## wpiea2019038 - 10. Sensitivity Analysis I: Sensitivity to Different Prior Selections

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---

### Introduction and context
- Adverse health shocks in low-income countries expose households to large out-of-pocket medical bills and financial risk; in Africa out-of-pocket spending accounts for 50 percent of total healthcare expenditure and 30 percent of households cope through borrowing and selling assets.
- Community Based Health Insurance (CBHI) aims to increase access, protect from financial risk, and improve health; Rwanda’s CBHI (Mutuelle de santé) started in 2004 and covered about 86 percent of the population in 2008.
- Concerns: a subsidized flat premium of about U$2 per person may still be too high for the poor; transport and nonessential drug costs may increase out-of-pocket spending for the poor.
- Data used: three rounds of Rwandan Integrated Household Living Conditions Survey (EICV) data from 2000, 2005, and 2010.

### Theoretical framework and empirical approach
- Utility framework: U(c_i, h_i) = u(c_i)^α h_i^(1−α); medical spending m_i = r I_i + p(I_i) V_i; health h_i = f(V_i, h_i0, Δ_i).
- Optimal out-of-pocket medical expenditure expression:
  - m_i(I_i^*, V_i^*; Y_i, r, γ, δ, μ, Δ) = r I^* + p(I^*) V^* = Y − [α (1−γ) (1−α) p(I^*) f(V^*, μ, Δ) / f_v(V^*, μ, Δ)]
- Main empirical issues: non-normal expenditure distribution with many zeros (corner solutions) and endogeneity from self-selection into CBHI.
- Estimation model: Extended Two-Part Model (ETPM) of Deb, Munkin, and Trivedi (2006), a three-equation system jointly modeling:
  - selection into CBHI d_i,
  - hurdle (probability of positive spending) s_i,
  - conditional amount of spending m_i.
- Latent-variable system:
  - d_i^* = β_d w_i + ε_id
  - s_i^* = β_s x_i + γ_s d_i + ε_is
  - m_i^* = β_m x_i + γ_m d_i + ε_im
- Error structure: ε_i ~ n(0, Σ_3×3) with first two diagonal elements = 1; conditional error decomposition:
  - ε_is = σ_ds ε_id + v_is
  - ε_im = σ_dm ε_id + v_im
- Identification: exclusion restriction using membership in microfinance institutions (z_i) as instrument excluded from x_i; balancing test indicates covariates balanced between microfinance members and non-members.

### Data and sample
- Three EICV rounds: 2000, 2005, 2010; household-level expenditure, individual-level insurance.
- Households “treated” if any member enrolled in CBHI.
- Final pooled sample: 26,195 households (6,390 from 2000; 6,259 from 2005; 13,546 from 2010 surveys) — note source tables report No. of households as 6,390; 6,900; 14,308 in years.
- CBHI coverage in surveys: 0% in 2000/2001, 41.9% in 2005/2006, 75.8% in 2010/2011.
- Other formal insurance: n.a. in 2000/2001, 9.1% in 2005/2006, 5.3% in 2010/2011.
- Uninsured: 100% in 2000/2001, 51.0% in 2005/2006, 22.2% in 2010/2011.
- Expenditures aggregated to annual per capita and deflated to year 2000 prices using National Institute of Statistics regional CPI.
- Controls: household demographics, number ill, consumption and wealth quartiles, occupation, education, 29 district dummies.

### Estimation details and priors
- Slope parameter priors: non-informative normal N(μ0=0, V0=10 I_K).
- Variance σ_m0^2 prior: inverse gamma 푖푔(ν/2, (c/2)^{-1}) with ν = 10 and c = 5.
- Priors on covariance parameters σ_ds and σ_dm: informative normal with mean zero and variance 0.5 (N(0, 1/2 I)); sensitivity checked with tighter priors N(0, 1/5 I) and N(0, 1/8 I).
- Estimation via Bayesian MCMC with data augmentation.
- Hypothesis testing for endogeneity: Savage-Dickey Bayes Factor (test H0: σ_ds = σ_dm = 0).

