## 1. Optimal Linear Measure

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### Major methodological components
- Objective: Use satellite-recorded nighttime lights to illuminate uncertainty in official measures of real GDP per capita, jointly estimate measurement-error distributions and the lights–GDP relationship, and construct improved measures of real GDP per capita.
- Key sections: Optimal Linear Measure; Semiparametric Conditional Mean; Data (Nighttime Lights; Administrative Maps and Country Location; GDP, Population, and Statistical Capacity); Results (Nighttime Lights’ Elasticity of Real GDP per capita; Uncertainty in Real GDP per capita; New Measures of Real GDP per capita; Official vs. New Measures); Appendices A–C (identification, sieve MLE, simulations, robustness).

### Data and empirical scope
- Nighttime lights datasets:
  - DMSP/OLS (NOAA composites), 1992–2013; pixel radiance values 0–63; majority of sample.
  - VIIRS (Suomi-NPP), available since 2013; monthly data averaged to annual.
- Country panels and sample sizes:
  - DMSP/OLS panel: unbalanced panel of 182 countries and 3,870 observations.
  - VIIRS panel: 184 countries and 920 observations.
  - Discrepancy due to availability of statistical capacity ratings (two countries: Barbados and the Czech Republic).
- Variables and observables in empirical i.i.d. setup {z_i, y_i, s_i, l_i}_i=1^n:
  - y_{i,t}: reported (observed) log real GDP per capita.
  - y^*_{i,t}: true (latent) log real GDP per capita.
  - s_{i,t}: statistical capacity.
  - z_{i,t}: log nighttime lights per capita.
  - l_i: latitude (centroid of largest contiguous land block; discretized indicator for tropics vs. outside).
- Night light intensity defined as (log) total nighttime lights per capita (radiance sum divided by population and logged).
- Statistical capacity: World Bank Statistical Capacity Indicator (scale 0–100); high-income countries assigned score 100 and treated separately.

### Statistical framework and identification
- Measurement-error model:
  - Reported GDP: y_{i,t} = y^*_{i,t} + ε^y_{i,t}(s_{i,t}) — error distribution may vary with s.
  - Nighttime lights production function: z_{i,t} = m(y^*_{i,t}) + ε^z_{i,t}(l_i) — error distribution may vary with l.
- Production function m(·) informed by data; empirical implementation assumes quadratic parametric form for m(·).
- Error term distributions ε^y and ε^z allowed nonparametrically and to vary with s and l respectively.
- Identification (Theorem 1): Under Assumptions 1–6, the observed distribution f(z,y,s,l) uniquely determines the joint distribution f(z,y,y^*,s,l) = f(z | y^*, l) f(y | y^*, s) f(y^*, s, l), enabling nonparametric identification of conditional distributions and the nonlinear lights–GDP relationship.
- Normalization and completeness conditions specified (Assumptions 4–6); high-income group handled via additivity/independence or simple specification when required.

### Estimation approach — sieve maximum likelihood
- Parameter vector α ≡ (θ′, f_{y^*|s,l}, f_{ε^y|s}, f_{ε^z|l})′ with m(·;θ) parametric (quadratic) and other elements nonparametric densities.
- Log-joint likelihood (observation i): log{ ∫ f_1(y^* | s_i,l_i) f_2(y_i − y^* | s_i) f_3(z_i − m(y^*;θ) | l_i) dy^* } + log f(s_i,l_i).
- Finite-dimensional sieve spaces A_n approximate functional spaces; Hermite orthogonal polynomials used as basis functions.
- Number of sieve terms chosen via simulation guidance: 6 for DMSP/OLS sample; 4 for VIIRS sample.
- Asymptotic results (under Assumptions 7–19):
  - Consistency: α̂_n → α_0 in appropriate norm (Lemma 1).
  - Convergence rates for nonparametric components: ‖α̂_n − α_0‖_2 = O_P(n^{−γ/(2γ+1)}) = o_P(n^{−1/4}), with γ ≡ min{γ_1/2, γ_2/2, γ_3/2} > 1/2 (Theorem 2).
  - Asymptotic normality of θ̂_n and semiparametric efficiency results for θ̂_n (Theorem 3), with explicit definitions of semiparametric score and information.

### Predictive regression and optimal linear combination
- Predictive regression used for light-predicted GDP per capita:
  - y_{i,t} = β z_{i,t} + δ_i D_{c_i} + γ_t D_{y_t} + η_{i,t}.
  - Nighttime light–predicted real GDP per capita: ˆy_{i,t} = β z_{i,t} + δ_i D_{c_i} + γ_t D_{y_t}.
  - Predictive model preferred linear form for simplicity; explains more than 98% of variation in official real GDP per capita (see Table 5).
- Optimal linear combination:
  - New measure ˆy^*_{i,t} = λ_{i,t} ˆy_{i,t} + (1−λ_{i,t}) y_{i,t}.
  - Conditional MSE decomposition: E[(ˆy^*_{i,t} − y^*_{i,t})^2 | z_{i,t}, s_{i,t}, l_{i}] = λ^2 E[(ˆy_{i,t} − y^*_{i,t})^2 | z_{i,t}, s_{i,t}, l_{i}] + (1−λ)^2 E[(ε_y)^2 | s_{i,t}].
  - Closed-form optimal weight:
    - λ_{i,t} = E[(ε_y)^2 | s_{i,t}] / { E[(ˆy_{i,t} − y^*_{i,t})^2 | z_{i,t}, s_{i,t}, l_{i}] + E[(ε_y)^2 | s_{i,t}] }.
  - λ_{i,t} ∈ [0,1]; varies by country-year depending on relative uncertainty.

### Semiparametric conditional mean (alternative estimator)
- Conditional mean estimator:
  - E[y^*_{i,t} | y_{i,t}, z_{i,t}, s_{i,t}, l_{i}] = ∫ y^* f(y^* | s_{i,t}, l_{i}) f(y_{i,t} − y^* | s_{i,t}) f(z_{i,t} − m(y^*; θ) | l_{i}) dy^* / ∫ f(y^* | s_{i,t}, l_{i}) f(y_{i,t} − y^* | s_{i,t}) f(z_{i,t} − m(y^*; θ) | l_{i}) dy^*.
- Advantage: uses full conditional distribution information.
- Drawback: less robust in sparse regions of the empirical support; can be volatile when observations are sparse. Authors favor the optimal linear measure for robustness.

