Cross-sectional dependence (CD) tests for panel residuals

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("pairwise", "drop.incomplete.times"),
  ...
)

# 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("pairwise", "drop.incomplete.times"),
  ...
)

# 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. Seeded calls restore the caller's RNG state; NULL uses the current RNG stream.
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" removes time periods with any missing observations to create a balanced panel for CD*; "pairwise" (default) uses pairwise correlations for CD; unbalanced CDw/CDw+ requests error and CD* warns.
x: An object of class cd_test.
digits: Number of digits to print (default 3).

Value

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

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, 2022): Independent Rademacher weights $w_i \in \{-1,1\}$ are applied by unit. For a balanced residual panel, the statistic is $$CD_W = \left(\frac{1}{NT}\sum_{i,t}w_i^2 e_{it}^2\right)^{-1} \sqrt{\frac{2}{TN(N-1)}} \sum_t\sum_{i<j}w_i e_{it}w_j e_{jt}.$$ The first factor is the inverse pooled residual variance. One set of random weights is drawn per call; use seed for reproducibility.
CDw+ (Juodis and Reese, 2022; Fan, Liao, and Yao, 2015): The power-enhanced statistic is $$CD_{W+} = CD_W + \sum_{i<j}|\rho_{ij}| 1\left\{|\rho_{ij}| > 2\sqrt{\log(N)/T}\right\}.$$ Here $\rho_{ij}$ is the ordinary residual correlation, without multiplication by $\sqrt{T}$. The nonnegative screening term is asymptotically zero under the conditions of the null hypothesis.
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.

CD* requires a nondegenerate bias-correction denominator. Near-zero denominators can produce severe size distortions, including proportional loading/error-scale designs after standardization. Numerical rank checks do not establish the validity of the asymptotic approximation.

Missing data and balance

Time periods containing no finite residual for any retained unit are outside the effective residual sample and are always removed before balance is assessed. Partially observed periods are handled according to na.action.

CD: Uses pairwise-complete observations by default. Each pairwise correlation uses available overlaps.
CDw, CDw+: Require a balanced sample; explicitly select complete times if desired.
CD*: Requires a balanced panel. Explicitly setting 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.

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")
#> Cross-sectional dependence tests
#> N = 10, T = 10
#> 
#>        statistic p.value
#> CD        -1.325   0.185
#> CDw       -1.184   0.237
#> CDw+      -1.184   0.237
#> CDstar     0.081   0.935
cd_test(E_dep, type = "CD")
#> Cross-sectional dependence tests
#> N = 10, T = 10
#> 
#>    statistic p.value
#> CD    21.213   0.000

# Specific test with parameters
cd_test(E_indep, type = "CDstar", n_pc = 2)
#> Cross-sectional dependence tests
#> N = 10, T = 10
#> 
#>        statistic p.value
#> CDstar    -0.303   0.762

# 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")
#> Cross-sectional dependence tests
#> N = 10, T = 38
#> 
#>        statistic p.value
#> CD        -3.581   0.000
#> CDw        2.322   0.020
#> CDw+       4.464   0.000
#> CDstar    -3.153   0.002