# Länder aging–growth convergence audit

## Status and evidentiary boundary

This is an **unregistered scratch diagnostic**, not a result in paper v0.8.1 and not a causal estimate. It uses the exact 48 Land-decade rows from the frozen window panel at commit `0546db331aa8e3bad671cc2ff997359216103233`, release `laender-preanalysis-b86a097d5159fb29`. The source file is [`laender_window_preanalysis.csv`](https://github.com/Schym/IP-CURRENT-RESEARCH/blob/0546db331aa8e3bad671cc2ff997359216103233/Hayashi_Germany_Research_Package/Hayashi_Germany_Research_Package/outputs/releases/laender-preanalysis-b86a097d5159fb29/laender_window_preanalysis.csv).

The displayed coefficient is the estimated difference in **percentage points of annualized log growth** associated with a **10% higher period-average ratio of residents aged 50–79 to residents aged 20–49**. It is computed as the log-ratio slope multiplied by `log(1.1)`. It is not a 10-percentage-point change in the ratio, not a ten-year cumulative growth effect, and not a causal effect.

All intervals and p-values below use restricted, null-imposed WCR11 bootstrap-t inference with CRV1 studentization, 16 Land clusters where the full sample is retained, 9,999 Rademacher draws, and seed 20260813. Small-sample inference does not cure confounding.

## What the existing paper already controls

The v0.8.1 benchmark is not an unconditional scatter. It regresses annualized growth on period-average log old-to-young composition and controls for:

- the log outcome level at the start of the decade (the standard conditional-beta-convergence term);
- period-average log child dependency; and
- decade fixed effects for 1991–2001, 2001–2011, and 2011–2021.

The remaining issue is functional form: one common initial-income slope may not capture the exceptional reunification-era path of the five eastern non-city Länder.

## Preferred convergence diagnostic

The most defensible small-panel diagnostic is:

\[
g_{rw}=\alpha+\beta\bar a_{rw}+\theta\bar c_{rw}+\lambda_w
+\sum_w \rho_w\left(\log y_{r,w0}-\overline{\log y_{w0}}\right)1\{w\}
+\sum_w\delta_w East_r1\{w\}+u_{rw}.
\]

It retains the paper’s decade effects and child-dependency control, but allows the convergence slope to differ by decade and absorbs an arbitrary eastern non-city growth component in every decade. Berlin remains a city state, not part of the eastern non-city group. This is deliberately demanding: it asks whether an aging slope remains after removing the variation most plausibly tied to catch-up and East-specific restructuring.

## Results

### Real workplace GDP per resident aged 20–79

| Specification | Effect for 10% higher ratio (pp/year) | WCR11 95% CI | p-value |
|---|---:|---:|---:|
| Paper benchmark: common convergence slope | -0.317 | [-0.565, -0.068] | 0.011 |
| Convergence slope allowed to differ by decade | -0.187 | [-0.432, 0.052] | 0.168 |
| East-specific growth component by decade | -0.084 | [-0.360, 0.189] | 0.447 |
| **Preferred stringent diagnostic: both adjustments** | **-0.163** | **[-0.436, 0.113]** | **0.172** |
| Preferred diagnostic plus city-state-by-decade components | -0.285 | [-0.570, 0.004] | 0.052 |
| Land and decade fixed effects | +0.227 | [-0.080, 0.528] | 0.129 |
| Benchmark, 2001–2021 only | -0.096 | [-0.402, 0.204] | 0.464 |
| Benchmark, eastern non-city Länder deleted | -0.093 | [-0.586, 0.368] | 0.586 |

The preferred correction cuts the benchmark magnitude from -0.317 to -0.163 pp/year and the interval crosses zero. Adding only East-by-decade components reduces the point estimate further, to -0.084. Deleting the first post-unification decade or deleting the five eastern non-city Länder produces similarly small point estimates. The negative benchmark is therefore **not robust to flexible convergence adjustment**.

### Real workplace GDP per workplace-employed person

| Specification | Effect for 10% higher ratio (pp/year) | WCR11 95% CI | p-value |
|---|---:|---:|---:|
| Paper benchmark: common convergence slope | -0.367 | [-0.616, -0.116] | 0.002 |
| Convergence slope allowed to differ by decade | -0.165 | [-0.371, 0.043] | 0.142 |
| East-specific growth component by decade | -0.104 | [-0.407, 0.195] | 0.399 |
| **Preferred stringent diagnostic: both adjustments** | **-0.150** | **[-0.412, 0.121]** | **0.231** |
| Benchmark, 2001–2021 only | -0.042 | [-0.129, 0.044] | 0.621 |
| Benchmark, eastern non-city Länder deleted | -0.095 | [-0.628, 0.424] | 0.568 |

The same attenuation appears with the workplace-employment denominator. That makes the convergence warning broader than the residence/workplace denominator mismatch.

## Why the correction matters empirically

In 1991–2001, the eastern non-city Länder began at roughly EUR 18.9k of real workplace GDP per resident aged 20–79, compared with EUR 43.2k in the western non-city Länder. Their mean annual growth was 4.71%, versus 0.88% in the west. By 2011–2021, the eastern group had a much older average composition (old-to-young ratio about 1.22 versus 1.00 in the west) while still growing faster (1.73% versus 0.83%). Aging, low initial income, and catch-up therefore occupy much of the same regional and temporal variation.

The benchmark controls explain 81% of the variation in the composition regressor; the preferred stringent controls explain about 90%. Only about 10% of the original treatment variation remains for the aging slope. The loss of precision is substantive information about limited identifying variation, not a reason to prefer the statistically significant specification.

## Recommended interpretation

Public wording:

> In the benchmark model, a 10% higher old-to-young ratio is associated with about 0.32 percentage points less annual growth. This comparison already controls for initial output, but it assumes one common convergence path across all Länder and decades. Once the model allows catch-up to differ by decade and separates East German restructuring, the estimate is roughly halved to 0.16 points and is no longer statistically resolved. The data therefore show a negative composition pattern, but they do not yet separate demographic aging from migration, regional sorting, and post-unification convergence.

Do not write that “correcting for convergence proves aging has no effect.” The stringent model may absorb genuine demographic variation and is low-powered. It shows fragility of the benchmark interpretation, not evidence of a zero structural effect.

## Recommended article graphic

Use a two-panel Frisch–Waugh–Lovell residual plot:

1. benchmark-adjusted growth versus benchmark-adjusted aging; and
2. stringent convergence-adjusted growth versus stringent-adjusted aging.

Residualize **both** variables on the model controls. Scale the horizontal axis in residual 10%-ratio units, so the fitted slope is directly readable in pp/year per 10% higher ratio. Label it “conditional residual association,” never “effect.” The exact plotting rows are in `convergence_fwl_residuals.csv`.

## Reproducibility files

- `convergence_audit.py`: model construction and WCR11 execution
- `convergence_audit_results.csv`: complete two-outcome specification grid
- `convergence_fwl_residuals.csv`: exact residualized plotting data
- `convergence_audit_manifest.json`: frozen source and inference metadata

