Using median gamma: 0.003341, 245 rows (of 4,655 total)
Flexible Damage Functions
0.1 Labor: Low Risk Country Collapsed
Flexible damage function parameters at the Country level.
Outcome units: mins_per_worker_per_day. Change in productive minutes per worker per day for low-heat-exposure workers attributable to climate change, rebased to 2005 baseline. Negative = productivity loss; positive = gain. The IR-level nc4 attribute mislabels this as portion but the values match the projection system’s mins/worker/day pop-aggregated convention.
\[D_{it} = (\alpha_i T_t + \beta_i T_t^2) \cdot Y_{it}^{\gamma}\]
where \(\gamma\) is the income elasticity (fitted globally), \(\alpha_i\) is the linear coefficient, and \(\beta_i\) is the quadratic coefficient.
1 Global Estimation
1.1 Income Elasticity Estimation
The income elasticity \(\gamma\) is estimated using a fixed-effects regression:
\[y_{it} = \gamma \cdot \log(Y_{it}) + \mu_{g(i,T)} + \nu_t + \varepsilon_{it}\]
where \(\mu_{g(i,T)}\) are region-by-temperature-bin fixed effects and \(\nu_t\) are year fixed effects.
| Statistic | Value |
|---|---|
| Income elasticity (\(\gamma\)) | 0.0033 |
| Standard error | 7.53e-03 |
| 95% CI | [-0.0114, 0.0181] |
| R-squared | 0.9883 |
| Observations | 83,790 |
| Regions | 245 |
| Gamma quantiles | 19 |
2 Parameter Distributions
2.1 8-Panel Summary
Row 1: gamma, alpha, beta, rsqr1. Row 2: rho, zeta, eta, rsqr2.
2.2 Projection Equation and Parameter Definitions
The estimated parameters are used to project damages via Monte Carlo sampling:
\[D_{it}^k = (\hat{\alpha}_{ik} T_t + \hat{\beta}_{ik} T_t^2) Y_{it}^{\hat{\gamma}_k} + \hat{\theta}_{ik} T_t Y_{it}^{\hat{\gamma}_k} + \hat{\phi}_{it}^k\]
where \(k\) indexes the Monte Carlo draw. The parameters in each row of the output CSV control distinct components of this equation:
- gamma: income elasticity \(\hat{\gamma}_k\), one of 19 quantile values drawn from \(N(\hat{\gamma}, SE(\hat{\gamma}))\)
- alpha, beta: linear and quadratic temperature coefficients; \(\hat{\alpha}_{ik}\) and \(\hat{\beta}_{ik}\) are drawn from the joint normal defined by the VCV below
- sigma11, sigma12, sigma22: variance-covariance matrix of \((\alpha, \beta)\), used for joint uncertainty sampling
- rho: correlation between regional and global polynomial residuals \(\rho_i\), used to maintain spatial covariance across regions in Monte Carlo draws
- zeta: temperature-dependent error scale \(\zeta_{ik}\); the run-specific error term \(\hat{\theta}_{ik}\) is drawn from \(N(0, \zeta_{ik})\)
- eta: residual noise standard deviation \(\eta_{ik}\); the annual noise \(\hat{\phi}_{it}^k\) is drawn from \(N(0, \eta_{ik})\)
- rsqr1, rsqr2: polynomial fit quality and error model fit, respectively
2.3 Summary Statistics
| Parameter | Mean | Median | Std | Min | Max | N |
|---|---|---|---|---|---|---|
| alpha | -0.07109 | 0.02088 | 0.2399 | -0.9994 | 0.1578 | 245 |
| beta | -0.03784 | -0.03661 | 0.02395 | -0.1027 | 0 | 245 |
| rho | 0.3815 | 0.3846 | 0.184 | -0.1478 | 0.753 | 245 |
| zeta | 0.0195 | 0.01896 | 0.007879 | 0.009328 | 0.05157 | 245 |
| eta | 0.02693 | 0.024 | 0.01597 | 0.008004 | 0.09126 | 245 |
| rsqr1 | 0.8471 | 0.9447 | 0.1927 | 0.266 | 0.9916 | 245 |
| rsqr2 | 0.6946 | 0.6901 | 0.08404 | 0.4748 | 0.8814 | 245 |
3 Spaghetti Curves
Regional damage function curves showing D(T) = αT + βT² for sampled regions.