### Participation, selection and endogeneity — empirical findings
- Determinants of CBHI enrollment (posterior means significant):
  - Older and married household heads more likely to enroll.
  - Male-headed households, larger family size, and no education less likely to enroll.
  - Number of individuals with major illnesses significantly affects enrollment.
  - Higher income and wealth increase probability of enrollment (well-to-dos more likely to enroll).
  - Microfinance membership statistically significant predictor of enrollment.
  - District and year dummies significant: households in year 2010 more likely to enroll than in 2000.
- Evidence of selection on observables: variables like number of sick individuals and income affect both enrollment and spending.
- Savage-Dickey Bayes Factor results:
  - For prior variance v0 = 1/2, data strongly reject H0: σ_ds = σ_dm = 0 for overall, outpatient, and drug spending models.
  - Tests consistent for priors v0 = 2; v0 = 1 and for more informative selections v0 = 1/2; v0 = 1/5; v0 = 1/8, yielding Savage-Dickey ratios of zero or very close to zero for these models.
  - For inpatient spending model, data could not reject H0; Savage-Dickey ratio = 22.2, implying CBHI membership can be considered exogenous for inpatient spending.

### Treatment effects (ATEs) — exact reported values (2000 prices, RWF)
- Table 9 (ETPM vs TPM):
  - Overall:
    - ETPM: Prob. = 0.3179*; Amount = -3,609*
      - (standard errors: (0.019); (230))
    - TPM: Prob. = 0.1963*; Amount = 210*
      - ((0.007); (71))
  - Spending net of premium:
    - ETPM: Prob. = 0.2862*; Amount = -4,106*
      - ((0.028); (320))
    - TPM: Prob. = 0.1028*; Amount = -816*
      - ((0.007); (96))
  - Outpatient:
    - ETPM: Prob. = 0.345*; Amount = 346*
      - ((0.041); (91))
    - TPM: Prob. = 0.1249*; Amount = 29*
      - ((0.007); (14))
  - Inpatient:
    - ETPM: Prob. = 0.0483; Amount = 140
      - ((0.021); (159))
    - TPM: Prob. = 0.0325*; Amount = 49
      - ((0.004); (30))
  - Drug:
    - ETPM: Prob. = 0.0078; Amount = -3,553*
      - ((0.043); (484))
    - TPM: Prob. = -0.051; Amount = -976*
      - ((0.007); (99))
- Predicted expenditures (2000 prices):
  - Overall predicted expenditure (whole sample) = 2,312 (41)
  - Overall predicted expenditure (insured) = 1,410 (28)
  - Overall predicted expenditure (uninsured) = 5,019 (226)
  - Outpatient predicted expenditure (whole sample) = 385 (12)
  - Outpatient predicted expenditure (insured) = 689 (85)
  - Outpatient predicted expenditure (uninsured) = 343 (14)
  - Drug predicted expenditure (whole sample) = 1,855 (64)
  - Drug predicted expenditure (insured) = 612 (35)
  - Drug predicted expenditure (uninsured) = 4,166 (469)

### Covariance, variance and Bayes Factor (selected posterior means, Table 10)
- Covariance w/hurdle part:
  - -0.2783 (2-prior), -0.280 (1-prior), -0.2734 (1/5), -0.2794 (1/8)
  - (standard errors: (0.038), (0.041), (0.040), (0.044) respectively)
- Covariance w/expenditure part:
  - 0.8863, 0.8866, 0.8855, 0.8841
  - ((0.014), (0.013), (0.014), (0.015))
- Variance:
  - 1.707, 1.7073, 1.7079, 1.7091
  - ((0.024), (0.024), (0.026), (0.026))
- Bayes Factor reported as 0.0E+00 across prior selections in Table 10 for the overall model.

### Heterogeneity in treatment effects
- Individual-level treatment effects show substantial heterogeneity (Figures 2a–2e):
  - Treatment effects on overall and drug spending mainly negative but vary across households.
  - Treatment effects on outpatient and inpatient spending span both negative and positive domains.
- ATE variation by observed factors (Figures 3–6):
  - Overall spending ATE exhibits a “u-shape” by age with higher reduction for household heads aged between 20 and 60.
  - ATEs increase with households’ position in the income distribution (favoring better off households) for overall spending.
  - Outpatient spending shows the reverse pattern (CBHI increases outpatient spending).
  - Drug spending reductions concentrated among certain age/income/wealth percentiles.

