---
title: "Introduction to csdm: Panel Data Models with Cross-Sectional Dependence"
author: "Joao Claudio Macosso"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Introduction to csdm}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  warning = FALSE,
  message = FALSE
)
library(csdm)
```

## Overview

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).

## Methodology: Four Estimators


### Model Specification

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.

### 1. Mean Group (MG) Estimator

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.

### 2. Common Correlated Effects (CCE) Estimator

The Common Correlated Effects (CCE) estimator augments the model with cross-sectional averages of regressors $\bar{x}_t = N^{-1} \sum_{i=1}^{N} x_{it}$:

$$y_{it} = \alpha_i + \beta_i x_{it} + \gamma_i \bar{x}_t + \gamma_i \bar{y}_t + v_{it}$$

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.

### 3. Dynamic CCE (DCCE) Estimator

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.

### 4. Cross-Sectionally Augmented ARDL (CS-ARDL)

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.

## Data: Penn World Table Subset

The `PWT_60_07` dataset contains macroeconomic indicators for 93 countries covering 1960–2007 (48 years). Key variables include:

- `id`: Country identifier
- `year`: Calendar year (1960–2007)
- `log_rgdpo`: Log real GDP per capita
- `log_hc`: Log human capital index
- `log_ck`: Log capital stock
- `log_ngd`: Log government debt (control variable)

```{r load-data}
data(PWT_60_07, package = "csdm")
head(PWT_60_07, 10)
str(PWT_60_07)

# For computational speed in this vignette, use a subset: 
# first 15 countries, 1970-2007
first_15_ids <- unique(PWT_60_07$id)[1:15]
df <- subset(PWT_60_07, id %in% first_15_ids & year >= 1970 & year <= 2007)
```

The panel is relatively balanced. We will use growth regressions: modeling log-GDP (`log_rgdpo`) as a function of human capital (`log_hc`), capital stock (`log_ck`), and government debt (`log_ngd`), and test cross-sectional dependence in residuals.

## Package installation
To install the `csdm` package from CRAN, run:
```{r install-package, eval = FALSE}
install.packages("csdm")
```

To install the latest development version from GitHub, run:
```{r install-github, eval = FALSE}
install.packages("remotes")
remotes::install_github("Macosso/csdm")
```

## Model Estimation: Four Examples


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.

### Example 1: Mean Group (MG) Estimation

```{r mg-fit}
# 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)
```

**Interpretation**: The MG estimate suggests that on average across countries, increases in human capital, capital stock, and changes in debt are associated with changes in real GDP. The standard errors reflect cross-country heterogeneity in these relationships.

### Example 2: Common Correlated Effects (CCE)

```{r cce-fit}
# 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)
```

**Interpretation**: After accounting for global shocks (via cross-sectional averages of all variables), the CCE coefficients and standard errors may differ from MG. This indicates whether common factors (e.g., technology, energy prices) are a major source of cross-sectional dependence.

### Example 3: Dynamic CCE (DCCE)

```{r dcce-fit}
# 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)
```

**Interpretation**: The DCCE model includes lagged GDP to capture dynamic adjustment. The lagged coefficient typically lies between 0.8–0.95, indicating strong income persistence. The coefficients on other variables represent short-run elasticities; to compute long-run effects, divide by $(1 - \text{lag coefficient})$.

### Example 4: Cross-Sectionally Augmented ARDL (CS-ARDL)

```{r csardl-fit}
# 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)
```

**Interpretation**: The CS-ARDL model returns both short-run coefficients (immediate response to shocks) and long-run coefficients (equilibrium effect after full adjustment). The long-run elasticities are often larger than short-run responses, consistent with gradual accumulation effects in capital and human capital.

## Cross-Sectional Dependence Testing

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.

### Four CD Test Types

All CD tests have null hypothesis: **residuals are cross-sectionally independent**.

#### 1. Pesaran CD Test

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 $T \to \infty$.

#### 2. Pesaran CD Weighted (CDw)

The CDw statistic applies unit-level random sign weights to the cross-sectional correlations:

$$CD_w = \sqrt{\frac{2}{N(N-1)}} \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} w_i w_j \hat{\rho}_{ij} \sqrt{T}$$

where weights $w_i \in \{-1,1\}$ are independent random sign flips assigned at the unit level and held fixed within a replication. This random-weighting scheme improves the behavior of the test in the presence of heteroskedasticity.

#### 3. Pesaran CD Weighted Plus (CDw+)

CDw+ uses the same unit-level random sign weights but applies them to a bias-adjusted version of the CD statistic:

$$CD_w^+ = \sqrt{\frac{2}{N(N-1)}} \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} w_i w_j \hat{\rho}_{ij}^+ \sqrt{T}$$

where $\hat{\rho}_{ij}^+$ denotes the adjusted cross-sectional correlation. CDw+ is designed to improve robustness in large panels with heteroskedasticity.

#### 4. Pesaran CD*, Fan-Liao-Yao (FLY)

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.

### Running CD Tests with Seed Selection

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.

```{r cd-tests}
# 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)

# Test DCCE residuals for CSD
set.seed(1234)
cd_dcce <- cd_test(fit_dcce, type = "CDw")
print(cd_dcce)

# Test CS-ARDL residuals for CSD
set.seed(1234)
cd_csardl <- cd_test(fit_csardl, type = "all")
print(cd_csardl)
```

**Interpreting Results**: 

- **CD statistic p-value < 0.05**: Reject null of CSD independence; residuals are correlated across units.
- **CDw, CDw+, CD* variants**: Provide robustness checks; if all reject the null, CSD is strongly evidenced.
- **Magnitude**: Large $|CD|$ statistics (e.g., $|CD| > 3$) indicate substantial and economically meaningful dependence.

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.

## References

Chudik, A., & Pesaran, M. H. (2013). Common correlated effects estimation of heterogeneous dynamic panel data models with weakly exogenous regressors. *Journal of Business and Economic Statistics*, 33(2), 232–247.

Ditzen, J. (2021). Estimating dynamic common-correlated effects in Stata. *The Stata Journal*, 21(1), 39–59.

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.

---

*For further details on the theoretical foundations and implementation of CSD-robust methods, see the documentation for `?csdm`, `?cd_test`, and `?summary.csdm_fit`.*
