📘 MORISTAT: Advanced Econometrics

What Are Limited Dependent Variables?

Limited dependent variables are outcomes that are not continuous and unbounded. Common types include:

Type	Description	Examples
Binary	Two outcomes	Buy/Don't buy, Employed/Unemployed
Multinomial	Multiple unordered outcomes	Mode of transport (Car/Bus/Train)
Ordered	Multiple ordered outcomes	Satisfaction (Low/Medium/High)
Censored	Partially observed	Hours worked (≥0), Expenditure
Truncated	Selectively observed	Income (only above poverty line)
Count	Non-negative integers	Number of patents, Doctor visits

Why Can't We Use OLS?

Problem	Issue	Consequence
Non-normal errors	Binary outcome → heteroskedastic errors	Inefficient estimates
Non-linear relationship	Effect of X on P(Y=1) is non-linear	OLS mis-specifies the relationship
Predicted probabilities outside [0,1]	OLS can predict negative probabilities	Nonsensical predictions
Heteroskedasticity	Var(Y|X) = P(1-P) depends on X	Invalid inference

The Binary Choice Framework

We want to model:

Where:

• Y = Binary outcome (0 or 1)
• X = Vector of explanatory variables
• F = Cumulative distribution function (CDF)

The Latent Variable Interpretation

Assume there is an unobserved (latent) variable Y*:

We observe:

Then:

The Two Main Models

Model	Distribution of ε	Link Function	F(z)
Logit	Logistic	log[P/(1-P)] = X'β	Λ(z) = e^z/(1+e^z)
Probit	Normal	Φ⁻¹(P) = X'β	Φ(z)

Estimation: Maximum Likelihood

The likelihood for a binary choice model is:

The log-likelihood:

Marginal Effects

For Logit:

For Probit:

Interpretation Guidelines

Method	Interpretation
Logit coefficients	Change in log-odds for a 1-unit increase in X
Odds ratios	exp(βⱼ) — multiplicative effect on odds
Marginal effects	Change in probability for a 1-unit increase in X
Probit coefficients	Change in the latent variable index

Comparing Logit and Probit

Aspect	Logit	Probit
Distribution	Logistic	Normal
Tails	Heavier	Lighter
Coefficient Ratio	β_Logit ≈ 1.6 × β_Probit	β_Probit ≈ 0.625 × β_Logit
Interpretation	Odds ratios	Latent index
Computation	Easier	Slightly harder
When to Use	Default choice	When normality is preferred

Multinomial Logit (MNL)

When the outcome has more than two unordered categories.

Model:

Normalization: β_J = 0 (base category)

Interpretation: Relative risk ratios.

Conditional Logit

When the choices have attributes that vary by alternative.

Model:

Ordered Logit/Probit

When the outcome has ordered categories.

Model:

Where:

• αⱼ = cut points (thresholds)
• F = Logistic (ordered logit) or Normal (ordered probit)

Interpretation: Coefficients indicate the direction of the effect on the underlying latent variable.

When to Use Tobit

The Tobit model is used when the dependent variable is censored — we observe the value only within a certain range.

Examples:

• Hours worked (≥ 0, with many zeros)
• Expenditure on luxury goods (≥ 0)
• Test scores (capped at 100%)
• Donation amounts (≥ 0)

The Tobit Model

Latent variable:

Observed variable:

The Likelihood Function

For observations with Y > 0:

For observations with Y = 0:

Interpretation of Tobit

Two components:

1. Probability of being uncensored: P(Y > 0 | X) = Φ(X'β/σ)
2. Expected value conditional on being uncensored: E(Y | Y > 0, X) = X'β + σ × λ(X'β/σ)

Where λ(z) = φ(z)/Φ(z) is the inverse Mills ratio.

Marginal Effects in Tobit

Effect	Formula
On all observations	∂E(Y|X)/∂Xⱼ = βⱼ × Φ(X'β/σ)
On uncensored observations	∂E(Y|Y>0,X)/∂Xⱼ = βⱼ × [1 - λ(z)(z + λ(z))]
On probability of censoring	∂P(Y>0|X)/∂Xⱼ = (βⱼ/σ) × φ(X'β/σ)

What Is Count Data?

Count data are non-negative integers measuring the number of times an event occurs.

Examples:

• Number of patents a firm holds
• Number of doctor visits in a year
• Number of accidents at a workplace
• Number of publications by a researcher

Zero-Inflated Models

Problem: Many count data have "too many" zeros.

Zero-Inflated Poisson (ZIP):

Zero-Inflated Negative Binomial (ZINB): Similar but with Negative Binomial instead of Poisson.

What Is Duration Data?

Duration data measure the time until an event occurs.