### Sensitivity analysis I: sensitivity to different prior selections (Table 10) — exact reported values
- Priors evaluated: v0 = 2; v0 = 1; v0 = 1/5; v0 = 1/8.
- ATE on probability of spending:
  - 0.317 (v0=2); 0.317 (v0=1); 0.315 (v0=1/5); 0.317 (v0=1/8)
  - (standard errors: (0.016), (0.017), (0.017), (0.018) respectively)
- ATE on amount of spending (RWF):
  - -3,624; -3,625; -3,639; -3,605 across the four prior choices
  - (standard errors: (215), (221), (222), (231) respectively)
- Covariance and variance estimates remain stable across priors.
- Bayes Factor reported as 0.0E+00 across all prior selections for the overall model — indicates strong rejection of exogeneity for overall out-of-pocket expenditure.
- Conclusion: magnitude and direction of treatment effects remain stable regardless of prior selection; Savage-Dickey ratios support rejecting exogeneity for overall spending.

### Interpretation and key policy-relevant findings
- Non-linear effects of CBHI:
  - Increases probability of any healthcare spending by about 31.8 percentage points (ETPM).
  - Decreases conditional amount of total out-of-pocket expenditure by about RWF3,609 (approximately US$12 in 2000 prices and exchange rate as noted in source), representing about 83 percent of the average per capita healthcare expenditure in 2000 prices.
- Component-level effects:
  - Outpatient: increases probability by 34.5 percentage points and increases amount by RWF346.
  - Drug: reduces annual per capita drug spending by RWF3,553 (about 83 percent reduction relative to baseline uninsured average).
  - Inpatient: no statistically significant impact detected in ETPM.
- Accounting for premiums (overall net of premium): CBHI increases probability by 28.6 percentage points and decreases amount net of premium by RWF4,106.
- Ignoring endogeneity (TPM vs ETPM) changes sign and magnitude — e.g., TPM overall Amount = 210* vs ETPM Amount = -3,609*, indicating positive selection into CBHI on spending if endogeneity ignored.
- Heterogeneity and equity concerns:
  - Reductions in out-of-pocket spending are larger for households in higher income and wealth percentiles, raising equity concerns about whether CBHI favors the poor.
- Robustness caveats:
  - Results robust to alternative prior specifications for covariance parameters.
  - Potential bias from measurement error in self-reported expenditure if errors systematically differ by CBHI status.

### Appendix A — Estimation algorithm and MCMC implementation (key steps and settings)
- MCMC algorithm (Box 1) steps (1–9): draw d_i^*, s_i^*, m_i^*, draw β_d, draw θ_s = [β_d, γ_s], draw σ_ds, draw θ_m = [β_m, γ_m], draw σ_dm, draw σ_m^{-2}.
- Conditional distributions provided for each draw (truncated normal and normal forms; see steps for μ and V expressions).
- MCMC settings:
  - 10,000 MCMC simulations with the first 5,000 draws dropped as burn-ins.
- Convergence assessment:
  - trace plots
  - formal convergence diagnostic test developed by Geweke (1992).
- Estimation code implemented in Matlab and tested on simulated data before application to real data.

### Appendix B — Covariate balancing test (Microfinance) — selected entries and summary
- Number of obs. 5,486 (Member) and 20,757 (Non-Member).
- Selected variable means and balance indicators (Member | Non-Member | %bias | t | p>t):
  - HHAgeHH: 44.325 | 44.521 | -1.3 | -0.71 | 0.479
  - HHSize: 5.410 | 5.409 | 0.1 | 0.03 | 0.975
  - HHAge: 22.629 | 22.758 | -1.2 | -0.71 | 0.476
  - HHmale: 0.775 | 0.770 | 1.1 | 0.58 | 0.56
  - HHmarried: 0.645 | 0.641 | 0.8 | 0.45 | 0.655
  - HHIllness: 1.021 | 1.021 | 0 | -0.02 | 0.987
  - Urban: 0.131 | 0.140 | -2.5 | -1.41 | 0.158
- Interpretation: p>t values range from 0.158 to 0.987 for reported covariates, supporting balance between microfinance members and non-members on listed observables.

*Source: wpiea2019038 - 10. Sensitivity Analysis I: Sensitivity to Different Prior Selections (excerpt from IMF working paper PDF).*

### References .............................................................................................................

