Joint Estimation of Latent Groups and Group-Specific Coefficients in Panel Data Models.

PAGFL

Unobservable group structures are a common challenge in panel data analysis. Disregarding group-level heterogeneity can introduce bias. Conversely, estimating individual coefficients for each cross-sectional unit is inefficient and may lead to high uncertainty.

This package efficiently addresses the issue of unobservable group structures by implementing the pairwise adaptive group fused Lasso (PAGFL) by Mehrabani (2023). PAGFL is a regularizer that identifies latent group structures and estimates group-specific coefficients in a single step. On top of that, we extend the PAGFL to time-varying functional coefficients.

The PAGFL package makes this powerful procedure easy to use. On top of that, we extend the PAGFL to time-varying functional coefficients.

Installation

Always stay up-to-date with the development version of PAGFL (1.1.1) from GitHub:

# install.packages("devtools")
devtools::install_github("Paul-Haimerl/PAGFL")
library(PAGFL)

The stable version (1.1.0) is available on CRAN:

install.packages("PAGFL")

Data

The PAGFL packages includes a function that automatically simulates a panel data set with a group structure in the slope coefficients:

# Simulate a simple panel with three distinct groups and two exogenous explanatory variables
set.seed(1)
sim <- sim_DGP(N = 20, n_periods = 150, p = 2, n_groups = 3)
data <- sim$data

$$y_{it} = \beta_i^\prime x_{it} + \eta_i + u_{it}, \quad i = 1, \dots, N, \quad t = 1, \dots, T,$$ where $y_{it}$ is a scalar dependent variable, $x_{it}$ a $p \times 1$ vector of explanatory variables, and $\eta_i$ reflects a fixed effect. The slope coefficients are subject to the group structure

$$\beta_{i} = \sum_{k = 1}^K \alpha_k \boldsymbol{1} {i \in G_k },$$ with $\cup_{k = 1}^K G_k = {1, \dots, N }$, and $G_k \cap G_j = \emptyset$ as well as $|| \alpha_k \neq \alpha_j ||$ for any $k \neq j$, $k,j = 1, \dots, K$ (see Mehrabani 2023, sec. 2).

sim_DGP also nests, among other, all DGPs employed in the simulation study of Mehrabani (2023, sec. 6). I refer to the documentation of sim_DGP or Mehrabani (2023, sec. 6) for more details.

Applying PAGFL

To execute the PAGFL procedure, simply pass the dependent and independent variables, the number of time periods, and a penalization parameter $\lambda$.

estim <- pagfl(y ~ X1 + X2, data = data, n_periods = 150, lambda = 20)
summary(estim)
#> Call:
#> pagfl(formula = y ~ X1 + X2, data = data, n_periods = 150, lambda = 20)
#> 
#> Balanced panel: N = 20, T = 150, obs = 3000
#> 
#> Convergence reached:
#> TRUE (49 iterations)
#> 
#> Information criterion:
#>        IC    lambda 
#>  1.353997 20.000000 
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -4.47230 -0.72086 -0.00120  0.76214  4.31838 
#> 
#> 2 groups:
#>  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 
#>  1  1  2  1  1  1  1  2  1  1  2  2  2  1  1  1  1  1  2  1 
#> 
#> Coefficients:
#>                X1       X2
#> Group 1 -0.36838  1.61275
#> Group 2 -0.49489 -1.23534
#> 
#> Residual standard error: 1.15012 on 2978 degrees of freedom
#> Mean squared error 1.31307
#> Multiple R-squared: 0.65845, Adjusted R-squared: 0.65605

pagfl() returns an object of type pagfl which holds

1. model: A data.frame containing the dependent and explanatory variables as well as individual and time indices (if provided).
2. coefficients: A $K \times p$ matrix of the post-Lasso group-specific parameter estimates.
3. groups: A list containing (i) the total number of groups $\hat{K}$ and (ii) a vector of estimated group memberships $(\hat{g}_1, \dots, \hat{g}_N)$, where $\hat{g}_i = k$ if $i$ is assigned to group $k$.
4. residuals: A vector of residuals of the demeaned model.
5. fitted: A vector of fitted values of the demeaned model.
6. args: A list of additional arguments.
7. IC: A list containing (i) the value of the IC, (ii) the employed tuning parameter $\lambda$, and (iii) the mean squared error.
8. convergence: A list containing (i) a logical variable if convergence was achieved and (ii) the number of executed ADMM algorithm iterations.
9. call: The function call.

Furthermore, pagfl objects can be used in a variety of useful generic methods like summary(), fitted(), resid(), df.residual, formula, and coef().

estim_fit <- fitted(estim)

Selecting a $\lambda$ value a priori can be tricky. For instance, it seems like lambda = 20 is too high since the number of groups $K$ is underestimated. We suggest iterating over a comprehensive range of candidate values to trace out the correct model. To specify a suitable grid, create a logarithmic sequence ranging from 0 to a penalty parameter that induces an entirely homogeneous model (i.e., $\widehat{K} = 1$). The resulting $\lambda$ grid vector can be passed in place of any specific value, and a BIC IC selects the best-fitting parameter.

