The csdm package implements econometric methods for
panel data with cross-sectional dependence (CSD). In many applications,
observations across units (e.g., countries, firms, regions) are not
independent—macroeconomic shocks, trade relationships, or spillovers
create correlation across cross-sectional units. The csdm
package provides robust estimators that account for this dependence
structure, plus diagnostic tests to detect and characterize it.
This vignette demonstrates four core estimation methods and related inference tools on real panel data from the Penn World Table (PWT).
Consider a panel model with \(T\) time periods and \(N\) cross-sectional units (e.g., countries):
\[y_{it} = \alpha_i + \beta_i x_{it} + u_{it}, \quad i = 1, \ldots, N; \quad t = 1, \ldots, T\]
where: - \(y_{it}\) is the outcome variable for unit \(i\) at time \(t\) - \(\alpha_i\) is a unit-specific intercept - \(\beta_i\) is a unit-specific slope (heterogeneous across units) - \(x_{it}\) is explanatory variable(s) - \(u_{it}\) is the idiosyncratic error term
The key feature is heterogeneity in slopes (\(\beta_i\) varies by unit), which allows each unit to have its own relationship between \(x\) and \(y\). Four estimators are available to fit this model under different assumptions about cross-sectional dependence.
The Mean Group (MG) estimator fits unit-specific regressions separately and averages the results:
\[\hat{\beta}_{MG} = \frac{1}{N} \sum_{i=1}^{N} \hat{\beta}_i\]
Interpretation: The MG coefficient is the simple average of individual unit slopes. It is consistent under mild regularity conditions and allows arbitrary cross-sectional dependence in errors \(u_{it}\).
Use case: When dependence is present but you only care about average effects. MG is robust to forms of CSD that would break other methods.
The Common Correlated Effects (CCE) estimator augments the model with cross-sectional averages of regressors \(\bar{x}_t = \frac{1}{N} \sum_{i=1}^{N} x_{it}\):
\[y_{it} = \alpha_i + \beta_i x_{it} + \gamma_i \bar{x}_t + \gamma_i \bar{y}_t + v_{it}\]
By including these cross-sectional mean terms, the estimator controls for common factors (unobserved shocks that affect all units similarly). The CCE approach is more efficient than MG when common factor structure is strong, while retaining robustness to CSD.
Interpretation: After accounting for common shocks, the \(\beta_i\) coefficients represent unit-specific sensitivities net of factor loadings \(\gamma_i\).
Use case: When cross-sectional dependence arises primarily from common shocks or latent factors.
The Dynamic Common Correlated Effects (DCCE) estimator extends CCE to include lagged dependent variable:
\[y_{it} = \alpha_i + \lambda_i y_{it-1} + \beta_i x_{it} + \gamma_i \bar{x}_t + \gamma_i \bar{y}_t + v_{it}\]
where \(\lambda_i\) is the unit-specific autoregressive coefficient. DCCE is ideal for dynamic panel models (e.g., when studying persistence of outcomes over time).
Interpretation: \(\lambda_i\) captures dynamic adjustment within units, \(\beta_i\) measures the long-run effect after accounting for dynamics, and \(\gamma_i\) adjusts for common factors.
Use case: When the outcome has substantial persistence (lagged effects) and cross-sectional dependence is suspected.
The CS-ARDL model extends the ARDL framework with cross-sectional augmentation:
\[\Delta y_{it} = \alpha_i + \lambda_i (y_{it-1} - \theta_i x_{it-1}) + \beta_i \Delta x_{it} + \hat{\gamma}_i \Delta \bar{x}_t + v_{it}\]
This model combines autoregressive and distributed lag dynamics. It separates short-run effects (\(\beta_i\)) from long-run cointegrating relationships (\(\theta_i\)), all while controlling for common factors.
Interpretation: - \(\theta_i\) is the long-run equilibrium relationship (cointegrating coefficient) - \(\beta_i\) is the short-run adjustment to shocks - \(\lambda_i\) governs speed of reversion to equilibrium
Use case: When studying long-run relationships in non-stationary panels with complex short-run dynamics.
To install the csdm package from CRAN, run:
install.packages("csdm")To install the latest development version from GitHub, run:
install.packages("remotes")
remotes::install_github("Macosso/csdm")All models are fitted with csdm(), which automatically
detects the input structure and applies the appropriate methodology. The
key arguments are id and time to specify the
cross-sectional and time-period identifiers, and model to
choose the estimator. For CCE and DCCE, additional arguments
(csa and lr) specify treatment of
cross-sectional averages and dynamics.
