Estimating heterogeneous panels with csdm

Scope

csdm estimates heterogeneous panel models in which unobserved common factors may generate dependence across units. The package implements:

mean group (MG) estimation (Pesaran & Smith, 1995);
static common correlated effects (CCE) (Pesaran, 2006);
dynamic CCE (DCCE) (Chudik & Pesaran, 2015); and
a cross-sectionally augmented ARDL fit with implied adjustment and long-run parameters.

These estimators share a mean-group structure: a separate regression is fitted for every eligible unit and the unit-level coefficients are averaged. CCE-based models augment those regressions with cross-sectional averages that proxy the latent common-factor space.

The package adopts parts of the model structure used by Stata’s xtdcce2 (Ditzen, 2018), but it does not claim complete command or option parity. Pooled restrictions, estimation weights, CS-DL, CS-ECM, alternative fit-level covariance estimators, and prediction on new data are currently unavailable.

Data and sample

The bundled PWT_60_07 data contain 93 countries observed annually from 1960 through 2007. The variables follow the original xtdcce2 example:

log_rgdpo: log real output;
log_hc: log human capital;
log_ck: log physical capital; and
log_ngd: log population growth plus a 5% break-even investment rate.

The 93 values of log_ngd in 1960 are missing because its construction uses a growth rate. The worked example starts in 1970 and uses 15 countries to keep the vignette quick to build.

data(PWT_60_07, package = "csdm")

keep_ids <- unique(PWT_60_07$id)[1:15]
dat <- subset(PWT_60_07, id %in% keep_ids & year >= 1970)

dim(dat)
#> [1] 570   6
range(dat$year)
#> [1] 1970 2007

The examples below are levels regressions. Calling them growth regressions would require a differenced dependent variable, which is not part of the formula used here.

Estimators

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

$y_{it}=\alpha_i+\boldsymbol{\beta}_i'\mathbf{x}_{it}+u_{it}.$

Mean group

MG estimates the equation separately for each unit and averages the identified unit coefficients:

$\widehat{\boldsymbol{\beta}}_{MG} =\frac{1}{N_e}\sum_{i\in\mathcal E}\widehat{\boldsymbol{\beta}}_i,$

where $\mathcal E$ is the common set of eligible units and $N_e$ is its size. The reported covariance is the cross-unit sample covariance of the coefficient vectors divided by $N_e$ . Inference uses a large- $N$ normal approximation.

MG permits slope heterogeneity but does not model common-factor dependence.

Common correlated effects

CCE augments each unit equation with cross-sectional averages:

$y_{it}=\alpha_i+\boldsymbol{\beta}_i'\mathbf{x}_{it} +\boldsymbol{\gamma}_i'\bar{\mathbf z}_t+e_{it},$

where $\bar{\mathbf z}_t$ commonly contains the averages of the dependent variable and regressors. Under the CCE assumptions, these averages span the relevant common-factor space and allow consistent estimation of the unit slopes (Pesaran, 2006). Adding averages does not by itself guarantee that the factor space is adequately represented; rank, dimensions, and the choice of averages still matter.

Dynamic CCE

DCCE adds dynamics and lags of the cross-sectional averages:

$y_{it}=\alpha_i +\sum_{p=1}^{P}\phi_{ip}y_{i,t-p} +\sum_{q=0}^{Q}\boldsymbol{\beta}_{iq}'\mathbf{x}_{i,t-q} +\sum_{s=0}^{S}\boldsymbol{\delta}_{is}'\bar{\mathbf z}_{t-s} +e_{it}.$

csdm_lr(type = "ardl", ylags = P, xdlags = Q) controls model lags, and csdm_csa(lags = S) controls lags of the averages. Dynamic mean-group estimates can have short- $T$ bias; CCE augmentation does not remove that bias.

CS-ARDL output

model = "cs_ardl" fits the same unit-level levels ARDL used by DCCE and then transforms its coefficients. For unit $i$ , define

$d_i=1-\sum_{p=1}^{P}\phi_{ip},\qquad \varphi_i=-d_i,\qquad \boldsymbol{\theta}_i= \frac{\sum_{q=0}^{Q}\boldsymbol{\beta}_{iq}}{d_i}.$

The package reports $\varphi_i$ as the adjustment coefficient and $\boldsymbol{\theta}_i$ as the long-run ratio before computing their mean-group summaries. Ratios are undefined when $d_i$ is numerically zero. The implementation also stores AR-root diagnostics, but it does not automatically discard unstable units.

