MORISTAT: Chapter 5 — Applied Econometrics Case Studies

5.1 Case Study 1: Returns to Education (IV/2SLS)

📊 Research Question

What is the causal effect of education on earnings?

This classic question in labor economics suffers from endogeneity: education is correlated with ability, which is unobserved and affects earnings.

Dataset: Simulated dataset based on Angrist & Krueger (1991)

Variable	Description	Type
wage	Hourly wage (USD)	Continuous
education	Years of schooling	Continuous
experience	Years of work experience	Continuous
ability	Cognitive ability (unobserved)	Latent
qob	Quarter of birth (1-4)	Categorical
smsa	Metropolitan area (0/1)	Binary
married	Marital status (0/1)	Binary

Sample Size: N = 1,000

The Endogeneity Problem

True Model: log(wage) = β₀ + β₁·education + β₂·experience + β₃·ability + ε

Problem: ability is omitted and correlated with education → Cov(education, ε) ≠ 0 → OLS is biased and inconsistent

The Solution: Instrumental Variables

Instrument: Quarter of Birth (QOB)

Why QOB is a good instrument:

Relevant: QOB affects education through compulsory schooling laws
Exogenous: QOB is randomly assigned and unrelated to ability

Model Specification

First Stage: education = π₀ + π₁·Q2 + π₂·Q3 + π₃·Q4 + π₄·experience + v Second Stage: log(wage) = β₀ + β₁·education_hat + β₂·experience + ε

Results: OLS vs 2SLS Comparison

Variable	OLS	2SLS	Difference
Education	0.089*** (0.012)	0.132** (0.018)	0.043**
Experience	0.026*** (0.008)	0.024*** (0.009)	-0.002
Intercept	1.234*** (0.456)	0.987** (0.512)	-0.247

Diagnostics

Test	Statistic	p-value	Conclusion
Hausman-Wu	6.78	0.009	Endogeneity detected
Sargan-Hansen J	2.34	0.311	Instruments valid
First-Stage F	18.56	0.000	Strong instruments

Interpretation:

An additional year of education increases earnings by 13.2% (2SLS) compared to 8.9% (OLS). The OLS estimate is biased downward due to ability bias. The 2SLS estimate gives the causal effect of education on earnings.

Policy Implications:

Each additional year of education increases earnings by approximately 13%
This justifies investment in education (both private and public)
The effect is larger than OLS suggests → ability bias masks the true returns

5.2 Case Study 2: Production Function Estimation (Panel Data)

🏭 Research Question

What is the elasticity of output with respect to capital, labor, and materials?

Dataset: Simulated firm-level panel data

Variable	Description	Type
output	Value added (millions USD)	Continuous
capital	Capital stock (millions USD)	Continuous
labor	Number of employees	Continuous
materials	Materials cost (millions USD)	Continuous
firm_id	Firm identifier	Categorical
year	Year (2000-2010)	Time

Sample Size: N = 5,000 firms × T = 10 years = 50,000 observations

The Endogeneity Problem

True Model: log(output) = β₁·log(capital) + β₂·log(labor) + β₃·log(materials) + μᵢ + εᵢₜ

Problem: Input choices are correlated with productivity (μᵢ) → Cov(inputs, μᵢ) ≠ 0 → Pooled OLS is biased

Results: Pooled OLS vs FE vs GMM

Variable	Pooled OLS	FE	System GMM
Capital	0.12*** (0.003)	0.08*** (0.004)	0.09*** (0.005)
Labor	0.45*** (0.008)	0.38*** (0.009)	0.40*** (0.010)
Materials	0.35*** (0.006)	0.32*** (0.007)	0.33*** (0.008)
Lagged Output	—	—	0.15*** (0.020)
N	50,000	50,000	45,000

Diagnostics

Test	Statistic	p-value	Conclusion
Hausman	145.67	0.000	Use FE (not RE)
AR(1)	-3.45	0.001	Expected
AR(2)	-1.23	0.219	No serial correlation
Sargan J	12.34	0.145	Instruments valid

Interpretation:

The results show constant returns to scale (sum of elasticities ≈ 0.82). The OLS estimates are biased upward due to unobserved productivity. FE and GMM give more conservative estimates.

