Package 'csdm'

Cross-sectional dependence (CD) tests for panel residuals

Description

Computes Pesaran CD, CDw, CDw+, and CD* tests for cross-sectional dependence in panel residuals. The implementation supports residual matrices or fitted csdm_fit objects and provides consistent handling of unbalanced panels.

Usage

cd_test(object, ...)

## Default S3 method:
cd_test(
  object,
  type = c("CD", "CDw", "CDw+", "CDstar", "all"),
  n_pc = 4L,
  seed = NULL,
  min_overlap = 2L,
  na.action = c("drop.incomplete.times", "pairwise"),
  ...
)

## S3 method for class 'csdm_fit'
cd_test(
  object,
  type = c("CD", "CDw", "CDw+", "CDstar", "all"),
  n_pc = 4L,
  seed = NULL,
  min_overlap = 2L,
  na.action = c("drop.incomplete.times", "pairwise"),
  ...
)

## S3 method for class 'cd_test'
print(x, digits = 3, ...)

Arguments

object	A csdm_fit model object or a numeric matrix of residuals (N x T).
...	Additional arguments passed to methods.
type	Which test(s) to compute: one of "CD", "CDw", "CDw+", "CDstar", or "all" (default: "CD").
n_pc	Number of principal components for CD* (default 4).
seed	Integer seed for weight draws in CDw/CDw+ (default NULL = no seed set).
min_overlap	Minimum number of overlapping time periods required for a unit pair to be included in CD/CDw/CDw+ (default 2).
na.action	How to handle missing data: "drop.incomplete.times" (default) removes time periods with any missing observations to create a balanced panel for CD*; "pairwise" uses pairwise correlations for CD/CDw/CDw+ and warns for CD*.
x	An object of class cd_test.
digits	Number of digits to print (default 3).

Details

Notation

Let $E$ be the residual matrix with $N$ cross-sectional units and $T$ time periods. For each unit pair $(i,j)$, let $T_{ij}$ be the number of overlapping time periods and $\rho_{ij}$ the pairwise correlation.

Test statistics

CD (Pesaran, 2015)

$CD = \sqrt{\frac{2}{N(N-1)}} \sum_{i<j} \sqrt{T_{ij}} \, \rho_{ij}$

CDw (Juodis and Reese, 2021)

Random sign flips $w_i \in \{-1,1\}$ are applied to residuals before computing correlations. The statistic is CD applied to the sign-flipped data.

CDw+ (Fan, Liao, and Yao, 2015)

Power enhancement adds a sparse thresholding term to CDw. The threshold is

$c_N = \sqrt{\frac{2 \log(N)}{T}}$

and the power term sums $\sqrt{T_{ij}} |\rho_{ij}|$ for pairs exceeding the threshold.

CD* (Pesaran and Xie, 2021)

CD is computed on residuals after removing n_pc principal components from $E$. This provides a bias-corrected test under multifactor errors.

Missing data and balance

CD, CDw, CDw+

Always use pairwise-complete observations. Each pairwise correlation uses available overlaps.

CD*

Requires a balanced panel. By default, na.action = "drop.incomplete.times" removes any time period with missing observations. With na.action = "pairwise", CD* returns NA and a warning when missing values are present.

Value

An object of class cd_test with fields tests, type, N, T, na.action, and call. The tests list contains one or more test results, each with statistic and p.value.

References

Pesaran MH (2015). “Testing weak cross-sectional dependence in large panels.” Econometric Reviews, 34(6-10), 1089–1117.

Pesaran MH (2021). “General diagnostic tests for cross-sectional dependence in panels.” Empirical Economics, 60(1), 13–50.

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.

Fan J, Liao Y, Yao J (2015). “Power Enhancement in High-Dimensional Cross-Section Tests.” Econometrica, 83(4), 1497–1541.

Pesaran MH, Xie Y (2021). “A bias-corrected CD test for error cross-sectional dependence in panel models.” Econometric Reviews, 41(6), 649–677.