These transformations do not fit a separate error-correction model and do not establish cointegration. The levels component is the fitted levels ARDL coefficient vector, rather than the complete transformed short-run ECM vector.

Specifying and fitting models

The formula contains the contemporaneous economic regressors. Two specification objects add CCE terms and dynamics:

csdm_csa(vars, lags) selects variables whose cross-sectional averages enter the model and their maximum lags;
csdm_lr(type, ylags, xdlags) selects lags of the dependent variable and regressors.

With vars = "_all", averages are constructed from the evaluated response and economic model-matrix columns, excluding intercepts and unit trends. An explicit character vector instead refers to numeric columns in data. vars = "_none" turns off CCE augmentation where the chosen estimator permits it.

form <- log_rgdpo ~ log_hc + log_ck + log_ngd
csa_vars <- c("log_rgdpo", "log_hc", "log_ck", "log_ngd")

static_csa <- csdm_csa(vars = csa_vars)
dynamic_csa <- csdm_csa(vars = csa_vars, lags = 3)
ardl_1_0 <- csdm_lr(type = "ardl", ylags = 1, xdlags = 0)

MG

fit_mg <- csdm(
  form, data = dat, id = "id", time = "year", model = "mg"
)
summary(fit_mg)
#> csdm summary: Mean Group Model (MG)
#> Formula: log_rgdpo ~ log_hc + log_ck + log_ngd
#> N: 15, T: 38
#> Number of obs: 570
#> R-squared (mg): 0.9449
#> CD = 3.2379, p = 0.0012
#> (For additional CD diagnostics, use cd_test())
#> 
#> Mean Group:
#>              Coef. Std. Err.      z  P>|z| Signif. CI 2.5% CI 97.5%
#> (Intercept) 6.5609    0.8648 7.5867 0.0000     ***  4.8659   8.2558
#> log_hc      0.1725    0.8530 0.2022 0.8398         -1.4993   1.8443
#> log_ck      0.3056    0.1359 2.2485 0.0245       *  0.0392   0.5720
#> log_ngd     0.7800    0.3777 2.0652 0.0389       *  0.0398   1.5203
#> 
#> Mean Group Variables: log_hc, log_ck, log_ngd
#> Cross Sectional Averaged Variables: none (lags=0)
#> 
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

CCE

fit_cce <- csdm(
  form, data = dat, id = "id", time = "year", model = "cce",
  csa = static_csa
)
summary(fit_cce)
#> csdm summary: Static Common Correlated Error Model (CCE)
#> Formula: log_rgdpo ~ log_hc + log_ck + log_ngd
#> N: 15, T: 38
#> Number of obs: 570
#> R-squared (mg): 0.9777
#> CD = -2.6758, p = 0.0075
#> (For additional CD diagnostics, use cd_test())
#> 
#> Mean Group:
#>               Coef. Std. Err.       z  P>|z| Signif. CI 2.5% CI 97.5%
#> (Intercept)  1.9003    2.1195  0.8965 0.3700         -2.2540   6.0545
#> log_hc      -1.4921    1.0152 -1.4697 0.1416         -3.4819   0.4977
#> log_ck       0.1367    0.0956  1.4298 0.1528         -0.0507   0.3240
#> log_ngd      0.8075    0.2972  2.7175 0.0066      **  0.2251   1.3899
#> 
#> Mean Group Variables: log_hc, log_ck, log_ngd
#> Cross Sectional Averaged Variables: log_rgdpo, log_hc, log_ck, log_ngd (lags=0)
#> 
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