### wpiea2019038 - References

### Figures
- 1. Histogram of Log of Out-of-Pocket Healthcare Spending on Overall, Outpatient, Inpatient, and Drug Expenditures ........................................................................................19
- 2. Kernel Density of Treatment Effects ...................................................................................20
- 3. ATE on Overall Out-Of-Pocket Expenditure by Age, Income, and Wealth Percentiles .....21
- 4. ATE on Outpatient Expenditure by Age, Income and Wealth Percentiles ..........................21
- 5. ATE on Inpatient Expenditure by Age, Income, and Wealth Percentiles ...........................22
- 6. ATE on Drug Expenditure by Age, Income, and Wealth Percentiles .................................22

### Tables
- 1. Health Insurance Status........................................................................................................23
- 2. Per Capita Out-of-Pocket Healthcare Spending (RWF) in 2000 Prices ..............................23
- 3. Descriptive Statistics of Control Variables ..........................................................................24
- 4. Overall Healthcare Expenditure: ETPM Posterior Means and Standard Deviation ............25
- 5. Outpatient Expenditure: ETPM Posterior Means and Standard Deviations ........................26
- 6. Inpatient Expenditure: ETPM Posterior Means and Standard Deviations...........................27
- 7. Drug Expenditure: ETPM Posterior Means and Standard Deviations.................................28
- 8. Overall Healthcare Expenditure Net of Premium: ETPM Posterior Means and Standard Deviations .............................................................................................................29
- 9. Average Treatment Effects on the Probability and the Amount of Healthcare Spending ...............................................................................................................................30

*Content unit: wpiea2019038 - References; Source PDF filename: wpiea2019038 - References .............................................................................................................; Canonical source URL: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019038.pdf*

### 10. Sensitivity Analysis I: Sensitivity to Different Prior Selections ........................................30

### 10. Sensitivity Analysis I: Sensitivity to Different Prior Selections

### Introduction and context
- Adverse health shocks in low-income countries expose households to large out-of-pocket medical bills and financial risk; in Africa out-of-pocket spending accounts for 50 percent of total healthcare expenditure and 30 percent of households cope through borrowing and selling assets.
- Community Based Health Insurance (CBHI) aims to increase access, protect from financial risk, and improve health; Rwanda’s CBHI (Mutuelle de santé) started in 2004 and covered about 86 percent of the population in 2008.
- Concerns exist that a subsidized flat premium of about U$2 per person may still be too high for the poor; transport and nonessential drug costs may increase out-of-pocket spending for the poor.
- This paper investigates causal effects of CBHI on out-of-pocket healthcare spending in Rwanda using three rounds of Rwandan Integrated Household Living Conditions Survey (EICV) data from 2000, 2005, and 2010.

### Theoretical framework and empirical approach
- Utility framework: U(c_i, h_i) = u(c_i)^α h_i^(1−α); medical spending m_i = r I_i + p(I_i) V_i; health h_i = f(V_i, h_i0, Δ_i).
- Optimal out-of-pocket medical expenditure expression:
  - m_i(I_i^*, V_i^*; Y_i, r, γ, δ, μ, Δ) = r I^* + p(I^*) V^* = Y − [α (1−γ) (1−α) p(I^*) f(V^*, μ, Δ) / f_v(V^*, μ, Δ)]
- Main empirical issues: non-normal expenditure distribution with many zeros (corner solutions) and endogeneity from self-selection into CBHI.
- Model used: Extended Two-Part Model (ETPM) of Deb, Munkin, and Trivedi (2006), a three-equation system jointly modeling:
  - selection into CBHI d_i,
  - hurdle (probability of positive spending) s_i,
  - conditional amount of spending m_i.
- Latent-variable system:
  - d_i^* = β_d w_i + ε_id
  - s_i^* = β_s x_i + γ_s d_i + ε_is
  - m_i^* = β_m x_i + γ_m d_i + ε_im
- Error structure: ε_i ~ n(0, Σ_3×3) with first two diagonal elements = 1; conditional error decomposition:
  - ε_is = σ_ds ε_id + v_is
  - ε_im = σ_dm ε_id + v_im
- Identification strategy: exclusion restriction using membership in microfinance institutions (z_i) as instrument excluded from x_i; balancing test indicates covariates balanced between microfinance members and non-members.

### Data
- Three rounds of EICV surveys: 2000, 2005, 2010; household-level expenditure, individual-level insurance.
- Households categorized as “treated” if any member enrolled in CBHI.
- Final pooled sample: 26,195 households (6,390 from 2000; 6,259 from 2005; 13,546 from 2010 surveys) — note Table 1 reports No. of households as 6,390; 6,900; 14,308 in source tables for years.
- CBHI coverage in surveys: 0% in 2000/2001 (baseline), 41.9% in 2005/2006, 75.8% in 2010/2011; Other formal insurance: n.a. in 2000/2001, 9.1% in 2005/2006, 5.3% in 2010/2011; Uninsured: 100% in 2000/2001, 51.0% in 2005/2006, 22.2% in 2010/2011.
- Expenditure measures aggregated to annual per capita spending and deflated to year 2000 prices using National Institute of Statistics regional CPI.
- Controls include household demographics, number of individuals with illnesses, consumption and wealth quartiles, occupation, education, 29 district dummies.