### Main empirical findings (four key findings)
- Nonlinearity of lights–GDP relationship:
  - Elasticity of nighttime lights with respect to real GDP per capita varies with income and is nonconstant.
  - Baseline DMSP/OLS estimates (quadratic m(y^*) = θ_0 + θ_1 y^* + θ_2 (y^*)^2):
    - θ_0 point estimate 0.398, standard error (0.176).
    - θ_1 point estimate 1.234, standard error (0.139).
    - θ_2 point estimate −0.244, standard error (0.049).
  - Estimated elasticity: about 2.5 for low income countries and close to 0 for high income countries (DMSP/OLS).
  - VIIRS estimates (less precise; linear adequate):
    - θ_0 point estimate 0.225, standard error (0.170).
    - θ_1 point estimate 0.964, standard error (0.248).
    - θ_2 point estimate −0.019, standard error (0.157).
- Measurement-error heterogeneity by income/statistical capacity:
  - High income countries: measurement errors concentrated at zero (high precision).
  - Low and middle income countries: fat-tailed measurement-error distributions (larger errors).
  - Among low/middle income countries, distinction between high and low statistical capacity is blurred in estimated errors.
- Usefulness of nighttime lights for augmenting GDP measures:
  - Nighttime lights most useful in low and middle income countries.
  - Optimal weight on light-predicted GDP reaches 70% for middle income countries in some country-year observations (general statement in Results).
  - Pattern of λ_{i,t} (DMSP/OLS 1992–2013) broadly bell-shaped across income:
    - Very low or very high income countries receive small weights on light-predicted GDP.
    - Middle-income countries receive higher weights.
  - Representative λ by World Bank income status (Table 6):
    - Low Income λ = 0.24
    - Lower Middle Income λ = 0.34
    - Upper Middle Income λ = 0.28
    - High Income λ = 0.01
  - Mean squared error (Table 6):
    - Low Income 0.02
    - Lower Middle Income 0.02
    - Upper Middle Income 0.07
    - High Income 0.01
  - #observations by income classification (Table 6): Low Income 121, Lower Middle Income 121, Upper Middle Income 216, High Income 738, Total observations 3,852.
- Insights on conflicts and political disruptions:
  - New measures often reveal that countries disrupted by conflicts and political instability underestimated economic deterioration during downturns and recovery in official GDP measures.
  - Illustrated country cases (Figure 7): Democratic Republic of the Congo, Djibouti, Kenya, Sierra Leone — new measures show worse downturns and stronger recoveries than official series indicate.

### Predictive regression and fit statistics (selected table results)
- Table 5 (1992–2013, N = 3,870) predictive regressions:
  - Column (1) baseline: (log) Nighttime Lights per capita coefficient 0.218, standard error (0.013), adj. R^2 = 0.981.
  - Column (2) with squared term: coefficient on (log) Nighttime Lights per capita 0.265 (0.031); squared term coefficient 0.005 (0.003); adj. R^2 = 0.981.
  - Column (3) with country interactions: coefficient 0.373 (0.110); adj. R^2 = 0.986.
- Authors use prediction based on column (1) to retain cross-country information in nighttime lights.

### Robustness checks, simulations, and alternative specifications
- Simulation design (B.1):
  - DGP like empirical model with quadratic m(·); six equal-sized groups based on statistical capacity (s_i = 1,2,3) and location (l_i = 1,2).
  - Group-specific true GDP distributions are mixtures of two normals; measurement error variances σ(s) and σ(l) differ by group.
  - Quadratic m(·) in simulations uses same coefficients as DMSP/OLS point estimates.
  - Simulation draws: 400 samples with sample size n = 4000.
- Table 7 (parameterization in simulations) — group entries preserved exactly:
  - Group 1: y^* = 0.6 N(−0.8,0.6^2) + 0.4 N(0.8,0.7^2); σ(s) = 0.4; σ(l) = 1
  - Group 2: y^* = 0.7 N(−1,0.9^2) + 0.3 N(0.7,0.8^2); σ(s) = 0.4; σ(l) = 0.8
  - Group 3: y^* = 0.4 N(−0.1,0.6^2) + 0.6 N(0.2,0.8^2); σ(s) = 0.3; σ(l) = 1
  - Group 4: y^* = 0.2 N(−0.4,0.7^2) + 0.8 N(0.6,0.7^2); σ(s) = 0.3; σ(l) = 0.8
  - Group 5: y^* = 0.3 N(−0.2,0.8^2) + 0.7 N(1,0.6^2); σ(s) = 0.1; σ(l) = 1
  - Group 6: y^* = 0.5 N(−1,1) + 0.5 N(−0,4,0.6^2); σ(s) = 0.1; σ(l) = 0.8
- Sieve term guidance from simulations (Table 8):
  - True θ values: θ_0 = 0.398, θ_1 = 1.234, θ_2 = −0.244.
  - Estimates accurate around k = 6 orthogonal Hermite terms (motivating k = 6 for DMSP/OLS; k = 4 for VIIRS).
- Alternative specifications:
  - Linear vs. quadratic vs. cubic m(·): quadratic term statistically significant; cubic not significant and unstable.
  - Parsimonious error-structure (errors differ only by scale σ(s), σ(l)) yields similar results but less robust to sieve choice (Table 9: θ_0 = 0.326 (0.160), θ_1 = 1.261 (0.090), θ_2 = −0.212 (0.032)).
  - Full parametric Gaussian specification vulnerable to outliers; Table 10 reports counterintuitive σ_y3 when including small high-lighting city-states; excluding Hong Kong SAR, Macau SAR, Singapore resolves the anomaly (σ_{y3} = 0.050 (0.006) when excluding).
- Gas flaring robustness check (B.3):
  - τ = fraction of nighttime lights within gas flare shapefiles (average 1992–2013).
  - Top τ country entries (Table 11): GNQ 0.75; GAB 0.69; NGA 0.57; LBY 0.36; IRQ 0.32; KWT 0.31; OMN 0.31; DZA 0.30; COG 0.26; QAT 0.26; KAZ 0.24; YEM 0.20; RUS 0.17; AGO 0.16; ECU 0.15; TKM 0.15; PNG 0.13; COD 0.11; UZB 0.09; IRN 0.10; IDN 0.09; VEN 0.08; ARE 0.06; SAU 0.05; EGY 0.04; ARG 0.04; SDN 0.03; COL 0.03; CHL 0.03; PER 0.02; TUN 0.02; CAN 0.01; CHN 0.01; ...; USA 0.00; ZAF 0.00; BRA 0.00; NOR 0.00.
  - Dropping countries with τ ≥ 0.02 (34 countries) and reestimating yields no statistically significant difference in production-function coefficients.

### Constructed measures and practical use
- Two constructed measures using identified distributions:
  - Optimal linear combination (baseline): convex, observation-specific weights λ_{i,t} combining official GDP and light-predicted GDP; robust across countries and sparse-data regions.
  - Semiparametric conditional mean: uses full conditional distribution; makes fuller use of information but requires greater data density and can be volatile in sparse regions.
- Role and interpretation:
  - New measures complement official GDP measures rather than substitute them; construction depends on existence of official GDP.
  - Discrepancy between new and official measures averages to zero over a long period by construction; not a direct measure of absolute size of informal economy but can reflect changes in informal activity.
  - Empirical link to informal economy proxy:
    - Using agricultural share deviation from long-run average for low and middle income countries, correlation between discrepancy and agricultural-share deviation is 0.20 (positive relationship; each blue dot in Figure 8 is a country-year).
- Use cases:
  - Detecting uncertainty and potential mismeasurement in official real GDP.
  - Highlighting periods of underestimated deterioration or recovery due to conflicts or political instability.