DCCE

fit_dcce <- csdm(
  form, data = dat, id = "id", time = "year", model = "dcce",
  csa = dynamic_csa,
  lr = ardl_1_0
)
summary(fit_dcce)
#> csdm summary: Dynamic Common Correlated Error Model (DCCE)
#> Formula: log_rgdpo ~ log_hc + log_ck + log_ngd
#> N: 15, T: 38
#> Number of obs: 525
#> R-squared (mg): 0.9861
#> CD = -2.392, p = 0.0168
#> (For additional CD diagnostics, use cd_test())
#> 
#> Mean Group:
#>                  Coef. Std. Err.       z  P>|z| Signif. CI 2.5% CI 97.5%
#> (Intercept)     9.2888    8.0386  1.1555 0.2479         -6.4666  25.0441
#> log_hc         -1.9558    1.5659 -1.2490 0.2117         -5.0249   1.1132
#> log_ck          0.6666    0.2424  2.7499 0.0060      **  0.1915   1.1417
#> log_ngd        -0.3178    1.1271 -0.2819 0.7780         -2.5268   1.8913
#> lag1_log_rgdpo -0.0745    0.0555 -1.3424 0.1795         -0.1834   0.0343
#> 
#> Mean Group Variables: log_hc, log_ck, log_ngd, lag1_log_rgdpo
#> Cross Sectional Averaged Variables: log_rgdpo, log_hc, log_ck, log_ngd (lags=3)
#> 
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

CS-ARDL

fit_cs_ardl <- csdm(
  form, data = dat, id = "id", time = "year", model = "cs_ardl",
  csa = dynamic_csa,
  lr = ardl_1_0
)

summary(fit_cs_ardl)
#> csdm summary: Cross-Sectional ARDL (CS-ARDL)
#> Formula: log_rgdpo ~ log_hc + log_ck + log_ngd
#> N: 15, T: 38
#> Number of obs: 525
#> R-squared (mg): 0.9861
#> 
#> CD = -2.392, p = 0.0168
#> (For additional CD diagnostics, use cd_test())
#> 
#> Levels ARDL Est.
#>                  Coef. Std. Err.       z  P>|z| Signif. CI 2.5% CI 97.5%
#> (Intercept)     9.2888    8.0386  1.1555 0.2479         -6.4666  25.0441
#> log_hc         -1.9558    1.5659 -1.2490 0.2117         -5.0249   1.1132
#> log_ck          0.6666    0.2424  2.7499 0.0060      **  0.1915   1.1417
#> log_ngd        -0.3178    1.1271 -0.2819 0.7780         -2.5268   1.8913
#> lag1_log_rgdpo -0.0745    0.0555 -1.3424 0.1795         -0.1834   0.0343
#> 
#> Adjust. Term
#>                Coef. Std. Err.        z P>|z| Signif. CI 2.5% CI 97.5%
#> lr_log_rgdpo -1.0745    0.0555 -19.3513     0     *** -1.1834  -0.9657
#> 
#> Long Run Est.
#>              Coef. Std. Err.       z  P>|z| Signif. CI 2.5% CI 97.5% n_used
#> lr_log_hc  -1.8378    1.4743 -1.2465 0.2126         -4.7274   1.0518     15
#> lr_log_ck   0.6106    0.2076  2.9409 0.0033      **  0.2037   1.0175     15
#> lr_log_ngd -0.4098    1.2364 -0.3314 0.7403         -2.8330   2.0134     15
#> 
#> Mean Group Variables: lag1_log_rgdpo, log_hc, log_ck, log_ngd
#> Cross Sectional Averaged Variables: log_rgdpo, log_hc, log_ck, log_ngd (lags=3)
#> Long Run Variables: log_hc, log_ck, log_ngd
#> Cointegration variable(s): log_rgdpo
#> 
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
coef(fit_cs_ardl, component = "long_run")
#>  lr_log_hc  lr_log_ck lr_log_ngd 
#> -1.8377835  0.6105883 -0.4097798

The four calls use the same levels formula, so their coefficient meanings are directly comparable only after accounting for their different dynamic and CCE terms. In particular, a contemporaneous coefficient in DCCE is not a long-run effect.

