Marginal Effects for Censored Models¶

Quick Reference

Method: model.marginal_effects(at="overall", which="conditional") Applies to: PooledTobit, RandomEffectsTobit, PanelHeckman Stata equivalent: margins, predict(ystar(0,.)) or margins, dydx(*) R equivalent: margins::margins()

Overview¶

In nonlinear models like Tobit and Heckman, the estimated coefficients \(\hat{\beta}\) do not directly represent marginal effects. Unlike linear regression where \(\partial E[y|X] / \partial x_j = \beta_j\), censored and selection models have non-constant marginal effects that depend on where in the distribution you evaluate them.

For the Tobit model, there are three distinct types of marginal effects, each answering a different economic question. Understanding which one to report is critical for correct interpretation of results.

Three Types of Marginal Effects in Tobit¶

1. Conditional Marginal Effect¶

\[\frac{\partial E[y \mid y > c, X]}{\partial x_j}\]

Question: Among observations that are uncensored (above the boundary), how does a change in \(x_j\) affect the expected outcome?

Formula (left censoring at \(c\)):

\[\text{ME}^{cond}_j = \beta_j \left[ 1 - \lambda(z) \cdot (z + \lambda(z)) \right]\]

where \(z = (X'\beta - c)/\sigma\) and \(\lambda(z) = \phi(z)/\Phi(z)\) is the Inverse Mills Ratio.

Use when: You care about the intensive margin -- how the outcome changes for those already above the boundary.

2. Unconditional Marginal Effect¶

\[\frac{\partial E[y \mid X]}{\partial x_j}\]

Question: How does a change in \(x_j\) affect the overall expected outcome, including the probability of being at the boundary?

Formula (left censoring at \(c\)):

\[\text{ME}^{uncond}_j = \beta_j \cdot \Phi(z)\]

where \(\Phi(z)\) is the probability of being uncensored. This is the most commonly reported marginal effect.

Use when: You want the total effect combining both the probability of being uncensored and the conditional effect.

3. Probability Marginal Effect¶

\[\frac{\partial P(y > c \mid X)}{\partial x_j}\]

Question: How does a change in \(x_j\) affect the probability of being uncensored?

Formula (left censoring at \(c\)):

\[\text{ME}^{prob}_j = \frac{\beta_j}{\sigma} \cdot \phi(z)\]

Use when: You care about the extensive margin -- how the probability of crossing the boundary changes.

McDonald & Moffitt (1980) Decomposition¶

The total (unconditional) marginal effect decomposes into two components:

\[\frac{\partial E[y|X]}{\partial x_j} = \underbrace{P(y > c) \cdot \frac{\partial E[y|y>c, X]}{\partial x_j}}_{\text{intensive margin}} + \underbrace{E[y|y>c, X] \cdot \frac{\partial P(y>c)}{\partial x_j}}_{\text{extensive margin}}\]

This decomposition separates the effect into:

Intensive margin: among the uncensored, how does the level change?
Extensive margin: how does the probability of being uncensored change, weighted by the expected level?

Computing Marginal Effects in PanelBox¶

Tobit Marginal Effects¶

Both PooledTobit and RandomEffectsTobit support the marginal_effects() method:

from panelbox.models.censored import PooledTobit

# Fit model
model = PooledTobit(endog=y, exog=X, groups=entity, censoring_point=0.0)
result = model.fit()

# Average Marginal Effects (AME) — conditional
ame_cond = result.marginal_effects(at="overall", which="conditional")

# Average Marginal Effects — unconditional
ame_uncond = result.marginal_effects(at="overall", which="unconditional")

# Average Marginal Effects — probability
ame_prob = result.marginal_effects(at="overall", which="probability")

# Marginal Effects at Means (MEM)
mem = result.marginal_effects(at="mean", which="conditional")

Parameters¶

Parameter	Values	Description
`at`	`"overall"`	AME: Average over all observations
	`"mean"`	MEM: Evaluate at sample means
`which`	`"conditional"`	\(\partial E[y \mid y > c, X] / \partial x\)
	`"unconditional"`	\(\partial E[y \mid X] / \partial x\)
	`"probability"`	\(\partial P(y > c \mid X) / \partial x\)
`varlist`	list of str	Compute for specific variables only

AME vs. MEM¶

Method	Computation	When to use
AME (`at="overall"`)	Compute ME for each obs, then average	Preferred for policy analysis; robust to nonlinearity
MEM (`at="mean"`)	Evaluate ME at \(\bar{X}\)	Simpler to compute; may not represent any actual observation

AME is generally preferred because it averages over the actual distribution of covariates rather than evaluating at a potentially unrealistic "average" observation.