### Estimation details and priors
- Slope parameter priors: non-informative normal N(μ0=0, V0=10 I_K).
- Variance σ_m0^2 prior: inverse gamma 푖푔(ν/2, (c/2)^{-1}) with ν = 10 and c = 5.
- Priors on covariance parameters σ_ds and σ_dm: informative normal with mean zero and variance 0.5 (i.e., N(0, 1/2 I)); sensitivity checked with tighter priors N(0, 1/5 I) and N(0, 1/8 I).
- Estimation via Bayesian MCMC with data augmentation; hypothesis testing for endogeneity uses Savage-Dickey Bayes Factor (testing H0: σ_ds = σ_dm = 0).

### Participation, selection and endogeneity (key empirical findings)
- Determinants of CBHI enrollment (posterior means significant):
  - Older and married household heads more likely to enroll.
  - Male-headed households, larger family size, and no education less likely to enroll.
  - Number of individuals with major illnesses significantly affects enrollment.
  - Higher income and wealth increase probability of enrollment (well-to-dos more likely to enroll).
  - Microfinance membership statistically significant predictor of enrollment.
  - District and year dummies significant: households in year 2010 more likely to enroll than in 2000.
- Evidence of selection on observables: variables like number of sick individuals and income affect both enrollment and spending.
- Test of endogeneity (Savage-Dickey Bayes Factor):
  - For prior variance v0 = 1/2, data strongly reject H0: σ_ds = σ_dm = 0 for overall, outpatient, and drug spending models.
  - Test consistent for priors v0 = 2; v0 = 1 and for more informative selections v0 = 1/2; v0 = 1/5; v0 = 1/8, yielding Savage-Dickey ratios of zero or very close to zero for these models.
  - For inpatient spending model, data could not reject H0; Savage-Dickey ratio = 22.2, implying CBHI membership can be considered exogenous for inpatient spending.

### Treatment effects (ATEs) — exact reported values (2000 prices, RWF)
- Table 9 (ETPM vs TPM):
  - Overall:
    - ETPM: Prob. = 0.3179*; Amount = -3,609*
      - (standard errors reported in table: (0.019) for Prob., (230) for Amount)
    - TPM: Prob. = 0.1963*; Amount = 210*
      - ((0.007); (71))
  - Spending net of premium:
    - ETPM: Prob. = 0.2862*; Amount = -4,106*
      - ((0.028); (320))
    - TPM: Prob. = 0.1028*; Amount = -816*
      - ((0.007); (96))
  - Outpatient:
    - ETPM: Prob. = 0.345*; Amount = 346*
      - ((0.041); (91))
    - TPM: Prob. = 0.1249*; Amount = 29*
      - ((0.007); (14))
  - Inpatient:
    - ETPM: Prob. = 0.0483; Amount = 140
      - ((0.021); (159))
    - TPM: Prob. = 0.0325*; Amount = 49
      - ((0.004); (30))
  - Drug:
    - ETPM: Prob. = 0.0078; Amount = -3,553*
      - ((0.043); (484))
    - TPM: Prob. = -0.051; Amount = -976*
      - ((0.007); (99))
- Predicted expenditures (2000 prices), Table 4 and Table 7:
  - Overall predicted expenditure (whole sample) = 2,312 (41)
  - Overall predicted expenditure (insured) = 1,410 (28)
  - Overall predicted expenditure (uninsured) = 5,019 (226)
  - Outpatient predicted expenditure (whole sample) = 385 (12)
  - Outpatient predicted expenditure (insured) = 689 (85)
  - Outpatient predicted expenditure (uninsured) = 343 (14)
  - Drug predicted expenditure (whole sample) = 1,855 (64)
  - Drug predicted expenditure (insured) = 612 (35)
  - Drug predicted expenditure (uninsured) = 4,166 (469)
- Covariances and variances (selected posterior means, Table 10 sensitivity):
  - Covariance w/hurdle part = -0.2783 (2-prior), -0.280 (1-prior), -0.2734 (1/5), -0.2794 (1/8)
    - (standard errors: (0.038), (0.041), (0.040), (0.044) respectively)
  - Covariance w/expenditure part = 0.8863, 0.8866, 0.8855, 0.8841
    - ((0.014), (0.013), (0.014), (0.015))
  - Variance = 1.707, 1.7073, 1.7079, 1.7091
    - ((0.024), (0.024), (0.026), (0.026))
  - Bayes Factor reported as 0.0E+00 across prior selections in Table 10.