### Caveats, limitations, and policy-relevant implications
- Caveats:
  - Country-specific nocturnal lighting norms vary; country fixed effects absorb time-invariant differences but not structural changes.
  - Assumption that nighttime lights measurement errors depend on centroid latitude may be strong for geographically large countries.
  - Lights–GDP relationship may change over time (energy conservation, climate change awareness).
  - Semiparametric conditional mean less robust in sparse data regions; optimal linear measure preferred for baseline.
- Policy-relevant results:
  - Official real GDP per capita measures are less precise for low income countries; nighttime lights can substantially improve measurement there.
  - New measures identify instances where official statistics may understate downturns and recoveries, informing targeted statistical-investment or validation efforts.
  - Methodology applicable to other remote-sensing indicators that are conditionally independent of official data, enabling broader augmentation of national accounts.

*Source: wpiea2019077 - 1. Optimal Linear Measure (IMF Working Paper PDF chapter).*

### 1.    Optimal Linear Measure .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    13

### 1.    Optimal Linear Measure

### Major methodological components
- Optimal Linear Measure (section title, page 13)
- Semiparametric Conditional Mean (page 15)
- Data (page 16)
  - Nighttime Lights (page 16)
  - Administrative Maps and Country Location (page 17)
  - GDP, Population, and Statistical Capacity (page 18)

### Results and empirical focus (section V, page 20)
- Nighttime Lights’ Elasticity of Real GDP per capita (page 20)
- Uncertainty in Real GDP per capita (page 21)
- New Measures of Real GDP per capita (page 22)
- Official vs. New Measures (page 25)

### Conclusion
- Conclusion (page 28)

### Appendices and technical materials
- References (page 29)
- Appendices (starting page 31)
  - A. Mathematical Proofs (page 31)
    - A.1. Assumptions (page 31)
    - A.2. Nonparametric Identification (page 31)
    - A.3. Proofs (page 33)
    - A.4. Identification of the Simple Specification (page 35)
    - A.5. Sieve Maximum Likelihood Estimation (page 36)
    - A.6. Consistency (page 38)
    - A.7. Convergence Rates and Asymptotic Normality (page 38)
      - A.7.1. Convergence Rates of Nonparametric Part (page 39)
      - A.7.2. Asymptotic Normality of Parametric Part (page 40)
  - B. Robustness Checks: Simulations, Specifications, and Estimates (page 43)
    - B.1. Simulations (page 43)
    - B.2. Alternative Specifications (page 45)
    - B.3. Gas Flaring in the Data (page 48)
    - B.4. Optimal Linear Measure for More Countries (page 49)
    - B.5. Semiparametric Conditional Mean for Robustness Check (page 50)
  - C. More Descriptive Details on Data (page 50)
    - C.1. Nighttime Lights vs. Real GDP per capita (page 50)
    - C.2. Statistical Capacity and Latitude (page 51)

### Empirical inputs and outputs (listed tables and figures with exact labels and pages)
- Tables (selected)
  - 1. Summary Statistics (page 19)
  - 2. Summary Statistics (page 19)
  - 3. Estimated Light Production Function (page 20)
  - 4. Estimated Light Production Function (page 20)
  - 5. Official Real GDP per capita and Nighttime Lights 1992-2013 (page 23)
  - 6. Optimal Weights and Mean Squared Error by Income Status (page 25)
  - 7. Parameterization in Simulations (page 43)
  - 8. Simulation Results (page 44)
  - 9. Estimated Light Production Function: Parsimonious Error Structure (page 47)
  - 10. Full Parametric Specification (page 48)
  - 11. Fraction of Nighttime Lights in Gas Flare Shapefiles (page 50)

- Figures (selected)
  - 1. Examples of Satellite Images of Nighttime Lights (page 6)
  - 2. Nighttime Lights vs. GDP (1992-2017) (page 11)
  - 3. Map of Nighttime Lights in 2010 (page 17)
  - 4. Estimation Results with Quadratic Light Production Function (page 21)
  - 5. Distribution of Measurement Errors of Real GDP per capita (page 22)
  - 6. Optimal Weights and Real GDP per capita: DMSP/OLS 1992-2013 (page 24)
  - 7. Real GDP per capita: Economic Disruption and Restoration (page 26)
  - 8. New Measures and the Informal Economy: 1992-2013 (page 27)
  - 9. Example of Density Function Approximation (page 44)
  - 10. Semiparametric Conditional Mean and Optimal Linear Measure Performance in Simulations (page 46)
  - 11. Linear Specification (page 47)
  - 12. Quadratic Specification with More Parsimonious Error Structure (page 47)
  - 13. Gas Flares in Nigeria (page 49)
  - 14. New Measures for Low Income Countries (page 51)
  - 15. Semiparametric Conditional Mean and Optimal Linear Measure (page 52)
  - 16. Nighttime Lights vs. Real GDP per capita (1992-2013) (page 53)
  - 17. Real GDP per capita vs. Statistical Capacity by Location (1992-2013) (page 54)
  - 18. Centroids of Selected European Countries (page 55)

*Source: wpiea2019077 - 1. Optimal Linear Measure (PDF chapter), canonical URL provided with content.*

### 19.    Real GDP per capita vs. Latitude by Statistical Capacity (1992-2013).  .  .  .  .  .  .    56

### 19.    Real GDP per capita vs. Latitude by Statistical Capacity (1992-2013).

### Major themes and goals
- Use satellite-recorded nighttime lights to illuminate uncertainty in official measures of real GDP per capita.
- Jointly uncover the distribution of measurement errors in official GDP measures and nighttime lights and estimate their functional relationship.
- Construct improved measures of real GDP per capita by optimally combining official GDP, nighttime lights information, statistical capacity, and geographic location (latitude).
- Rely on nonclassical and nonlinear measurement error models and recent identification results for measurement error models.

### Data and empirical scope
- Two nighttime lights datasets are used: one spanning 1992 to 2013 and the other 2013 to 2017, recorded by different satellite systems.
- Geographic location is proxied by a country’s centroid latitude because the satellites recording nighttime lights are polar orbiting.
- Observables in the empirical i.i.d. setup: {z_i, y_i, s_i, l_i}_i=1^n where:
  - y_i,t: reported (observed) log real GDP per capita
  - y*_i,t: true (latent) log real GDP per capita
  - s_i,t: statistical capacity of country i at time t
  - z_i,t: log nighttime lights per capita
  - l_i: latitude

### Statistical framework and model specification
- Additive measurement-error specification for reported GDP (error distribution may vary with statistical capacity):
  - y_i,t = y*_i,t + ε^y_i,t(s_i,t)
- Production-function specification for nighttime lights with additive error (error distribution may vary with geographic location):
  - z_i,t = m(y*_i,t) + ε^z_i,t(l_i)
- The production function m(·) is informed by the data and, for tractability in estimation, assumed to be quadratic in the empirical implementation.
- Error term distributions ε^y and ε^z are allowed to be nonparametric and to vary with s and l respectively.
- Identification claims:
  - Under sufficient conditions (detailed in appendices), the nonlinear relationship between nighttime lights and GDP and the distributions of measurement errors are nonparametrically identified using cross-country variation in statistical capacity and latitude.
  - For high income countries identification relies on additivity and independence assumptions.