Examples

# Simulate independent and dependent panels
set.seed(1)
E_indep <- matrix(rnorm(100), nrow = 10)
E_dep <- matrix(rnorm(10), nrow = 10, ncol = 10, byrow = TRUE)

# Compute all tests
cd_test(E_indep, type = "all")
cd_test(E_dep, type = "all")

# Specific test with parameters
cd_test(E_indep, type = "CDstar", n_pc = 2)

# From a fitted csdm model
data(PWT_60_07, package = "csdm")
df <- PWT_60_07
ids <- unique(df$id)[1:10]
df_small <- df[df$id %in% ids & df$year >= 1970, ]
fit <- csdm(
  log_rgdpo ~ log_hc + log_ck + log_ngd,
  data = df_small,
  id = "id",
  time = "year",
  model = "cce",
  csa = csdm_csa(vars = c("log_rgdpo", "log_hc", "log_ck", "log_ngd"))
)
cd_test(fit, type = "all")

---

Extract model coefficients from a fitted csdm model

Description

Returns estimated mean-group coefficients from a csdm_fit object. For model = "cs_ardl", the returned vector includes short-run mean-group coefficients, the adjustment coefficient (named lr_<y>), and long-run coefficients when available.

Usage

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

Arguments

object	A fitted object of class csdm_fit.
...	Currently unused.

Value

A named numeric vector of estimated coefficients.

See Also

summary.csdm_fit(), vcov.csdm_fit()

---

Panel Model Estimation with Cross-Sectional Dependence

Description

Estimate heterogeneous panel data models with optional cross-sectional augmentation and dynamic structure. The interface supports Mean Group (MG), Common Correlated Effects (CCE), Dynamic CCE (DCCE), and Cross-Sectionally Augmented ARDL (CS-ARDL) estimators with a consistent specification workflow for cross-sectional averages, lag structure, and variance-covariance estimation.

Usage

csdm(
  formula,
  data,
  id,
  time,
  model = c("mg", "cce", "dcce", "cs_ardl", "cs_ecm", "cs_dl"),
  csa = csdm_csa(),
  lr = csdm_lr(),
  pooled = csdm_pooled(),
  trend = c("none", "unit", "pooled"),
  fullsample = FALSE,
  mgmissing = FALSE,
  vcov = csdm_vcov(),
  ...
)

Arguments

formula	Model formula of the form y ~ x1 + x2.
data	A data.frame (or plm::pdata.frame) containing the variables in formula.
id, time	Column names (strings) for the unit and time indexes. If data is a pdata.frame, these are taken from its index and the provided values are ignored.
model	Estimator to fit. One of "mg", "cce", "dcce", or "cs_ardl".
csa	Cross-sectional-average specification, created by csdm_csa().
lr	Long-run or dynamic specification, created by csdm_lr().
pooled	Pooled specification (reserved for future use), created by csdm_pooled().
trend	One of "none" or "unit" (adds a linear unit trend). "pooled" is reserved and not implemented.
fullsample	Logical; reserved for future extensions.
mgmissing	Logical; reserved for future extensions.
vcov	Variance-covariance specification, created by csdm_vcov().
...	Reserved for future extensions.

Details

Let $i = 1, \ldots, N$ index cross-sectional units and $t = 1, \ldots, T$ index time. A baseline heterogeneous panel model is

$y_{it} = \alpha_i + \beta_i^T x_{it} + u_{it}.$

Here $\alpha_i$ is a unit-specific intercept, $x_{it}$ is a vector of regressors, $\beta_i$ is a vector of unit-specific slopes, and $u_{it}$ is an error term that may exhibit cross-sectional dependence.

Cross-sectional averages are specified through csdm_csa() and dynamic or long-run structure is specified through csdm_lr(). This keeps the model interface consistent across estimators while allowing the degree of cross-sectional augmentation and lag structure to vary by application.

Implemented estimators

MG (Pesaran and Smith, 1995)

The Mean Group estimator fits separate regressions for each unit and averages the resulting coefficients:

$\hat{\beta}_{MG} = \frac{1}{N}\sum_{i=1}^N \hat{\beta}_i.$

This estimator accommodates slope heterogeneity but does not explicitly model cross-sectional dependence.