5.3 Case Study 3: Credit Default Prediction (Logit/Lasso)

💳 Research Question

What factors predict credit default, and how accurately can we predict it?

Dataset: Simulated credit data

Variable	Description	Type
default	Defaulted on loan (0/1)	Binary
income	Annual income (USD)	Continuous
age	Age (years)	Continuous
debt_ratio	Debt-to-income ratio	Continuous
credit_score	Credit score (300-850)	Continuous
employment	Years employed	Continuous
homeowner	Owns home (0/1)	Binary
education	Years of education	Continuous

Sample Size: N = 10,000

Results: Logit Coefficient Estimates

Variable	Coefficient	Std Error	z-stat	p-value	Sig.
Intercept	-2.145	0.456	-4.70	0.000	***
Income	0.089	0.012	7.42	0.000	***
Age	-0.023	0.008	-2.88	0.004	**
Debt Ratio	1.234	0.156	7.91	0.000	***
Credit Score	-0.567	0.089	-6.37	0.000	***
Employment	-0.045	0.023	-1.96	0.050	*
Homeowner	-0.234	0.098	-2.39	0.017	**
Education	-0.012	0.018	-0.67	0.503

Lasso Variable Selection

Variable	Retained	Coefficient
Debt Ratio	Yes	0.987
Credit Score	Yes	-0.432
Income	Yes	0.067
Age	Yes	-0.018
Employment	No	0.000
Homeowner	No	0.000
Education	No	0.000

Confusion Matrix

	Predicted Default	Predicted No Default
Actual Default	850	150
Actual No Default	200	8,800

Performance Metrics

Metric	Value
Accuracy	96.5%
Precision	80.9%
Recall	85.0%
F1 Score	82.9%
AUC	0.942

Interpretation:

Strongest predictors: Debt ratio and credit score
Lasso selected: 4 of 7 variables (parsimonious model)
Model performance: Excellent (AUC = 0.942)

5.4 Case Study 4: Healthcare Utilization (Count Data)

🏥 Research Question

What factors affect the number of doctor visits?

Dataset: Simulated health data

Variable	Description	Type
visits	Number of doctor visits (0-30+)	Count
insured	Has health insurance (0/1)	Binary
age	Age (years)	Continuous
income	Annual income (USD)	Continuous
chronic	Chronic condition (0/1)	Binary
female	Female (0/1)	Binary
distance	Distance to clinic (miles)	Continuous

Sample Size: N = 5,000

Model Comparison

Model	Log-Likelihood	AIC	BIC
Poisson	-8,234	16,482	16,520
Negative Binomial	-6,789	13,594	13,638
ZIP	-6,512	13,042	13,098

Negative Binomial Coefficients

Variable	Coefficient	Std Error	z-stat	p-value	Sig.
Intercept	1.234	0.089	13.86	0.000	***
Insured	0.456	0.045	10.13	0.000	***
Age	0.023	0.003	7.67	0.000	***
Income	-0.012	0.004	-3.00	0.003	**
Chronic	0.789	0.056	14.09	0.000	***
Female	0.234	0.043	5.44	0.000	***
Distance	-0.067	0.012	-5.58	0.000	***

Expected Visits (ZIP)

	Insured	Uninsured
Chronic	4.2 visits/year	2.8 visits/year
No Chronic	1.8 visits/year	0.9 visits/year

Interpretation:

Negative Binomial fits better than Poisson (AIC lower)
ZIP fits best (handles excess zeros)
Insurance increases visits by ~60%
Chronic condition is the strongest predictor
Distance reduces utilization

5.5 Case Study 5: Economic Growth Convergence (Panel + Spatial)

📈 Research Question

Is there evidence of economic growth convergence across countries?