### Estimation approach
- A seminonparametric sieve maximum likelihood estimator (sieve MLE) is developed:
  - Parameter vector α ≡ (θ′, f_{y*|s,l}, f_{ε^y|s}, f_{ε^z|l})′ where m(·;θ) is the parametric (quadratic) production function and other elements are nonparametric densities.
  - Log-joint likelihood for α uses the integral over the latent y*:
    - `(for each observation) log{ ∫ f_{1}(y*|s_i,l_i) f_{2}(y_i−y*|s_i) f_{3}(z_i−m(y*;θ)|l_i) dy* } + log f(s_i,l_i)`
  - Finite-dimensional sieve spaces A_n approximate the functional spaces; the sieve MLE α̂_n maximizes the sample log-likelihood over A_n.
- Asymptotic results established (under assumptions in appendices):
  - Consistency of α̂_n for α_0 and convergence rates for nonparametric components.
  - Asymptotic normality of θ̂_n around θ_0; when m(·;θ) is misspecified, θ̂_n is asymptotically normal around a pseudo-true value and is semiparametrically efficient for θ_0.

### Main empirical findings (four key findings)
- Nonlinearity of lights–GDP relationship:
  - The elasticity of nighttime lights with respect to real GDP per capita varies with income and is nonconstant.
  - Baseline estimate: elasticity is around 2.5 for low income countries and close to 0 for high income countries.
  - Elasticity steadily decreases as real GDP per capita increases, reflecting different development modes at different stages.
- Measurement-error heterogeneity by income/statistical capacity:
  - Measurement errors in real GDP per capita are larger for lower-income countries.
  - Sharp distinction between high income countries and the rest: high income countries have measurement errors concentrated at zero; low and middle income countries exhibit fat tails in the distribution of measurement errors.
  - Distinction between high and low statistical capacity among low and middle income countries is blurred.
- Usefulness of nighttime lights for augmenting GDP measures:
  - Nighttime lights are most useful in low and middle income countries.
  - The optimal weight on light-predicted GDP in the new linear combined measure reaches 70% for middle income countries.
  - The optimal weight declines for countries at either income extreme:
    - For extremely low real GDP per capita countries, darkness and limited electricity access increase uncertainty in light-predicted GDP.
    - For high real GDP per capita countries, sensor saturation and post-industrial activities reduce the informativeness of nighttime lights.
- Insights on conflicts and political disruptions:
  - New measures often reveal that countries disrupted by conflicts and political instability underestimated economic deterioration during downturns and subsequent recovery in official GDP measures.
  - Possible mechanisms: disruptions to tax registration and emergence of informal economy that escape national accounts.

### Constructed measures of GDP and practical use
- Two new measures constructed using the identified distributions:
  - Optimal linear combination: convex combination of official GDP and light-predicted GDP with observation-specific optimal weight; robust across countries.
  - Semiparametric conditional mean: uses the full conditional distribution of true GDP given observables; makes fuller use of information but requires more data density and may be less robust where observations are sparse.
- Role of new measures:
  - Complement official GDP measures rather than substitute them; construction depends on the existence of official GDP measures.
  - Can provide useful insights into the size and dynamics of economies where official measures are uncertain.

### Methodological and literature contributions
- First paper to estimate distributions of measurement errors in official GDP and nighttime lights directly from data.
- Advances prior literature by allowing:
  - Nonlinear and income-varying elasticity between lights and GDP (vs. constant elasticity in some prior works).
  - Observation-specific optimal weighting in linear combinations (vs. constant weights).
- Builds on nonclassical measurement error identification literature (e.g., Hu and Schennach (2008); Carroll, Chen, and Hu (2010); Schennach and Hu (2013)) and applies sieve MLE estimation methods.
- Broader applicability: method can be applied to other remote sensing data with measurement errors and nonlinear relationships to economic variables.

*Source: IMF Working Paper content unit wpiea2019077 — "19.    Real GDP per capita vs. Latitude by Statistical Capacity (1992-2013)."*

### 1.   Optimal Linear Measure

### 1.   Optimal Linear Measure

### Real GDP per capita predicted by nighttime lights
- Predictive regression: y_i,t = β z_i,t + δ_i D_c_i + γ_t D_y_t + η_i,t; nighttime light–predicted real GDP per capita defined as ˆy_i,t = β z_i,t + δ_i D_c_i + γ_t D_y_t.
- Regression specification uses country and year fixed effects; authors prefer the linear form in equation (3)/(4) for simplicity and to avoid overfitting.
- The predictive model explains more than 98% of variation in official real GDP per capita (see Table 5).

### Optimal linear combination of official and light-predicted GDP
- New measure: ˆy*_{i,t} = λ_{i,t} ˆy_{i,t} + (1−λ_{i,t}) y_{i,t}, where ˆy_{i,t} is nighttime light–predicted GDP per capita, y_{i,t} is official GDP per capita, and λ_{i,t} is the weight.
- Optimal weight λ_{i,t} minimizes conditional mean squared error: λ_{i,t} = arg min_λ E[(ˆy*_{i,t} − y*_{i,t})^2 | z_{i,t}, s_{i,t}, l_{i}].
- Conditional MSE decomposes into uncertainty from ˆy_{i,t} and official GDP measurement error: E[(ˆy*_{i,t} − y*_{i,t})^2 | z_{i,t}, s_{i,t}, l_{i}] = λ^2 E[(ˆy_{i,t} − y*_{i,t})^2 | z_{i,t}, s_{i,t}, l_{i}] + (1−λ)^2 E[(ε_y)^2 | s_{i,t}].
- Closed-form optimal weight: λ_{i,t} = E[(ε_y)^2 | s_{i,t}] / { E[(ˆy_{i,t} − y*_{i,t})^2 | z_{i,t}, s_{i,t}, l_{i}] + E[(ε_y)^2 | s_{i,t}] }.
- λ_{i,t} ∈ [0,1]; weights vary by country-year depending on relative uncertainty of the two measures.

### Semiparametric conditional mean (alternative estimator)
- Conditional mean estimator: E[y*_{i,t} | y_{i,t}, z_{i,t}, s_{i,t}, l_{i}] = ∫ y* f(y* | s_{i,t}, l_{i}) f(y_{i,t} − y* | s_{i,t}) f(z_{i,t} − m(y*; θ) | l_{i}) dy* / ∫ f(y* | s_{i,t}, l_{i}) f(y_{i,t} − y* | s_{i,t}) f(z_{i,t} − m(y*; θ) | l_{i}) dy*.
- Advantage: uses full information in conditional distributions.
- Drawback: nonparametric feature makes it less robust and requires large sample size; can be volatile in sparse regions of (y_{i,t}, z_{i,t}, s_{i,t}, l_{i}). Hence authors choose the optimal linear measure for robustness.