CCE (Pesaran, 2006)

Regressions are augmented with cross-sectional averages to proxy unobserved common factors:

$y_{it} = \alpha_i + \beta_i^T x_{it} + \gamma_i^T \bar{z}_{t} + v_{it}.$

A common choice is

$\bar{z}_t = (\bar{y}_t, \bar{x}_t),$

with

$\bar{x}_t = \frac{1}{N}\sum_{i=1}^N x_{it}, \qquad \bar{y}_t = \frac{1}{N}\sum_{i=1}^N y_{it}.$

More generally, $\bar{z}_t$ collects the cross-sectional averages specified in csa.

DCCE (Chudik and Pesaran, 2015)

Dynamic CCE extends CCE by allowing lagged dependent variables and lagged cross-sectional averages:

$y_{it} = \alpha_i + \sum_{p=1}^{P} \phi_{ip} y_{i,t-p} + \beta_i^T x_{it} + \sum_{q=0}^{Q} \delta_{iq}^T \bar{z}_{t-q} + e_{it}.$

In the package implementation, lagged dependent variables and distributed lags of regressors are controlled through lr, while contemporaneous and lagged cross-sectional averages are controlled through csa.

CS-ARDL (Chudik and Pesaran, 2015)

In the package 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 unit-specific coefficients into adjustment and long-run parameters.

The underlying unit-level regression is of the form

$y_{it} = \alpha_i + \sum_{p=1}^{P} \phi_{ip} y_{i,t-p} + \sum_{q=0}^{Q} \beta_{iq}^T x_{i,t-q} + \sum_{s=0}^{S} \omega_{is}^T \bar{z}_{t-s} + e_{it}.$

From this dynamic specification, the package recovers the implied error-correction form

$\Delta y_{it} = \alpha_i + \varphi_i \left(y_{i,t-1} - \theta_i^T x_{i,t-1}\right) + \sum_{j=1}^{P-1} \lambda_{ij} \Delta y_{i,t-j} + \sum_{j=0}^{Q-1} \psi_{ij}^T \Delta x_{i,t-j} + \sum_{s=0}^{S} \tilde{\omega}_{is}^T \bar{z}_{t-s} + e_{it},$

where $\varphi_i$ is the adjustment coefficient and $\theta_i$ is the implied long-run relationship. In the current implementation, these quantities are computed from the estimated lag polynomials rather than from a direct ECM regression.

Identification and assumptions

MG requires sufficient time-series variation within each unit.

CCE relies on cross-sectional averages acting as proxies for latent common factors, together with adequate cross-sectional and time dimensions.

DCCE additionally requires enough time periods to support lagged dependent variables, distributed lags, and lagged cross-sectional averages.

CS-ARDL requires sufficient time length for the distributed-lag structure and is intended for applications where both short-run dynamics and long-run relationships are of interest in the presence of common factors.

Value

An object of class csdm_fit containing estimated coefficients, residuals, variance-covariance estimates, model metadata, and diagnostics. Use summary(), coef(), residuals(), vcov(), and cd_test() to access standard outputs.

References

Pesaran MH, Smith R (1995). “Estimating long-run relationships from dynamic heterogeneous panels.” Journal of Econometrics, 68(1), 79–113.

Pesaran MH (2006). “Estimation and inference in large heterogeneous panels with multifactor error structure.” Econometrica, 74(4), 967–1012.

Chudik A, Pesaran MH (2015). “Common correlated effects estimation of heterogeneous dynamic panel data models with weakly exogenous regressors.” Journal of Econometrics, 188(2), 393–420.

Examples

library(csdm)
data(PWT_60_07, package = "csdm")
df <- PWT_60_07

# Keep examples fast but fully runnable
keep_ids <- unique(df$id)[1:10]
df_small <- df[df$id %in% keep_ids & df$year >= 1970, ]

# Mean Group (MG)
mg <- csdm(
  log_rgdpo ~ log_hc + log_ck + log_ngd,
  data = df_small, id = "id", time = "year", model = "mg"
)
summary(mg)

# Common Correlated Effects (CCE)
cce <- csdm(
  log_rgdpo ~ log_hc + log_ck + log_ngd,
  data = df_small, id = "id", time = "year", model = "cce",
  csa = csdm_csa(vars = c("log_rgdpo", "log_hc", "log_ck", "log_ngd"))
)
summary(cce)

