7 Worked Example: Ditzen (2018) Growth Regression
We replicate the core results from Ditzen (2018), Section 7, using the
pwt8 dataset (Penn World Tables 8, N = 93 countries, T = 48
years, 1960–2007). The model is a Solow-type growth regression:
\[ \Delta \log y_{it} = \alpha_i + \phi_i \log y_{i,t-1} + \beta_{1i} \log h_{it} + \beta_{2i} \log k_{it} + \beta_{3i} \log(n_{it} + g + \delta) + \text{CSD correction} + u_{it}, \]
where \(y\) is real GDP (output-side), \(h\) is the human capital index, \(k\) is the physical capital stock, \(n+g+\delta\) aggregates population growth and depreciation, and \(\phi_i = \rho_i - 1\) is the speed of adjustment.
7.1 Step 1: Detect CSD in naive residuals
dat <- na.omit(pwt8[, c("country", "year", "d_log_rgdpo",
"log_rgdpo", "log_hc", "log_ck", "log_ngd")])
fit_ols2 <- lm(d_log_rgdpo ~ log_hc + log_ck + log_ngd + log_rgdpo, data = dat)
cd_naive <- pcd_test(residuals(fit_ols2), data = dat,
unit_index = "country", time_index = "year",
test = "pesaran")
cat(sprintf("CD statistic (OLS): %.2f p-value: %.4f\n",
cd_naive$statistics$statistic[1],
cd_naive$statistics$p_value[1]))
#> CD statistic (OLS): 35.38 p-value: 0.0000Result: The CD statistic is large and significant, confirming strong CSD.
7.2 Step 2: Mean Group (no CSD correction)
fit_mg2 <- dcce(
data = pwt8,
unit_index = "country",
time_index = "year",
formula = d_log_rgdpo ~ L(log_rgdpo, 1) + log_hc + log_ck + log_ngd,
model = "mg",
cross_section_vars = NULL
)
cd_mg <- pcd_test(fit_mg2, test = "pesaran")
cat(sprintf("MG: phi = %.3f, log_ck = %.3f\n",
coef(fit_mg2)["L(log_rgdpo,1)"], coef(fit_mg2)["log_ck"]))
#> MG: phi = -0.200, log_ck = 0.039
cat(sprintf("CD after MG: %.2f p-value: %.4f\n",
cd_mg$statistics$statistic[1], cd_mg$statistics$p_value[1]))
#> CD after MG: 30.86 p-value: 0.0000MG residuals still exhibit significant CSD because MG does not correct for common factors.
7.3 Step 3: Dynamic CCE with 3 CSA lags (Ditzen 2018 Example 7.3)
fit_dcce2 <- dcce(
data = pwt8,
unit_index = "country",
time_index = "year",
formula = d_log_rgdpo ~ L(log_rgdpo, 1) + log_hc + log_ck + log_ngd,
model = "dcce",
cross_section_vars = ~ log_rgdpo + log_hc + log_ck + log_ngd,
cross_section_lags = 3
)
cd_dcce2 <- pcd_test(fit_dcce2, test = "pesaran")
cat(sprintf("DCCE: phi = %.3f, log_ck = %.3f\n",
coef(fit_dcce2)["L(log_rgdpo,1)"], coef(fit_dcce2)["log_ck"]))
#> DCCE: phi = -0.614, log_ck = 0.210
cat(sprintf("CD after DCCE(3): %.2f p-value: %.4f\n",
cd_dcce2$statistics$statistic[1], cd_dcce2$statistics$p_value[1]))
#> CD after DCCE(3): 1.40 p-value: 0.1601After applying DCCE with three CSA lags, the CD statistic is no longer significant, indicating that the estimator has successfully absorbed the common factors.
7.4 Step 4: Summary table
# Coefficient table comparing MG and DCCE
vars <- c("L(log_rgdpo,1)", "log_hc", "log_ck", "log_ngd")
tab <- data.frame(
Variable = vars,
MG_coef = round(coef(fit_mg2)[vars], 3),
MG_se = round(fit_mg2$se[vars], 3),
DCCE_coef = round(coef(fit_dcce2)[vars], 3),
DCCE_se = round(fit_dcce2$se[vars], 3),
row.names = NULL
)
tab$MG_sig <- ifelse(abs(coef(fit_mg2)[vars] / fit_mg2$se[vars]) > 1.96, "*", "")
tab$DCCE_sig <- ifelse(abs(coef(fit_dcce2)[vars] / fit_dcce2$se[vars]) > 1.96, "*", "")
print(tab)
#> Variable MG_coef MG_se DCCE_coef DCCE_se MG_sig DCCE_sig
#> 1 L(log_rgdpo,1) -0.200 0.018 -0.614 0.031 * *
#> 2 log_hc 0.044 0.085 -1.287 0.388 *
#> 3 log_ck 0.039 0.015 0.210 0.049 * *
#> 4 log_ngd -0.021 0.060 0.005 0.100Interpretation:
- The speed of adjustment (\(\hat\phi\)) is negative and significant, confirming mean-reverting dynamics in log GDP. The DCCE estimate (\(\approx -0.6\)) implies \(\hat\rho \approx 0.4\), plausible for this sample.
- Physical capital (
log_ck) has a positive, significant effect in both specifications, consistent with the Solow model. - Population growth (
log_ngd) is negative, also in line with the Solow model. - The shift in coefficient magnitudes from MG to DCCE reflects the bias from CSD in the MG estimator.
7.5 Step 5: Unit-level heterogeneity
One advantage of the Mean Group approach is access to unit-level estimates. We can examine the distribution of the speed-of-adjustment across countries:
b_unit <- coef(fit_dcce2, type = "unit")
ar_coefs <- b_unit$estimate[b_unit$term == "L(log_rgdpo,1)"]
rho_implied <- ar_coefs + 1 # rho = (phi + 1)
cat(sprintf("Implied rho: mean = %.3f, median = %.3f, [min, max] = [%.3f, %.3f]\n",
mean(rho_implied), median(rho_implied),
min(rho_implied), max(rho_implied)))
#> Implied rho: mean = 0.386, median = 0.392, [min, max] = [-0.341, 0.964]
cat(sprintf("Fraction with 0 < rho < 1 (stationarity): %.1f%%\n",
100 * mean(rho_implied > 0 & rho_implied < 1)))
#> Fraction with 0 < rho < 1 (stationarity): 87.1%
hist(
rho_implied,
breaks = 20,
main = "Unit-level AR(1) coefficient (rho = phi + 1)",
xlab = expression(hat(rho)[i]),
col = "steelblue",
border = "white"
)
abline(v = median(rho_implied), col = "firebrick", lwd = 2, lty = 2)
legend("topright", legend = "Median", col = "firebrick", lty = 2, lwd = 2, bty = "n")
The distribution is concentrated in \((0, 1)\), confirming stationary dynamics for most countries, with the median \(\hat\rho \approx 0.4\).
