Advanced Standard Errors in Time Series & Panel Data

1. TSCS & Panel Challenges: Beck & Katz vs. Parks

In Time-Series Cross-Sectional (TSCS) data (N units over T periods): y_{it} = X_{it}' \beta + u_{it}, \quad \mathbb{E}[u u'] = \Omega \quad (NT \times NT)

The Parks-Kmenta (1967) FGLS Fallacy

• Idea: Full FGLS estimating unit-specific AR(1) coefficients \rho_i, heteroskedasticity \sigma_i^2, and cross-sectional covariances \sigma_{ij}.
• The Problem: Requires estimating N(N+1)/2 covariance parameters. When T is not vastly larger than N, sample variability in \hat{\Sigma} causes FGLS standard errors to be severely underestimated (by 50%–80%), leading to extreme over-rejection of H_0.

Beck & Katz (1995) Panel-Corrected Standard Errors (PCSE)

• Retains OLS coefficient estimates \hat{\beta}_{\text{OLS}} and replaces the standard error formula with: \hat{\Sigma}_{\text{PCSE}} = \frac{E' E}{T} \quad (N \times N \text{ matrix of OLS residuals } E)
• Trade-off: Works very well for T > N. However, if N > T, E'E is rank-deficient/singular and PCSE fails completely.

2. Multi-Way Clustering (Cameron, Gelbach, & Miller 2011)

When errors are correlated across multiple non-nested or nested dimensions (e.g., entity i and period t):

Inclusion-Exclusion Principle

\hat{V}_{\text{Two-Way}} = \hat{V}_{i} + \hat{V}_{t} - \hat{V}_{i \cap t}

Where: * \hat{V}_i: One-way cluster covariance matrix by entity i (handles within-entity serial correlation). * \hat{V}_t: One-way cluster covariance matrix by period t (handles aggregate cross-sectional shocks). * \hat{V}_{i \cap t}: Cluster covariance matrix by (i, t) cell (subtracted to avoid double-counting heteroskedasticity).

Asymptotic & Practical Requirements

• Requires N_{\text{cluster1}} \to \infty AND N_{\text{cluster2}} \to \infty (typically \ge 30–50 clusters per dimension).
• Flexible: Extends easily to 3-way or multi-way clustering (e.g., firm, worker, time).

3. Pesaran (2006) Common Correlated Effects (CCE)

Instead of adjusting standard errors post hoc, CCE models cross-sectional dependence directly using an interactive fixed effects / factor structure:

y_{it} = X_{it}' \beta_i + \gamma_i' f_t + e_{it}

where f_t is an unobserved m \times 1 vector of common aggregate factors with unit-specific factor loadings \gamma_i.

The Pesaran (2006) Innovation

Filter out unobserved factors f_t by including cross-sectional averages of y_{it} and X_{it} as proxies: \bar{y}_t = \frac{1}{N} \sum_{i=1}^N y_{it}, \quad \bar{X}_t = \frac{1}{N} \sum_{i=1}^N X_{it}

Operational CCE Regression Model

y_{it} = X_{it}' \beta + d_{1i} \bar{y}_t + d_{2i}' \bar{X}_t + e_{it}

• Pooled CCE (CCEP): Constrains \beta_i = \beta across all units.
• Mean Group CCE (CCEMG): Estimates \hat{\beta}_i unit-by-unit and averages across i: \hat{\beta}_{\text{CCEMG}} = \frac{1}{N}\sum_{i=1}^N \hat{\beta}_i.

4. Parameter Interpretation in CCE: Structural vs. Nuisance

Structural Parameters (\beta)

• Pooled CCE (\hat{\beta}_{\text{CCEP}}): Represents the within-unit partial effect of X_{it} on y_{it}, holding constant time-invariant heterogeneity (\alpha_i) and unobserved common shocks (f_t).
• Mean Group CCE (\hat{\beta}_{\text{CCEMG}}): Represents the Average Partial Effect (APE) across heterogeneous entities.

Cross-Sectional Averages (d_{1i} and d_{2i}): Pure Nuisance Parameters

Critical Pitfall: d_{1i} and d_{2i} must NEVER be interpreted as peer effects, neighborhood spillovers, or network coefficients.

• They are purely a mathematical projection device.
• Because \bar{y}_t \approx \bar{X}_t \bar{\beta} + \bar{\gamma}' f_t, solving for f_t yields: \gamma_i' f_t \approx d_{1i} \bar{y}_t + d_{2i}' \bar{X}_t
• The coefficients d_{1i} and d_{2i} are complex, non-linear combinations of unit loadings \gamma_i, average loadings \bar{\gamma}, and average slopes \bar{\beta}, stripped of direct economic meaning.