### Data — Nighttime lights, locations, GDP and sample
- Nighttime lights sources:
  - DMSP/OLS (NOAA composites) available 1992–2013; each pixel is 30 arc-second grid; radiance value from 0 to 63.
  - VIIRS (Suomi-NPP) available since 2013; monthly data averaged to annual series for analysis.
- Country maps: use GADM administrative shapefiles; sum radiance within country borders to obtain country-year nighttime lights panel.
- Country location: centroid of largest contiguous land block used as country latitude; locations discretized into binary indicator of whether centroid latitude is between the Northern and Southern Tropic or outside.
- GDP, population, statistical capacity:
  - GDP per capita (PPP, constant 2011 international dollars) and population from the World Bank.
  - Night light intensity = (log) total nighttime lights per capita (radiance sum divided by population and logged).
  - Statistical capacity ratings used to group countries: high income; low/middle income below median statistical capacity rating; low/middle income above median.
- Sample sizes:
  - DMSP/OLS panel: unbalanced panel of 182 countries and 3,870 observations (majority 1992–2013).
  - VIIRS panel: 184 countries and 920 observations.
  - Discrepancy due to availability of statistical capacity ratings (two countries: Barbados and the Czech Republic).
- Sieve MLE implementation:
  - Hermite orthogonal polynomials used as basis functions.
  - Number of sieve terms chosen via simulation: 6 for DMSP/OLS sample; 4 for VIIRS sample.

### Results — Light production functions and elasticities
- DMSP/OLS estimated quadratic light production function: m(y*) = θ_0 + θ_1 y* + θ_2 (y*)^2
  - θ_0 point estimate 0.398, standard error (0.176)
  - θ_1 point estimate 1.234, standard error (0.139)
  - θ_2 point estimate −0.244, standard error (0.049)
  - Standard errors based on 400 sample bootstraps.
  - Quadratic term non-zero: elasticity of nighttime lights with respect to real GDP per capita is not constant; slope decreases as real GDP per capita increases.
  - Estimated elasticity: about 2.5 for low income countries and close to 0 for high income countries (DMSP/OLS).
- VIIRS estimated production function:
  - θ_0 point estimate 0.225, standard error (0.170)
  - θ_1 point estimate 0.964, standard error (0.248)
  - θ_2 point estimate −0.019, standard error (0.157)
  - Estimates less precise; linear production function appears adequate; estimated elasticity close to 1 across countries.
- Interpretation: DMSP/OLS overpass (7pm–9pm) better captures economic activity than VIIRS (after midnight), consistent with lower elasticities for VIIRS.

### Results — Measurement error distributions and uncertainty
- Distribution of measurement errors of real GDP per capita:
  - High income countries: measurement errors concentrated at zero (high precision).
  - Low and middle income countries: distributions with fat tails (larger measurement errors).
  - Among low/middle income countries, distinction between high and low statistical capacity is not clear-cut; both data sets suggest similar measurement errors across those subgroups.
- Table 3/4/5 findings:
  - Regression of (log) real GDP per capita on (log) nighttime lights per capita and dummies (DMSP/OLS) yields adjusted R^2 above 0.98 (Table 5).
  - Table 5 estimates (1992–2013, N = 3,870):
    - Column (1) baseline: (log) Nighttime Lights per capita coefficient 0.218, standard error (0.013), adj. R^2 = 0.981.
    - Column (2) with squared term: coefficient on (log) Nighttime Lights per capita 0.265 (0.031); squared term coefficient 0.005 (0.003); adj. R^2 = 0.981.
    - Column (3) with country interactions: coefficient 0.373 (0.110); adj. R^2 = 0.986.
  - Authors use prediction based on column (1) to retain cross-country information in nighttime lights.

### Results — Optimal weights and performance of new measure
- Optimal weights λ_{i,t} computed from equation (8); pattern (DMSP/OLS 1992–2013) broadly bell-shaped across income:
  - Very low or very high income countries receive small weights on light-predicted GDP.
  - Middle-income countries receive comparatively higher weights.
- Examples:
  - Malawi: night light intensity almost zero; optimal weight on nighttime light–predicted GDP below 0.05.
  - High income countries (e.g., United States): almost zero weights on nighttime light–predicted GDP.
  - Middle and low income examples (Sierra Leone, China, Brazil, Indonesia, Pakistan): optimal weights on nighttime light–predicted GDP range from 0.2 to 0.6.
- Absolute accuracy (Table 6: by World Bank income status):
  - λ by income status:
    - Low Income λ = 0.24
    - Lower Middle Income λ = 0.34
    - Upper Middle Income λ = 0.28
    - High Income λ = 0.01
  - Mean squared error:
    - Low Income 0.02
    - Lower Middle Income 0.02
    - Upper Middle Income 0.07
    - High Income 0.01
  - #observations by income classification: Low Income 121, Lower Middle Income 121, Upper Middle Income 216, High Income 738, Total observations 3,852.
  - Interpretation: mean squared error of light-predicted GDP per capita for low income countries is almost twice as it is for high income countries; greater discrepancy between light-predicted GDP per capita and official figures for low income countries.

### Official vs. new measures — illustrative country cases
- For countries affected by conflict and political instability, the optimal linear measure often indicates:
  - Underestimation of economic deterioration during downturns in official figures.
  - Underestimation of recovery/upturns in official figures.
- Examples illustrated (Figure 7): Democratic Republic of the Congo, Djibouti, Kenya, Sierra Leone.
  - Democratic Republic of the Congo: new measures show worse conditions during 1992–2001 deterioration and higher living standards after recovery than official figures indicate.
  - Djibouti, Kenya, Sierra Leone: similar patterns of official overestimation during downturns and underestimation during recovery, consistent with difficulty tracking disrupted economies and expansion of informal activity during restoration.

*Source: wpiea2019077 - 1.   Optimal Linear Measure (IMF PDF chapter).*

### Appendix B.4 presents the official measures of real GDP per capita and our optimal linear

### wpiea2019077 - Appendix B.4 presents the official measures of real GDP per capita and our optimal linear

### Relation to the Informal Economy
- Literature documents large informal economy and its strong linkage to agriculture (La Porta and Shleifer (2014); Medina and Schneider (2018)).
- The optimal linear measure, by construction through equations (4) and (5), implies the discrepancy between the new and official measures should be zero on average over a long period of time.
- Therefore, the discrepancy:
  - Does not reflect the absolute size of the informal economy.
  - Could reflect changes in the size of the informal economy over time.
- Empirical exercise for low and middle income countries:
  - Uses the agricultural share of the economy as a proxy for the informal economy (footnote 15: Data are from the World Bank).
  - Figure 8 contrasts the discrepancy between the new and official measures of real GDP per capita against the deviation of a country’s agricultural share from its long-run average; each blue dot represents a country-year observation.
  - There is a positive relationship: years when the agricultural share is above its own average, the discrepancy between the two GDP measures tends to increase.
  - The correlation is 0.20, indicating other dimensions beyond the informal economy drive the discrepancy.
- The precise mechanisms driving the discrepancy are left for future research.