Samples, missing values, and identification

csdm() requires unique, nonmissing unit-time keys. By default, numeric time indexes advance in steps of one; set time_step when the intended grid differs. Lags respect gaps rather than treating the previous observed row as the previous period.

subset is evaluated before model construction. The supported missing-value policies are na.omit, na.exclude, and na.fail. Unit regressions must retain positive residual degrees of freedom, and every economic coefficient must be identified after projecting out the CCE terms. Units that fail these checks are listed with reasons in fit$units. At least two eligible units are required.

For csa = csdm_csa("_all"), the source sample for cross-sectional averages is the complete base formula sample after subsetting and before dynamic lag trimming. It therefore need not equal the final set of fitted observations. Set fullsample = TRUE to calculate each average from all finite observations of that variable in the selected sample. This is useful when the averaging variables have different missing-value patterns and mirrors the fullsample option used in xtdcce2.

Inference and R model methods

Standard methods expose the fitted model and its sample:

coef(fit_cce)
#> (Intercept)      log_hc      log_ck     log_ngd 
#>   1.9002762  -1.4921059   0.1366750   0.8075283
sqrt(diag(vcov(fit_cce)))
#> (Intercept)      log_hc      log_ck     log_ngd 
#>  2.11954323  1.01523715  0.09559111  0.29715332
nobs(fit_cce)
#> [1] 570
head(model.frame(fit_cce))
#>   log_rgdpo    log_hc   log_ck   log_ngd
#> 1  8.016315 0.7679234 11.79632 -2.727981
#> 2  8.035557 0.7790388 11.84772 -2.712420
#> 3  8.035343 0.7901542 11.90875 -2.702157
#> 4  8.060844 0.8012696 11.97760 -2.697793
#> 5  8.075463 0.8123851 12.03988 -2.701199
#> 6  8.034805 0.8235005 12.07300 -2.709801

head(residuals(fit_cce, format = "long"))
#>    id year .row    residual
#> 11  1 1970    1  0.01979243
#> 12  1 1971    2 -0.00607203
#> 13  1 1972    3 -0.03194126
#> 14  1 1973    4 -0.03727097
#> 15  1 1974    5  0.00548517
#> 16  1 1975    6  0.02278676
head(fitted(fit_cce, format = "vector"))
#>       11       12       13       14       15       16 
#> 7.996523 8.041629 8.067284 8.098115 8.069978 8.012019

For CS-ARDL, coef() and vcov() accept component = "levels", "adjustment", "long_run", or "all". Every component uses a common eligible-unit sample for its coefficients and covariance. Exact algebraic relationships can make the combined "all" covariance singular.

The package also supplies tidy(), glance(), and augment() methods:

broom::tidy(fit_cce, conf.int = TRUE)
broom::glance(fit_cce)
broom::augment(fit_cce)

modelsummary::modelsummary(
  list(MG = fit_mg, CCE = fit_cce, DCCE = fit_dcce),
  statistic = "std.error"
)

Cross-sectional dependence in residuals and variables

Let $e_{it}$ denote the fitted residual and $T_{ij}$ the number of overlapping finite observations for units $i$ and $j$ . The classical statistic implemented by cd_test() is

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

It uses pairwise-complete correlations by default. Large absolute values are evidence against the null represented by the standard-normal reference approximation. Depending on the theoretical formulation, that null is stated as cross-sectional independence or sufficiently weak dependence (Pesaran, 2015, 2021).

For a balanced panel, CDw draws one independent Rademacher weight $w_i\in\{-1,1\}$ per unit, independently of the data. Write $u_{it}=e_{it}-\bar e_i$ for the input demeaned within unit. Equation (30) of the published Juodis-Reese paper (Juodis & Reese, 2022, p. 1197) is

$\widehat s_W^2=\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}w_i^2u_{it}^2, \qquad CD_W=(\widehat s_W^2)^{-1} \sqrt{\frac{2}{TN(N-1)}} \sum_{t=1}^{T}\sum_{i=2}^{N}\sum_{j=1}^{i-1}w_i u_{it}w_j u_{jt}.$

The paper denotes the residuals by $\widehat\varepsilon_{it}$ ; its unit intercepts make their time means zero. The within-unit demeaning above also allows the package to accept raw variables. Since $w_i^2=1$ , $\widehat s_W^2$ is the pooled mean squared demeaned input. The inverse pooled variance is essential: CDw uses weighted covariances with this common scale, rather than individually standardized correlations. The shorthand $\sum_{i<j}$ used elsewhere denotes the same unordered pairs as the displayed double sum. Set seed to reproduce the random weights without changing the caller’s random-number state.