Marginal Effects in the Heckman Model¶

For the Panel Heckman model, marginal effects must account for the selection correction. The prediction types provide the building blocks:

from panelbox.models.selection import PanelHeckman

model = PanelHeckman(
    endog=y, exog=X, selection=d, exog_selection=Z,
    method="two_step",
)
results = model.fit()

# Unconditional prediction: E[y*] = X'beta
y_uncond = results.predict(type="unconditional")

# Conditional prediction: E[y|selected] = X'beta + rho*sigma*lambda
y_cond = results.predict(type="conditional")

Unconditional Marginal Effect (Heckman)¶

\[\frac{\partial E[y^*]}{\partial x_j} = \beta_j\]

The unconditional marginal effect equals the coefficient -- the same as in a linear model -- because the latent outcome is linear in \(X\).

Conditional Marginal Effect (Heckman)¶

\[\frac{\partial E[y \mid d=1, X, W]}{\partial x_j} = \beta_j + \rho \sigma_\varepsilon \frac{\partial \lambda}{\partial x_j}\]

If \(x_j\) appears only in the outcome equation (not in the selection equation), the conditional effect simplifies to \(\beta_j\). If \(x_j\) also appears in the selection equation, there is an additional indirect effect through the IMR.

Interpreting Results¶

Coefficients vs. Marginal Effects¶

A common mistake is to interpret Tobit coefficients as marginal effects:

	Linear (OLS)	Tobit
\(\beta_j\) represents	\(\partial E[y]/\partial x_j\) directly	Effect on latent \(y^*\), not observed \(y\)
Marginal effect	\(= \beta_j\) (constant)	\(\neq \beta_j\) (varies with \(X\))
Scale	One unit of \(y\) per unit of \(x\)	Dampened by \(\Phi(z)\)

The Dampening Factor¶

For the unconditional ME, the coefficient is dampened by \(\Phi(z)\), the probability of being uncensored:

\[\text{ME}^{uncond}_j = \beta_j \cdot \Phi\!\left(\frac{X'\beta - c}{\sigma}\right)\]

Since \(\Phi(z) \in (0, 1)\), the marginal effect is always smaller in magnitude than the coefficient \(\beta_j\). The closer to the censoring point, the stronger the dampening.

Practical Comparison¶

# Compare coefficients vs marginal effects
print("Variable | Coefficient | AME (uncond) | Ratio")
print("-" * 55)

ame = result.marginal_effects(at="overall", which="unconditional")
for j, (coef, me) in enumerate(zip(result.beta, ame.effects)):
    ratio = me / coef if abs(coef) > 1e-10 else float('nan')
    print(f"x_{j:<6d} | {coef:>11.4f} | {me:>12.4f} | {ratio:.3f}")

The ratio \(\text{ME}/\beta\) tells you how much the censoring dampens the raw coefficient. A ratio near 1 means little censoring; a ratio near 0 means heavy censoring.

Tutorials¶

Tutorial	Description	Link
Censored Models	Marginal effects computation and interpretation

References¶

McDonald, J. F., & Moffitt, R. A. (1980). The uses of Tobit analysis. Review of Economics and Statistics, 62(2), 318-321.
Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2^nd ed.). MIT Press. Chapter 17.
Cameron, A. C., & Trivedi, P. K. (2005). Microeconometrics: Methods and Applications. Cambridge University Press.
Greene, W. H. (2012). Econometric Analysis (7^th ed.). Prentice Hall. Chapter 19.

Marginal Effects for Censored Models¶

Overview¶

Three Types of Marginal Effects in Tobit¶

1. Conditional Marginal Effect¶

2. Unconditional Marginal Effect¶

3. Probability Marginal Effect¶

McDonald & Moffitt (1980) Decomposition¶

Computing Marginal Effects in PanelBox¶

Tobit Marginal Effects¶

Parameters¶

AME vs. MEM¶

Marginal Effects in the Heckman Model¶

Unconditional Marginal Effect (Heckman)¶

Conditional Marginal Effect (Heckman)¶

Interpreting Results¶

Coefficients vs. Marginal Effects¶

The Dampening Factor¶

Practical Comparison¶

Tutorials¶

See Also¶

References¶