6. Dynamic CCE (Chudik & Pesaran, 2015)

When the model includes lagged dependent variables: y_{it} = \rho y_{i,t-1} + X_{it}' \beta + \gamma_i' f_t + e_{it}

The Failure of Standard CCE

In dynamic models, contemporaneous cross-sectional averages \bar{z}_t = (\bar{y}_t, \bar{X}_t')' remain correlated with e_{i,t-1} at finite T. Standard CCE produces inconsistent estimates for \rho and \beta.

The Chudik & Pesaran Solution

To restore consistency, include p_T lags of the cross-sectional averages alongside contemporaneous averages: y_{it} = \rho_i y_{i,t-1} + X_{it}' \beta_i + \sum_{\ell=0}^{p_T} \delta_{i,\ell}' \bar{z}_{t-\ell} + e_{it}

• Lag Selection Rule: Set p_T = \lfloor T^{1/3} \rfloor (e.g., p_T \approx 3 or 4 for T = 30–60).
• Requirement: Requires N, T \to \infty jointly.

7. Applied Synergy: Combining Mundlak CRE & CCE

In empirical settings with short/unbalanced panels, micro-level selection, and macro shocks, researchers increasingly combine both projection devices:

y_{it} = X_{it}' \beta + \underbrace{\bar{X}_i' \delta}_{\text{Mundlak (CRE for } \alpha_i)} + \underbrace{d_{1i} \bar{y}_t + d_{2i}' \bar{X}_t + \sum_{\ell=1}^{p_T} \delta_{i,\ell}' \bar{z}_{t-\ell}}_{\text{Dynamic CCE for } f_t} + \varepsilon_{it}

Why Use Both?

• Mundlak Component (\bar{X}_i): Controls for time-invariant selection bias and correlated random effects, retaining the ability to estimate coefficients on time-invariant regressors (e.g., gender, location, distance).
• CCE Component (\bar{z}_{t-\ell}): Controls for unobserved time-varying macro trends, interactive fixed effects, and spatial dependence.

8. Master Econometric Taxonomy Matrix

9. Software Implementation: Canonical Grunfeld Application

We apply these panel estimators to the canonical Grunfeld Panel Data (N=10 firms, T=20 years, 1935–1954). * Dependent Variable (y): inv (Gross Investment) * Regressors (X): value (Market Value of Firm), capital (Stock of Plant & Equipment)

library(plm)
library(fixest)
library(pcse)
library(dplyr)

# Load canonical Grunfeld dataset (N=10 firms, T=20 years)
data("Grunfeld", package = "plm")

# 1. Two-Way Fixed Effects & Two-Way Clustering (fixest)
tw_mod <- feols(inv ~ value + capital | firm + year, 
               data = Grunfeld, 
               cluster = ~firm + year)
summary(tw_mod)

OLS estimation, Dep. Var.: inv
Observations: 200
Fixed-effects: firm: 10,  year: 20
Standard-errors: Clustered (firm & year) 
        Estimate Std. Error  t value   Pr(>|t|)    
value   0.117716   0.011266 10.44897 2.4797e-06 ***
capital 0.357916   0.045191  7.92003 2.3983e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
RMSE: 47.5     Adj. R2: 0.943118
             Within R2: 0.720145

# 2. Beck & Katz PCSE (pcse)
ols_mod <- plm(inv ~ value + capital, data = Grunfeld)
summary(ols_mod, vcov = vcovBK)

Oneway (individual) effect Within Model

Note: Coefficient variance-covariance matrix supplied: vcovBK

Call:
plm(formula = inv ~ value + capital, data = Grunfeld)

Balanced Panel: n = 10, T = 20, N = 200

Residuals:
    Min.  1st Qu.   Median  3rd Qu.     Max. 
-184.009  -17.643    0.563   19.192  250.710 

Coefficients:
        Estimate Std. Error t-value  Pr(>|t|)    
value   0.110124   0.011844  9.2976 < 2.2e-16 ***
capital 0.310065   0.040282  7.6973 7.651e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Total Sum of Squares:    2244400
Residual Sum of Squares: 523480
R-Squared:      0.76676
Adj. R-Squared: 0.75311
F-statistic: 64.3086 on 2 and 9 DF, p-value: 4.678e-06

# 3. Pesaran Static CCE (plm)
pgrun <- pdata.frame(Grunfeld, index = c("firm", "year"))
ccep_mod  <- pcce(inv ~ value + capital, data = pgrun, model = "p")  # Pooled CCE
ccemg_mod <- pcce(inv ~ value + capital, data = pgrun, model = "mg") # Mean Group CCE

summary(ccep_mod)