With reps = G, the test draws $G$ independent sets of unit weights and combines the resulting statistics using equation (33) of Juodis and Reese:

$\overline{CD}_W = \frac{1}{\sqrt{G}}\sum_{g=1}^{G}CD_W^{(g)}.$

The default reps = 1 preserves the single-draw statistic. The authors suggest keeping the number of draws modest, for example 30, since increasing $G$ can amplify lower-order terms.

CDw+ follows equation (32) of Juodis and Reese, which applies the power-enhancement principle of Fan, Liao, and Yao (Fan et al., 2015):

$CD_{W+}=CD_W+ \sum_{i<j}|\widehat\rho_{ij}|\, 1\left\{|\widehat\rho_{ij}|>2\sqrt{\log(N)/T}\right\}.$

The threshold is applied to the ordinary residual correlation, without a $\sqrt{T}$ multiplier. The enhancement is nonnegative, so CDw+ is not a second independent random-sign test. With multiple draws, CDw+ adds this screening term once to $\overline{CD}_W$ .

CD* uses the Pesaran-Xie correction algebra (Pesaran & Xie, 2021, 2026), with a standardized-PCA implementation variant. For matrix and data inputs, write $A_{ti}=(e_{it}-\bar e_i)/s_i$ , where $s_i$ is the unit’s sample standard deviation. Extract n_pc principal components from $A$ , normalizing the loading matrix so that $\widehat\Gamma'\widehat\Gamma/N=I$ . Set $\widehat F=A\widehat\Gamma/N$ and $r=A-\widehat F\widehat\Gamma'$ . The implemented correction is

$\widehat\sigma_i=\sqrt{T^{-1}\sum_t r_{it}^{2}},\qquad \widehat\varphi=N^{-1}\sum_i\frac{\widehat\gamma_i}{\widehat\sigma_i},\qquad \widehat a_i=1-\widehat\sigma_i\widehat\varphi'\widehat\gamma_i,$

$\widehat\theta=1-N^{-1}\sum_i\widehat a_i^{2},\qquad CD^*=\frac{CD(r)+\sqrt{T/2}\,\widehat\theta}{1-\widehat\theta}.$

Pesaran-Xie estimate the PCs from unstandardized observations. Standardizing before PCA can change the factor space when unit variances differ, so this package’s variant should not be described as an exact reproduction of their PCA procedure. Setting n_pc = 0 returns classical CD on the supplied panel. The correction requires a nondegenerate denominator; a numerically computable result alone does not establish that the asymptotic assumptions are suitable.

For fitted models, CD* starts from the partial residual $\widehat v_{it}=y_{it}-x_{it}'\widehat\beta_i$ , retaining the fitted common-factor component. Here $x_{it}$ includes the economic regressors and any intercept, unit trend, and constructed economic lags. The CSA contribution is retained for PCA, following the input construction in Pesaran-Xie’s regression procedure. CD, CDw, and CDw+ instead use the full fitted residuals. Passing residuals(fit) directly to CD* therefore tests a different input from passing the fit itself. CD* records its input and sample exclusions in tests$CDstar; in an "all" result, its sample dimensions are in tests$CDstar$N and tests$CDstar$T.

All reported p-values are two-sided standard-normal approximations, $2\Phi(-|\text{statistic}|)$ . The implemented tests do not estimate a serial-correlation correction or establish validity for nonstationary series. The papers impose time-series, factor, and panel-dimension conditions; accepting a numeric variable does not establish those conditions. In particular, classical CD can be biased on CCE residuals after estimating common time parameters. CDw addresses that problem under Juodis-Reese’s assumptions, including $\sqrt{T}/N\to0$ . Omitting pairs or using very short overlaps can also affect classical CD calibration; min_overlap = 2 is a computational minimum.