### Nighttime Lights as an Information Source
- Nighttime lights can improve official real GDP data by:
  - Detecting uncertainty in official data.
  - Detecting potential mismeasurement of real GDP.
- Systematic differences between the new and official measures may warrant further investigation into contributing factors.
- The method is not limited to nighttime lights; any measures of real GDP conditionally independent of official data can be used similarly.
- The paper constructs new measures of real GDP per capita and real GDP growth using a semi-nonparametric Sieve MLE based on estimated distributions.

### Caveats and Limitations
- Country-specific social norms and nocturnal lighting habits vary; if time-invariant, country fixed effects in the optimal linear measure account for them.
- If a country undergoes structural transformation dissimilar to others, the new measures may not adequately capture such changes.
- The model assumes measurement errors in nighttime lights depend on the latitude of a country’s centroid; this assumption may be strong for countries with large geographical coverage.
- Nighttime lights–GDP relationship may change over time, especially given increasing global awareness of energy conservation and climate change.
- The new measures are viewed as more useful and relevant for countries with an average size.

### Key Findings and Policy-Relevant Results
- Official real GDP per capita measures are less precise for low income countries; nighttime lights can play a bigger role in improving such measures.
- Comparing new measures with official measures:
  - Countries disrupted by conflicts and political instability often underestimated the deterioration of the economy during downturns and its recovery afterwards.
- The statistical framework provides sufficient conditions under which the joint distribution of observables and latent true GDP is uniquely determined by the distribution of observables.
- The methodology is expected to have broad impact on measuring GDP using additional information and to augment official statistics in gauging the economy.

*wpiea2019077 - Appendix B.4 presents the official measures of real GDP per capita and our optimal linear*

### REFERENCES

### wpiea2019077 - REFERENCES (and Appendices A–C)

### Key bibliographic references
- Alesina, Alberto; Stelios Michalopoulos; Elias Papaioannou, 2016, “Ethnic Inequality,” Journal of Political Economy, Vol. 124, No. 2, pp. 428–488.
- Aruoba, S Boraĝan; Francis X Diebold; Jeremy Nalewaik; Frank Schorfheide; Dongho Song, 2016, “Improving GDP Measurement: A Measurement-Error Perspective,” Journal of Econometrics, Vol. 191, No. 2, pp. 384–397.
- Carroll, Raymond J; Xiaohong Chen; Yingyao Hu, 2010, “Identification and Estimation of Nonlinear Models Using Two Samples With Nonclassical Measurement Errors,” Journal of Nonparametric Statistics, Vol. 22, No. 4, pp. 379–399.
- Donaldson, Dave; Adam Storeygard, 2016, “The View from Above: Applications of Satellite Data in Economics,” Journal of Economic Perspectives, Vol. 30, No. 4, pp. 171–98.
- Elvidge, Christopher D; Kimberly E Baugh; Eric A Kihn; Herbert W Kroehl; Ethan R Davis, 1997, “Mapping City Lights With Nighttime Data from the DMSP Operational Linescan System,” Photogrammetric Engineering and Remote Sensing, Vol. 63, No. 6, pp. 727–734.
- Elvidge, Christopher D; Edward H Erwin; Kimberly E Baugh; Daniel Ziskin; Benjamin T Tuttle; Tilottama Ghosh; Paul C Sutton, 2009, “Overview of DMSP Nightime Lights and Future Possibilities,” in Urban remote sensing event, 2009 joint, pp. 1–5 (IEEE).
- Feng, Shuaizhang; Yingyao Hu, 2013, “Misclassification Errors and the Underestimation of the US Unemployment Rate,” American Economic Review, Vol. 103, No. 2, pp. 1054–70.
- Freyberger, Joachim, 2017, “On Completeness and Consistency in Nonparametric Instrumental Variable Models,” Econometrica, Vol. 85, No. 5, pp. 1629–1644.
- Ghosh, Tilottama; Rebecca L Powell; Christopher D Elvidge; Kimberly E Baugh; Paul C Sutton; Sharolyn Anderson, 2010, “Shedding Light on the Global Distribution of Economic Activity,” Open Geography Journal, Vol. 3, pp. 147–160.
- Henderson, J Vernon; Tim Squires; Adam Storeygard; David Weil, 2018, “The Global Distribution of Economic Activity: Nature, History, and the Role of Trade,” The Quarterly Journal of Economics, Vol. 133, No. 1, pp. 357–406.
- Henderson, J Vernon; Adam Storeygard; David N Weil, 2012, “Measuring Economic Growth from Outer Space,” American Economic Review, Vol. 102, No. 2, pp. 994–1028.
- Hu, Yingyao, 2017, “The Econometrics of Unobservables: Applications of Measurement Error Models in Empirical Industrial Organization and Labor Economics,” Journal of Econometrics, Vol. 200, No. 2, pp. 154–168.
- Hu, Yingyao; Susanne M Schennach, 2008, “Instrumental Variable Treatment of Nonclassical Measurement Error Models,” Econometrica, Vol. 76, No. 1, pp. 195–216.
- La Porta, Rafael; Andrei Shleifer, 2014, “Informality and development,” Journal of Economic Perspectives, Vol. 28, No. 3, pp. 109–26.
- Medina, Leandro; Friedrich Schneider, 2018, “Shadow Economies Around the World: What Did We Learn Over the Last 20 Years?” IMF Working Paper 18/17.
- Pinkovskiy, Maxim; Xavier Sala-i Martin, 2016, “Lights, Camera . . . Income! Illuminating the National Accounts-Household Surveys Debate,” The Quarterly Journal of Economics, Vol. 131, No. 2, pp. 579–631.
- Schennach, Susanne M; Yingyao Hu, 2013, “Nonparametric Identification and Semiparametric Estimation of Classical Measurement Error Models Without Side Information,” Journal of the American Statistical Association, Vol. 108, No. 501, pp. 177–186.
- Shen, Xiaotong; Wing Hung Wong, 1994, “Convergence Rate of Sieve Estimates,” The Annals of Statistics, pp. 580–615.
- Storeygard, Adam, 2016, “Farther on down the Road: Transport Costs, Trade and Urban Growth in Sub-Saharan Africa,” The Review of Economic Studies, Vol. 83, No. 3, pp. 1263–1295.
- WORLD BANK, 2017, “Growth Out of the Blue,” South Asia Economic Focus.