# Dynamic CCE (DCCE)
dcce <- csdm(
  log_rgdpo ~ log_hc + log_ck + log_ngd,
  data = df_small, 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)
)
summary(dcce)

# CS-ARDL
cs_ardl <- csdm(
  log_rgdpo ~ log_hc + log_ck + log_ngd,
  data = df_small, 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 = 0)
)
summary(cs_ardl)

---

Specification: Cross-sectional averages (CSA)

Description

Specification: Cross-sectional averages (CSA)

Usage

csdm_csa(
  vars = "_all",
  lags = 0,
  scope = c("estimation", "global", "cluster"),
  cluster = NULL
)

Arguments

vars	Character. One of "_all", "_none", or a character vector of variable names.
lags	Integer. Either a scalar integer >= 0 applied to all CSA variables, or a named integer vector giving per-variable maximum lags.
scope	Character vector. One or more of c("estimation","global","cluster").
cluster	Reserved for future use.

Value

A spec object (list) used by csdm().

Examples

# Cross-sectional averages (CSA) configuration for DCCE
csa <- csdm_csa(
  vars = c("log_rgdpo", "log_hc", "log_ck", "log_ngd"),
  lags = 3
)
csa

---

Specification: Long-run configuration

Description

Specification: Long-run configuration

Usage

csdm_lr(
  vars = NULL,
  type = c("none", "ecm", "ardl", "csdl"),
  ylags = 0,
  xdlags = 0,
  options = list()
)

Arguments

vars	Reserved for future use.
type	One of c("none","ecm","ardl","csdl").
ylags	Integer >= 0. Within-unit lags of the dependent variable to include when supported by the chosen model/type.
xdlags	Integer >= 0. Scalar distributed lags to apply to each RHS regressor when supported by the chosen model/type.
options	Reserved for future use.

Value

A spec object (list) used by csdm().

Examples

# Long-run / dynamic configuration (ARDL-style lags)
lr <- csdm_lr(type = "ardl", ylags = 1)
lr

# Minimal end-to-end DCCE example (kept small for speed)
data(PWT_60_07, package = "csdm")
df <- PWT_60_07
keep_ids <- unique(df$id)[1:10]
df_small <- df[df$id %in% keep_ids & df$year >= 1970, ]
fit <- csdm(
  log_rgdpo ~ log_hc + log_ck + log_ngd,
  data = df_small,
  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)
)
summary(fit)

---

Specification: Pooled constraints (stub)

Description

Specification: Pooled constraints (stub)

Usage

csdm_pooled(vars = NULL, constant = FALSE, trend = FALSE)

Arguments

vars	Reserved for future use.
constant	Logical; pooled constant.
trend	Logical; pooled trend.

Value

A spec object (list) used by csdm().

---

Specification: Variance-covariance for MG output (stub)

Description

Specification: Variance-covariance for MG output (stub)

Usage

csdm_vcov(type = c("mg", "np", "nw", "wpn", "ols"), ...)

Arguments

type	One of c("mg","np","nw","wpn","ols").
...	Reserved for future use.

Value

A spec object (list) used by csdm().

---

Predict method for csdm models

Description

Produces fitted values (index "xb") when available, or returns model residuals. Prediction on new data is not yet implemented.

Usage

## S3 method for class 'csdm_fit'
predict(object, newdata = NULL, type = c("xb", "residuals"), ...)

Arguments

object	A fitted object of class csdm_fit.
newdata	Optional new data (not yet supported).
type	One of "xb" for fitted values or "residuals".
...	Currently unused.

Value

A numeric matrix of fitted values or residuals, depending on type.

See Also

residuals.csdm_fit(), summary.csdm_fit()

---

Compact print method for fitted csdm models

Description

Prints a concise overview of a fitted csdm_fit object, including the model type, formula, panel dimensions, and a coefficient table with standard errors when available.

Usage

## S3 method for class 'csdm_fit'
print(x, digits = 4, ...)

Arguments

x	A fitted object of class csdm_fit.
digits	Number of printed digits.
...	Currently unused.

Value

Invisibly returns x.