### Heterogeneity in treatment effects
- Individual-level treatment effects show substantial heterogeneity (Figures 2a–2e):
  - Treatment effects on overall and drug spending mainly negative but vary across households.
  - Treatment effects on outpatient and inpatient spending span both negative and positive domains.
- ATE variation by observed factors (Figures 3–6):
  - Overall spending ATE exhibits a “u-shape” by age with higher reduction for household heads aged between 20 and 60.
  - ATEs increase with households’ position in the income distribution (favoring better off households) for overall spending.
  - Outpatient spending shows the reverse pattern (CBHI increases outpatient spending).
  - Drug spending reductions concentrated among certain age/income/wealth percentiles (Figures quantify patterns in source).

### Sensitivity analysis I: sensitivity to different prior selections (Table 10)
- Priors evaluated: v0 = 2; v0 = 1; v0 = 1/5; v0 = 1/8.
- ATE on probability of spending:
  - 0.317 (v0=2); 0.317 (v0=1); 0.315 (v0=1/5); 0.317 (v0=1/8)
  - (standard errors: (0.016), (0.017), (0.017), (0.018) respectively)
- ATE on amount of spending (RWF):
  - -3,624; -3,625; -3,639; -3,605 across the four prior choices
  - (standard errors: (215), (221), (222), (231) respectively)
- Covariance and variance estimates remain stable across priors (see exact values above).
- Bayes Factor reported as 0.0E+00 across all prior selections for the overall model — indicating strong rejection of exogeneity for overall out-of-pocket expenditure.
- Conclusion from sensitivity: magnitude and direction of treatment effects remain stable regardless of prior selection; Savage-Dickey ratios remain supportive of rejecting exogeneity for overall spending.

### Interpretation and key policy-relevant findings
- CBHI has a non-linear effect:
  - Increases probability of any healthcare spending by about 31.8 percentage points (ETPM).
  - Decreases conditional amount of total out-of-pocket expenditure by about RWF3,609 (approximately US$12 in 2000 prices and exchange rate as noted in source), representing about 83 percent of the average per capita healthcare expenditure in 2000 prices.
- Component-level effects:
  - Outpatient: increases probability by 34.5 percentage points and increases amount by RWF346.
  - Drug: reduces annual per capita drug spending by RWF3,553 (about 83 percent reduction relative to baseline uninsured average).
  - Inpatient: no statistically significant impact detected in ETPM.
- Accounting for premiums (overall less premium): CBHI increases probability by 28.6 percentage points and decreases amount net of premium by RWF4,106.
- Ignoring endogeneity (comparing TPM vs ETPM) changes sign and magnitude of estimates — e.g., TPM estimates positive ATE on overall spending (210 RWF) whereas ETPM ATE is -3,609 RWF, indicating positive selection into CBHI on spending if endogeneity ignored.
- Heterogeneous impacts: reductions in out-of-pocket spending are larger for households in higher income and wealth percentiles, raising equity concerns about whether CBHI favors the poor in reducing out-of-pocket burden.
- Robustness caveats:
  - Results robust to alternative prior specifications for covariance parameters.
  - Potential bias remains from measurement error in self-reported expenditure if errors systematically differ by CBHI status.

### Concluding remarks (policy implications)
- CBHI in Rwanda increases utilization (probability of spending) while reducing conditional out-of-pocket spending overall and for drugs, but not for inpatient services.
- The pattern that richer households experience larger reductions in out-of-pocket spending suggests that current subsidy and premium structures may be less favorable to the poorest, raising equity concerns for CBHI design.
- Proper accounting for selection on observables and unobservables is crucial; ignoring endogeneity leads to biased policy conclusions.
- Sensitivity checks indicate findings are robust to reasonable prior choices for covariance parameters, but measurement error in self-reported spending remains an important limitation to consider in policy interpretation.