### Appendix A — Mathematical proofs, identification, and estimation methodology
- Assumptions summary (exact formulations preserved):
  - Assumption 1: f(z_{i,t} | y^*_{i,t}, y_{i,t}, s_{i,t}, l_{i}) = f(z_{i,t} | y^*_{i,t}, l_{i}). Interpretation: nighttime lights relate to true real GDP per capita and geographic location, but not to measurement method or statistical capacity.
  - Assumption 2: f(y_{i,t} | y^*_{i,t}, s_{i,t}, l_{i}) = f(y_{i,t} | y^*_{i,t}, s_{i,t}). Interpretation: statistical capacity captures accuracy of GDP measurement regardless of location.
  - Assumption 3: f(z,y,y^*,s,l) is positive, bounded on support Z×Y×Y^*×S×L, and continuous in (z,y,y^*)∈Z×Y×Y^*.
  - Assumption 4: (i) bounded completeness of f(y | y^*, s): ∫ f(y|y^*,s) h(y^*) dy^* = 0 for all y implies h ≡ 0; (ii) for each (s,l), ∫ f(z,y|s,l) h(y) dy = 0 implies h ≡ 0. (Part (ii) directly testable under some restrictions.)
  - Assumption 5: Define κ_{jk}(y^*) as in equation (11). For any y^*_1 ≠ y^*_2, there exist j,k such that κ_{jk}(y^*_1) ≠ κ_{jk}(y^*_2) and sup_{y^*∈Y^*} κ_{jk}(y^*) < ∞. Interpretation: distribution of real GDP varies with statistical capacity and location.
  - Assumption 6: Normalization — for some s_j, either (i) mean E[y | y^*, s_j] = y^*; or (ii) mode arg max_y f(y | y^*, s_j) = y^*; or (iii) median = y^*.
- Theorem 1 (nonparametric identification):
  - Suppose Assumptions 1–6 hold. Then the observed distribution f(z,y,s,l) uniquely determines the joint distribution f(z,y,y^*,s,l) satisfying f(z,y,y^*,s,l) = f(z | y^*, l) f(y | y^*, s) f(y^*, s, l). (Equation (12))
  - Interpretation: conditional distributions of nighttime lights and official measures of real GDP per capita are identified from the joint observed distribution, enabling consistent estimation in parametric, semiparametric, or nonparametric specifications.
- Identification strategy (operator-based):
  - Defines integral operators L_{y|y^*,s_a}, L_{y,z|s_a,l_j}, L_{z|y^*,l_j}, and diagonal operator D_{y^*|s_a,l_j}.
  - Shows operator equivalences (equations (15)–(19)) and diagonalization (equation (18)), connecting observed operators to diagonal operator D_{jk}^{y^*} whose diagonal entries are κ_{jk}(y^*).
  - Uses uniqueness of diagonal decomposition (Dunford and Schwartz (1971)) and Assumption 6 (ordering normalization) to pin down exact y^* values for each eigenfunction and thereby fully identify f(y | y^*, s).
- Simple specification identification:
  - When errors ε_y and ε_z are mutually independent with zero mean and jointly independent of y^*, characteristic functions of y and z do not vanish on real line, and m(·) is monotone and continuously differentiable (not in a particular parametric family including linear), then distributions of y^*, ε_y, ε_z and function m(.) are identified (Theorem 1 in Schennach and Hu (2013) applied).
  - Practical note: World Bank Statistical Capacity Indicator missing for high-income countries; an assigned high-income category is used but requires the simple specification for identification of that category’s error distribution.
- Sieve Maximum Likelihood Estimation (seminonparametric estimator):
  - Parameter vector α_0 = (θ_0^T, f_{01}, f_{02}, f_{03})^T ≡ (θ_0^T, f_{y^*|s,l}, f_{ε_y|s}, f_{ε_z|l})^T.
  - Sieve MLE α̂_n maximizes sample log-likelihood over finite-dimensional sieve spaces A_n = Θ × F^n_1 × F^n_2 × F^n_3.
  - Uses Hölder spaces Λ_γ(V) and linear sieve basis p_{k_n}(·) (power series, splines, Fourier, orthogonal polynomials, etc.) to approximate densities.
  - Log-joint likelihood for α: Σ_{i=1}^n log{ ∫ f_1(y^* | s_i, l_i) f_2(y_i − y^* | s_i) f_3(z_i − m(y^*; θ) | l_i) dy^* } + log f(s_i, l_i).
- Consistency and asymptotics — assumptions and results (exact numeric/formal conditions preserved):
  - Assumption 7: Conditions on smoothness γ_1 > 1/2, γ_2 > 1, γ_3 > 1 for f_{y^*|s,l}, f_{ε_y|s}, f_{ε_z|l}, respectively.
  - Assumption 8: i.i.d. sample {z_i, y_i, s_i, l_i}; m(y^*; θ) continuous in θ; Θ compact; θ_0 unique solution of E[z | y^*, l] = m(y^*; θ).
  - Assumption 9–10: Regularity on likelihood, sieve basis growth conditions: min{k_{1,n},k_{2,n},k_{3,n}}→∞ and max{k_{1,n},k_{2,n},k_{3,n}}/n → 0.
  - Lemma 1: Under Assumptions 7–10, ‖α̂_n − α_0‖_s = o_p(1).
  - Convergence rates: define pseudo-metric ‖·‖_2; with γ ≡ min{γ_1/2, γ_2/2, γ_3/2} > 1/2 and k choices k_{1,n} = O(n^{1/(γ_1+1)}), k_{2,n} = O(n^{1/(γ_2+1)}), k_{3,n} = O(n^{1/(2γ_3+1)}), then ‖α̂_n − α_0‖_2 = O_P(n^{−γ/(2γ+1)}) = o_P(n^{−1/4}). (Theorem 2)
  - Asymptotic normality for θ̂_n:
    - Define μ^* (solutions μ^*_k ∈ U minimizing specified quadratic forms), V^* as in equation (21).
    - Semiparametric efficient score S_{θ_0} = ∂`/∂θ^T − ∂`/∂h [μ^*]; semiparametric information I^* = E[S_{θ_0}^T S_{θ_0}] = V^* (equation (23)).
    - Under Assumptions 11–19, √n (θ̂_n − θ_0) d→ N(0, V_*^{-1} I_* V_*^{-1}). (Theorem 3)