# MG: Separate regression per country, then average coefficients
fit_mg <- csdm(
log_rgdpo ~ log_hc + log_ck + log_ngd,
data = df,
id = "id",
time = "year",
model = "mg"
)
print(fit_mg)
summary(fit_mg)# CCE: Add cross-sectional means to control for common shocks
fit_cce <- csdm(
log_rgdpo ~ log_hc + log_ck + log_ngd,
data = df,
id = "id",
time = "year",
model = "cce",
csa = csdm_csa(vars = c("log_rgdpo", "log_hc", "log_ck", "log_ngd"))
)
print(fit_cce)
summary(fit_cce)# DCCE: Include dynamics and cross-sectional means
# Use lagged dependent variable to capture dynamic adjustment
fit_dcce <- csdm(
log_rgdpo ~ log_hc + log_ck + log_ngd,
data = df,
id = "id",
time = "year",
model = "dcce",
csa = csdm_csa(
vars = c("log_rgdpo", "log_hc", "log_ck", "log_ngd"),
lags = 3
),
lr = csdm_lr(type = "ardl", ylags = 1, xdlags = 0)
)
print(fit_dcce)
summary(fit_dcce)# CS-ARDL: Separate short-run and long-run dynamics
# Includes lagged dependent and lagged regressors
fit_csardl <- csdm(
log_rgdpo ~ log_hc + log_ck + log_ngd,
data = df,
id = "id",
time = "year",
model = "cs_ardl",
csa = csdm_csa(
vars = c("log_rgdpo", "log_hc", "log_ck", "log_ngd"),
lags = 3
),
lr = csdm_lr(type = "ardl", ylags = 1, xdlags = 1)
)
print(fit_csardl)
summary(fit_csardl)After fitting a model, we can test whether residuals exhibit cross-sectional dependence using the Pesaran CD test and related variants. CSD tests detect whether residuals \(u_{it}\) are correlated across units—a key assumption violation that can bias standard errors.
All CD tests have null hypothesis: residuals are cross-sectionally independent.
The Pesaran CD statistic is:
\[CD = \sqrt{\frac{2}{N(N-1)}} \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} \hat{\rho}_{ij} \sqrt{T}\]
where \(\hat{\rho}_{ij}\) is the cross-sectional correlation between residuals of units \(i\) and \(j\). The test statistic is approximately standard normal under the null.
Interpretation: Large \(|CD|\) rejects independence; both positive and negative correlations are flagged. This is the most general CD test and works even when \(N\) is fixed and \(T \to \infty\).
The CDw statistic uses unit-level random sign flips to form a wild-type version of the CD test:
\[CD_w = \sqrt{\frac{2}{N(N-1)}} \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} w_i w_j \, \hat{\rho}_{ij} \sqrt{T},\]
where \((w_1,\ldots,w_N)\) are independent random weights with \(w_i \in \{-1,1\}\) applied at the unit level. This statistic can be used in randomization-based or simulation-based inference procedures.
CDw+ applies an alternative unit-level random sign-flip scheme:
\[CD_w^+ = \sqrt{\frac{2}{N(N-1)}} \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} w_i^{(+)} w_j^{(+)} \, \hat{\rho}_{ij} \sqrt{T},\]
where \((w_1^{(+)},\ldots,w_N^{(+)})\) are again independent random weights with \(w_i^{(+)} \in \{-1,1\}\) (typically a separate draw from that used for \(CD_w\)).
The CD* statistic is a semiparametric refinement for large \(N\) and \(T\):
\[CD^* = \frac{1}{\sqrt{N(N-1)}} \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} (\hat{\rho}_{ij}^2 - \tau_T)\]
where \(\tau_T\) is a variance adjustment. FLY-type tests are designed for large panel dimensions and provide robustness against certain forms of weak cross-sectional dependence.
The cd_test() function accepts the fitted model and
computes all test variants. Tests use a random seed to
initialize pseudo-random computations (for cdw and
cdw+); setting a seed ensures reproducibility
of numerical results across runs.
# Test MG residuals for CSD
cd_mg <- cd_test(fit_mg, type = "CD")
print(cd_mg)
# Test CCE residuals for CSD
set.seed(1234)
cd_cce <- cd_test(fit_cce, type = "all")
print(cd_cce)Interpreting Results:
In practice, models that do not account for cross-sectional dependence (like MG without augmentation) typically show significant CD test rejections, justifying the use of CSD-robust methods like CCE and DCCE.
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Fan, J., Liao, Y., & Yao, J. (2015). Power-enhanced simultaneous test for high-dimensional covariance matrix. Journal of the American Statistical Association, 110(510), 325–337.
Juodis, A., & Reese, S. (2022). The role of the N/T ratio in large N, large T panel time-series models. Econometric Reviews, 41(2), 221–261.
Pesaran, M. H. (2007). A simple unit root test in the presence of cross-section dependence. Journal of Applied Econometrics, 22(2), 265–312.
Pesaran, M. H., & Xie, Y. (2021). A bias-adjusted LM test of error cross-section independence. Econometric Reviews, 40(1), 7–24.