Common Correlated Effects Pooled model

Call:
pcce(formula = inv ~ value + capital, data = pgrun, model = "p")

Balanced Panel: n = 10, T = 20, N = 200

Residuals:
     Min.   1st Qu.    Median   3rd Qu.      Max. 
-154.0750   -7.3557   -0.3129    6.5690  199.9604 

Coefficients:
        Estimate Std. Error z-value Pr(>|z|)   
value   0.124329   0.045580  2.7277 0.006378 **
capital 0.094986   0.071405  1.3303 0.183434   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Total Sum of Squares: 9359900
Residual Sum of Squares: 191200
HPY R-squared: 0.89756

# 4. Dynamic CCE & Mundlak CRE Synergy
# Calculate unit time-means (Mundlak) and period cross-sectional means (CCE)
Grunfeld_aug <- Grunfeld %>%
  group_by(firm) %>%
  mutate(val_mean = mean(value), cap_mean = mean(capital)) %>%
  group_by(year) %>%
  mutate(inv_bar = mean(inv), val_bar = mean(value), cap_bar = mean(capital)) %>%
  ungroup() %>%
  pdata.frame(index = c("firm", "year"))

# Dynamic CCE with Lagged Dependent Variable + CS Averages
dcce_mod <- feols(inv ~ lag(inv, 1) + value + capital + 
                    val_mean + cap_mean + 
                    inv_bar + val_bar + cap_bar + 
                    lag(inv_bar, 1) + lag(val_bar, 1) | firm,
                  data = Grunfeld_aug, 
                  cluster = ~firm + year)
summary(dcce_mod)

OLS estimation, Dep. Var.: inv
Observations: 200
Fixed-effects: firm: 10
Standard-errors: Clustered (firm & year) 
              Estimate Std. Error    t value  Pr(>|t|)    
lag(inv, 1)  1.000e+00      1e-08  1.000e+08 < 2.2e-16 ***
value        1.440e-16      1e-08  1.440e-08         1    
capital     -1.140e-16      1e-08 -1.140e-08         1    
inv_bar      1.292e-15      1e-08  1.292e-07         1    
val_bar      1.080e-16      1e-08  1.080e-08         1    
cap_bar     -8.750e-16      1e-08 -8.750e-08         1    
... 4 variables were removed because of collinearity (val_mean, cap_mean
and 2 others [full set in $collin.var])
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
RMSE: 1.022e-13     Adj. R2: 1
                  Within R2: 1

* ====================================================================
* Canonical Grunfeld Application in Stata
* ====================================================================

* Load canonical Grunfeld dataset
webuse grunfeld, clear
xtset firm year

* 1. Two-Way Fixed Effects with Two-Way Clustering (reghdfe)
reghdfe inv value capital, absorb(firm year) vce(cluster firm year)

* 2. Beck & Katz PCSE (xtpcse)
xtpcse inv value capital, correlation(ar1) hett

* 3. Pesaran Static CCE (xtcce package)
* ssc install xtcce
xtcce inv value capital, pooled                 // Pooled CCE (CCEP)
xtcce inv value capital, mg                     // Mean Group CCE (CCEMG)

* 4. Dynamic CCE (Chudik & Pesaran 2015 via xtcce)
* Fits dynamic model with 1 lag of cross-sectional averages (p_T = 1)
xtcce inv value capital, pooled cr(inv value capital) cr_lags(1)

* 5. Mundlak CRE + Dynamic CCE Synergy
bysort firm: egen val_mean = mean(value)
bysort firm: egen cap_mean = mean(capital)

xtcce inv L.inv value capital val_mean cap_mean, pooled cr(inv value capital) cr_lags(1)

10. Applied Decision Rules for Econometricians

                              Is N > T or T > N?
                                      │
            ┌─────────────────────────┴─────────────────────────┐
            ▼                                                   ▼
       Short Panel (N >> T)                               Long Panel (T >> N or T ≈ N)
            │                                                   │
    Are units correlated                               Does spatial correlation decay
    across space/time?                                  with physical distance?
      ┌─────┴─────┐                                       ┌─────┴─────┐
      ▼           ▼                                       ▼           ▼
     NO          YES                                     YES          NO
      │           │                                       │           │
   Cluster     Two-Way                                 Conley       Driscoll-
    (id)       Cluster                                 Spatial       Kraay /
            (id + time)                                  SEs          PCSE
                                                                        │
                                                            Unobserved Macro Shocks?
                                                                        │
                                                                   Pesaran CCE
                                                                        │
                                                             Dynamic Model (y_{t-1})?
                                                                        │
                                                                   Dynamic CCE
                                                              (Chudik & Pesaran 2015)