See Also

summary.csdm_fit(), coef.csdm_fit(), residuals.csdm_fit()

---

Print method for csdm summary objects

Description

Formats and prints a summary.csdm_fit object. Output adapts to model type and includes coefficient tables, selected goodness-of-fit diagnostics, and compact model metadata.

Usage

## S3 method for class 'summary.csdm_fit'
print(x, digits = 4, ...)

Arguments

x	A summary.csdm_fit object.
digits	Number of digits to print.
...	Further arguments passed to methods.

Details

The printout includes classic Pesaran CD diagnostics from the summary object. For a full CD diagnostic panel (CD, CDw, CDw+, CD*), use cd_test() on the fitted model.

Value

Invisibly returns x.

See Also

summary.csdm_fit(), cd_test()

---

Penn World Tables panel (93 countries, 1960-2007)

Description

A panel of 93 countries (unit id) observed annually over 1960-2007 (time/year), with the log-transformed variables used in xtdcce2-style examples.

Usage

PWT_60_07

Format

A data frame with 4464 rows and 6 variables:

id

Unit identifier (country id).

year

Time identifier (year, 1960-2007).

log_rgdpo

Log real GDP (output).

log_hc

Log human capital index.

log_ck

Log capital stock.

log_ngd

Log (net) government debt (or similar), used as a covariate/control.

Source

Penn World Table (PWT). This dataset is included as a small, convenient panel for examples and tests.

---

Extract residual matrix from a fitted csdm model

Description

Returns residuals as an $N x T$ matrix (rows are units, columns are time). This method is designed for panel diagnostics and downstream tools such as cd_test().

Usage

## S3 method for class 'csdm_fit'
residuals(object, type = c("e", "u"), ...)

Arguments

object	A fitted object of class csdm_fit.
type	Residual type. Currently only "e" is implemented.
...	Currently unused.

Value

A numeric matrix of residuals with dimensions $N x T$.

See Also

get_residuals(), cd_test(), predict.csdm_fit()

---

Summarize csdm model estimation results

Description

Computes post-estimation summaries for csdm_fit objects, including mean-group coefficient inference, model-level diagnostics, and model-specific summary tables (for example, short-run and long-run blocks for CS-ARDL).

Usage

## S3 method for class 'csdm_fit'
summary(object, digits = 4, ...)

Arguments

object	A fitted model object of class csdm_fit.
digits	Number of digits to print.
...	Further arguments passed to methods.

Details

Reported inference

For each coefficient $\hat\beta_k$, the summary reports standard errors, $z$-statistics, and two-sided normal-approximation p-values:

$z_k = \frac{\hat\beta_k}{\operatorname{se}(\hat\beta_k)}, \qquad p_k = 2\{1-\Phi(|z_k|)\}.$

Diagnostics

The printed summary shows the classic Pesaran CD diagnostic by default. Extended diagnostics (CDw, CDw+, CD*) are available through cd_test().

Value

An object of class summary.csdm_fit with core metadata (call/formula/model/N/T), coefficient tables, fit statistics, and model-specific components for printing and downstream inspection.

See Also

print.summary.csdm_fit(), cd_test(), coef.csdm_fit(), vcov.csdm_fit()

Examples

data(PWT_60_07, package = "csdm")
df <- PWT_60_07
ids <- unique(df$id)[1:10]
df_small <- df[df$id %in% ids & df$year >= 1970, ]
fit <- csdm(
  log_rgdpo ~ log_hc + log_ck + log_ngd,
  data = df_small,
  id = "id",
  time = "year",
  model = "cce",
  csa = csdm_csa(vars = c("log_rgdpo", "log_hc", "log_ck", "log_ngd"))
)
s <- summary(fit)
s

---

Extract coefficient covariance matrix from a fitted csdm model

Description

Extract coefficient covariance matrix from a fitted csdm model

Usage

## S3 method for class 'csdm_fit'
vcov(object, ...)

Arguments

object	A fitted object of class csdm_fit.
...	Currently unused.

Value

A numeric variance-covariance matrix aligned with coef(object) for models where this is available.

See Also

coef.csdm_fit(), summary.csdm_fit()

---