Statsmodels

# Discrete Choice Models Reference

This document provides comprehensive guidance on discrete choice models in statsmodels, including binary, multinomial, count, and ordinal models.

## Overview

Discrete choice models handle outcomes that are:
- **Binary**: 0/1, success/failure
- **Multinomial**: Multiple unordered categories
- **Ordinal**: Ordered categories
- **Count**: Non-negative integers

All models use maximum likelihood estimation and assume i.i.d. errors.

## Binary Models

### Logit (Logistic Regression)

Uses logistic distribution for binary outcomes.

**When to use:**
- Binary classification (yes/no, success/failure)
- Probability estimation for binary outcomes
- Interpretable odds ratios

**Model**: P(Y=1|X) = 1 / (1 + exp(-Xβ))

```python
import statsmodels.api as sm
from statsmodels.discrete.discrete_model import Logit

# Prepare data
X = sm.add_constant(X_data)

# Fit model
model = Logit(y, X)
results = model.fit()

print(results.summary())
```

**Interpretation:**
```python
import numpy as np

# Odds ratios
odds_ratios = np.exp(results.params)
print("Odds ratios:", odds_ratios)

# For 1-unit increase in X, odds multiply by exp(β)
# OR > 1: increases odds of success
# OR < 1: decreases odds of success
# OR = 1: no effect

# Confidence intervals for odds ratios
odds_ci = np.exp(results.conf_int())
print("Odds ratio 95% CI:")
print(odds_ci)
```

**Marginal effects:**
```python
# Average marginal effects (AME)
marginal_effects = results.get_margeff(at='mean')
print(marginal_effects.summary())

# Marginal effects at means (MEM)
marginal_effects_mem = results.get_margeff(at='mean', method='dydx')

# Marginal effects at representative values
marginal_effects_custom = results.get_margeff(at='mean',
                                              atexog={'x1': 1, 'x2': 5})
```

**Predictions:**
```python
# Predicted probabilities
probs = results.predict(X)

# Binary predictions (0.5 threshold)
predictions = (probs > 0.5).astype(int)

# Custom threshold
threshold = 0.3
predictions_custom = (probs > threshold).astype(int)

# For new data
X_new = sm.add_constant(X_new_data)
new_probs = results.predict(X_new)
```

**Model evaluation:**
```python
from sklearn.metrics import (classification_report, confusion_matrix,
                             roc_auc_score, roc_curve)

# Classification report
print(classification_report(y, predictions))

# Confusion matrix
print(confusion_matrix(y, predictions))

# AUC-ROC
auc = roc_auc_score(y, probs)
print(f"AUC: {auc:.4f}")

# Pseudo R-squared
print(f"McFadden's Pseudo R²: {results.prsquared:.4f}")
```

### Probit

Uses normal distribution for binary outcomes.

**When to use:**
- Binary outcomes
- Prefer normal distribution assumption
- Field convention (econometrics often uses probit)

**Model**: P(Y=1|X) = Φ(Xβ), where Φ is standard normal CDF

```python
from statsmodels.discrete.discrete_model import Probit

model = Probit(y, X)
results = model.fit()

print(results.summary())
```

**Comparison with Logit:**
- Probit and Logit usually give similar results
- Probit: symmetric, based on normal distribution
- Logit: slightly heavier tails, easier interpretation (odds ratios)
- Coefficients not directly comparable (scale difference)

```python
# Marginal effects are comparable
logit_me = logit_results.get_margeff().margeff
probit_me = probit_results.get_margeff().margeff

print("Logit marginal effects:", logit_me)
print("Probit marginal effects:", probit_me)
```

## Multinomial Models

### MNLogit (Multinomial Logit)

For unordered categorical outcomes with 3+ categories.

**When to use:**
- Multiple unordered categories (e.g., transportation mode, brand choice)
- No natural ordering among categories
- Need probabilities for each category

**Model**: P(Y=j|X) = exp(Xβⱼ) / Σₖ exp(Xβₖ)

```python
from statsmodels.discrete.discrete_model import MNLogit

# y should be integers 0, 1, 2, ... for categories
model = MNLogit(y, X)
results = model.fit()

print(results.summary())
```

**Interpretatio