Marginal Effects for Discrete Choice Models

Quick Reference

Method: results.marginal_effects(at="overall", method="dydx") Stata equivalent: margins, dydx(*) R equivalent: margins::margins()

Overview

In nonlinear models such as logit, probit, and multinomial logit, the estimated coefficients \(\beta\) do not directly represent the effect of a one-unit change in \(x_k\) on the probability of the outcome. This is because the probability function is nonlinear:

\[P(y = 1 \mid X) = F(X'\beta)\]

The marginal effect of variable \(x_k\) on the probability is:

\[\frac{\partial P(y = 1)}{\partial x_k} = f(X'\beta) \cdot \beta_k\]

where \(f(\cdot) = F'(\cdot)\) is the density function. This depends on the values of all covariates through \(f(X'\beta)\), so the marginal effect varies across observations.

PanelBox provides three approaches to summarize marginal effects, along with standard errors computed via the delta method.

Types of Marginal Effects

Average Marginal Effect (AME)

Compute the marginal effect for each observation, then average:

\[\text{AME}_k = \frac{1}{N} \sum_{i=1}^{N} f(X_i'\hat{\beta}) \cdot \hat{\beta}_k\]

This is the most commonly reported measure and reflects the average effect across the actual sample distribution.

me = model.marginal_effects(at="overall", method="dydx")

Marginal Effect at the Mean (MEM)

Evaluate the marginal effect at the sample mean of all covariates:

\[\text{MEM}_k = f(\bar{X}'\hat{\beta}) \cdot \hat{\beta}_k\]

This gives the effect for a "representative" individual at the mean of all variables.

me = model.marginal_effects(at="mean", method="dydx")

Marginal Effect at Representative Values (MER)

Evaluate at user-specified covariate values:

# For PooledLogit / PooledProbit
me = model.marginal_effects(
    at="overall",
    method="dydx",
    representative={"education": 16, "experience": 10}
)

AME vs MEM: Which to Use?

Feature	AME (at="overall")	MEM (at="mean")
Computation	Average over all observations	Evaluate at \(\bar{X}\)
Interpretation	Population-level average effect	Effect for "average" person
Robustness	More robust to distribution shape	Sensitive to where \(\bar{X}\) falls
Recommended	Preferred in most applications	Quick summary measure

Prefer AME

The AME is generally preferred because (1) the "average person" (at \(\bar{X}\)) may not actually exist in the data, (2) AME accounts for the full distribution of covariates, and (3) AME is more robust when the link function is highly nonlinear.

Marginal Effects by Model Type

Binary Models (Logit / Probit)

For PooledLogit, the marginal effect of \(x_k\) is:

\[\frac{\partial P}{\partial x_k} = \Lambda(X'\beta) \cdot [1 - \Lambda(X'\beta)] \cdot \beta_k\]

For PooledProbit:

\[\frac{\partial P}{\partial x_k} = \phi(X'\beta) \cdot \beta_k\]

from panelbox.models.discrete.binary import PooledLogit, PooledProbit

# Logit
logit = PooledLogit("employed ~ education + experience + age", data, "id", "year")
logit_res = logit.fit(cov_type="cluster")
me_logit = logit.marginal_effects(at="overall", method="dydx")
print(me_logit)

# Probit
probit = PooledProbit("employed ~ education + experience + age", data, "id", "year")
probit_res = probit.fit(cov_type="cluster")
me_probit = probit.marginal_effects(at="overall", method="dydx")
print(me_probit)

Logit vs Probit Marginal Effects

Even though logit and probit coefficients differ in scale, AME values are typically very similar between the two models. This makes AME a useful basis for comparing model specifications.