*Source: wpiea2019038 - 10. Sensitivity Analysis I: Sensitivity to Different Prior Selections (excerpt from IMF working paper PDF).*

### Appendix A. Estimation Algorithm of Two-Part Model with Binary Selection

### Appendix A. Estimation Algorithm of Two-Part Model with Binary Selection

### Estimation algorithm (Box 1)
- Step 1: draw the latent variable d_i^* from its conditional truncated normal distribution
- Step 2: draw the latent variable s_i^* from its conditional truncated normal distribution
- Step 3: for i=1,...,n< N such that s_i=0, draw the latent variable m_i^* from its conditional normal distribution, otherwise set y_i^* = ln(y_i)
- Step 4: draw β_d for its conditional normal distribution
- Step 5: draw θ_s = [β_d, γ_s] from the joint conditional normal distribution
- Step 6: Draw σ_ds from the conditional truncated normal distribution
- Step 7: draw θ_m = [β_m, γ_m] from the joint conditional normal distribution
- Step 8: Draw σ_dm from the conditional normal distribution
- Step 9: draw σ_m^{-2} from the conditional gamma distribution

### Conditional draws and posterior updating (steps 1--9 detailed)
- For i=1,...,N, draw the latent variable d_i^* from its conditional truncated normal distribution given by
  - d_i^* | · ~ { TN(−∞,0](μ_d, V_d) if d_i = 0; TN(0,∞)(μ_d, V_d) if d_i = 1 }
  - where μ_d = β_d w_i + V_d[ σ_ds (s_i^* − β_s x_i − γ_s d_i) + σ_dm σ_m^2^{-1} (m_i^* − β_m x_i − γ_m d_i) ], and V_d = (1 + σ_ds^2 + σ_dm^2 σ_m^{-2})^{-1}

- For i=1,...,N, draw the latent variable s_i^* from its conditional truncated normal distribution given by
  - s_i^* | · ~ { TN(−∞,0](μ_s, 1) if s_i = 0; TN(0,∞)(μ_s, 1) if s_i = 1 }
  - where μ_s = β_s x_i + γ_s d_i + σ_ds ε_i^d

- For i=1,...,n < N such that s_i = 0, draw the latent variable m_i^* from its conditional normal distribution given by
  - m_i^* | · ~ { N(μ_m, σ_m^2) if s_i = 0; m if s_i = 1 }
  - where μ_m = β_m x_i + γ_m d_i + σ_dm ε_i^d

- Draw β_d from its conditional normal distribution given by
  - β_d | · ~ N(μ_{β_d}, V_{β_d})
  - where μ_{β_d} = V_{β_d}[ V_0^{-1} μ_0 + ∑ w_i′{ d_i^*(1+σ_ds^2+σ_dm^2 σ_m^{-2}) − σ_ds (s_i^* − β_s x_i − γ_s d_i) − ∑_{i=1}^N σ_dm σ_m^{-2} (m_i^* − β_m x_i − γ_m d_i) } ], and V_{β_d} = [ V_0^{-1} + ∑ w_i w_i′ (1+σ_ds^2+σ_dm^2 σ_m^{-2}) ]^{-1}

- Draw θ_s = [β_s, γ_s] from the joint conditional normal distribution given by
  - θ_s | · ~ N(μ_{θ_s}, V_{θ_s})
  - where μ_{θ_s} = V_{θ_s}[ (V_0^{-1} μ_0, V_γ0^{-1} μ_γ0) + ∑ X_i′(s_i^* − σ_ds ε_i^d) ], and V_{θ_s} = [ [V_0^{-1} 0; 0 V_γ0^{-1}] + ∑ X_i′ X_i ]^{-1}

- Draw σ_ds from the conditional truncated normal distribution
  - σ_ds | · ~ TN[−0.9,0.9](μ_{σ_ds}, V_{σ_ds})
  - where μ_{σ_ds} = V_{σ_ds}(V_{σ0}^{-1} μ_{σ0} + ∑ ε_i^d (s_i^* − β_s x_i − γ_s d_i) ), and V_{σ_ds} = [ V_{σ0}^{-1} + ∑ ε_i^{d′} ε_i^d ]^{-1}