Examples:

• Time until unemployment ends
• Time until a firm fails
• Time until a patient dies
• Time until a machine breaks down

Survival Functions

Survival function: S(t) = P(T > t) (probability of surviving past time t)

Hazard function: h(t) = lim_{Δt→0} P(t ≤ T < t+Δt | T ≥ t) / Δt

Relationship: h(t) = -d ln S(t) / dt

Parametric Survival Models

Model	Hazard Function	Distribution
Exponential	h(t) = λ	Constant hazard
Weibull	h(t) = λp t^{p-1}	Monotonic hazard
Log-normal	h(t) = λp t^{p-1}	Non-monotonic hazard
Log-logistic	h(t) = λp t^{p-1}/(1+λt^p)	Non-monotonic hazard

Kaplan-Meier Estimator

Non-parametric estimate of the survival function:

Where:

• dⱼ = number of events at time tⱼ
• nⱼ = number at risk at time tⱼ

The ML-Econometrics Interface

Machine Learning	Econometrics	Purpose
Prediction	Forecasting	Out-of-sample prediction
Regularization	Variable selection	Shrinkage (Ridge, Lasso)
Cross-validation	Model selection	Choosing model complexity
Ensemble methods	Combining models	Improving predictions
Feature engineering	Specification search	Finding functional forms

Elastic Net

Combines Ridge and Lasso:

Post-Lasso / Double ML

1. Use Lasso for variable selection
2. Estimate the selected model with OLS (post-Lasso)
3. Or use Double ML for causal inference

Machine Learning Diagnostics

Concept	Description
Overfitting	Model fits training data too well, performs poorly on test data
Cross-validation	Splitting data to evaluate model performance
Bias-Variance Tradeoff	Simpler models have higher bias, complex models have higher variance
Hyperparameters	Parameters that control model complexity (λ, α)

What Is Spatial Econometrics?

Spatial econometrics accounts for spatial dependence and spatial heterogeneity in data.

Spatial Weight Matrix

Definition: W is an N×N matrix where w_ij captures the spatial relationship between observation i and j.

Common Specifications:

• Contiguity: w_ij = 1 if i and j share a border
• Distance decay: w_ij = 1/d_ij²
• k-nearest neighbors: w_ij = 1 if j is among k nearest neighbors

Spatial Durbin Model (SDM)

Where WXθ captures the effect of neighboring X variables.

Moran's I Test

Testing for spatial autocorrelation:

Interpretation:

• I > 0: Positive spatial autocorrelation (clustering)
• I < 0: Negative spatial autocorrelation (dispersion)
• I ≈ 0: No spatial autocorrelation

What Are Mixture Models?

Mixture models assume the data come from multiple subpopulations (mixture components).

Examples:

• Different market segments
• Multiple regimes in business cycles
• Heterogeneous treatment effects

Finite Mixture Model

Density:

Where:

• K = number of components
• πₖ = mixing proportion (Σπₖ = 1)
• fₖ = component-specific density

Latent Class Analysis

Interpretation: Each class has a different relationship between X and Y.

Mixture of Regressions

Interpretation: Different regression coefficients for each component.

Model Selection for Mixture Models

Criterion	Description
AIC	Akaike Information Criterion
BIC	Bayesian Information Criterion
ICL	Integrated Completed Likelihood
Entropy	Measures classification quality

Checklist for Advanced Models

For Binary Choice (Logit/Probit):

• Check for separation (perfect prediction)
• Consider marginal effects vs coefficients
• Test for heteroskedasticity
• Check for outliers (influence statistics)

For Multinomial Models:

• Choose appropriate base category
• Check the IIA assumption (Hausman test)
• Consider nested logit if IIA is violated

For Count Data:

• Test for overdispersion
• Consider zero-inflation if many zeros
• Check for outliers

For Tobit:

• Verify censoring mechanism
• Test for heteroskedasticity
• Consider Heckman selection if censoring is endogenous

Common Mistakes to Avoid

Mistake	Why It's Wrong	What To Do Instead
Interpreting coefficients as marginal effects	Coefficients are not probabilities	Compute marginal effects
Ignoring overdispersion in count data	Inefficient estimates	Use Negative Binomial
Using Tobit without censoring	Inconsistent estimates	Use OLS if no censoring
Ignoring spatial dependence	Inefficient inference	Use spatial models
Overfitting with complex models	Poor out-of-sample performance	Use cross-validation

Binary Choice Summary

Model	Link Function	CDF	Interpretation
Logit	log[P/(1-P)] = X'β	Λ(z) = e^z/(1+e^z)	Log-odds, odds ratios
Probit	Φ⁻¹(P) = X'β	Φ(z)	Latent index

Count Data Summary

Model	Mean	Variance	When to Use
Poisson	λ	λ	Equidispersion
Negative Binomial	λ	λ + αλ²	Overdispersion
ZIP	(1-p)λ	(1-p)λ(1+pλ)	Excess zeros

Tobit Summary

Spatial Models Summary

Model	Equation	Interpretation
SAR	Y = ρWY + Xβ + ε	Spatial spillovers in Y
SEM	Y = Xβ + u, u = λWu + ε	Spatial error correlation
SDM	Y = ρWY + Xβ + WXθ + ε	Spillovers in X and Y

Key Takeaways

1. Binary choice models (Logit/Probit) are used for binary outcomes — interpret via marginal effects
2. Multinomial/Ordered models extend binary choice to multiple categories
3. Tobit models handle censored data — two-part interpretation
4. Count data models (Poisson/NB) handle non-negative integers — check for overdispersion
5. Duration models analyze time-to-event data — use survival functions and hazard ratios
6. Machine learning methods (Ridge, Lasso) are useful for prediction and variable selection
7. Spatial models account for spatial dependence — use Moran's I to test for spatial correlation
8. Mixture models capture heterogeneity across subpopulations

Before Proceeding

• Understand the data type (binary, count, censored, etc.)
• Choose the appropriate model for your outcome
• Know how to interpret coefficients vs marginal effects
• Understand the diagnostic tests for each model
• Ready for Software Implementation (Part 2)