cd_test(fit_mg, type = "CD")
#> Cross-sectional dependence tests
#> N = 15, T = 38
#> 
#>    statistic p.value
#> CD     3.238   0.001
cd_test(fit_cce, type = "all", seed = 42)
#> Cross-sectional dependence tests
#> N = 15, T = 38
#> 
#>        statistic p.value
#> CD        -2.676   0.007
#> CDw       -0.714   0.475
#> CDw+       2.247   0.025
#> CDstar     2.950   0.003
cd_test(fit_cce, type = "CDw+", reps = 30, seed = 42)
#> Cross-sectional dependence tests
#> N = 15, T = 38
#> 
#>      statistic p.value
#> CDw+     4.490   0.000

To test panel variables directly, select them explicitly through .... Bare column names, quoted names, and character vectors are supported. For a plain data frame, supply named id and time column names. All test controls must also be named because they follow ... in the data methods.

variable_tests <- cd_test(
  dat, log_rgdpo, "log_hc", id = "id", time = "year"
)
variable_tests
#> Cross-sectional dependence tests by variable
#> 
#>   variable test statistic p.value  N  T
#>  log_rgdpo   CD    18.823   0.000 15 38
#>     log_hc   CD    52.602   0.000 15 38
variable_tests$log_rgdpo$tests$CD
#> $statistic
#> [1] 18.82336
#> 
#> $p.value
#> [1] 4.860322e-79
#> 
#> $N
#> [1] 15
#> 
#> $T
#> [1] 38
#> 
#> $pairs_used
#> [1] 105

vars <- c("log_rgdpo", "log_hc")
cd_test(dat, vars, id = "id", time = "year", type = "CDw",
        reps = 30, seed = 42)
#> Cross-sectional dependence tests by variable
#> 
#>   variable test statistic p.value  N  T
#>  log_rgdpo  CDw     4.896   0.000 15 38
#>     log_hc  CDw     1.895   0.058 15 38

Each variable is tested separately on its own available sample, with no regression fitted. The result is always a named list of class cd_test_list, even for one variable; printing it shows a combined table with the sample dimensions for each variable. Seeded tests apply the seed separately to each variable, making results independent of selection order. A pdata.frame provides the indexes automatically, including when its index columns have been removed:

panel_dat <- plm::pdata.frame(dat, index = c("id", "year"), drop.index = TRUE)
cd_test(panel_dat, log_rgdpo, log_hc, type = "CDw+", reps = 30, seed = 42)

Periods with no finite residual for any retained unit are outside the effective residual sample and are removed automatically. Partially observed periods are handled as follows:

classical CD remains pairwise-complete under the default na.action = "pairwise";
CDw and CDw+ require a balanced matrix and otherwise error;
CD* requires a balanced matrix and otherwise returns NA with a warning.

To evaluate all diagnostics on one common time sample, use:

cd_test(
  fit_cce,
  type = "all",
  seed = 42,
  na.action = "drop.incomplete.times"
)

A rejection concerns the variable or residual dependence targeted by the selected test; it does not by itself identify the source of misspecification. A non-rejection does not prove residual independence, particularly in small samples or weak-power settings.

Practical limits

Large- $N$ normal inference does not correct short- $T$ dynamic bias.
CCE validity requires enough informative averages and suitable factor and loading conditions.
Long-run ratios require a stable, nonzero AR denominator and an economically defensible long-run interpretation.
Reported mean-group fit statistics summarize unit regressions; they are not pooled-regression goodness-of-fit measures.
Saved fits from older releases should be refitted because corrected samples, transformations, and covariance calculations can change results.

References

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. https://doi.org/10.3982/ECTA12749

Juodis, A., & Reese, S. (2022). The incidental parameters problem in testing for remaining cross-section correlation. Journal of Business and Economic Statistics, 40(3), 1191–1203. https://doi.org/10.1080/07350015.2021.1906687

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

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 data models with latent factors (Working Paper No. 9234). CESifo. https://www.ifo.de/DocDL/cesifo1_wp9234.pdf

Pesaran, M. H., & Xie, Y. (2026). How to detect network dependence in latent factor models? A bias-corrected CD test. Econometric Theory, 1–41. https://doi.org/10.1017/S0266466625100212

Joao Claudio Macosso