamic panel data models, with or without endogenous regressors.

Usage

grouped_plm(
  formula,
  data,
  groups,
  index = NULL,
  n_periods = NULL,
  method = "PLS",
  Z = NULL,
  bias_correc = FALSE,
  rho = 0.07 * log(N * n_periods)/sqrt(N * n_periods),
  verbose = TRUE,
  parallel = TRUE,
  ...
)

## S3 method for class 'gplm'
print(x, ...)

## S3 method for class 'gplm'
formula(x, ...)

## S3 method for class 'gplm'
df.residual(object, ...)

## S3 method for class 'gplm'
summary(object, ...)

## S3 method for class 'gplm'
coef(object, ...)

## S3 method for class 'gplm'
residuals(object, ...)

## S3 method for class 'gplm'
fitted(object, ...)

Arguments

formula

a formula object describing the model to be estimated.

data

a data.frame or matrix holding a panel data set. If no index variables are provided, the panel must be balanced and ordered in the long format \bold{Y}=(Y_1^\prime, \dots, Y_N^\prime)^\prime, Y_i = (Y_{i1}, \dots, Y_{iT})^\prime with Y_{it} = (y_{it}, x_{it}^\prime)^\prime. Conversely, if data is not ordered or not balanced, data must include two index variables that declare the cross-sectional unit i and the time period t of each observation.

groups

a numerical or character vector of length N that indicates the group membership of each cross-sectional unit i.

index

a character vector holding two strings. The first string denotes the name of the index variable identifying the cross-sectional unit i, and the second string represents the name of the variable declaring the time period t. In case of a balanced panel data set that is ordered in the long format, index can be left empty if the the number of time periods n_periods is supplied.

n_periods

the number of observed time periods T. If an index is passed, this argument can be left empty.

method

the estimation method. Options are

"PLS"

for using the penalized least squares (PLS) algorithm. We recommend PLS in case of (weakly) exogenous regressors (Mehrabani, 2023, sec. 2.2).

"PGMM"

for using the penalized Generalized Method of Moments (PGMM). PGMM is required when instrumenting endogenous regressors, in which case a matrix \bold{Z} containing the necessary exogenous instruments must be supplied (Mehrabani, 2023, sec. 2.3).

Default is "PLS".

Z

a NT \times q matrix or data.frame of exogenous instruments, where q \geq p, \bold{Z}=(z_1, \dots, z_N)^\prime, z_i = (z_{i1}, \dots, z_{iT})^\prime and z_{it} is a q \times 1 vector. Z is only required when method = "PGMM" is selected. When using "PLS", the argument can be left empty or it is disregarded. Default is NULL.

bias_correc

logical. If TRUE, a Split-panel Jackknife bias correction following Dhaene and Jochmans (2015) is applied to the slope parameters. We recommend using the correction when working with dynamic panels. Default is FALSE.

rho

a tuning parameter balancing the fitness and penalty terms in the IC. If left unspecified, the heuristic \rho = 0.07 \frac{\log(NT)}{\sqrt{NT}} of Mehrabani (2023, sec. 6) is used. We recommend the default.

verbose

logical. If TRUE, helpful warning messages are shown. Default is TRUE.

parallel

logical. If TRUE, certain operations are parallelized across multiple cores. Default is TRUE.

...

ellipsis

x

of class gplm.

object

of class gplm.

Details

Consider the grouped panel data model

y_{it} = \gamma_i + \beta^\prime_{i} x_{it} + \epsilon_{it}, \quad i = 1, \dots, N, \; t = 1, \dots, T,

where y_{it} is the scalar dependent variable, \gamma_i is an individual fixed effect, x_{it} is a p \times 1 vector of explanatory variables, and \epsilon_{it} is a zero mean error. The coefficient vector \beta_i is subject to the observed group pattern

\beta_i = \sum_{k = 1}^K \alpha_k \bold{1} \{i \in G_k \},

with \cup_{k = 1}^K G_k = \{1, \dots, N\}, G_k \cap G_j = \emptyset and \| \alpha_k - \alpha_j \| \neq 0 for any k \neq j, k = 1, \dots, K.

Using PLS, the group-specific coefficients for group k are obtained via OLS

\hat{\alpha}_k = \left( \sum_{i \in G_k} \sum_{t = 1}^T \tilde{x}_{it} \tilde{x}_{it}^\prime \right)^{-1} \sum_{i \in G_k} \sum_{t = 1}^T \tilde{x}_{it} \tilde{y}_{it},

where \tilde{a}_{it} = a_{it} - T^{-1} \sum_{t=1}^T a_{it}, a = \{y, x\} to concentrate out the individual fixed effects \gamma_i (within-transformation).

In case of PGMM, the slope coefficients are derived as

\hat{\alpha}_k = \left( \left[ \sum_{i \in G_k} T^{-1} \sum_{t = 1}^T z_{it} \Delta x_{it} \right]^\prime W_k \left[ \sum_{i \in G_k} T^{-1} \sum_{t = 1}^T z_{it} \Delta x_{it} \right] \right)^{-1}

\quad \quad \left[ \sum_{i \in G_k} T^{-1} \sum_{t = 1}^T z_{it} \Delta x_{it} \right]^\prime W_k \left[ \sum_{i \in G_k} T^{-1} \sum_{t = 1}^T z_{it} \Delta y_{it} \right],

where W_k is a q \times q p.d. symmetric weight matrix and \Delta denotes the first difference operator \Delta x_{it} = x_{it} - x_{it-1} (first-difference transformation).

Value

An object of class gplm holding

model

a data.frame containing the dependent and explanatory variables as well as cross-sectional and time indices,

coefficients

a K \times p matrix of the group-specific parameter estimates,

groups

a list containing (i) the total number of groups K and (ii) a vector of group memberships g_1, \dots, g_N), where g_i = k if i is assigned to group k,

residuals

a vector of residuals of the demeaned model,

fitted

a vector of fitted values of the demeaned model,

args

a list of additional arguments,

IC

a list containing (i) the value of the IC and (ii) the MSE,

call

the function call.

A gplm object has print, summary, fitted, residuals, formula, df.residual, and coef S3 methods.

Author(s)

Paul Haimerl

References

Dhaene, G., & Jochmans, K. (2015). Split-panel jackknife estimation of fixed-effect models. The Review of Economic Studies, 82(3), 991-1030. doi:10.1093/restud/rdv007. Mehrabani, A. (2023). Estimation and identification of latent group structures in panel data. Journal of Econometrics, 235(2), 1464-1482. doi:10.1016/j.jeconom.2022.12.002.

Examples

# Simulate a panel with a group structure
sim <- sim_DGP(N = 20, n_periods = 80, p = 2, n_groups = 3)
y <- sim$y
X <- sim$X
groups <- sim$groups
df <- cbind(y = c(y), X)

# Estimate the grouped panel data model
estim <- grouped_plm(y ~ ., data = df, groups = groups, n_periods = 80, method = "PLS")
summary(estim)

# Lets pass a panel data set with explicit cross-sectional and time indicators
i_index <- rep(1:20, each = 80)
t_index <- rep(1:80, 20)
df <- data.frame(y = c(y), X, i_index = i_index, t_index = t_index)
estim <- grouped_plm(
  y ~ ., data = df, index = c("i_index", "t_index"), groups = groups, method = "PLS"
)
summary(estim)