Furthermore, it is also possible to supply a data.frame with named variables and choose a specific formula that selects the variables in that data.frame. If the explanatory variables in X are named, these names also appear in the output.

colnames(data)[-1] <- c("a", "b")

lambda_set <- exp(log(10) * seq(log10(1e-4), log10(10), length.out = 10))
estim_set <- pagfl(y ~ a + b, data = data, n_periods = 150, lambda = lambda_set)
summary(estim_set)
#> Call:
#> pagfl(formula = y ~ a + b, data = data, n_periods = 150, lambda = lambda_set)
#> 
#> Balanced panel: N = 20, T = 150, obs = 3000
#> 
#> Convergence reached:
#> TRUE (51 iterations)
#> 
#> Information criterion:
#>        IC    lambda 
#> 1.1287693 0.2154435 
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -3.47858 -0.66283 -0.02688  0.72880  3.77812 
#> 
#> 3 groups:
#>  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 
#>  1  1  2  3  1  3  3  2  3  3  2  2  2  1  1  1  3  1  2  3 
#> 
#> Coefficients:
#>                 a        b
#> Group 1 -0.95114  1.61719
#> Group 2 -0.49489 -1.23534
#> Group 3  0.24172  1.61613
#> 
#> Residual standard error: 1.03695 on 2978 degrees of freedom
#> Mean squared error 1.06738
#> Multiple R-squared: 0.72236, Adjusted R-squared: 0.7204

When, as above, the specific estimation method is left unspecified, pagfl defaults to penalized Least Squares (PLS) method = 'PLS' (Mehrabani, 2023, sec. 2.2). PLS is very efficient but requires weakly exogenous regressors. However, even endogenous predictors can be accounted for by employing a penalized Generalized Method of Moments (PGMM) routine in combination with exogenous instruments $\boldsymbol{Z}$.

Specify a slightly more elaborate endogenous and dynamic panel data set and apply PGMM. When encountering a dynamic panel data set, we recommend using a Jackknife bias correction, as proposed by Dhaene and Jochmans (2015).

# Generate a panel where the predictors X correlate with the cross-sectional innovation, 
# but can be instrumented with q = 3 variables in Z. Furthermore, include GARCH(1,1) 
# innovations, an AR lag of the dependent variable, and specific group sizes
sim_endo <- sim_DGP(N = 20, n_periods = 200, p = 2, n_groups = 3, group_proportions = c(0.3, 0.3, 0.4), 
error_spec = 'GARCH', q = 2, dynamic = FALSE)
data_endo <- sim_endo$data
Z <- sim_endo$Z

# Note that the method PGMM and the instrument matrix Z needs to be passed
estim_endo <- pagfl(y ~ ., data = data_endo, n_periods = 200, lambda = 2, method = "PGMM", Z = Z, bias_correc = TRUE, max_iter = 50e3)
summary(estim_endo)
#> Call:
#> pagfl(formula = y ~ ., data = data_endo, n_periods = 200, lambda = 2, 
#>     method = "PGMM", Z = Z, bias_correc = TRUE, max_iter = 50000)
#> 
#> Balanced panel: N = 20, T = 200, obs = 3980
#> 
#> Convergence reached:
#> TRUE (14632 iterations)
#> 
#> Information criterion:
#>       IC   lambda 
#> 1.971293 2.000000 
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -4.87011 -0.90055  0.01193  0.90767  5.54203 
#> 
#> 3 groups:
#>  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 
#>  1  2  3  3  3  2  2  3  1  3  2  2  1  1  2  1  3  1  3  3 
#> 
#> Coefficients:
#>                X1       X2
#> Group 1  0.55337 -1.22836
#> Group 2 -0.88484 -0.89231
#> Group 3  1.60547 -1.43718
#> 
#> Residual standard error: 1.38812 on 3958 degrees of freedom
#> Mean squared error 1.91621
#> Multiple R-squared: 0.87079, Adjusted R-squared: 0.8701

Furthermore, pagfl lets you select a minimum group size, adjust the efficiency vs. accuracy trade-off of the iterative estimation algorithm, and modify a list of further settings. Visit the documentation ?pagfl() for more information.

The Time-varying PAGFL

The development version of the package also includes the functions sim_tv_DGP()and tv_pagfl(), which generate and estimate a grouped panel data models with the time-varying coefficients $\beta_{it} = \beta_i \left( \frac{t}{T} \right)$. Just like in the static case, the functional coefficients admit to a group structure $\beta_{it} = \sum_{k = 1}^K \alpha_k \left( \frac{t}{T} \right) 1 {i \in G_k }$. Following Su et al. (2019), the time-varying coefficients are estimated using polynomial B-spline functions employing a penalized sieve estimation (PSE).