4 Zero Crossings
The zero crossing (extremum) of the parabola occurs at \(T^* = -\alpha / (2\beta)\).
| Category | Count | Percentage |
|---|---|---|
| β = 0 (no crossing) | 16 | 6.5% |
| T > 20°C (beyond graph) | 4 | 1.6% |
| T < 0°C (negative crossing) | 107 | 43.7% |
| Valid crossings (0-20°C) | 118 | 48.2% |
5 Slope Analysis
Maximum slope between 0 and 10°C: \(\frac{dM}{dT} = \alpha + 2\beta T\)
The maximum occurs at either T=0 or T=10 (endpoints of interval).
Convexity analysis omitted (beta constraint active).
6 R-squared Analysis
6.1 Polynomial Fit Quality (rsqr1)
6.2 Error Model R-squared (rsqr2) Quantiles
| 0% (Min) | 25% | 50% (Median) | 75% | 100% (Max) |
|---|---|---|---|---|
| 0.4748 | 0.6413 | 0.6901 | 0.7569 | 0.8814 |
7 Modelled Variance
Modelled variance statistic: \(1 - \frac{\sum_i \eta_i^2}{\sum_i D_i^2}\)
| Statistic | Value |
|---|---|
| Modelled variance | 0.9989 |
| Sum(η²) | 0.2398 |
| Sum(D²) at T=3.0°C | 222.3114 |
| N regions | 245 |
8 Best- and Worst-Fitting Regions
The 3 worst- and 3 best-fitting regions by R-squared, with raw simulation data overlaid on the fitted polynomial curve. Red rows = worst fits, green rows = best fits.
Top 3 best fit (R²) Top 3 worst fit (R²)
| Region | α | β | η | R² |
|---|---|---|---|---|
| MRT | -0.896 | -0.0327 | 0.06 | 0.992 |
| SAU | -0.703 | -0.0281 | 0.0493 | 0.991 |
| BEN | -0.342 | -0.0836 | 0.0401 | 0.991 |
| PRK | 0.102 | -0.0211 | 0.0171 | 0.266 |
| BTN | 0.0927 | -0.0207 | 0.0141 | 0.278 |
| VUT | 0.0568 | -0.012 | 0.0093 | 0.28 |
9 Regional Parameter Maps
Maps of key parameters at the country level. Red = negative (damage increases with T), Blue = positive.
9.1 Alpha (Linear Coefficient)
Alpha (\(\alpha\)) represents the linear sensitivity to temperature. Regions with negative alpha experience damage that increases with the first degree of warming.
9.2 Beta (Quadratic Coefficient)
Beta (\(\beta\)) represents the curvature of the damage function.
The concavity (\(\beta \leq 0\)) constraint is enforced for this sector, meaning optimal temperature exists, damages accelerate beyond it.
9.3 R-squared (Fit Quality)
\(R^2\) measures the polynomial fit quality. Higher values indicate that the quadratic form captures more of the variance in the data.
10 F2: Flex vs Raw Comparison
Comparison of flexible damage function predictions against raw simulation means.