Fixed Effects Logit

The FE logit estimates conditional log-odds ratios. Because entity fixed effects \(\alpha_i\) are not estimated, standard marginal effects on \(P(y = 1)\) are not computable. However, the coefficients have a conditional odds ratio interpretation:

\[\exp(\beta_k) = \text{odds ratio for a one-unit change in } x_k\]

from panelbox.models.discrete.binary import FixedEffectsLogit

model = FixedEffectsLogit("employed ~ education + experience", data, "id", "year")
results = model.fit()

# Odds ratios (exponentiated coefficients)
import numpy as np
odds_ratios = np.exp(results.params)
print("Odds ratios:", odds_ratios)

No Marginal Effects for FE Logit

Fixed Effects Logit does not support marginal_effects() because individual effects are not estimated. Use odds ratios or consider Random Effects Probit if marginal effects are essential.

Random Effects Probit

The RE probit integrates over the distribution of \(\alpha_i\):

\[P(y_{it} = 1 \mid X_{it}) = \int \Phi(X_{it}'\beta + \alpha_i) \, \phi(\alpha_i / \sigma_\alpha) \, d\alpha_i\]

Marginal effects can be computed as population-averaged effects (integrating out \(\alpha_i\)):

from panelbox.models.discrete.binary import RandomEffectsProbit

model = RandomEffectsProbit("employed ~ education + experience + female",
                             data, "id", "year", quadrature_points=12)
results = model.fit()

# Marginal effects (integrating over RE distribution)
me = model.marginal_effects(at="mean", method="dydx")
print(me)

Ordered Models

For ordered logit/probit, marginal effects are computed per category. Increasing \(x_k\) can:

• Increase the probability of some categories
• Decrease the probability of others
• The effects across all categories sum to zero

For an Ordered Logit with \(J\) categories:

\[\frac{\partial P(y = j)}{\partial x_k} = [f(\kappa_{j-1} - X'\beta) - f(\kappa_j - X'\beta)] \cdot \beta_k\]

where \(f(\cdot)\) is the logistic PDF.

from panelbox.models.discrete.ordered import OrderedLogit
import numpy as np

model = OrderedLogit(endog=y, exog=X, groups=entity, time=time)
model.fit()

# Get probabilities at the mean
probs = model.predict_proba()

# Manual marginal effects for ordered models:
# The sign of beta determines the direction of shift
# Positive beta: increases probability of higher categories,
#                decreases probability of lower categories
print("Coefficients:", model.beta)
print("Average predicted probabilities:", probs.mean(axis=0))

Intermediate Categories

For intermediate categories, the sign of the marginal effect can be ambiguous -- a positive coefficient does not always increase the probability of a middle category. Always compute the full set of category-specific marginal effects.

Multinomial Logit

In multinomial logit, marginal effects form a \((J, K)\) matrix. The effect of \(x_k\) on \(P(y = j)\) is:

\[\frac{\partial P(y = j)}{\partial x_k} = P(y = j) \left[ \beta_{jk} - \sum_{m=0}^{J-1} P(y = m) \, \beta_{mk} \right]\]

where \(\beta_{0k} = 0\) for the base alternative. Marginal effects sum to zero across alternatives for each variable.

from panelbox.models.discrete.multinomial import MultinomialLogit

model = MultinomialLogit(endog=y, exog=X, n_alternatives=3, base_alternative=0)
result = model.fit()

# Marginal effects: (J, K) matrix
me = result.marginal_effects(at="mean")  # At sample means
print("Marginal effects at mean:")
print(me)

# Average marginal effects
me_overall = result.marginal_effects(at="overall")

# For a specific variable
me_var = result.marginal_effects(at="mean", variable=0)

# Standard errors for marginal effects
me_se = result.marginal_effects_se(at="mean")

Visualization

Use the built-in plotting for multinomial marginal effects:

result.plot_marginal_effects(variable=0, at="mean")

Dynamic Binary Panel

Marginal effects in dynamic models have two components:

1. Short-run effect: the immediate effect of \(x_k\) on \(P(y_{it} = 1)\)
2. Long-run effect: the accumulated effect through the feedback loop \(y_{it} \to y_{i,t+1} \to \cdots\)

The short-run marginal effect is \(f(\gamma y_{i,t-1} + X'\beta + \alpha_i) \cdot \beta_k\).

from panelbox.models.discrete.dynamic import DynamicBinaryPanel

model = DynamicBinaryPanel(endog=y, exog=X, entity=entity, time=time,
                            initial_conditions="wooldridge", effects="random")
result = model.fit()

# Marginal effects at the mean
me = result.marginal_effects()
print(me)

Discrete vs Continuous Variables

For continuous variables, the marginal effect is the partial derivative \(\partial P / \partial x_k\).

