Estimate a time-varying panel data model with a latent group structure using the pairwise adaptive group fused lasso (time-varying PAGFL). The time-varying PAGFL jointly identifies the latent group structure and group-specific time-varying functional coefficients. The time-varying coefficients are modeled as polynomial B-splines. The function supports both static and dynamic panel data models.
tv_pagfl(
formula,
data,
index = NULL,
n_periods = NULL,
lambda,
d = 3,
M = floor(length(y)^(1/7) - log(p)),
min_group_frac = 0.05,
const_coef = NULL,
kappa = 2,
max_iter = 50000,
tol_convergence = 1e-10,
tol_group = 0.001,
rho = 0.04 * log(N * n_periods)/sqrt(N * n_periods),
varrho = 1,
verbose = TRUE,
parallel = TRUE,
...
)# S3 method for tvpagfl
summary(object, ...)
# S3 method for tvpagfl
formula(x, ...)
# S3 method for tvpagfl
df.residual(object, ...)
# S3 method for tvpagfl
print(x, ...)
# S3 method for tvpagfl
coef(object, ...)
# S3 method for tvpagfl
residuals(object, ...)
# S3 method for tvpagfl
fitted(object, ...)
An object of class tvpagfl holding
modela data.frame containing the dependent and explanatory variables as well as cross-sectional and time indices,
coefficientslet \(p^{(1)}\) denote the number of time-varying coefficients and \(p^{(2)}\) the number of time constant parameters. 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 post-Lasso estimates.
groupsa 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 \(k\),
residualsa vector of residuals of the demeaned model,
fitteda vector of fitted values of the demeaned model,
argsa list of additional arguments,
ICa list containing (i) the value of the IC, (ii) the employed tuning parameter \(\lambda\), and (iii) the MSE,
convergencea list containing (i) a logical variable if convergence was achieved and (ii) the number of executed ADMM algorithm iterations,
callthe function call.
An object of class tvpagfl has print, summary, fitted, residuals, formula, df.residual and coef S3 methods.
a formula object describing the model to be estimated.
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.
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 \(t\). The data is automatically sorted according to the variables in index, which may produce errors when the time index is a character variable. 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.
the number of observed time periods \(T\). If an index character vector is passed, this argument can be left empty. Default is Null.
the tuning parameter determining the strength of the penalty term. Either a single \(\lambda\) or a vector of candidate values can be passed. If a vector is supplied, a BIC-type IC automatically selects the best fitting \(\lambda\) value.
the polynomial degree of the B-splines. Default is 3.
the number of interior knots of the B-splines. If left unspecified, the default heuristic \(M = \text{floor}((NT)^{\frac{1}{7}} - \log(p))\) is used. Note that \(M\) does not include the boundary knots and the entire sequence of knots is of length \(M + d + 1\).
the minimum group cardinality as a fraction of the total number of individuals \(N\). In case a group falls short of this threshold, each of its members is allocated to one of the remaining groups according to the MSE. Default is 0.05.
a character vector containing the variable names of explanatory variables that enter with time-constant coefficients.
the a non-negative weight used to obtain the adaptive penalty weights. Default is 2.
the maximum number of iterations for the ADMM estimation algorithm. Default is \(5*10^4\).
the tolerance limit for the stopping criterion of the iterative ADMM estimation algorithm. Default is \(1*10^{-10}\).
the tolerance limit for within-group differences. Two individuals are assigned to the same group if the Frobenius norm of their coefficient vector difference is below this threshold. Default is \(1*10^{-3}\).
the tuning parameter balancing the fitness and penalty terms in the IC that determines the penalty parameter \(\lambda\). If left unspecified, the heuristic \(\rho = 0.07 \frac{\log(NT)}{\sqrt{NT}}\) of Mehrabani (2023, sec. 6) is used. We recommend the default.
the non-negative Lagrangian ADMM penalty parameter. For the employed penalized sieve estimation PSE, the \(\varrho\) value is trivial. We recommend the default 1.
logical. If TRUE, helpful warning messages are shown. Default is TRUE.
logical. If TRUE, certain operations are parallelized across multiple cores. Default is TRUE.
ellipsis
of class tvpagfl.
of class tvpagfl.