### Appendix B — Robustness checks, simulations, specifications, and estimates
- Simulation design (B.1):
  - Data-generating process like equations (1) and (2) with quadratic nighttime light production function m(·).
  - Six equal-sized groups of countries based on statistical capacity (s_i = 1,2,3) and location (l_i = 1,2).
  - Each group’s true GDP distribution follows a mixture of two normal distributions.
  - Measurement errors in GDP per capita and nighttime lights follow normal distributions with variances σ(s) and σ(l) differing by group.
  - Quadratic function m(·) assumed to have same coefficients as point estimates in DMSP/OLS data.
  - Simulation draws: 400 samples with sample size n = 4000.
- Table 7: Parameterization in simulations (exact entries preserved)
  - Group 1: y^* = 0.6 N(−0.8,0.6^2) + 0.4 N(0.8,0.7^2); σ(s) = 0.4; σ(l) = 1
  - Group 2: y^* = 0.7 N(−1,0.9^2) + 0.3 N(0.7,0.8^2); σ(s) = 0.4; σ(l) = 0.8
  - Group 3: y^* = 0.4 N(−0.1,0.6^2) + 0.6 N(0.2,0.8^2); σ(s) = 0.3; σ(l) = 1
  - Group 4: y^* = 0.2 N(−0.4,0.7^2) + 0.8 N(0.6,0.7^2); σ(s) = 0.3; σ(l) = 0.8
  - Group 5: y^* = 0.3 N(−0.2,0.8^2) + 0.7 N(1,0.6^2); σ(s) = 0.1; σ(l) = 1
  - Group 6: y^* = 0.5 N(−1,1) + 0.5 N(−0,4,0.6^2); σ(s) = 0.1; σ(l) = 0.8
- Simulation-based guidance on sieve terms:
  - Table 8 (simulation results for m(y^*) = θ_0 + θ_1 y^* + θ_2 (y^*)^2) reports true values and estimates for different k (number of orthogonal Hermite terms): true values θ_0 = 0.398, θ_1 = 1.234, θ_2 = −0.244.
  - Estimates (k = 4,5,6,8) show coefficients close to true values; coefficients "accurately estimated around k = 6", motivating k = 6 for DMSP/OLS and k = 4 for VIIRS (less observations).
  - Standard errors based on 400 sample bootstraps.
- Comparative measures of true GDP:
  - Two proposed measures: the optimal linear measure (chosen baseline) and the semiparametric conditional mean (uses full conditional distribution information).
  - Simulation results: optimal linear measure slightly smaller mean squared error but close to semiparametric conditional mean; semiparametric conditional mean is less robust in sparse regions of empirical distribution f(y,z,s,l).
  - Empirical kernel-density-based diagnostics (Figure 10): difference between semiparametric conditional mean and optimal linear measure decreases as empirical density increases.
- Alternative specifications examined (B.2):
  - Linear vs. quadratic vs. cubic m(·): Quadratic term statistically significant; linear misspecification leads to misattribution of deviations to GDP measurement errors; cubic not significantly different from zero and results unstable.
  - Parsimonious error-structure specification: assume ε_{y,i,t}(s_i) = σ(s) ε_{y,i,t} and ε_{z,i,t}(l_i) = σ(l) ε_{z,i,t} (errors differ only by scale). Using k = 6 sieve terms yields similar results (Table 9) but less robust to choice of sieve terms.
  - Full parametric specification (quadratic m(·); y^*, errors Gaussian): computationally lightweight but vulnerable to outliers; with full DMSP/OLS sample yields counterintuitive result that high-income economies have highest measurement error (Table 10). Excluding Hong Kong SAR, Macau SAR, and Singapore gives more intuitive results (σ_y3 = 0.050 when excluding those).
- Table 9 (parsimonious error structure, DMSP/OLS): point estimates m(y^*) = θ_0 + θ_1 y^* + θ_2 (y^*)^2 — point estimate θ_0 = 0.326, θ_1 = 1.261, θ_2 = −0.212; standard errors (0.160), (0.090), (0.032) respectively.
- Table 10 (full parametric specification, DMSP/OLS):
  - All Data: θ_0 = 0.336 (0.047), θ_1 = 1.352 (0.034), θ_2 = −0.256 (0.041), σ_{y1} (low capacity) = 0.458 (0.025), σ_{y2} (high capacity) = 0.474 (0.020), σ_{y3} (high income) = 0.791 (0.056).
  - Excluding Hong Kong SAR, Macau SAR, and Singapore: θ_0 = 0.316 (0.012), θ_1 = 1.354 (0.015), θ_2 = −0.268 (0.008), σ_{y1} = 0.458 (0.019), σ_{y2} = 0.471 (0.015), σ_{y3} = 0.050 (0.006).
  - Data re-centered at zero. Standard errors from 400 bootstraps.
- Gas flaring robustness check (B.3):
  - Use gas flare shapefiles provided by NOAA to compute τ = fraction of nighttime lights within gas flare shapefiles for each country (average 1992–2013).
  - Example: Nigeria map with gas flare polygons; τ computed as sum of lights within flare polygons divided by total lights for country.
  - Table 11: countries ranked by τ (top entries preserved):
    - GNQ 0.75; GAB 0.69; NGA 0.57; LBY 0.36; IRQ 0.32; KWT 0.31; OMN 0.31; DZA 0.30; COG 0.26; QAT 0.26; KAZ 0.24; YEM 0.20; RUS 0.17; AGO 0.16; ECU 0.15; TKM 0.15; PNG 0.13; COD 0.11; UZB 0.09; IRN 0.10; IDN 0.09; VEN 0.08; ARE 0.06; SAU 0.05; EGY 0.04; ARG 0.04; SDN 0.03; COL 0.03; CHL 0.03; PER 0.02; TUN 0.02; CAN 0.01; CHN 0.01; CMR 0.01; MYS 0.00; USA 0.00; ZAF 0.00; BRA 0.00; NOR 0.00; TTO 0.00; PHL 0.00; AUS 0.01; etc.
  - Practical result: dropping countries with τ ≥ 0.02 (34 countries) and reestimating the model yields no statistically significant difference in estimated coefficients of the nighttime light production function compared to full sample.
- Additional robustness and empirical notes:
  - Combining DMSP/OLS and VIIRS by assuming same distribution for GDP measurement errors across datasets yields similar production function estimates and little change in new measures.
  - Semiparametric conditional mean used as robustness check; difference between it and optimal linear measure falls with higher empirical density; for high-income countries difference almost zero because identifiable measurement errors are small there.
  - Optimal linear measure favored in baseline due to robustness in sparse empirical-distribution regions and similar predictive performance to semiparametric conditional mean.

### Appendix C — Data descriptive details
- Nighttime lights vs. real GDP per capita:
  - Figure 16 shows country placements on nighttime lights and real GDP per capita (1992–2013); selected countries span GDP per capita spectrum (figures not reproduced here).
- Statistical Capacity and latitude:
  - Statistical Capacity Indicator (World Bank) used for developing countries — composite score (methodology, data sources, periodicity and timeliness), scale 0–100 based on 25 criteria; average score during sample period used; high-income countries assigned score 100 and treated as separate group.
  - Latitude computed as centroid of largest contiguous block of country polygon (to avoid overseas territories). Figures illustrate centroids for selected European countries.
  - Empirical relationships:
    - Figure 17: positive relationship between real GDP per capita and statistical capacity; high-income countries located at top-right with statistical capacity 100.
    - Figure 19: real GDP per capita varies with latitude — countries at high latitudes in Northern and Southern Hemispheres tend to be richer, low-latitude countries poorer (North-South divide).
  - These patterns support Assumption 5 that distribution of real GDP varies with statistical capacity and geographic location.

*Italic: Source — wpiea2019077 - REFERENCES (pdf content and appended Appendices A–C).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019077.pdf_