For discrete (binary/categorical) variables, the marginal effect is the discrete change in probability:

\[\Delta P = P(y = 1 \mid x_k = 1) - P(y = 1 \mid x_k = 0)\]

PanelBox computes this automatically when using method="dydx" based on variable type detection:

# For continuous variables: partial derivative
# For discrete/binary variables: discrete change
me = model.marginal_effects(at="overall", method="dydx")

Interpretation Guide

Comparing Marginal Effects Across Models

Model	ME Computation	Notes
PooledLogit	\(\Lambda' \cdot \beta_k\)	Standard AME or MEM
PooledProbit	\(\phi \cdot \beta_k\)	AME similar to logit AME
FE Logit	Not available	Use odds ratios: \(\exp(\beta_k)\)
RE Probit	Integrated over \(\alpha_i\)	Population-averaged effects
Ordered	Per-category, sums to zero	\(J\) effects per variable
Multinomial	\((J, K)\) matrix, sums to zero	Per-alternative effects
Dynamic	Short-run effect	Long-run requires simulation

Common Pitfalls

Interpretation Mistakes to Avoid

1. Comparing raw coefficients across models: Logit and probit coefficients have different scales. Use marginal effects for comparison.
2. Ignoring the evaluation point: MEM depends on \(\bar{X}\), which may not be representative. AME is safer.
3. Sign of intermediate categories: In ordered models, the marginal effect on middle categories can have the opposite sign of \(\beta\).
4. Base alternative confusion: In multinomial logit, coefficients are relative to the base. Marginal effects are absolute (summing to zero).
5. FE Logit marginal effects: These do not exist in the standard sense because \(\alpha_i\) is not estimated.

Summary Table

Model	at="overall"	at="mean"	SE Method	Command
PooledLogit	AME	MEM	Delta method	model.marginal_effects(at=..., method="dydx")
PooledProbit	AME	MEM	Delta method	model.marginal_effects(at=..., method="dydx")
FE Logit	N/A	N/A	N/A	Use np.exp(results.params) for odds ratios
RE Probit	Integrated	Integrated	Delta method	model.marginal_effects(at=..., method="dydx")
Ordered Logit/Probit	Per-category	Per-category	Numerical	Via predict_proba()
Multinomial Logit	\((J,K)\) matrix	\((J,K)\) matrix	Numerical Hessian	result.marginal_effects(at=...)
Dynamic Binary	Short-run	Short-run	Numerical	result.marginal_effects()

Tutorials

Tutorial	Description	Link
Marginal Effects Guide	Computing and comparing MEs across models

See Also

• Binary Choice Models -- Logit and Probit models
• Ordered Choice Models -- Marginal effects for ordinal outcomes
• Multinomial and Conditional Logit -- Marginal effects in \((J, K)\) form
• Dynamic Binary Panel -- Short-run vs long-run effects

References

• Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. 2^nd ed. MIT Press.
• Cameron, A. C. and Trivedi, P. K. (2005). Microeconometrics: Methods and Applications. Cambridge University Press. Section 14.3.
• Greene, W. H. (2018). Econometric Analysis. 8^th ed. Pearson. Chapter 17.
• Bartus, T. (2005). "Estimation of Marginal Effects Using margeff." The Stata Journal, 5(3), 309-329.
• Long, J. S. and Freese, J. (2014). Regression Models for Categorical Dependent Variables Using Stata. 3^rd ed. Stata Press.