For each scenario (RCP × SSP × Model), we compute at the final year:
- Flex predicted: \((\alpha \cdot T + \beta \cdot T^2) \cdot Y^\gamma\)
- Raw actual: The outcome variable (y) from the source data
Year 2099: 1,012 rows
Correlation: 0.9675 | RMSE: 1.016230 | Sign agreement: 96.3%
Top 5 best predicted Top 5 worst predicted
Year 2099 · Tmean across these regions: 1.96°C
| Region | Raw | Flex | Residual |
|---|---|---|---|
| KOR | 0.0756 | 0.0754 | -1.32e-4 |
| LCA | -0.0282 | -0.0275 | 6.55e-4 |
| WSM | -0.0715 | -0.0684 | 0.00312 |
| ALB | -0.132 | -0.137 | -0.00524 |
| VGB | -0.0205 | -0.0264 | -0.00591 |
| MLI | -7.06 | -1.99 | 5.07 |
| NER | -7.01 | -2.09 | 4.91 |
| SDN | -6.32 | -2.1 | 4.22 |
| BFA | -6.05 | -1.9 | 4.15 |
| TCD | -5.96 | -1.96 | 4 |
Correlation: 0.9471 | RMSE: 1.236544 | Sign agreement: 94.7%
Top 5 best predicted Top 5 worst predicted
Year 2099 · Tmean across these regions: 1.96°C
| Region | Raw | Flex | Residual |
|---|---|---|---|
| PER | -0.0868 | -0.087 | -1.31e-4 |
| TWN | 0.0288 | 0.0224 | -0.00646 |
| UMI | -0.0734 | -0.0819 | -0.0085 |
| AZE | -0.461 | -0.47 | -0.00979 |
| ZAF | -0.0822 | -0.0922 | -0.00999 |
| NER | -8.24 | -2.08 | 6.16 |
| MLI | -8.03 | -1.98 | 6.05 |
| BFA | -7.92 | -1.89 | 6.03 |
| TCD | -7.44 | -1.95 | 5.49 |
| MRT | -6.85 | -1.93 | 4.92 |
Correlation: 0.9655 | RMSE: 3.707582 | Sign agreement: 95.1%
Top 5 best predicted Top 5 worst predicted
Year 2099 · Tmean across these regions: 4.20°C
| Region | Raw | Flex | Residual |
|---|---|---|---|
| TWN | -0.158 | -0.159 | -6.25e-4 |
| SVN | -0.199 | -0.206 | -0.00701 |
| ABW | -0.117 | -0.134 | -0.0171 |
| AUS | -0.154 | -0.13 | 0.0243 |
| BMU | -0.232 | -0.199 | 0.0331 |
| NER | -21.7 | -4.7 | 17 |
| MLI | -21.5 | -4.84 | 16.6 |
| SDN | -20.1 | -4.8 | 15.3 |
| BFA | -19.6 | -4.61 | 15 |
| TCD | -18.4 | -4.47 | 13.9 |
Correlation: 0.9514 | RMSE: 4.631861 | Sign agreement: 96.3%
Top 5 best predicted Top 5 worst predicted
Year 2099 · Tmean across these regions: 4.20°C
| Region | Raw | Flex | Residual |
|---|---|---|---|
| ABW | -0.14 | -0.134 | 0.00568 |
| DEU | 0.0407 | 0.0506 | 0.0099 |
| NLD | 0.165 | 0.182 | 0.0173 |
| FJI | -0.129 | -0.149 | -0.0193 |
| AUT | 4.46e-4 | -0.0206 | -0.021 |
| NER | -24.9 | -4.67 | 20.2 |
| BFA | -24.5 | -4.58 | 20 |
| MLI | -24.4 | -4.83 | 19.6 |
| TCD | -22.9 | -4.44 | 18.5 |
| SDN | -23 | -4.78 | 18.2 |
10.1 Scenario Summary
| RCP | SSP | Model | N | Corr | RMSE | Sign% |
|---|---|---|---|---|---|---|
| rcp45 | SSP3 | high | 245 | 0.9675 | 1.01623 | 96.3% |
| rcp45 | SSP3 | low | 245 | 0.9471 | 1.23654 | 94.7% |
| rcp85 | SSP3 | high | 245 | 0.9655 | 3.70758 | 95.1% |
| rcp85 | SSP3 | low | 245 | 0.9514 | 4.63186 | 96.3% |
11 Data Reference
11.1 File Locations
| Item | Path |
|---|---|
| Regional CSV | /project/cil/gcp/flex_damage_funcs/parameters/labor__low_risk_country_collapsed__regional_parameters.csv |
| Global JSON | /project/cil/gcp/flex_damage_funcs/parameters/labor__low_risk_country_collapsed__global_results.json |
| Metadata JSON | /project/cil/gcp/flex_damage_funcs/parameters/labor__low_risk_country_collapsed__metadata.json |
11.2 Column Definitions
| Column | Description |
|---|---|
region |
Region identifier (hierarchical code, first 3 chars = country ISO3) |
gamma |
Income elasticity quantile value |
alpha |
Linear temperature coefficient |
beta |
Quadratic temperature coefficient |
sigma11 |
Var(alpha) |
sigma12 |
Cov(alpha, beta) |
sigma22 |
Var(beta) |
rho |
Correlation with global residual process |
zeta |
Temperature-dependent heteroskedasticity |
eta |
Residual standard deviation |
rsqr1 |
R-squared of polynomial fit |
rsqr2 |
R-squared of error model |
Report generated with FlexDamage v1.0.0