Paul Haimerl
Consider the grouped time-varying panel data model $$y_{it} = \gamma_i + \beta^\prime_{i} (t/T) x_{it} + \epsilon_{it}, \quad i = 1, \dots, N, \; t = 1, \dots, T,$$ 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 \(\beta_{i} (t/T)\) is subject to the latent group pattern $$\beta_i \left(\frac{t}{T} \right) = \sum_{k = 1}^K \alpha_k \left( \frac{t}{T} \right) \bold{1} \{i \in G_k \},$$ 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 \(k \neq j\), \(k = 1, \dots, K\).
The time-varying coefficient functions are estimated as polynomial B-splines using the penalized sieve-technique. To this end, let \(B(v)\) denote a \(M + d +1\) vector basis functions, where \(d\) denotes the polynomial degree and \(M\) the number of interior knots. Then, \(\beta_{i}(t/T)\) and \(\alpha_{k}(t/T)\) are approximated by forming linear combinations of the basis functions \(\beta_{i} (t/T) \approx \pi_i^\prime B(t/T)\) and \(\alpha_{i}(t/T) \approx \xi_k^\prime B(t/T)\), where \(\pi_i\) and \(\xi_i\) are \((M + d + 1) \times p\) coefficient matrices.
The explanatory variables are projected onto the spline basis system, which results in the \((M + d + 1)p \times 1\) vector \(z_{it} = x_{it} \otimes B(v)\). Subsequently, the DGP can be reformulated as $$y_{it} = \gamma_i + z_{it}^\prime \text{vec}(\pi_{i}) + u_{it},$$ where \(u_{it} = \epsilon_{it} + \eta_{it}\) and \(\eta_{it}\) reflects a sieve approximation error. We refer to Su et al. (2019, sec. 2) for more details on the sieve technique.
Inspired by Su et al. (2019) and Mehrabani (2023), the time-varying PAGFL jointly estimates the functional coefficients and the group structure by minimizing the criterion $$Q_{NT} (\bold{\pi}, \lambda) = \frac{1}{NT} \sum^N_{i=1} \sum^{T}_{t=1}(\tilde{y}_{it} - \tilde{z}_{it}^\prime \text{vec}(\pi_{i}))^2 + \frac{\lambda}{N} \sum_{i = 1}^{N - 1} \sum_{j > i}^N \dot{\omega}_{ij} \| \pi_i - \pi_j \|$$ with respect to \(\bold{\pi} = (\text{vec}(\pi_i)^\prime, \dots, \text{vec}(\pi_N)^\prime)^\prime\). \(\tilde{a}_{it} = a_{it} - T^{-1} \sum^{T}_{t=1} a_{it}\), \(a = \{y, z\}\) to concentrate out the individual fixed effects \(\gamma_i\). \(\lambda\) is the penalty tuning parameter and \(\dot{w}_{ij}\) denotes adaptive penalty weights which are obtained by a preliminary non-penalized estimation. \(\| \cdot \|\) represents the Frobenius norm. The solution criterion function is minimized via the iterative alternating direction method of multipliers (ADMM) algorithm proposed by Mehrabani (2023, sec. 5.1).
Two individuals are assigned to the same group if \(\| \text{vec} (\hat{\pi}_i - \hat{\pi}_j) \| \leq \epsilon_{\text{tol}}\), where \(\epsilon_{\text{tol}}\) is determined by tol_group. Subsequently, the number of groups follows as the number of distinct elements in \(\hat{\bold{\pi}}\). Given an estimated group structure, it is straightforward to obtain post-Lasso estimates \(\hat{\bold{\xi}}\) using group-wise least squares (see grouped_tv_plm).
We recommend identifying a suitable \(\lambda\) parameter by passing a logarithmically spaced grid of candidate values with a lower limit close to 0 and an upper limit that leads to a fully homogeneous panel. A BIC-type information criterion then selects the best fitting \(\lambda\) value.
In case of an unbalanced panel data set, the earliest and latest available observations per group define the start and end-points of the interval on which the group-specific time-varying coefficients are defined.
Mehrabani, A. (2023). Estimation and identification of latent group structures in panel data. Journal of Econometrics, 235(2), 1464-1482. tools:::Rd_expr_doi("10.1016/j.jeconom.2022.12.002").
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. tools:::Rd_expr_doi("10.1080/07350015.2017.1340299").
# Simulate a time-varying panel with a trend and a group pattern
set.seed(1)
sim <- sim_tv_DGP(N = 10, n_periods = 50, intercept = TRUE, p = 1)
df <- data.frame(y = c(sim$y))
# Run the time-varying PAGFL
estim <- tv_pagfl(y ~ ., data = df, n_periods = 50, lambda = 10, parallel = FALSE)
summary(estim)
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