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.
The csdm() interface estimates heterogeneous panel data
models with optional cross-sectional augmentation and dynamic structure.
A baseline heterogeneous panel model is:
\[ y_{it} = \alpha_i + \beta_i' x_{it} + u_{it}, \qquad i = 1, \ldots, N\; t = 1, \ldots, T \]
where:
The inner product \(\beta_i'
x_{it}\) is scalar-valued. Heterogeneous slopes allow each unit
to respond differently to the regressors. In many applications,
cross-sectional dependence arises because the error term contains
unobserved common factors. The estimators implemented in
csdm() differ in how they handle this dependence and
whether they allow for dynamic adjustment.
The Mean Group estimator fits separate regressions for each unit and averages the resulting coefficients:
\[ \hat{\beta}_{MG} = \frac{1}{N}\sum_{i=1}^N \hat{\beta}_i \]
Key idea: Estimation is performed unit by unit, with no pooling of slope coefficients across cross-sectional units.
Interpretation:
Properties:
Use case: A natural benchmark when the main concern is heterogeneous slopes and no explicit factor structure is imposed.
The CCE estimator augments each unit regression with cross-sectional averages to proxy unobserved common factors:
\[ y_{it} = \alpha_i + \beta_i' x_{it} + \gamma_i' \bar{z}_t + v_{it} \]
where ({z}_t) collects the cross-sectional averages specified through
csdm_csa(), for example
\[ \bar{z}_t = (\bar{y}_t, \bar{x}_t), \qquad \bar{x}_t = \frac{1}{N}\sum_{i=1}^N x_{it}, \qquad \bar{y}_t = \frac{1}{N}\sum_{i=1}^N y_{it}. \]
Key idea: Cross-sectional averages serve as proxies for latent common factors that induce dependence across units.
Interpretation:
Properties:
csaUse case: When dependence across units is believed to reflect common unobserved factors.
The DCCE estimator extends CCE to dynamic settings by including lagged dependent variables, optional distributed lags of regressors, and lagged cross-sectional averages:
\[ y_{it} = \alpha_i + \sum_{p=1}^{P} \phi_{ip} y_{i,t-p} + \sum_{q=0}^{Q} \beta_{iq}' x_{i,t-q} + \sum_{s=0}^{S} \delta_{is}' \bar{z}_{t-s} + e_{it} \]
where the dynamic structure is controlled through
csdm_lr() and the cross-sectional averages and their lags
are controlled through csdm_csa().
Key idea: Dynamics are introduced directly in the unit equation, while lagged cross-sectional averages help absorb common factor dependence over time.
Interpretation:
Properties:
Use case: When the outcome is persistent over time and cross-sectional dependence remains important.
In the current csdm() implementation,
model = "cs_ardl" is obtained by first estimating a
cross-sectionally augmented ARDL-style regression in levels, using the
same dynamic specification as model = "dcce", and then
transforming the estimated unit-specific coefficients into adjustment
and long-run parameters.
The underlying unit-level regression is
\[ y_{it} = \alpha_i + \sum_{p=1}^{P} \phi_{ip} y_{i,t-p} + \sum_{q=0}^{Q} \beta_{iq}' x_{i,t-q} + \sum_{s=0}^{S} \omega_{is}' \bar{z}_{t-s} + e_{it} \]
From this dynamic specification, the implied error-correction form is
\[ \Delta y_{it} = \alpha_i + \varphi_i \left( y_{i,t-1} - \theta_i' x_{i,t-1} \right) + \sum_{j=1}^{P-1} \lambda_{ij} \Delta y_{i,t-j} + \sum_{j=0}^{Q-1} \psi_{ij}' \Delta x_{i,t-j} + \sum_{s=0}^{S} \tilde{\omega}_{is}' \bar{z}_{t-s} + e_{it} \]
where the dynamic structure is controlled through
csdm_lr() and the cross-sectional averages are supplied
through csdm_csa().
Key idea: cs_ardl reports the implied
short-run and long-run quantities from a cross-sectionally augmented
ARDL fit.
Interpretation:
Properties:
Use case: When the objective is to study long-run relationships together with heterogeneous short-run adjustment in panels affected by common factors.
Two helper specifications control the main extensions in
csdm():
csdm_csa() defines which variables enter as
cross-sectional averages and how many lags of those averages are
includedcsdm_lr() defines the dynamic or long-run structure,
such as lagged dependent variables and distributed lagsThis design keeps the estimation interface consistent across the four estimators while allowing the model specification to vary by application.
| Estimator | Heterogeneous Slopes | Cross-Sectional Averages | Dynamics | Long-Run Structure |
|---|---|---|---|---|
| MG | Yes | No | No | No |
| CCE | Yes | Yes | No | No |
| DCCE | Yes | Yes | Yes | No |
| CS-ARDL | Yes | Yes | Yes | Yes |
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.
Chudik, A., & Pesaran, M. H. (2013). Large panel data models with cross-sectional dependence: A survey [Globalization Institute Working Papers]. Federal Reserve Bank of Dallas, (153).
Chudik, A., & Pesaran, M. H. (2015). Common correlated effects estimation of heterogeneous dynamic panel data models with weakly exogenous regressors. Journal of Econometrics, 188(2), 393–420.
Ditzen, J. (2018). Estimating dynamic common-correlated effects in STATA. The STATA Journal, 18(3), 585–617. https://doi.org/10.1177/1536867X1801800306
Fan, J., Liao, Y., & Yao, J. (2015). Power enhancement in high-dimensional cross-section tests. Econometrica, 83(4), 1497–1541.
Juodis, A., & Reese, S. (2021). The incidental parameters problem in testing for remaining cross-sectional correlation. Journal of Business and Economic Statistics, 40(3), 1191–1203.
Pesaran, M. H. (2006). Estimation and inference in large heterogeneous panels with multifactor error structure. Econometrica, 74(4), 967–1012.
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. (2015). Testing weak cross-sectional dependence in large panels. Econometric Reviews, 34(6-10), 1089–1117.
Pesaran, M. H. (2021). General diagnostic tests for cross-sectional dependence in panels. Empirical Economics, 60(1), 13–50.
Pesaran, M. H., & Smith, R. (1995). Estimating long-run relationships from dynamic heterogeneous panels. Journal of Econometrics, 68(1), 79–113.
Pesaran, M. H., & Xie, Y. (2021). A bias-corrected CD test for error cross-sectional dependence in panel models. Econometric Reviews, 41(6), 649–677.