- Draw θ_m = [β_m, γ_m] from the joint conditional normal distribution given by
  - θ_m | · ~ N(μ_{θ_m}, V_{θ_m})
  - where μ_{θ_m} = V_{θ_m}[ (V_0^{-1} μ_0, V_γ0^{-1} μ_γ0) + ∑ X_i′ σ_m^{-2} (m_i^* − σ_dm ε_i^d) ], and V_{θ_m} = [ [V_0^{-1} 0; 0 V_γ0^{-1}] + ∑ X_i′ σ_y^{-2} X_i ]^{-1}

- Draw σ_dm from the conditional truncated normal distribution
  - σ_dm | · ~ TN[−0.9,0.9](μ_{σ_dm}, V_{σ_dm})
  - where μ_{σ_dm} = V_{σ_dm}(V_{σ0}^{-1} μ_{σ0} + ∑ ε_i^d (m_i^* − β_m x_i − γ_m d_i) ), and V_{σ_dm} = [ V_{σ0}^{-1} + ∑ ε_i^{d′} ε_i^d σ_m^2 ]^{-1}

- Draw σ_m^{-2} from the conditional gamma distribution given by
  - σ_m^{-2} ~ g( (ν + N)/2, [ c/2 + ∑ (m_i^* − β_m x_i − γ_m d_i − σ_dm ε_i^d)^2 / 2 ]^{-1} )

### MCMC implementation and convergence
- Cycling through steps 1--9 until convergence provides the posterior parameter estimates.
- Estimation code implemented in Matlab and tested on simulated data before application to real data.
- MCMC settings:
  - 10,000 MCMC simulations with the first 5,000 draws dropped as burn-ins.
- Convergence assessment:
  - trace plots
  - formal convergence diagnostic test developed by Geweke (1992).

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### Appendix B. Additional Table — Table B.1: Covariate Balancing Test (Microfinance)

- Number of obs. 5,486 (Member) and 20,757 (Non-Member)

- Variable | Member | Non-Member | %bias | t | p>t
  - HHAgeHH | 44.325 | 44.521 | -1.3 | -0.71 | 0.479
  - HHSize | 5.410 | 5.409 | 0.1 | 0.03 | 0.975
  - HHAge | 22.629 | 22.758 | -1.2 | -0.71 | 0.476
  - HHmale | 0.775 | 0.770 | 1.1 | 0.58 | 0.56
  - HHmarried | 0.645 | 0.641 | 0.8 | 0.45 | 0.655
  - HHIllness | 1.021 | 1.021 | 0 | -0.02 | 0.987
  - AlcoholUse | 0.405 | 0.398 | 1.5 | 0.79 | 0.43
  - CigaretteUse | 0.182 | 0.183 | -0.3 | -0.16 | 0.875
  - HHU5 | 0.988 | 0.981 | 0.8 | 0.42 | 0.672
  - HHElderly | 0.131 | 0.136 | -1.3 | -0.69 | 0.49
  - HHChild | 2.004 | 1.999 | 0.3 | 0.15 | 0.879
  - FirstConsQuart | 0.160 | 0.162 | -0.6 | -0.35 | 0.728
  - SeconConsQuart | 0.241 | 0.235 | 1.3 | 0.7 | 0.486
  - ThirdConsQuart | 0.306 | 0.302 | 0.7 | 0.36 | 0.721
  - FourtConsQuart | 0.293 | 0.300 | -1.4 | -0.71 | 0.475
  - FirstWealthQuart | 0.175 | 0.182 | -1.6 | -0.91 | 0.36
  - SeconWealthQuart | 0.239 | 0.236 | 0.6 | 0.32 | 0.751
  - ThirdWealthQuart | 0.297 | 0.296 | 0.1 | 0.07 | 0.942
  - FourtWealthQuart | 0.289 | 0.286 | 0.8 | 0.4 | 0.688
  - HeadPrimary | 0.588 | 0.584 | 1 | 0.51 | 0.608
  - HeadSecVocUniv | 0.092 | 0.095 | -1.1 | -0.51 | 0.613
  - HeadNoEduc | 0.253 | 0.254 | -0.2 | -0.11 | 0.909
  - WageEarner | 1.119 | 1.134 | -1.2 | -0.61 | 0.541
  - Urban | 0.131 | 0.140 | -2.5 | -1.41 | 0.158

- Interpretation (from table content):
  - The covariate balance table reports Member vs Non-Member means, %bias, t-statistic, and p-value (p>t) for each covariate. The listed p>t values range from 0.158 to 0.987 for the reported covariates, with Number of obs. 5,486 and 20,757.

*Source: Appendix A and Appendix B of the provided content.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019038.pdf_