# Simulate a time-varying panel with a trend and a group pattern
N <- 20
n_periods <- 100
tv_sim <- sim_tv_DGP(N = N, n_periods = n_periods, sd_error = 1, intercept = TRUE, p = 1)
tv_data <- tv_sim$data

tv_estim <- tv_pagfl(y ~ 1, data = tv_data, n_periods = n_periods, lambda = 5)
summary(tv_estim)
#> Call:
#> tv_pagfl(formula = y ~ 1, data = tv_data, n_periods = n_periods, 
#>     lambda = 5)
#> 
#> Balanced panel: N = 20, T = 100, obs = 2000
#> 
#> Convergence reached:
#> TRUE (212 iterations)
#> 
#> Information criterion:
#>      IC  lambda 
#> 1.16747 5.00000 
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -3.57761 -0.68826  0.00820  0.70118  3.40708 
#> 
#> 3 groups:
#>  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 
#>  1  1  1  2  2  2  1  3  3  3  2  2  3  1  3  1  2  3  2  3 
#> 
#> Residual standard error: 1.02901 on 1974 degrees of freedom
#> Mean squared error 1.04509
#> Multiple R-squared: 0.74213, Adjusted R-squared: 0.73886

tv_pagfl() returns an object of class tvpagfl which contains

1. model: A data.frame containing the dependent and explanatory variables as well as individual and time indices (if provided).
2. coefficients: A list holding (i) a $T \times p^{(1)} \times \hat{K}$ array of the post-Lasso group-specific functional coefficients and (ii) a $K \times p^{(2)}$ matrix of time-constant parameter estimates (when running a mixed time-varying panel data model).
3. groups: A list containing (i) the total number of groups $\hat{K}$ and (ii) a vector of estimated group memberships $(\hat{g}_1, \dots, \hat{g}_N)$, where $\hat{g}_i = k$ if $i$ is assigned to group $k$.
4. residuals: A vector of residuals of the demeaned model.
5. fitted: A vector of fitted values of the demeaned model.
6. args: A list of additional arguments.
7. IC: A list containing (i) the value of the IC, (ii) the employed tuning parameter $\lambda$, and (iii) the mean squared error.
8. convergence: A list containing (i) a logical variable if convergence was achieved and (ii) the number of executed ADMM algorithm iterations.
9. call: The function call.

Again, tvpagfl objects have generic summary(), fitted(), resid(), df.residual, formula, and coef() methods.

In empirical applications, it is commonplace to encounter unbalanced panel data sets. In such instances, time-varying coefficient functions can be estimated nonetheless. The nonparametric spline functions simply interpolate missing periods. However, when using unbalanced datasets it is required to provide explicit indicator variables that declare the cross-sectional individual and time period each observation belongs to.

Lets delete 30% of observations, add indicator variables, and run tv_pagfl() again.

# Draw some observations to be omitted
delete_index <- as.logical(rbinom(n = N * n_periods, prob = 0.7, size = 1))
# Construct cross-sectional and time indicator variables
tv_data$i_index <- rep(1:N, each = n_periods)
tv_data$t_index <- rep(1:n_periods, N)
# Delete some observations
tv_data <- tv_data[delete_index,]
# Apply the time-varying PAGFL to an unbalanced panel
tv_estim_unbalanced <- tv_pagfl(y ~ 1, data = tv_data, index = c("i_index", "t_index"), lambda = 5)
summary(tv_estim_unbalanced)
#> Call:
#> tv_pagfl(formula = y ~ 1, data = tv_data, index = c("i_index", 
#>     "t_index"), lambda = 5)
#> 
#> Unbalanced panel: N = 20, T = 64-75, obs = 1379
#> 
#> Convergence reached:
#> TRUE (950 iterations)
#> 
#> Information criterion:
#>     IC lambda 
#> 1.1915 5.0000 
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -3.43491 -0.69055 -0.00812  0.68488  3.63894 
#> 
#> 3 groups:
#>  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 
#>  1  1  1  2  2  2  1  2  3  3  2  2  3  1  3  1  2  3  2  2 
#> 
#> Residual standard error: 1.04387 on 1353 degrees of freedom
#> Mean squared error 1.06912
#> Multiple R-squared: 0.73683, Adjusted R-squared: 0.73197

Furthermore, tv_pagfl lets you specify a lot more optionalities than shown here. For example, it is possible to adjust the polyomial degree and the number of interior knots in the spline basis system, or estimate a panel data model with a mix of time-varying and time-constant coefficients. See ?tv_pagfl() for details.

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

• Schumaker, L. (2007). Spline functions: basic theory. Cambridge university press. DOI: 10.1017/CBO9780511618994

• Su, L., Wang, X., & Jin, S. (2019). Sieve estimation of time-varying panel data models with latent structures. Journal of Business & Economic Statistics, 37(2), 334-349. DOI: 10.1080/07350015.2017.1340299