Dataset: Simulated country-year panel data

Variable	Description	Type
gdp_growth	Annual GDP growth rate	Continuous
initial_gdp	GDP per capita (initial year)	Continuous
investment	Investment/GDP ratio	Continuous
education	Years of schooling	Continuous
population	Population growth rate	Continuous
trade	Trade/GDP ratio	Continuous
inflation	Inflation rate	Continuous
country	Country identifier	Categorical
year	Year (1990-2020)	Time

Sample Size: N = 100 countries × T = 30 years = 3,000 observations

Results: Panel FE vs Spatial SAR

Variable	Panel FE	Spatial SAR
Initial GDP	-0.018*** (0.003)	-0.015*** (0.003)
Investment	0.145*** (0.015)	0.132*** (0.014)
Education	0.087*** (0.012)	0.079*** (0.011)
Population	-0.023 (0.018)	-0.019 (0.017)
Trade	0.045** (0.016)	0.041** (0.015)
Rho (spatial)	—	0.234*** (0.045)
R²	0.65	0.71

Spatial Diagnostics

Test	Statistic	p-value	Conclusion
Moran's I	0.234	0.002	Spatial autocorrelation
LM-SAR	15.67	0.000	Use SAR
LM-Error	2.34	0.109	No spatial error

Interpretation:

Convergence: Negative coefficient on initial GDP → poorer countries grow faster
Investment: Important for growth (elasticity ≈ 0.13-0.15)
Education: Positive effect on growth
Spatial dependence: Countries affect neighbors (rho = 0.234)

5.6 Case Study 6: Firm Survival Analysis (Cox PH)

⏱️ Research Question

What factors determine firm survival?

Dataset: Simulated firm survival data

Variable	Description	Type
survival_time	Months until failure or censoring	Continuous
exit	Failed (1) or censored (0)	Binary
size	Number of employees	Continuous
age	Firm age (years)	Continuous
profitability	Profit margin (%)	Continuous
leverage	Debt/equity ratio	Continuous
industry	Industry sector	Categorical
location	Urban/Rural	Binary

Sample Size: N = 5,000 firms

Cox PH Coefficients

Variable	Coefficient	HR	Std Error	z-stat	Sig.
Size	-0.234	0.791	0.045	-5.20	***
Age	-0.189	0.828	0.032	-5.91	***
Profitability	-0.345	0.708	0.056	-6.16	***
Leverage	0.456	1.578	0.078	5.85	***
Industry (Tech)	-0.234	0.791	0.089	-2.63	**
Location (Urban)	-0.123	0.884	0.067	-1.84	*

Survival Probabilities (5-year)

Firm Type	1-Year Survival	3-Year Survival	5-Year Survival
Low Risk	98%	92%	85%
Medium Risk	95%	85%	72%
High Risk	90%	75%	60%

Interpretation:

Leverage increases hazard (HR = 1.58)
Profitability decreases hazard (HR = 0.71)
Size and age decrease hazard (firms become more stable)
Proportional hazards assumption holds ✓

5.7 Chapter Summary

Key Takeaways

1. Returns to Education (IV/2SLS)

IV/2SLS addresses endogeneity
QOB is a valid instrument
OLS underestimates returns by ~30%

2. Production Function (Panel Data)

Panel FE controls for unobserved productivity
Constant returns to scale
GMM addresses dynamics

3. Credit Default (Logit/Lasso)

Logit models probability of default
Lasso selects important variables
Excellent prediction accuracy

4. Healthcare Utilization (Count Data)

Count data requires specialized models
ZIP handles excess zeros
NB addresses overdispersion

5. Economic Growth (Panel + Spatial)

Panel FE + Spatial SAR
Evidence of convergence
Spatial spillovers matter

6. Firm Survival (Cox PH)

Cox PH handles censoring
Key risk factors identified
Survival probabilities estimated

5.8 What's Next?

📘 Chapter 6: Advanced Diagnostics and Robustness

We will cover:

Sensitivity Analysis — How robust are your results?
Specification Testing — Are you using the right model?
Outlier Detection — Influential observations
Bootstrap Inference — Non-parametric standard errors
Model Averaging — Combining multiple models
Publication-Ready Output — Professional reporting

📘 MORISTAT: Advanced Econometrics

Chapter 5: Applied Econometrics — Case Studies